Chapter 1 Reasoning About Quantities - Macmillan …

Chapter 1 Reasoning About Quantities - Macmillan … Part I Reasoning About Numbers and Quantities Much of what you see in these first eleven chapters will look familiar. However, you will be asked to begin understanding the mathematics of arithmetic at a more fundamental level than you may now do. Knowing how to compute is not sufficient for teaching; a teacher should understand the underlying reasons we compute as we do. An important focus of these chapters is on using numbers. We do this by helping you develop the skill of quantitative reasoning. This reasoning is fundamental to solving problems involving quantities (such as the speed of a car of the price of chips). You may recognize such problems as story problems. In Chapters1, 8, and 9 you will learn how to approach such problems, using quantitative reasoning. Our base ten system of numbers, introduced in Chapter 2, is the foundation of all of the procedures we use for computing with whole numbers. A teacher must acquire a deep understanding of the base ten system and the arithmetic procedures, which is the focus of Chapters 3, 4, and 5. We then turn to the study of fractions, decimals, and percents in Chapters 6 and 7. Many people in our society do not know how to compute with these numbers, or how these numbers are related. The emphasis on understanding these numbers and how to operate on them will give you a foundation to teach fractions, decimals, and percents well. Your students will benefit if they do not develop the belief that they cannot ―do‖ fractions. Teachers who can teach arithmetic in a manner that students understand will better prepare their students for the mathematics yet to come. Chapter 10 expands the type of numbers we used in previous chapters to include negative numbers, and discusses arithmetic operations on these numbers. Again, the focus is on understanding. Chapter 11, on number theory, provides different ways of thinking about and using numbers. Topics such as factors, multiples, prime numbers, and divisibility are found in this chapter. An underlying theme throughout these chapters is an introduction to children’s thinking about the mathematics they do. You will be surprised to learn the ways that children do mathematics, and we think that knowing how children reason mathematically will convince you that you need to know mathematics at a much deeper level than you likely do now. Prospective teachers are often fascinated with these glimpses into the ways children reason, particularly if they have a good foundation in their work with numbers. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 1 Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 2 This Word document follows the pagination Chapter 1 of the student pages in the Preliminary Reasoning About Quantities Edition as closely as possible. In some places, however, particularly on pages with Learning We encounter quantities of many kinds each day. In this chapter you will Exercises, the be asked to think about quantities and the manner in which we use them to pagination is not quite better understand our lives. In particular, you will encounter ―story the same. problems‖ and learn how to solve them using a powerful reasoning process, most likely aided by a drawing of some sort. Please read the Introduction in the 1.1 What Is a Quantity? Instructor Notes for information about the Consider the following questions: Reasoning about Numbers and Quantities course. You How long do humans live? will find it helpful in planning your course. How fast is the wind blowing? Then see INSTRUCTOR Which is more crowded, New York City or Mexico City? NOTE 1 for Chapter 1, which introduces this How big is this room? chapter. How far is it around the earth? The answer to each of these questions involves some quantity. A quantity is anything (an object, event, or quality thereof) that can be It is important that students measured or counted. The value of a quantity is its measure or the understand the difference number of items that are counted. A value of a quantity involves a between the value of a quantity and the quantity number and a unit of measure or number of units. itself. For example, the length of a room is a quantity. It can be measured. Suppose the measurement is 14 feet. Note that 14 is a number, and feet is a unit of measure: 14 feet is the value of the quantity, length of the room. The number of people in the bus is another example of a quantity. Suppose the count is 22 people. Note that 22 is a number, and the unit counted is people, so 22 is the value associated with the quantity ―number of people on the bus.‖ Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 3 A person‘s age, the speed of the wind, the population density of New York City, the area of this room, and the distance around the earth at the equator are all examples of quantities. ... What are some possible values of the five quantities Think About―Crowdedness‖ mentioned above? Notice that population alone will not be sufficient to could be measured address the question of how crowded a city or a country is. Why? by considering density, e.g., people Not all qualities of objects, events, or persons can be quantified. Consider per square mile. love. Young children sometimes attempt to quantify love stretching their arms out wide when asked, ―How much do you love me?,‖ but love is not a quantity. Love, anger, boredom, and interest are some examples of qualities that are not quantifiable. Feelings, in general, are not quantifiable—thus, they are difficult to assess. Other examples that Think About... Name some other ―things,‖ besides feelings, that are not are not feelings are quantities. justice, fairness, shrewdness, beauty, The fact that a quantity is not the same as a number should be clear to you. etc. In fact, one can think of a quantity without knowing its value. For example, the amount of rain fallen on a given day is a quantity, regardless of whether or not someone measured the actual number of inches of rain fallen. One can speak of the amount of rain fallen without knowing how many inches fell. Likewise, one can speak of a dog’s weight, a tank’s capacity, the speed of the wind, or the amount of time it takes to do a Discussion 1: Identifying chore (all quantities) without knowing their actual values. Quantities and Measures 1 a. height Discussion 1 Identifying Quantities and Measures b. rate of flow 1. Identify the quantity or quantities addressed in each of the c. gross national product or income per capita following questions. d. amount of damage, a. How tall is the Eiffel tower? number of casualties, etc. b. How fast does water come out of a faucet? 2. a. feet, meters, etc. c. Which country is wealthier, Honduras or Mozambique? b. any unit of volume per d. How much damage did the earthquake cause? any unit of time c. dollars or dollars per 2. Identify an appropriate unit of measure that can be used to person determine the value of the quantities involved in answering the d. number of dollars it questions in Problem 1. Is the wealth of a country measured the costs same way as the wealth of an individual? Explain. Discussion 2: Easy to Discussion 2 Easy To Quantify? Quantify? The idea of 1. Many attributes or qualities of objects are easily quantifiable. something being Others are not so straightforward. Of the following items, which quantifiable is important to the process of quantitative are easy to quantify and which aren‘t? reasoning. a. The weight of a newborn baby f. Infant mortality rate SEE INSTRUCTOR b. The gross national product g. Teaching effectiveness NOTE 1.1A. on using the c. Student achievement h. Human intelligence Activities and Discussions d. Blood pressure i. Air quality in this text. e. Livability of a city j. Wealth of a nation Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 4 2. How is each item in Problem 1 typically quantified? 3. What sorts of events and things do you think primitive humans felt a need to quantify? Make a list. How do you think primitive 5. Apgar scores for measuring health of societies kept track of the values of those quantities? babies at birth, 4. Name some attributes of objects (besides those listed in Problem 1) numerical scores for that are not quantifiable or that are hard to quantify. measuring health of gums, measures of 5. Name some quantities for which units of measure have been only proportion of body recently developed. weight due to fat, etc. The discussion Easy to Take-Away Message…In this introductory section you have learned to identify Quantify could be quantities and their values and to distinguish between the two. This understanding assigned as homework, is the basis for the quantitative analyses in the next section. 1.2 Quantitative Analysis SEE INSTRUCTOR There are no exercises In this section you will use what you have learned about quantities in NOTE 1.2A. for this introductory Section 1.1 to analyze problem situations in terms of their quantitative lesson. This lesson structure. Such analyses are essential to being skillful at solving leads directly into mathematical problems. Section 1.2, if desired. For the purposes of this course, to understand a problem situation means to understand the quantities embedded in the situation and how they are related to one another. Understanding a problem situation ―drives‖ the solution to the problem. Without such understanding, the only recourse a person has is to guess at the calculations that need to be performed. It is important that you work through this section with care and attention. Analyzing problem situations quantitatively is central to the remainder of this course and other courses that are part of your preparation to teach school mathematics. This problem will be Activity 1 The Hot Dog Problem trivial for some of your 31students, but surprisingly 21Albert ate hot dogs and Reba ate hot dogs. 24not for all. A prospective What part of all the hot dogs eaten did Albert eat? teacher, in an interview, spent about 30 minutes This fairly simple problem is given here to illustrate the process of using a on this problem, quantitative analysis to solve a problem. We first analyze the situation in guessing at which terms of the quantities it involves and how those quantities are related to operation was called for. Not until he drew a one another: its quantitative structure. To do so productively, it is diagram was he able to extremely important to be specific about what the quantities are. For solve the problem and example, it is not sufficient to say that ―hot dogs eaten‖ is a quantity in indicate that he this problem situation. If you indicated ―hot dogs eaten‖ as one of the understood the structure quantities and specified no more, then someone could ask: Which hot dogs? of the problem. The ones Albert ate? The ones Reba ate? The total number of hot dogs eaten? (Continue on next page) Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 5 Continuation of Activity 1 To understand the quantitative structure of this problem situation, you can do the following: 1. Name as many quantities as you can that are involved in this situation. Be aware that some quantities may not be explicitly stated although they are essential to the situation. Also, just because the value of a quantity may not known nor given, this does not mean that the quantity is not part of the situation‘s quantitative structure. 2. For each quantity, if the value is given, write it in the appropriate Quantities and values in space. If the value is not given, indicate that the value is unknown, Activity 1:Hot Dog and write the unit you would use to measure it. You may need more Problem: space than provided here. (Reminder: There should be no numbers a. Number of hot dogs eaten by Albert—2 ? appearing in the Quantity column.) Compare your list with those of b. Number of hot dogs others in your class. eaten by Reba—1 ? Value Quantity c. Number of hot dogs eaten all together--? 3 a. Number of hot dogs eaten by Albert 24d. Portion of hot dogs eaten by Albert--? b. etc. e. Portion of hot dogs 3. Make a drawing that illustrates this problem. Here, the outer eaten by Reba--? rectangle indicates the total number of hot dogs eaten, and the parts show the portions eaten by each person. (Note that drawings such as this one need not be drawn to scale. Here, the number of hot dogs eaten by Albert is shown only to be more than the hot dogs eaten by Many students believe that Reba.) 3 drawing a picture to 4Hot dogs eaten by Albert: 2 represent a problem is ―cheating.‖ Assure them, Hot dogs eaten 1over and over, that 2by Reba: 1 drawings are appropriate in coming to understand a problem. Notice that a continuous drawing is used though hot dogs are 4. Finally, use the drawing to solve the problem. The drawing shows discrete objects. The Issues 311424for Learning in Section 1.4 that the total number of hot dogs eaten is 2 + 1 or 4. What 3discuss why this is done. 24144part of the total was eaten by Albert? hot dogs, which 1117simplifies to of the hot dogs. This is about two-thirds of the hotdogs, a reasonable answer to this question. SEE INSTRUCTOR NOTE 1.2A. for an The purpose in Activity 1 to illustrate how one must analyze a situation in instructional terms of the quantities and the relationships present in the situation in suggestion for beginning order to gain an understanding that can lead to a meaningful solution. Quantitative Such an activity is called a quantitative analysis. On problems such as Analyses. this one, you may find that you don‘t really need to think through all four steps. However, more difficult problems, such as those in the next two activities, are more easily solved by undertaking a careful quantitative analysis. You will improve your skill at solving problems when you engage in such analyses because you will come to a better understanding of the problem. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 6 Activity 2 Sisters and Brothers Try this problem by undertaking a quantitative analysis before you read the solution. Then compare your solution to the one shown here. Two women, Alma and Beatrice, each had a brother, Alfred and Benito, respectively. The two women argued about which woman stood taller over her brother. It turned out that Alma won the argument by a 17-centimeter difference. Alma was 186 cm tall. Alfred was 87 cm tall. Beatrice was 193 cm tall. How tall was Benito? First, name the quantities involved in the problem. a. Alma‘s height b. Beatrice‘s height c. Alfred‘s height d. Benito‘s height e. The difference between Alma‘s and Alfred‘s heights f. The difference between Beatrice‘s and Benito‘s heights g. The difference between the differences in the heights of the sister and brother pairs Next, identify the values of the quantities. a. Alma‘s height 186 cm b. Beatrice‘s height 193 cm c. Alfred‘s height 87 cm d. Benito‘s height unknown e. The difference between Alma‘s and Alfred‘s heights unknown f. The difference between Beatrices‘s and Benito‘s heights unknown g. The difference between the differences in the heights of the sister and brother pairs (a crucial 17 cm (Note that Quantity g is a given quantity and crucial to solving the problem.) Draw a picture involving these quantities. Here is one possibility. Need to relateDiff Diffthe two differences B's A's(A's is 17 cm more(99 cm)than B's):186193 cm cm17 cm(9987 cmcm)So, 99-17 = 82 cm Sis BBro BSis ABro A ht ht ht ht Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 7 Or, perhaps this alternative drawing would help clarify the problem. Next to each quantity, place its value if known, and use these values known to find values unknown. Alma (186 cm) Difference A (99 cm) Alfred (87 cm) Difference of differences (17 cm) Beatrice (193 cm) Difference B ( ???) Benito (???) Finally, use the drawing and/or diagram to solve the problem. Notice that in this case the solution is shown in the first drawing, and can easily be found in the second drawing. But note that the question asked was Benito‘s height. According to the drawings, it is 193 cm – 82 cm = 111 cm. Often students begin a problem such as the one in Activity 2 by asking themselves ―What operations do I need to perform, with which numbers, and in what order?‖ Instead of these questions, you would be much better off asking yourself questions such as the following. Quantitative Analysis Questions This list of questions is • What do I know about this situation? extremely important. • What quantities are involved here? Which ones are critical? Whenever you find your students struggling to • Are there any quantities that are related to other quantities? If so, understand a problem how are they related? situation, come back to • Which quantities do I know the value of? this list and make sure that they are asking • Which quantities do I not know the value of? Are these related to themselves these types other quantities in the situation? Can these relationships enable me of questions and really to find any unknown values? trying to make sense of • Would drawing a diagram or acting out the situation help me the situation. answer any of these other questions? And so on. The point is that you need to be specific when doing a quantitative analysis of a given situation. How specific? It is difficult to say in general, because it will depend on the situation. Use your common sense in analyzing the situation. When you first read a problem, avoid trying to think about numbers and the operations of addition, subtraction, multiplication, and division. Instead, start out by posing the questions such as those listed above and try to answer them. Once you‘ve done that, you will have a better understanding of the problem, and thus you will be well on your way to solving it. Keep in mind that understanding the problem is the most difficult aspect of solving it. Once you understand the problem, what to do to solve it often follows quite easily. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 8 Activity 3 Down the Drain Here is another problem situation on which to practice quantitative Activity: Down the Drain. SEE INSTRUCTOR analysis. NOTE 1.2B for suggestions on this activity. Water is flowing from a faucet into an empty tub at 4.5 gallons per minute. An important quantity to After 4 minutes, a drain in the tub is opened, and the water begins to flow out at notice is the amount of water 6.3 gallons per minute. lost per minute after the a. Will the tub ever fill up completely without overflowing? initial 4 min, at which time 18 gal are in the tub. 6.3 - 4.5 b. Will it ever empty completely? is 1.8 gal is lost per minute, c. What if the faucet is turned off after 4 minutes? so that 10 minutes after the drain is opened, 18 gal. are d. What if the rates of flow in and out are reversed? lost and the tub is empty. e. What assumptions do we have to make in order to answer these a. The tub will fill up IF it questions? holds exactly18 gal. b. It will empty completely IF When you carry out a quantitative analysis of a situation, the questions the faucet and drain both run you ask yourself should be guided by your common sense. A sense-for 10 minutes. c. The tub is draining at the making approach to understanding a situation and then solving a problem rate of 6.3 gal/min, with 18 is much more productive than trying to decide right away which gal to empty, so it will be computations, formulas, or mathematical techniques you need in order to empty in about 2.9 min. solve the problem at hand. Using common sense may lead you to make d, After 4 minutes there will be 25.2 gal in the tub. Water some sort of a diagram. Never be embarrassed to use a diagram. Such will continue to increase in diagrams often enhance your understanding of the situation, because they the tub at 1.8 gal/min until help you to think more explicitly about the quantities that are involved. the tub overflows. This Deciding what operations you need to perform often follows naturally assumes that the tub holds from a good understanding of the situation, the quantities in it, and how 25.2 or more gallons. e. SEE INSTRUCTOR those quantities are related to one another. NOTE 1.2B. Take-Away Message . . . You should be able to determine the quantities and their Remember to bring closure to relationships within a given problem situation, and you should be able to use this class activities by discussion information, together with drawings as needed, to solve problems. These steps are them in class or in some way often useful: (1) List the quantities that are essential to the problem. (2) List known (such as an overhead values for these quantities. (3) Determine the relationships involved, which is transparency) sharing frequently done more easily with a drawing. (4) Use the knowledge of these answers. There are instances, relationships to solve the problem. This approach may not seem easy at first, but it however,, where you may becomes a powerful tool for understanding problem situations. This type of want students to think about a quantitative analysis can also be used with algebra story problems, and thus, when problem until the next class. used with elementary school students, prepares them for algebra. Students do not have the Learning Exercises for Section 1.2 answers for LE 1a, 1d, 2b, 1. Some problems are simple enough that the quantitative structure is 3, 5, and 8. Note that some obvious, particularly after a drawing is made. The following problems of these exercises are time-iconsuming and may need are from a fifth-grade textbook. For each problem given, make a class discussion. Be sure to drawing and then provide the answer to the problem. discuss 6 in class. Remind students that the i above a. The highest elevation in North America is Mt. McKinley, Alaska, the period after textbook which is 20,320 feet above sea level. The lowest elevation in North refers to a reference at the America is Death Valley, California, which is 282 feet below sea end of the chapter. level. What is the change in elevation from the top of Mt. McKinley to Death Valley? Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 9 b. The most valuable violin in the world is the Kreutzer, created in Italy in 1727. It was sold at auction for $1,516,000 in England in 1998. How old was the violin when it was sold? c. Two sculptures are similar. The height of one sculpture is four times the height of the other sculpture. The smaller sculpture is 2.5 feet tall. How tall is the larger sculpture? d. Aiko had $20 to buy candles. She returned 2 candles for which she had paid $4.75 each. Then she bought 3 candles for $3.50 each and 1 candle for $5.00. How much money did Aiko have then? e. In Ted‘s class, students were asked to name their favorite sport. 1Football was the response of of them. If 3 students said football, how 8 many students are in Ted‘s class? f. The first year of a dog‘s life equals 15 ―human years.‖ The second year equals 10 human years. Every year thereafter equals 3 human years. Use this formula to find a 6-year-old dog‘s age in human years. ii2. These problems are from a sixth-grade textbook from a different series. This time, undertake a full quantitative analysis to solve each of the problems. a. At Loud Sounds Music Warehouse, CDs are regularly priced at $9.95 and tapes are regularly priced at $6.95. Every day this month the store is offering a 10% discount on all CDs and tapes. Joshua and Jeremy go to Loud Sounds to buy a tape and a CD. They do not have much money, so they have pooled their funds. When they get to the store, they find that there is another discount plan just for that day—if they buy three or more items, they can save 20% (instead of 10%) on each item. If they buy a CD and a tape, how much money will they spend after the store adds a 6% sales tax on the discounted prices? b. Kelly wants to fence in a rectangular space in her yard, 9 meters by 7.5 meters. The salesperson at the supply store recommends that she put up posts every 1.5 meters. The posts cost $2.19 each. Kelly will also need to buy wire mesh to string between the posts. The wire mesh is sold by the meter from large rolls and costs $5.98 a meter. A gate to fit in one of the spaces between the posts costs $25.89. Seven staples are needed to attach the wire mesh to each post. Staples come in boxes of 50, and each box costs $3.99. How much will the materials cost before sales tax? 3. This would be a 3. All Aboard! Amtrail trains provide efficient, non-stop transportation good problem to do in between Los Angeles and San Diego. Train A leaves Los Angeles headed class if you want more towards San Diego at the same time that Train B leaves San Diego headed class problems. for Los Angeles, traveling on parallel tracks. Train A travels at a constant Students do not have access to the answer. speed of 84 miles per hour. Train B travels at a constant speed of 92 miles per hour. The two stations are 132 miles apart. How long after they leave their respective stations do the trains meet? 4. My brother and I walk the same route to school every day. My brother takes 40 minutes to get to school and I take 30 minutes. Today, my 4. This problem may iiiseem difficult, but brother left 8 minutes before I did. the solution is Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 10 actually quite intuitive. Note: Exercise 4 begins on page 8 in a. How long will it take me to catch up with him? 11. What b. Part of someone‘s work on this problem included ,304011 quantities do the two fractions in represent? ,3040 c. Suppose my brother‘s head start is 5 minutes instead of 8 minutes. Now how long does it take for me to catch up with him? 5. At one point in a Girl Scout cookie sales drive, region C had sold 1500 boxes of cookies, and region D had sold 1200 boxes of cookies. If region D tries harder, they can sell 50 more boxes of cookies every day than region C can. a. How many days will it take for region D to catch up? b. If sales are stopped after eight more days, can you tell how many total boxes each region sold? Explain. 6. The last part of one triathlon is a 10K (10 kilometers, or 10,000 meters) run. When runner Aña starts this last running part, she is 600 meters Problem 6, the behind runner Bea. But Aña can run faster than Bea: Aña can run (on triathlon problem, average) 225 meters each minute, and Bea can run (on average) 200 should definitely be discussed in class. The meters each minute. Who wins, Aña or Bea? If Aña wins, when does students have the she catch up with Bea? If Bea wins, how far behind is Aña when Bea answer but may not finishes? understand all of it. See INSTRUCTOR 7. Research on how students solve word problems contained the iv NOTE 1.2C for following incident.Dana, a seventh grader in a gifted program in information on mathematics, was asked to solve the following problem: difficulties students have with this problem A carpenter has a board 200 inches long and 12 inches wide. He makes 4 identical and the reason to avoid shelves and still has a piece of board 36 inches long left over. How long is each an algebraic solution that can hide a real shelf? understanding of (and Dana tried to solve the problem as follows: She added 36 and 4 , then teaching) this type of scratched it out, and wrote 200 12, but she thought that was too large ，the problem. so she scratched that out. Then she tried 2400 – 36 which was also too large and discarded it. Then she calculated 4 36 and subtracted that ， from 200, getting 56. She then subtracted 12, and got 44. Dana used a weak strategy called ―Try all operations and choose.‖ She obviously did not know what to do with this problem, although she was very good at solving one-step problems. Do a quantitative analysis of this problem situation, and use it to make sense of the problem in a way that Dana did not. Use your analysis to solve the problem. 8. The problems listed below, and in Exercise 9, are from a Soviet Grade 3 vtextbook. Solve the problems and compare their conditions and solutions: Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 11 (a) Two pedestrians left two villages simultaneously and walked towards each other, meeting after 3 hours. The first pedestrian walked 4 km in an hour, and the second walked 6 km. Find the distance between their villages: (b) Two pedestrians left two villages 27 km apart simultaneously and walked towards each other. The first one walked 4 km per hour, and the second walked 6 km per hour. After how many hours did the pedestrians meet? (c) Two pedestrians left two villages 27 km apart simultaneously and walked towards each other, meeting after 3 hours. The first pedestrian walked at a speed of 4 km per hour. At what speed did the second pedestrian walk? 9. Two trains simultaneously left Moscow and Sverdlovsk, and traveled towards each other. The first traveled at 48 km per hour, and the second at 54 km per hour. How far apart were the two trains 12 hours after departure if it is 1822 km from Moscow to Sverdlovsk? 1.3 Values of Quantities SEE INSTRUCTOR The value of a quantity may involve very large or very small NOTE 1.3A. for advice numbers. Furthermore, since the value is determined by on teaching this section, particularly metric units. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 12 Tape measures or rulers are needed in this section for both English and metric units. counting or other ways of measuring, it can involve any type of number— whole numbers, fractions, decimals. Consider the following quantities. Which ones would you Think About … expect to have large values? Small values? a. The distance between two stars Think About 1. The answers will depend to a b. The diameter of a snowflake certain extent on the unit c. The weight of an aircraft carrier of measure used in each, d. The national deficit e.g., the weight of an aircraft carrier in pounds e. The thickness of a sheet of paper has a much larger In a particular book the height of the arch in St. Louis is reported to be numerical value than the value if it is weighed in 630 feet. In another book the height is 192 meters. Why are the numbers tons. different? Think About . . . What determines the magnitude of the number that Think About 2. All are possible, but the denotes the value of a given quantity? Can we measure the speed of a car numerical values would in miles per day? Miles per year? Miles per century? Is it convenient to do increase dramatically with so? Explain. each new unit of measure, and would be too large to Discussion 3 Units of Measure make much sense. 1. What determines the appropriateness of the unit chosen to express the value of a quantity? Discussion 3 1. Leads to a reasonable value; the 2. For each quantity in parts (a)-(e) above, name a unit of measure precision that you need; that would be appropriate to measure the quantity. familiarity; culture. 3. Explain how you would determine the thickness of a sheet of paper. 3. A micrometer can measure small lengths. 4. What determines the ―precision‖ of the value of a quantity? Many students will want to measure the height of a Unless one has some appreciation and understanding of the magnitude of stack of paper and divide large numbers, it is impossible to make judgments about such matters as by the number of sheets. the impact of a promised five million dollars in relief funds after a 4. Precision relates to the catastrophic flood or earthquake, the level of danger of traveling in a specificity of the unit of country that has experienced three known terrorist attacks in a single year, measure, as opposed to accuracy, which relates to the personal consequences of the huge national debt, or the meaning of the degree to which the costly military mistakes. For example, people are shocked to hear that the given measurement Pentagon spent $38 for each simple pair of pliers bought from a certain approximates the actual defense contractor, yet they pay little attention to the cost of building the measurement. Stealth fighter or losing a jet fighter during testing. Activity 4 Jet Fighter Crashes Activity 4 This activity does not In the 1990s, a west coast newspaper carried a brief article saying that have one answer. But if a $50 million jet fighter crashed into the ocean off the California you have students work it in class, be sure to coast. How many students could go to your university tuition-free for follow up the activity one year with $50 million? with a brief discussion. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 13 Discussion 4 What Is Worth A Trillion Dollars? Suppose you hear a politician say "A billion dollars, a trillion Discussion 4 dollars, I don't care what it costs, we have got to solve the AIDS Some students tend to problem in this country." Would you agree? Is a billion dollars begin discussing ethical issues and the too much to spend on a national health crisis? A trillion dollars? need to solve the AIDS How do the numbers one billion and one trillion compare? problem, without addressing the size of Reminder: Values of quantities, like 16 tons or $64, involve units of the numbers in terms measure—ton, dollar—as well as numbers. of what is actually possible. ... Name several units of measure that you know and use. Think About What quantities are each used to measure? Where do units come from? This is a very brief Units can be arbitrary. Primary teachers have their students measure introduction to the metric system. It is lengths and distances with pencils or shoe lengths, or measure weights placed here because with plastic cubes. The intent of these activities is to give the children we use metric units in experience with the measuring process, so that later measurements will many problems. A make sense. As you know, there are different systems of standard units, more extensive like the English or ―ordinary‖ system (inches, pounds, etc.) and the metric treatment is in Part III: Reasoning about system (meters, kilograms, etc.), more formally known as SI (from Le Shapes and Système International d'Unités). Measurement. If your students have Virtually the rest of the world uses the metric system to denote values of some familiarity with quantities, so many are surprised that the United States, a large industrial metric units you may nation, has clung to the English system so long. Although the general decide to omit these public has not responded favorably to governmental efforts to mandate the paragraphs. metric system, international trade efforts are having the effect of forcing Discussion 5. Standard us to be knowledgeable about, and to use, the metric system. Some of our units allow us to largest industries have been the first to convert to the metric system from understand and share the English system. information. People are loath to give up a system with which they are familiar. Also, changing Discussion 5 Standard Units is costly. Why are standard units desirable? For what purposes are they A global economy necessary? Why has the public resisted adopting the metric system? necessitates much greater standardization. It is Scientists have long worked almost exclusively in metric units. As a result, neither economical nor you may have used metric units in your high school or college science practical (storage, classes. Part of the reason for this is that the rest of the world uses the inventory, display, etc.) metric system because it is a sensible system. A basic metric unit is to maintain two systems of measurement when carefully defined (for the sake of permanence and later reproducibility). one will do. Larger units and smaller sub-units are related to each other in a consistent fashion, so it is easy to work within the system. (In contrast, the English system unit, foot, might have been the length of a now-long-dead king‘s foot, and it is related to other length units in an inconsistent manner: 1 foot = 12 inches; 1 yard = 3 feet; 1 rod = 16.5 feet; 1 mile = 5280 feet; 1 1furlong = mile; 1 fathom = 6 feet. Quick!—how many rods are in a 8 Metric units are given attention here because we Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 14 use metric units throughout. However, a section on the metric system is included in a later section on measurement, so mile? A comparable question in the metric system is just a matter of adjusting a decimal point. To get a better idea of how the metric system works, let‘s consider something that we frequently measure in metric units—length. Length is a quality of most objects. We measure the lengths of boards and pieces of rope or wire. We also measure the heights of children; height and length Think About: Other refer to the same quality but the different words are used in different words for ―length‖ are contexts. width, depth, girth, thickness, perimeter, What are some other words that refer to the same quality Think About...radius, circumference, as do ―length‖ and ―height"? distance, base, altitude, etc. Students may The basic SI unit for length (or its synonyms) is the meter. (The official incorrectly suggest SI spelling is metre. You occasionally see ―metre‖ in U.S. books.) The area or volume units. meter is too long to show with a line segment here, but two sub-units fit easily, and illustrate a key feature of the metric system: Units larger and smaller than the basic unit are multiples or sub-multiples of powers of 10. In the printing process these lengths may be distorted. 0.1 meterCheck them for accuracy. 0.01 meter ecimeter, centimeter—which Furthermore, these sub-units have names—d are formed by putting a prefix on the word for the basic unit. The prefix ―deci-‖ means one-tenth, so ―decimeter‖ means 0.1 meter. Similarly, ―centi-‖ means one one-hundredth, so ―centimeter‖ means 0.01 meter. You have probably heard ―kilometer;‖ the prefix ―kilo-‖ means 1000, so ―kilometer‖ means 1000 meters. On reversing one's thinking, so to speak, there are 10 decimeters in 1 meter, there are 100 centimeters in 1 meter, and 1 meter is 0.001 kilometer. Another feature of SI is that there are symbols for the basic units—m for In some manuals of meter—and for the prefixes—d for deci-, c for centi-, and so on. By using style you may see the symbol for meter together with the symbol for a prefix a length periods for abbreviations in the measurement can be reported quite concisely: for example,18 cm and 2.3 English system, but dm. The symbols, cm and dm, do not have periods after them, nor do the periods are not use abbreviations in the English system: ft, mi, etc., except that for inch (in.). in SI. If you are new to the metric system, your first job will be to familiarize yourself with the prefixes so you can apply them to the basic units for other qualities. The table on the next page shows some of the other metric prefixes. Combining the symbol k for kilo- and the symbol m for meter gives km for kilometer. Students may Prefix Symbol Meaning of Prefix Applied to Length need a review of Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 15 the use of exponents. Have them see the Appendix ―A Review of Some Rules.‖ 3 km kilo- k 1 000 or 102hecto- h 100 or 10 hm 1deka- da 10 or 10 dam 0no prefix 1 or 10 m -1deci- d 0.1 or 10 dm -2centi- c 0.01 or 10 cm -3milli- m 0.001 or 10 mm Activity 5 It’s All in the Unit 1. Measure the width of your desk or table, in decimeters. Express that length in centimeters, millimeters, meters, and kilometers. 2. Measure the width of your desk or table, in feet. Express that length in inches, yards, and miles. 3. In which system are conversions easier? Explain why. Take-Away Message: The value of a quantity is expressed using a number and a unit of measure. Commerce depends upon having agreed-on sets of measures. Standard units accomplish this. The common system of measurement in the United States is the English system of measurement. We use the metric system, another measurement system, for science and for international trade. Most countries of the world use only the metric system. The metric system is based on powers of ten and on common prefixes, making the system an easier one to use. Learning Exercises for Section 1.3 LE 1.3 Students 1. Name an appropriate unit for measuring each given quantity. do not have answers to 1, 2c, a. the amount of milk that a mug will hold. 2d, 2e, 3, or 4. b. the height of the Empire State Building. c. the distance between San Francisco and New York. d. the capacity of the gas tank in your car. e. the safety capacity of an elevator. f. the amount of rainfall in one year. 2. a. What does it mean to say that ―a car gets good mileage?‖ 2b. In Europe mileage is often b. What unit is used to express gas mileage in the United States? measured in c.How could you determine the mileage you get from your car? Exercise 4 can be ―liters/100 km.‖ treated as an in-class d. Do you always get the same mileage from your car? List activity. If it is some factors that influence how much mileage you get. assigned for homework, then it e. How would you measure gas consumption? Is gas should be discussed consumption related to mileage? If so, how? in class. What 3. Explain how rainfall is quantified. You may need to use assumptions were made? Why are resources such as an encyclopedia or the internet. answers from 4. a. Calculate an approximation of the amount of time you have different people so spent sleeping since you were born. Explain your calculations. discrepant? What type of calculations Express your approximation in hours, in days, in years. did students use? b. What part of your life have you spent sleeping? Were they surprised? Are there Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 16 different ways of approaching this problem? c. On the average how many hours do you sleep each day? What fractional part of the day is this? d.How does your answer in part (b) compare to your answer in part (c)? 5. Name an item that can be used to estimate the following metric units: a. a centimeter b. a gram c. a liter d. a meter e. a kilometer f. a kilogram 6. There are some conversions from English to metric units that are commonly used, particularly for inch, mile, and quart. What are they? 1.4 Issues for Learning: Ways of Thinking About Solving Story Problems viIn solving story problems, how do children decide what to do? In one study students were found to use the following seven different strategies. The first six strategies are based on something other than understanding the problem. 1. Find the numbers in the problem and just do something to them, usually addition because that is the easiest operation. 2. Guess at the operation to be used, perhaps based on what has been most recently studied. Let the numbers ―tell‖ you what to do. (One student said, ―If it‘s like 3. 78 and maybe 54, then I‘d probably either add or multiply. But if the numbers are 78 and 3, it looks like a division because of the size of the numbers.‖) 4. Try all the operations and then choose the most reasonable answer. (This strategy often works for one-step problems, but rarely does for two-step problems.) 5. Look for ―key‖ words to decide what operation to use. For example, ―all together‖ means to add. (This strategy works sometimes, but not all the time. Also, words like ―of‖ and ―is‖ often signify multiplication and equals, but some students confuse the two.) 6. Narrow the choices, based on expected size of the answer. (For example, when a student used division on a problem involving reduction in a photocopy machine, he said he did because ‖it‘s reducing something, and that means taking it away or dividing it.‖) 7. Choose an operation based on understanding the problem. [Often students would make a drawing when they used this strategy. Unfortunately, though, few of the children (sixth and eighth graders, of average or above average ability in mathematics) used this strategy.] Only the last strategy was considered a mature strategy. These children understood the problem because they had undertaken a quantitative analysis of the problem, even though not so formally as introduced in this chapter. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 17 Making drawings or mental pictures can play an essential role in coming to understand a problem. Here is an excerpt from one interview with one of the children in this study: Emmy: I just pictured the post, how deep the water was...Sometimes I picture the objects in my mind that I‘m working with, if it‘s a hard problem... Interviewer: Does that help? Emmy: Yeh, it helps. That‘s just one way of, kind of cheating, I guess you‘d say. Unfortunately, too many students seem to believe that making drawings is cheating, or is juvenile, yet making a drawing is a valuable problem-solving process. viin another study, two researchers compared how drawings are used in I U.S. textbooks and in Japanese textbooks. They found that many elementary school students in the United States are not encouraged to make drawings that will help them understand a story problem. However, even just flipping through the pages of Japanese textbooks shows that drawings are used throughout. Teachers say to students, ―If you can draw a picture, you can solve the problem.‖ Think About... In what ways have drawings helped you so far with solving the problems in this first chapter? All too often adults try to avoid making a drawing because they think that the need for drawings is childish. But this is not the case. The ability to represent a problem with a drawing is an important component of problem solving, no matter what the age of the problem solver is. Young children often draw a picture to help them understand a problem. For example, for the problem ―There are 24 legs in the sheep pen where two men are shearing sheep. How many sheep are there?‖ one child might draw something similar to this: Yet another child might represent this problem as: | | | | | | | | | | | | | | | | | | | | | | | | There are, of course, multiple ways this problem could be represented with a drawing, but here perhaps the first is more appropriately called a picture, and the second would more appropriately be called a diagram. Students will quite naturally evolve from pictures to diagrams. ―A Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 18 diagram is a visual representation that displays information in a spatial ix Although diagrams are very useful in understanding the layout.‖ structure of a problem, students are often unable to produce an appropriate diagram without some assistance. Whether or not a picture or a diagram is appropriate depends on how well it represents the structure of the problem. A diagram is more abstract than a picture in that it often does not contain extraneous information. It takes less time to draw, but it continues to represent the problem. In the sheep problem, the objects are discrete, that is, they can be counted. But not all problems about discrete quantities need to be represented by Discussion 6. For individual objects because diagramming large numbers can become a problem 1, the student chore, and pictures would take far too long to produce. Thus we often use uses both drawings and lines or boxes to represent the problems, even those involving discrete a diagram. It appears quantities. that the drawing may have been drawn Discussion 6 Drawings and Diagrams without the markings, which were added 1. A fourth grader was asked to solve some story problems you saw in later. For the second the Learning Exercises for Section 1.2. Two problems are shown problem, she used a here. Before looking at the student‘s solutions, go back and see drawing, which could also be thought of as a how you solved the problems. diagram. 2. Discuss the types of drawings used by this student. How did they help her visualize the problems leading to solutions? Problem 1. (Modified from Section 1.2, Learning Exercise 1c) Two sculptures are similar. The height of one sculpture is two and one-half times the height of the other sculpture. The smaller sculpture is 3 feet tall. How tall is the larger sculpture? QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture. Problem 2. (From Section 1.2, Learning Exercise 7) A carpenter has a board 200 inches long and 12 inches wide. He makes 4 identical shelves and still has a piece of board 36 inches long left over. How long is each shelf? Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 19 QuickTime?and aTIFF (Uncompressed) decompressorare needed to see this picture. Drawing diagrams to represent and then solve problem situations are common in Singapore textbooks. They use diagrams they call strip xdiagrams. S. Beckmann has provided the following problems as examples. Example 1 is from a third-grade textbook: Mary made 686 biscuits. She sold some of them. If 298 were left over, how many biscuits did ixshe sell? (Primary Mathematics, volume 3A, p. 20, Problem 4) 686 number sold 298 Example 2 is from a fifth-grade textbook: Raju and Samy share $410 between them. Raju received $100 more than Samy. How much money did Samy receive? (Primary Mathematics, volume 5A, p. 23, ixproblem 1) Raju $410 Samy ? $100 2 units = $410 – $100 = $310 1 unit = $___ Samy received $___ Think About... Can the sheep problem be represented with a strip drawing? In this chapter we have used diagrams rather than pictures to represent problems. Analyzing situations quantitatively, which may include making Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 20 drawings, can help one understand the problems being solved. ―Quantitative reasoning is more than reasoning about numbers, and it is more than skilled calculating. It is about making sense of the situation to x which we apply numbers and calculations.‖ Learning Exercises for Section 1.4 LE 1.4. Students do Use a mature strategy and a strip diagram to work each of the following: not have answers to 5, 1. Kalia spent a quarter of her weekly allowance on a movie. The movie 6, 7, or 8. was $4.25. What is her weekly allowance? 2. Calle had five times as many stickers as Sara did, and twice as many as Juniper. Juniper had 25 stickers. How many stickers did they have altogether? 3. Nghiep gave his mother half of his weekly earnings, and then spent half of what was left on a new shirt. He then had $32. What were his weekly earnings? 4. Zvia‘s sweater cost twice as much as her hat, and a third as much as her coat. Her coat cost $114. How much did her sweater cost? 5. One number is 4 times as large as another number, and their sum is 5285. What are the two numbers? 6. Jinfa, upon receiving his paycheck, spent two-thirds of it on car repairs and then bought a $40 gift for his mother. He had $64 left. How much was his paycheck? 7. Rosewood Elementary School had 104 students register for the fourth grade. After placing 11 of the students in a mixed grade with fifth graders, the remaining students were split evenly into 3 classrooms. How many students were in each of these 3 classrooms? 8. Jacqui, Karen, and Lynn all collect stamps. Jacqui has 12 more stamps than Karen, and Karen has three times as many as Lynn. Together they have 124 stamps. How many does each person have? 9. Jo-Jo has downloaded 139 songs on his iPod. Of those songs, 36 are jazz, twice that are R & B, and the remaining are classical. How many classical songs has he downloaded? 1.5 Check Yourself In this first chapter, you have learned about the role quantities play in our lives and the ways we express quantities and their values. You have The questions and learned about dealing with problem situations by analyzing the problem in problems given here terms of its quantities and their relationships to one another. Quantitative represent the key items in analysis helps solve a problem in a meaningful, sense-making way. The this chapter. If you do not same kind of analysis can be applied to arithmetic problems and to algebra cover all sections, the questions may need to be problems. modified. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 21 You also learned how quantities are measured in terms of numbers of units, and how to express values of quantities in standard units, including metric units. You should be able to work problems like those assigned and to meet the following objectives. 1. Identify the quantities addressed by such questions as, How much damage did the flooding cause? 2. Name attributes of objects that cannot be quantified. 3. Distinguish between a quantity and its value. 4. Given a problem situation, undertake a quantitative analysis of the problem and use that analysis to solve the problem. 5. Determine appropriate units to measure quantities. 6. Discuss reasons why the metric system is used for measurement in most countries. 7. Discuss some incorrect ways that children solve story problems. 8. Discuss the importance of appropriate drawings in problem solving. 9. Make strip diagrams to represent simple arithmetic and algebraic problems. References for Chapter 1 i Maletsky, E. M., Andrews, A. G., Burton, G. M., Johnson, H. C., Luckie, L., Newman, V., Schultz, K. A., Scheer, J. K., & McLeod, J. C. (2002). Harcourt Math. Orlando, FL: Harcourt. iiLappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998). Connected mathematics: Bits and pieces II: Using rational numbers. Teachers Edition. Menlo Park, CA: Dale Seymour Publications. iiiKrutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren (J. Teller, Trans.). Chicago: University of Chicago Press. ivSowder, L. (1995). Addressing the story problem problem. In J. Sowder & B. Schappelle (Eds.), Providing a foundation for the teaching of mathematics in the middle grades (pp. 121-142). Albany, NY: SUNY Press. vRussian Grade 3 Mathematics. (1978). Translated by University of Chicago School Mathematics Project. viSowder, L. (1988). Children‘s solutions of story problems. Journal Of Mathematical Behavior, 7, 227-239. viiShigematsu, K., & Sowder, L. (1994). Drawings for story problems: Practices in Japan and the United States. Arithmetic Teacher, 41 (9), 544-547. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 22 viiDiezmann, C., & English, L. (2001). Promoting the use of diagrams as tools for thinking. In A. C. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 77-89). Reston, VA: NCTM ixBeckmann, S. (2004). Solving algebra and other story problems in simple diagrams: a method demonstrated in grade 4-6 texts used in Singapore. The Mathematics Educator, 14, 42-46. xThompson, P. (1995). Notation, convention, and quantity in elementary mathematics. In J. Sowder & B. Schappelle (Eds.), Providing a foundation for the teaching of mathematics in the middle grades (pp. 199-221). Albany, NY: SUNY Press. Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 23 (This page is purposely blank.) Reasoning About Numbers and Quantities Chapter 1 Instructor‘s Version p. 24 Chapter 2 Numeration Systems SEE INSTRUCTOR NOTE 2 for an overview of this chapter, including an Contrary to what you may believe, there are many ways of expressing interesting quote from numbers. Some of these ways are cultural and historical. Others are a mathematician about teaching primary different ways of thinking about what the digits of our conventional school mathematics. number system mean. For example, you probably think of 23 as meaning two tens and three ones. We think this way because we use a base ten system of counting. (How many fingers do you have?) But why not base five (using only five fingers)? What would 23 mean then? You are about SEE INSTRUCTOR to find out. NOTE 2.1A for an introduction to 2.1 Ways of Expressing Values of Quantities working with bases other than ten. The need to quantify and express the values of quantities led humans to invent numeration systems. Throughout history, people have found ways to express values of quantities they measured in several ways. A variety of Timing: This section words and special symbols, called numerals, have been used to is covered quickly communicate number ideas. How one expresses numbers using these because its message special symbols makes up a numeration system. Our Hindu-Arabic is quite simple—that we have many ways system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Virtually all present-day of expressing values. societies use the Hindu-Arabic numeration system. With the help of decimal points, fraction bars, and marks like square root signs, these ten Think About: π and e, for digits allow us to express almost any number and therefore the value of example, are exceptions. almost any quantity. Students may suggest i and ?. Think About...Why are numerals used so much? What are advantages of The conventional these special symbols over using just words to express numbers? What are numerals are much more some exceptions to representing numbers with digits? compact than the words for the same number Activity 1 You Mean People Didn’t Always Count the Way We Do? would be. This compactness makes it A glimpse of the richness of the history of numeration systems lies in easier to ―see‖ the looking at the variety of ways in which the number twelve has been number involved, and to expressed. In the different representations shown, see if you can calculate with it. deduce what each individual mark represents. Each representation Numerals for very large expresses this many: or small numbers are nonetheless also difficult to ―see‖ or to write, which is one reason that scientific notation is used with them. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 25 Ancient symbols and ,,XIIours: Some students Old Chinese Old Greek Roman Babylonianbecome very interested to learn about other symbol systems for numbers. In the ?Egyptian numerals, the ? 12 30•symbol for 10 was a Mayan Aztec Today, Today, heelprint; for 100, a base ten base fourcoiled rope; for 1000, a 4lotus flower; for 10, a pointing finger; for 5610, a tadpole; for 10, an astonished man. Think About... How would ten have been written in each of these earlier numeration systems? Some ancient cultures did not need many number words. For example, Note: Tell students that i they may have needed words only for ―one,‖ ―two,‖ and ―many.‖ When the superscript number after ―easily‖ and all larger quantities were encountered, they could be expressed by some sort superscript Roman of matching with pebbles or sticks or parts of the body, but without the numerals of this sort use of any distinct word or phrase for the number involved. For example, refer to references in a recently-discovered culture in Papua New Guinea, the same word provided at the end of the ―doro‖ was used for 2, 3, 4, 19, 20, and 21. But by pointing also to chapter. Numbers different parts of the hands, arms, and face when counting and saying referring to powers, such as powers of ten, will ―doro,‖ these people could tell which number is intended by the word. have larger superscripts This method of pointing allows the Papua New Guineans to express and use Arabic numerals. inumbers up through 22 easily. It was only when this culture came into contact with the outside world and began trading with other cultures that SEE INSTRUCTOR they needed to find ways of expressing larger numbers. NOTES 2.1B for additional information on the research in Papua Think About… Why do you think we use ten digits in our number New Guinea. system? Would it make sense to use twenty? Why or why not? Discussion 1. The types Discussion 1 Changing Complexity of Quantities Over Time of quantities get more What quantities, and therefore what number words, would you complex (e.g., from counting sabre-tooth expect a caveman to have found useful? (Assume that the tigers to measuring a caveman had a sufficiently sophisticated language.) A person in a container of grain to primitive agricultural society? A pioneer? An ordinary citizen measuring radiation), and living today? A person on Wall Street? An astronomer? A the sizes cover a greater subatomic physicist? range (small whole numbers to using scientific notation). Try to incorporate reference to a Take-Away Message…Mathematical symbols have changed over the years, and ratio into the discussion-- they may change in the future. Symbols used for numbers depend upon our need to e.g., so many beads for determine the value of the quantities with which we work. each container of corn. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 26 Learning Exercises for Section 2.1 1. Based on what you have seen of the old counting systems such as SEE INSTRUCTOR Greek, Chinese, Roman, Babylonian, Mayan, and Aztec, which NOTE 2.1C for references on the history of systems make the most sense to you? Explain. numberation and a possible 2. Symbols for five and for ten often have had special prominence in additional credit assignment. geographically and chronologically remote systems. Why? 3. Numbers can be expressed in a fascinating variety of ways. Different LE 2.1 In our experience languages, of course, use different words and different symbols to students can get bogged down in this section. Take represent numbers. Some counting words are given below. care that they don‘t. The Take-Away message is the major point made, and the English Spanish German French Japanese Swahili only one necessary to remember. zero cero null zero zero sifuri Students do not have one uno eins un ichi moja answers to 1, 2, 3, 4c, 5c, two dos zwei deux ni mbili 6c, 7 in this set of exercises. three tres drei trois san tatu Be sure to assign 2. four cuatro vier quatre shi nne five cinco fünf cinq go tano six seis sechs six roku sita seven siete sieben sept shichi saba eight ocho acht huit hachi nane nine nueve neun neuf kyu tisa ten diez zehn dix ju kumi For exercises 4, 5, and 6: Warning—Do not get Which two sets of these counting words most resemble one another? caught up in work with Roman numerals. Why do you think that is true? Do you know these numbers in yet Students should know another language? they exist but they need not be fluent in using 4. Roman numerals have survived to a degree, as in motion picture film them. credits and on cornerstones. Here are the basic symbols: I = one, V = five, X = ten, L = fifty, C = one hundred, D = five hundred, and M = one thousand. For example, CLXI is 100 + 50 + 10 + 1 = 161. What numbers does each of these represent? a. MMCXIII b. CLXXXV c. MDVII 5. How would each of the following be written in Roman numerals? For example, one thousand one hundred thirty would be MCXXX. a. two thousand sixty-six b. seventy-eight c. six hundred five Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 27 6. Other systems we have seen all involve addition of the values of the symbols. Roman numerals use a subtractive principle as well; when a symbol for a smaller value comes before the symbol for a larger value, the former value is subtracted from the latter. For example, IV means 5 -1 = 4, or four; XC means 100 - 10 = 90; and CD = 500 - 100 = 400. Note that no symbol appears more than three times together, because with four symbols we would use this subtractive property. What number does each of these represent? a. CMIII b. XLIX c. CDIX 7. Even within the same language, there are often several words for a given number idea. For example, both ―two shoes‖ and ―a pair of shoes,‖ refer to the same quantity. What are some other words for the idea of two-ness? 2.2 Place Value Place value is What does each 2 in 22,222 mean? The different 2s represent different foundational to all values because our Hindu-Arabic numeration system is a place-value work with whole numbers and decimal system. This system depends upon the powers of ten to tell us the numbers, and the lack meaning of each digit. Once this system is understood, arithmetic of understanding of operations are much easier to learn. Understanding of place value is a place value leads to fundamental idea underlying elementary school mathematics. most errors when computing with these But first, what does it mean to have a place-value system? numbers. Yet we often overlook its importance because working with base ten has become second In a place-value system, the value of a digit in a numeral is determined nature to us. It often by its position in the numeral. takes time for teachers to appreciate how Example 1: In 506.7, the 5 is in the hundreds place, so it represents important place value five hundred. The 0 in 506.7 is in the tens place, so it represents is and how to use this knowledge in teaching. zero tens, or just zero. The 6 is in the ones place, so it represents six ones, or six. And the 7 is in the tenths place, so it represents seven- tenths. The complete 506.7 symbol then represents the sum of those values: five hundred six and seven-tenths. Notice that we do not say ―five hundred, zero tens, six and seven-tenths,‖ although we could. This is symptomatic of the relatively late appearance, historically, of a symbol for zero. The advantage of having a symbol to say that nothing is there is apparently a difficult idea, but the idea is vital to a place-value system. Would 506.7 mean the same number if we omitted the 0 to get 56.7? The 0 may have evolved from some type of round mark written in clay by the Babylonians to show that there are zero groups of a particular place value needed. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 28 ... In the Hindu-Arabic place-value system, how many Think About different places (positions) can you name and write numerically? (Don‘t forget places to the right of the decimal point.) Before the use of numerals became widespread, much calculation was done with markers on lines for different place values. The lines could be on paper, or just drawn in sand, with small stones used as markers. (Our word ―calculate‖ comes from the Latin word for ―stones.‖) One device that no doubt was inspired by these methods of calculating is the abacus, which continues to be used in some parts of the world. The abacus shown is the Chinese version. Each of the two counters above the bar represents 5; the ones below, 1. The Japanese abacus (soroban) has only 1 counter above the bar and 4 below. Discussing how A Chinese Abacus numbers are shown on these is worthwhile. Even Notice that we have often used words to discuss the numbers instead of today the abacus is used the usual numerals. The reason is that the symbol ―12‖ is automatically in Asian countries, and associated with ―twelve‖ in our minds because of our familiarity with the Japanese parents may send their children to usual numeration system. We will find that the numeral ―12‖ could mean special classes at night to five or six, however, in other systems! (If no base is indicated, assume the learn abacus calculation. familiar base ten is intended.) In our base-ten numeration system, the whole-number place values result from groups of ten—ten ones, ten tens, ten hundreds, etc. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 work fine until we have ten of something. But there is no single digit that means ten. When we have ten ones, we think of them as one group of ten, without any left-over ones, and we take advantage of place value to write ―10,‖ one ten and zero left-over ones. Similarly, two place values are sufficient through nine tens and nine ones, but when we have ten tens, we then use the next place value and write ―100.‖ It is like replacing ten pennies with one dime, and trading ten dimes for a dollar. Example 2 A few of your students are likely to need to If I want to find the number of ten dollar bills I could get for $365, think through these examples. However, the answer is not just 6, it is 36. don‘t belabor the If I want to know how many dollars bills I could get from $365, the examples. answer is 365. If I want to know the number of dimes I could get from $365, the answer is 3650. But if I want to know how many tens are in 365, I could say either 36 or 36.5, depending on the context. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 29 If I have 365 bars of soap and I want to know how many full boxes of 10 I could pack, the answer would be 36. If I am buying 365 individual bars of soap priced at $6 per 10 bars, then I would have to pay 36.5 times $6. With a good understanding of place value, the problems like those in Example 2 can be easily solved without undertaking long division or multiplication by 10 or powers of 10. Children who do not understand place value will often try to solve the problem of how many tens are in 365 by using long division to divide by 10, rather than observing that the answer is obvious from the number. Discussion 2. This might be a good discussion in pairs. Students are not used Discussion 2 Money and Place Value to considering digits other than the one Explain your answers to each of the following: named. That is, for 2., 1. How many ten-dollar bills does the 6 in $657 represent? The 5? they answer 5 rather than 65. 2. How many tens are in 657? 1. 60, 5 3. How many one-hundred dollar bills can you get for $53,908? 2. 65 3. 539 4. How many one-hundreds are in 53,908? 4. 539 or 539.08 5. How many pennies can you get for $347? For $34.70? For $3.47? 5. 34700, 3470, 347 6. 347, 34 or 34.7 6. How many ones are in 347? In 34.70? In 3.47 (#6 does not follow the The decimal point indicates that we are beginning to break up the unit one pattern. Ask why.) into tenths, hundredths, thousandths, etc. But the number one, not the decimal point, is the focal point of this system. So 0.642 is 642 SEE INSTRUCTOR thousandths of one. Put another way, 0.6 is six tenths of one, while 6 is six NOTE 2.2A about ones, and 60 is six tens, or 60 ones. But just as 0.6 is six tenths of one, 6 is an important six tenths of 10, 60 is six tenths of one hundred, and so on up the line. Or student starting with smaller numbers, 0.006 is six tenths of 0.01, while 0.06 is six misconception about tenths of 0.1. Likewise, 6000 is 60 hundreds, 600 is 60 tens, 60 is 60 ones, decimal numbers. ii 6 is 60 tenths, 0.6 is 60 hundredths, 0.06 is 60 thousandths, and so on. While this at first might seem confusing, it becomes less so with practice and thought. Take-Away Message…Our base ten place-value numeration system is adequate for expressing all whole numbers and many decimal numbers. The value of each digit in a numeral is determined by the position of the digit in the numeral. Digits in different places have different values. Finally, the number 1, not the decimal point, serves as the focal point of decimal numbers. LE 2.2 Students do Learning Exercises for Section 2.2 not have answers for 1. a. How many tens are in 357? How many whole tens? 1b, d, f, j, k, l, 3a, 5, and 8. b. How many hundreds are in 4362? How many whole hundreds? c. How many tens are in 4362? How many whole tens? Be sure to assign 2, 6, and 9, and discuss 2. d. How many thousands are in 456,654? How many whole thousands? e. How many hundreds are in 456,654? How many whole hundreds? Some answers for Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 30 Exercise 1 if discussed in class- 1c. 436.2, 436 1g. 234.7, 234 1h. 23470, 23470 f. How many tens are in 456,654? How many whole tens? g. How many tenths are in 23.47? How many whole tenths? h. How many thousandths are in 23.47? How many whole thousandths? i. How many ones are in 23.47? How many whole ones? j. How many hundredths are in 23.47? How many whole hundredths? k. How many tenths are in 2347? How many whole tenths? l. How many tenths are in 234.7? How many whole tenths? 2. In 123.456, the hundreds place is in the third place to the left of the decimal point; is the hundredths place in the third place to the right of the decimal point? In a long numeral like 333331.333333, what separates the number into two parts that match in the way hundreds and hundredths do? 3. a. Is the statement ―For a set of whole numbers, the longest numeral will belong to the largest number‖ true or false? Why? b. Is the statement ―For a set of decimals, the longest numeral will belong to the largest number‖ true or false? Why? 4. Pronounce 3200 in two different ways. Do the two pronunciations have the same value? 5. Write in words the way you would pronounce each: a. 407.053 b. 30.04 c. 0.34 d. 200.067 e. 0.276 6. Each of the following represents work of students who did not understand place value. Find the errors made by these students, and explain their reasoning. Be sure to assign this exercise. Understanding a. b. c. 7 1the student errors, all of 15 55 48 which stem from lack of + 95 + 48 – 2 6 understanding of place 1010 913 1 1 value, will motivate work on place value both in base ten and in other d. e. bases. 36 36 743 x 8 42 2448 22 21 7. In base ten, 1635 is exactly ________ ones, is exactly _________ tens, is exactly _________ hundreds, is exactly _________ thousands; it is also exactly __________ tenths, or exactly _________ hundredths. 8. In base ten, 73.5 is exactly ________ ones, is exactly _________ tens, is exactly _________ hundreds, is exactly _________ thousands; it is also exactly __________ tenths, or exactly _________ hundredths. 9. Do you change the value of a whole number by placing zeros to the right of the number? To the left of the number? Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 31 2.3 Bases Other Than Ten SEE INSTRUCTOR NOTE 2.3A Too often children learn to operate on numbers without having a deep information on understanding of place value, the lack of which leads them to make many teaching bases other computational errors. The purpose of this section is to provide experiences than ten, and the with base numeration systems other than ten so you understand the amount of time you underlying structure of the base ten system of numeration. You are not should spend on this section—less than expected to become fluent in a base other than ten. Rather, you should be what students want. able to calculate in different bases to the extent that is needed to understand the role of place value in calculations. IMPORTANT: Think About…We use a base ten system of counting because we have ten Note that base fingers. Other cultures have used other bases. For example, some Eskimos materials are used in were found to count using base five. Why would that be? What other bases this section. Have might have been used for counting? your students bring the cutouts of base Cartoon characters often have three fingers and a thumb on each hand, a materials found in total of eight fingers (counting thumbs) instead of ten. Suppose that we Appendix D. live in this cartoon land and instead of having ten digits in our counting Counting in other system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) we have only eight digits (0, 1, 2, 3, 4, 5, bases is very new to 6, 7). Using this new counting system we write the number eight as most if not all of your 10, meaning 1 group of eight and 0 ones. Thus, we would write as we eightstudents. Spend time count in base eight: in class counting with them in different 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, ... bases, explaining why 10 means We read this list of numbers as: one, two, three, . . . , one-zero, one-one, different amounts in one-two,. . . , two-zero . . different bases. The ―humps‖ come at 20 and 100, and understanding what Activity 2 Place Value in Cartoon Land these numerals mean. 1. Show the value of each place in base eight by completing this pattern: Activity 2:Place Value... . . . _____ _____ _____ _____ _____ _____ Be sure you work out 543210 this or a similar example 8 8 8 8 8 8 in class. ? ? ? sixty-fours eights ones 2. What would follow 77 in base eight? 3. What would each digit indicate in the numeral 743 in base eight? Notice that the base-eight numeration system has eight digits, 0–7. Writing 6072 in base eight would require the use of the first four places to 01the left of the decimal point and represents 2 ones (8), 7 eights (8), 0 sets 23of eight squared (8), and 6 sets of eight cubed (8). The digits 6, 0, 7, and 2 would be placed in the Activity pattern in the four places to the left of the decimal point. We call this number ―six zero seven two, base eight‖ and write it as 6072. eight Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 32 ...If you had 602 chairs in an auditorium, how many Think Abouteight chairs would you have, written in base ten? Think About: 386 Discussion 3 Place Value in Base Three What are the place values in a base three system? What are the A major difficulty is digits, and how many do we need? (Rather than invent new breaking away from the symbols for digits, let's use whichever of the standard symbols we identification of a string of digits as always need.) Study the chart below. What should be in place of the having its base ten question marks? meaning, so we use the familiar word name to Items Name in base ten Name in base three Base three symbol indicate the number. zero zero 0 three one one 1 three two two 2 three three ??? ??? Naming three in base three is a key step in understanding base three. Since there are three single boxes above, they will be grouped to make one group of three, and the base three symbol is 10! Notice that in base three ―10‖ does not symbolize ten as we think about ten. In base three, ―10‖ means ―one group of three and zero left over.‖ Since it does not mean ten, we should not pronounce the numeral as ―ten.‖ The recommended pronunciation is ―one zero, base three,‖ saying just the name for each digit and for the base. Notice how this chart differs from the one above. Items Name in base ten Name in base three Base three symbol zero zero 0 three one one 1 three two two 2 threeContinuing in base three, 12 is three one-zero 10 threepronounced ―one two, base three,‖ and the next four one-one 11 threenumber is 20, or ―two zero, base If there are four boxes, as in the last line of this table, we can make one three.‖ group of three, and then there will be one left-over box, so in base three, four is written ―11‖. Because we have the strong link between the marks ―11‖ and eleven from all of our base ten experience, the notation 11is three often used for clarity to show that the symbols should be interpreted in Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 33 should be pronounced ―one one, base three,‖ base three. Recall that 11 three and not as ―eleven.‖ Activity 3 Count in Base Three and in Base Four Continue to draw more boxes and to write base three symbols. What do you write for five boxes? (Now you see why the symbol 12 might mean five.) Six? Seven? Eight? And, at another dramatic point, nine? Did you write ―100‖ for nine? What would 1000mean? threethree Check your counting skills by following along with counting in base four: 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, 200 . . . What does 1000mean? four Discussion 4. Working... Discussion 4 Working with Different Bases 1. Thirty-eight (in base ten) would be expressed as 123 1. What are the place values in base five? What digits are needed? in base five. How would thirty-eight (in base ten) be expressed in base five? Record the first fifteen counting numbers in base five: 1, 2, ... 2. Digits for 0, 1, 2, ..., b–1 are needed. What are the place values in a base b place-value system? What 2. digits are needed? 3. Eighteen would be 3. What are the place values in a base-two place-value system? How expressed as 10010 in base two. Simplistically, two-would eighteen (in base ten) be written in base two? The inner state systems—off/on, workings of computers use base two; do you see any reason for magnetized/ not—fit base this fact? two. 4. Perhaps surprisingly, there is a Duodecimal Society, which 4. Digits would be needed promotes the adoption of a base twelve numeration system. What to express ten and eleven. are the place values in a base twelve system? What new digits T and E can be used here as would have to be invented? needed. With several numeration systems possible, there can be many ―translations‖ among the symbols. For example, given a base ten numeral (or the usual word), find the base six (or four or twelve) numeral for the same number, and vice versa, given a numeral in some other base, find its base ten numeral (or the usual word). In each case, the key is knowing, and probably writing down, the place values in the unfamiliar system. 0 (Recall that any non-zero number to the 0 power is 1. Example: 5= 1. ) Example 3: Changing from a non-ten base to base ten: What does 2103 four represent in base ten? Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 34 Solution: has four digits. The first four place values in base four are 1. 2103four written here, and the given digits put in their places: ____1 ____ ____0 ___ ___3___ ____2_____ of four sixteens, of four fours, or of four ones, or 0 ones, or 4 Materials for different 321 or sixty-four, or 4 sixteen, or 4 four, or 4 bases, like blocks or software online are 3 2. What does the 2 tell us? The 2 stands for two of 4which is highly recommended. 2 64 = 128 in base ten. ，Many students need to manipulate things 2 3. What does the 1 tell us? The 1 stands for one 4which is (blocks of wood, 1 16 = 16 in base ten. ，representations on a screen) to fully 1 4. What does the 0 tell us? The 0 stands for zero of 4which is understand these 0 4 = 0 in base ten. exercises. Emphasize the ， importance of the place 0 5. What does the 3 tell us? The 3 indicates 4is used three times, values. Two-dimensional blocks can be found in 3 1 = 3 in base ten. ，Appendix D, and are introduced in this 6. Thus 2103 = (128 + 16 + 0 + 3) = 147 four tentenchapter. that is, 2103= 147 . fourten Example 4 Section 2.5 Suppose instead we want to change a number written in base ten, discusses the say 236, to a number written in another base, say base five. We use of bases to know that the places in base five are the following: understand place value. You may want to assign it to be . . . ___________ ___________ ___________ __________ read now. 210 one-hundred- twenty-fives (5) fives (5) ones (5) 3twenty fives (5) Solution: (You may find these steps easier to follow by dropping the ten subscript for now, for numbers in base ten.) 1. Look for the highest power of 5 in the base ten number; here it is 3 4 5because 5is 625 and 625 is larger than 236. Are tenten ten333 there any 5s in 236? Yes, just one 5 because 5 = 125, and ten there is only one 125 in 236. Place a 1 in the first place above to 3indicate one 5. Now you have ―used up‖ 125, so subtract: 236 – 125 = 111. tententen 22. The next place value of five is 5. Are there any twenty-fives in 2 111? There are 4, so place a 4 above 5.Now four twenty-ten fives, or 100, have been ―used," and 111 - 100 = 11.tententen Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 35 13. The next place value is 5 which is 5. How many fives are in 11 ? ten 1. It has two fives, so place 2 above 5 There is 1 one left, so place a 01 above 5. Thus 236= 1421. tenfive Working with different bases can be easier when one can physically move pieces that represent different values in a base system. Often, after doing physical manipulation, one can mentally picture the manipulation and work without physical objects. Multibase blocks are manipulatives that Multibase Blocks for have proven to be extremely useful in coming to understand any base students: An appendix system, but primarily base ten in elementary school. Multibase blocks are contains ―Multibase wooden or plastic blocks that can be used to demonstrate operations in Blocks‖ with small different bases. For base ten, a centimeter cube can be used to represent a blocks, the longs, and the unit or one; a long block one centimeter by one centimeter by ten flats in two dimensions, in bases two, four, five, centimeters (often marked in ones) would then represent ten; ten longs and ten. There are two together form a flat that is one cm by ten cm by ten cm and that represents pages for base five. They one hundred, ten flats form a ten cm by ten cm by ten cm cube that can be printed on card represents thousands. If the long is used for the unit, then the small cube stock for students to cut would represent one-tenth, the flat would represent ten, and so on. The out and use. multibase blocks can be used to strengthen place value understanding. SEE INSTRUCTOR NOTE 2.3B. on Multibase Blocks. If the multibase blocks are not available, then they can easily be sketched SEE INSTRUCTOR as shown below: NOTE 2.3C for a game that can be used to assess place value understanding. The materials are often called ―small cube, long, flat, big cube.‖ Any size of the multibase blocks can be used to represent one unit. Familiarize yourself with the multibase blocks by doing this section's Learning Exercises and making up more problems until you feel you are familiar with the blocks and their relationships. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 36 Example 5 The sketch below represents numbers with bases larger than five because there are five flats. If the little cube represents the unit one, the number here is 520 for any base larger than five. If the long represents one, then the number represented here is 52 in any base larger than five. If the flat represents one, then the number represented here is 5.2 in any base larger than five. Discussion 5 Representing Numbers with Multibase Drawings Discussion 5. 1. Here is a representation of a number: Possible answers include-- In base ten: If the flat is one unit then the number Which bases could use this representation if it is in the final represented is 2.52. If the form, with no more ―trades‖ possible? Why? What are some long is the unit, then the possible numbers that can be represented by this drawing? number represented is 25.2 In base six: If the small 2. In base eight, how many small cubes are in a long? How many cube is one unit then the in a flat? How many in a large cube? How many longs in a flat? number represented is How many flats in a large cube? Answer the same questions for 252. If the long is the sixbase ten; for base two. unit, then the number represented is 25.2. sixOne can also represent decimal numbers with base ten blocks or drawings. This drawing cannot You must first decide which block represents the unit. If the unit is the represent a number in a long, then the small block is one-tenth, the flat is ten, and the large block base less than six. is 100. Thus 2.3 in base ten could be represented as: Activity 4 Representing Numbers with Multibase Blocks For these problems, use your cutout blocks (from the appendix) or use Preparing for Activity 4. drawings such as shown above. Note that the drawings do not show Have your students bring the markings of the base that appears in the picture of the blocks, and to class the cutouts of thus do not clearly indicate the base in the way that multibase blocks base materials in the appendix. The markings do. on them assist the 1. Represent 2.3 in base ten using the long as one unit. Represent 2.3 student in understanding using another size of block as the unit. Compare your this activity better than the drawings. representation with a neighbor. in two different ways. 2. Use the base five blocks to represent 2.41 five Be sure to indicate which piece represents the unit in each case. Take-Away Message…We could just as easily have based our number system on something other than ten, but ten is a natural number to use because we have ten fingers. By working in bases other than ten, you have probably gained a new perspective on the structure and complexity of our place value system, particularly the importance of the value of each place. This understanding underlies all of the Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 37 procedures we use in calculating with numbers in base ten. As teachers, you will LE 2.3: need this knowledge to help students understand computational procedures. Students do not have answers for these Learning Exercises for Section 2.3 exercises: 3c, d, e, f, g; 1. If you have access to the internet, go to , 6d, e, f, 7, 8, 9d, e, and find Virtual Library, then Numbers and Operations, then 3-5, then f, 10, 13b, 14b, 15c,d,e, 16f, g, h, 17e, to Base Blocks. You cannot choose numbers to represent, but you can 18c,d,e,f, 21. set the base and you can set the number of decimal points. Practice doing this with the following: Be sure to assign 5, 7, and 13. a. whole numbers in base ten, b. decimal numbers in base ten, Exercise 1 will provide excellent practice with c. whole numbers in base five, bases, but students must have access to d. ―basimal‖ numbers in base five. the internet for this 2. Write ten (this many: ) in each given system. exercise. a. base four b. base five c. base eight 3. Write each of these. a. four in base four b. eight in base eight In Exercise 3, we continue (from c. twenty in base twenty d. b in base b Exercises 1) the 232e. b in base b f. b+ b in base b use of the term ―basimals,‖ the g. 29 in base three h. 115 in base five tentenbase b analogue to decimals in base i. 69, in base two j. 1728, in base twelve tententen. The term is again used in 4. Write the numerals for counting in base two, from one through twenty. Exercise 13. 5. How do you know that there is an error in each statement? a. ten = 24b. fifty-six = 107 threeseven c. thirteen and three-fourths = 25.3 four 6. Write each of these as a base ten numeral with the usual base ten words. 2For example, 111= (1 ， 2) + (1 ， 2) + (1 ， 1) = 7 and 31.2 twoten four25= (3 ， 4) + (1 ， 1) + = 12 + 1 + = 13.5, or thirteen and five-tenths. 410 a. 37 b. 37 c. 207.0024 twelvenineten d. 1000 e. 1,000,000 f. 221.2 twotwothree 7. For a given number, which base—two or twelve—will usually have a numeral with more digits? What are the exceptions? 8. In what bases would 4025 be a legitimate numeral? b 9. Compare these pairs of numbers by placing < or > or = in each box. a. 34 34 b. 4 4 c. 43 25five sixfivesixfive six d. 100 18 e. 111 7f. 23 23 five ninetwoten six five Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 38 10. On one of your space voyages, you uncover an alien document in which some ―one, two,...‖ counting is done: obi, fin, mus, obi na, obi obi, obi fin, obi mus. What base does this alien civilization apparently use? Continue counting through twenty in that system. 11. Hints of the influence of other bases remain in some languages. What base could have led to each of these? a. French for eighty is ―quatre-vingt.‖ b. The Gettysburg Address, ―Four score and seven years ago...‖ c. A gross is a dozen dozen. d. A minute has 60 seconds, and an hour has 60 minutes. mean? What is this number written in base ten? 12. What does 34.2 five 13. In each number, write the ―basimal‖ place values and then the usual base ten fraction or mixed number. We use the word ―basimal‖ in place of 21117 Example: 10.111= (2 + 0 + + + )= 2 (Recall: 4 = 2 ―decimal‖ in two ten 82483problems with and 8 = 2.) fractional parts because ―deci‖ refers a. 21.23b. 34.3 fourtwelveto the number 10. 14. Write each of these in ―basimal‖ notation. Thus ―basimal‖ can refer to any base. Example: three-fourths in base ten is what in base two? 311() = ( + )= 0.11 tenten two424 a. one-fourth, in base twelve b. three-fourths, in base twelve c. one-fourth, in base eight 15. Give the base ten numeral for each given number. a. 101010 b. 912 c. 425 two twelvesix d. 41.5 e. 1341 eightfive 16. Write this many ,,,,,,,,,,,, in each given base. (Note that there are 12 diamonds.) ten a. nine b. eight c. seven d. six e. five f. four g. three h. two 17. Write 100 in each given base. ten a. seven b. five c. eleven d. two e. thirty-one 18. Complete with the proper digits. a. 57 = ____ b. 86 = _____ tenfivenine ten c. 312 = ______ d. 237 = ____ four tenteneight e. 2101 = _____ f. 0.111 = ______three tentwo ten 19. Represent 34 in base ten, with the small block as the unit; with the long as the unit. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 39 20. a. Represent 234with the small cube as the unit. (Notice that 234 five does not mean two-hundred thirty-four here.) b. Represent 234with the small cube as the unit. six (If you have only base ten blocks available, then sketch drawings for these exercises.) 21. In base six, 5413 is _________ ones, is _________ sixes, is 23 _________ sixs; is __________ sixs. 22. Represent 2.34 in base ten with the flat as the unit. 23. Decide on a representation with base ten blocks for each number. a. 3542 b. 0.741 c. 11.11 24. Represent 5.4 and 5.21 with base ten blocks, using the same block as the unit. (What will you use to represent one?) Many school children say that 5.21 is larger than 5.4 because 21 is larger than 4. How would you try to correct this error using base ten blocks? 25. Someone said, ―A number can be written in many ways.‖ Explain that statement. 2.4 Operations in Different Bases Just as we can add, subtract, multiply, and divide in base ten, so can we perform these arithmetic operations in other bases. The standard algorithm for addition, depicted first below, is commonly used and is probably known to all of you. The expanded algorithms make the processes easier Algorithms: Some to understand. Once it is well understood, an expanded algorithm is easily students may not know adapted to become the standard algorithm. Not all standard algorithms in what ―algorithm‖ this country are used in other countries, so the word ―standard‖ is a means: A step-by-step relative one. procedure for solving a problem in a finite In base ten we could add 256 and 475 in these two ways, as shown here. number of steps. Both (There are other ways, of course.) The first way is called an expanded procedures used here are algorithms: The algorithm, and the second, called the standard algorithm, is probably the first is the standard one you were taught. algorithm, the second 11is an expanded 256 256 algorithm that more + 475 + 475 clearly indicates the reasoning, and which 11 (thinking 6 + 5) 731 can eventually lead to 120 (thinking 50 + 70) understanding the 600 (thinking 200 + 400) standard algorithm. 731 The expanded algorithm is now being taught in some schools as a preparation for the standard algorithm. Note how place value is attended to in the expanded algorithm: add the ones 6 + 5, then add the tens 50 + 70, then add the hundreds, 200 + 400, then add the resulting sums, 11 + 120 + 500. In the standard algorithm, each ―column‖ is treated the same: 6 + 5 in the column on the right, 5 + 7 +1 in the middle column, and 2 + 4 +1 in Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 40 the column to the left. Although the standard algorithm leads to the correct answer, students frequently do not know why each step is taken. But when the expanded algorithm is understood, it can be condensed into the standard algorithm as shown above. We can also use either method for adding in other bases, but the expanded algorithm is sometimes easier to follow until adding in another base is well understood. Example 6 Here is an example using both the standard and expanded algorithms to add the same two numbers in base ten and base eight. Make sure you can understand each way in each given base. 1 1 351 351 tenten + 250 +250 ten ten 601 1 thinking (1 + 0) tenten 100 thinking (50 + 50) ten 500 thinking (300 + 200) ten 601 thinking (1 + 100 + 500) ten 1 351 351eighteight + 250 eight + 250 eight 621 1eight thinking (1 + 0) eighteight 120 thinking (50 + 50) eighteight 500 thinking (300 + 200) eighteight 621 thinking (1 + 20 + 100 + 500)eighteight Activity 5 Adding in Base Four Activity: Adding... Add these two numbers in base four in both expanded and standard Answer: 1202 four. algorithms: 311 and 231(Drawings of base four pieces may fourfour. be helpful.) If we can add in different bases, we should be able to subtract in different bases. Here is an example of how to do this. Example 7 Find 321 – 132. fivefive One way to think about this problem is to regroup in base five just as we do in base ten, then use the standard way of subtracting in base ten. Solution: 321 - 132 Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 41 Step 1: We cannot remove 2 ones from 1 one, so we need to take one of the fives from 321 and trade it for five ones: five 321 , 300+ 20+ 1 , 300+ 10+ 11 fivefivefivefivefivefivefive Step 2: We can now take 2 ones from 11 ones (in base five) leaving 4 ones. (Notice how 321 has changed with 3 five squared, then 1 five, then 11 ones, from Step 1.) 1 1 3 2 1 five means 3 (five squared) + 1 five + 11 ones as in Step 1. -1 3 2 five 4 five Step 3: In the fives place: We cannot subtract 3 fives from 1 five, so we must change one five squared to five sets of five. This, together with the one five already in place, gives us 11 fives (or six fives). That is: 300+ 10 , 200 + 110 so fivefivefivefive, 2 11 1 1 3 2five means 11 fives, not 11 ones, so the 11 stands for 110 1 3 2 five and 11 fives minus 3 fives is 3 fives, or 110 – 30 is 30) 1 3 4five We now have 2 (five squared) from which 1 (five squared) is subtracted, leaving 1 (five squared). The answer is 1 five- squared plus 3 fives plus 4 ones which is 134. five Activity 6 Subtracting in Base Four SEE INSTRUCTOR Subtract 231 from 311 in base four. four fourNOTE 2.4A for an example of how to Subtracting in base four is similar to adding in base four. However, for subtract using both operations we can use base materials to help visualize adding and blocks: 124 – 65 in subtracting in other bases. We will do that next. You can cut out and use base ten. materials from an appendix on bases. As you use the base materials, notice how they support the symbolic work you did earlier in this section. Example 8 Suppose we want to add 231 and 311 using base four four four blocks, using the small block as the unit. We could first express the problem as Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 42 We have too many longs (in base four), so trade four longs for a flat. Now we have too many flats (each representing four squared). Trade four flats for a large cube (which represents four cubed). The answer is here represented: 1202The blocks represent four. one four cubed, two four squared, and two ones. Example 9 Suppose we want to subtract 23 from 3This time let us use four four. the long as the unit. 32is represented: four I cannot remove 3 longs (ones) until I change a flat to four longs (which means change one four into four ones). Remove two flats; three longs To take away 23 we must remove three longs (three ones), and four 2 flats ( 2 fours), and we are left with 3 as the difference. four Think About…If we had used the small block as the unit in the above subtraction example, would the numerical answer be different? Try it. Activity 7 Subtracting in Base Four Once again, subtract 231 from 311 in base four, this time using four four drawings. We can also multiply and divide in different bases. However, the intent here is to introduce you to different bases so that you have a better understanding of our own base ten system, and that you understand why children need time to learn to operate in base ten. Thus there are no examples or exercises provided here for multiplication and division in different bases, although it is certainly possible to carry out these operations. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 43 Take-Away Message… Arithmetic operations in other bases are undertaken in the same way as in base ten. However, because we have less familiarity with other bases, arithmetic operations in those bases take us longer than operations in base ten. For children not yet entirely familiar with base ten, time needed to complete arithmetic operations takes longer than it does for us. LE 2.4: Students do not have answers for: Learning Exercises for Section 2.4 3c, d. 4c, d; 7c; 8d. 1. Add 1111and 2102without drawings and then with drawings threethreeAgain, #3 provides in the ways illustrated above. Which way did you find it easier? excellent practice with bases and should be 2. Do these exercises in the designated bases, using the cardboard cutouts assigned if students in an appendix, or with drawings. have access to the internet. a. 341b. 101c. 321d. 296 fivetwofourten Block drawings are not + 220+ 110, 123, 28 fivetwofourtenprovided in the answers except for 4a. 3. Go to , then Numbers and Operations, then 3-5. Go to Base Blocks Decimals. You cannot choose numbers to add and subtract, but you can set the base and you can set the number of decimal points. Do the following: a. Practice adding and subtracting numbers in base ten using whole numbers. b. Practice adding and subtracting numbers using one decimal place. c. Practice adding and subtracting numbers in base four using whole numbers. d. Practice adding and subtracting numbers in base four using one ―decimal‖ place. 4. Add the following in the appropriate bases, without blocks unless you need them. a. 2431b. 351c. 643d. 99 fivenineseveneleven + 223 + 250 + 134 + 88 fivenineseveneleven 5. Subtract in different bases, without blocks unless you need them. a. 351b. 643c. 2431d. 772 ninesevenfiveeleven – 250 – 134 – 223 – 249 ninesevenfiveeleven 6. Do you think multiplying and dividing in different bases would be difficult? Why or why not? 7. Use the cut-outs from the appendix for the different bases to act out the following. As you act each out, record what would take place in the corresponding numerical work. a.232b. 232c. 232d. 101four five eight two 13 13 13 11 four five eight two 113+ 113 +113 +111 four five eight two Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 44 8. Use the cut-outs from an appendix for the different bases to act out the following. As you act each out, record what would take place in the corresponding numerical work. a. 200b. 200c. 200d. 100 four five eight two –13 –13 –13 –11four fiveeighttwo 9. Describe how cut-outs for base six would look. For base twelve. These notes reflect a 2.5 Issues for Learning: Understanding Place Value great deal of research on the The notion that ten ones and one ten give the same number is vital to important topic of understanding the usual numeration system, as are the later rethinking of place value. ten tens as one hundred, ten hundreds as one thousand, etc. Understanding place value is considered to be foundational to elementary school mathematics. But base ten for children might be as mysterious as base b may have been for you. (Admittedly, your extensive experience with base ten also gets in the way!) By working with other bases, you have had the opportunity to explore what it means to have a place-value system where each digit has a particular meaning, and thus come to a better understanding of our base ten system of writing numbers and calculating with numbers. One activity-centered primary program incorporates many activities involving grouping by twos, by threes, and so on, even before extensive work with base ten groupings, to accustom the children to counting not just one object at a time, but groups each made up of several objects. The teen numbers: The Ungrouping needs to be included also. That is, 132 could be regarded as words for our ―teen‖ numbers are somewhat one one-hundred, three tens, and 2 ones. Or, it could be regarded as one backward and puts our one-hundred and 32 ones. Here, the 3 tens are ―unbundled‖ to make 30 students at a ones. Regarding a group made up of several objects as one thing is a disadvantage compared major step that needs instructional attention. to students in cultures where the manner in The manner in which we vocalize numbers can sometimes cause problems which the teens are for students. For example, some young U.S. children will write 81 for spoken expresses the eighteen, whereas scarcely any Hispanic children (diez y ocho = eighteen) ten-ness first (e.g., Spanish: 16 = diez y seis, or Japanese children (ju hachi = eighteen) do so. (Some wishfully think 17 = diez y siete, 18 = we should say ―onetyeight‖ for eighteen in English.) What other numbers diez y ocho, 19 = diez y can cause the same sort of problem that eighteen does? nueve [the first few teens are irregular in Spanish: Place value instruction in schools is often superficial and limited to once (11), doce, trece, studying only the placement of digits. Thus, children are taught that the 7 catorce, quince]; in 7200 is in the thousands place, the 2 is in the hundreds place, a 0 is in Japanese is completely regular: ju ichi, ju ni, ju the tens place, and a 0 is in the ones place. But when asked how many san, ju shi, etc.). hundred dollar bills could be obtained from a bank account with $7200 in it, or how many boxes of ten golf balls could be packed from a container with 7200 balls, children almost always do long division, dividing by 100 or by 10. They do not read the number as 7200 ones, or 720 tens, or 72 hundreds, and certainly not as 7.2 thousands. But why not? These are all names for the same number, and the ability to rename in this way provides Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 45 a great deal of flexibility and insight when working with the number. (It is interesting that we later expect students to understand newspaper figures such as $3.2 billion. What does .2 billion mean here?) Over the years many different methods have been used to teach place value. An abacus with nine beads on each string is one type of device used to represent place value. The Base Ten Blocks pictured in Section 2.3 have been extensively used to introduce place value and operations on whole numbers and decimal numbers. One problem with these representations, however, is that students do not always make the connections between what is shown with the manipulative devices and what they write on paper. Our place value system of numeration extends to numbers less than 1 also. The naming of decimal numbers needs special attention. The place value name for 0.642 is six hundred forty-two thousandths. Compare this to reading 642, where we simply say six hundred forty-two, not 642 ones. This is a source of confusion that is compounded by the use of the tenths or hundredths with decimal numbers, the use of ten or hundred with whole numbers, and the additional digits in the whole number with a similar name. The number 0.642 is read 642 thousandths, meaning 642 thousandths of one, while 642,000 is read 642 thousand, meaning 642 thousand ones. That tens and tenths, hundreds and hundredths, etc., sound so much alike no doubt causes some children to lose sense-making when it comes to decimals. Some teachers resort to a digit-by-digit pronunciation—―two point one five‖ for 2.15—but that removes any sense for the number; it just describes the numeral. Plan to give an artificial emphasis to the -th sound when you are discussing decimals with children. (You can also say ―decimal numeral two and three-tenths‖ and ―mixed 3.) numeral two and three-tenths‖ to distinguish 2.3 and 210 To compare 0.45 and 0.6, students are often told to ―add a zero so the numbers are the same size.‖ (Try figuring out what this might mean to a student who does not understand decimal numbers in the first place!) The strategy works, in the sense that the student can then (usually) choose the larger number, but since it requires no knowledge of the size of the decimal numbers, it does not develop understanding of number size. Instead of annexing zeros, couldn't we expect students to recognize that six-tenths is more than forty-five hundredths because 45 hundredths has only 4 tenths and what is left is less than another tenth? But for students to do this naturally, they must have been provided with numerous opportunities to explore—and think about—place value. Comparing and operating on decimals, if presented in a non-rule oriented fashion, can provide these opportunities. If teachers postpone work with operations on decimals until students conceptually understand these numbers, students will be much more successful than if teachers attempt to teach iiicomputation too early. Some researchers have shown that once students have learned rote rules for calculating with decimals, it is extremely difficult for them to relearn how to calculate with decimals meaningfully. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 46 2.6 Check Yourself In this chapter you have explored the ways we express numbers. Historically, many numeration systems were used to express numbers in different ways. A place value numeration system such as the modern world now uses provides a far more efficient way to express numbers than ancient systems, such as the Roman numeral system. Our use of base ten is probably due to the fact that we have ten fingers. Other bases could be used. Because we are so familiar with base ten, however, working with other bases is useful in appreciating the difficulties children have in learning to use base ten, particularly when learning to operate with numbers in base ten. Understanding place value and its role in the elementary school mathematics curriculum is crucial. Too many teachers think that teaching place value is simply a matter of noting which digit is in the ones place, which is in the tens place, etc. But it is only when students have a deep understanding of place value that they can make sense of numbers larger than 10 and smaller than 1, and understand how to operate on these numbers. Most arithmetic errors (beyond careless errors) are due to a lack of understanding of place value. Unfortunately, the algorithms we teach usually treat digits in columns without attending to their values, and students who learn these algorithms without understanding the place value of each digit are far more likely to make computational errors. You should be able to work problems like those assigned and to meet the following objectives. 1. Discuss the advantages of a place value system over other ancient numeration systems. The questions and 2. Explain how the placement of digits determines the value of a problems given here number in base ten, on both sides of the decimal point. represent the key items in this chapter. 3. Explain how the placement of digits determines the value of a If you do not cover number in any base, such as base five or base twelve and answer all sections, the questions such as: What does 346.3 mean in base twelve? Convert questions may need that number to base ten. to be modified. 4. Given a particular base, write numbers in that system beginning with one. 5. Make a drawing with base materials that demonstrates a particular addition or subtraction problem, e.g., 35.7 + 24.7 or 35.7 – 24.7 in base ten. 6. Write base ten numbers in another base, such as 9 in base nine, or 33 in base two. 7. Add and subtract in different bases. 8. Understand the role of the unit, one, in reading and understanding decimal numbers. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 47 9. Discuss problems that children who do not have a good understanding of place value might have when they do computation problems. References for Chapter 2 iSaxe, G. B. (1981). Body parts as numerals: A developmental analysis of numeration among the Oksapmin in Papau New Guinea. Child Development, 52, 306–316. iiSowder, J. T. (1997). Place value as the key to teaching decimal operations. Teaching Children Mathematics, 3(8), 448-453. iiiHiebert, J. & Wearne, D. (1986). Procedures over concepts: The acquisition of decimal number knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 199-223). Hillsdale, NJ: Erlbaum. Reasoning About Numbers and Quantities Chapter 2 Instructor‘s Version p. 48 Chapter 3 Understanding Whole Number Operations SEE INSTRUCTOR NOTE 3 FOR AN OVERVIEW OF THIS CHAPTER. There are two major ideas interspersed in this chapter. The first is that an arithmetic operation such as subtraction can be modeled by many different situations. Teachers need to know what these situations are in order to After the study of understand and extend children‘s use of the operations. The second is that additive combinations the kind of procedures that children can develop for computing, when they and comparisons, have not yet been taught the usual standard algorithms, can demonstrate a student methods for deep conceptual understanding that might be lost if standard algorithms addition and are introduced too soon. The examples of student work given in this subtraction are examined. These section are all taken from published research, although in some cases the methods are often numbers have been changed. Most of the examples of nonstandard surprising to teachers, algorithms come from students who have been in classrooms where there but are well liked. is a strong emphasis on building on intuitive knowledge and on place Prospective and value understanding. The students demonstrate that they are able to practicing teachers have often not seen compute with ease. Some of the examples are from classrooms where the how students can students have calculators always available. However, they tend not to use operate on numbers calculators if they can do the calculation easily themselves using paper when they have been and pencil and/or mental computation. The use of calculators introduced provided opportunities these students to new ways of using numbers, including, in some cases, to come to a deep understanding of place using negative numbers to help them in their computation. value and have not been required to learn 3.1 Additive Combinations and Comparisons standard algorithms before they understand When does a problem situation call for adding? When does it call for place value. subtracting? Both types of situations are considered in this section. At times, knowing when and what to add or subtract is not at all easy. To SEE INSTRUCTOR solve more difficult problems, we turn once again to undertaking NOTE 3.1A for quantitative analyses of problems. information about addititive Activity 1 Applefest combinations and comparisons and Consider the following problem situation: for an appropriatae (but not unique) Tom, Fred, and Rhoda combined their apples for a fruit stand. Fred and drawing and solution Rhoda together had 97 more apples than Tom. Rhoda had 17 apples. for the Applefest i Tom had 25 apples.activity. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 49 Perform a quantitative analysis of this problem situation with these four steps. 1. Identify as many quantities as you can in this situation, including those for which you are not given a value. Can you make a drawing? Encourage students to 2. What does the 97 stand for in this situation? make a diagram representing the 3. How many apples did Fred and Rhoda have together? How many quantitative apples did all three of them have combined? relationships, and discuss the results. The 4. How many apples did Fred have? diagram shows how two line segments Note in the above problem that when Fred and Rhoda combined their joined together could apples, they had 97 more apples than Tom. The 97 apples does not refer to represent Tom and Fred and Rhoda‘s combined total, rather it refers to the difference between Rhoda‘s apples their combined total and Tom‘s number of apples. combined. Consider the following drawing that represents the applefest problem. Rhoda 17 apples Fred ?? apples Tom 25 apples difference of 97 apples Quantities are often combined additively (put together) and the new quantity has the value represented by the sum of the values of the quantities being combined. The sum of the number of Rhoda‘s apples and The use of ―additively‖ the number of Fred‘s apples is the value of the number of apples here may sound belonging to the two. Quantities can also be compared additively. In the redundant. ―Combining two quantities‖ is applefest problem, the quantities consisting of the value of the quantity generally understood in represented by Rhoda‘s and Fred‘s combined apples is compared to the the additive sense—not value of the quantity represented by Tom‘s apples. multiplicatively. Later we will distinguish between additive and multiplicative combinations. When quantities are combined additively, they are joined together, so In the next section we the appropriate arithmetic operation on their values is usually addition. describe ―missing addend‖ subtraction, This operation is called an additive combination. The result of an which must be additive combination is a sum of the values combined. Any time we distinguished from cases where addends are compare two quantities to determine how much greater or less one is known and can be than the other, we make an additive comparison. The difference of two combined through addition. quantities is the quantity by which one of them exceeds or falls short of the other. The appropriate arithmetic operation on their values is usually subtraction. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 50 The reason this comparison in this problem is called additive is that two quantities can also be compared multiplicatively. We will study multiplicative comparisons in a later section. Typically, subtraction is the mathematical operation used to find the difference between the known values of two quantities. We can think of the difference as the amount that has to be added to the lesser of the two to make it equal in value to the greater of the two. Thus, rather than use additive/subtractive comparison we use the shorter additive comparison. The following diagrams illustrate how one might think of a situation as either a difference of two quantities or as a combination of two quantities (even when, in the latter case, the value of one of the quantities being combined is unknown). Example 1 Julian wants to buy a bicycle. The bike costs $143.95. Julian has a total of $83.48 in cash and savings. How much more does he need? Two ways to conceive the situation The drawings help in seeing how the quantities are related in this situation. Although typically subtraction is the operation used to find a difference, this is not always the case. Consider the following example. Example 2 Jim is 15 cm taller than Sam. This difference is five times as great as the difference between Abe and Sam‘s heights. What is the difference between Abe and Sam‘s height? Solution: The difference between Abe‘s and Sam‘s heights is 15 ? 5 cm, or 3 cm. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 51 ...What would the diagram for Example 2 look like? Think About Example 2 illustrates that the additive comparison of two quantities does not automatically signal that one must subtract. How the quantities in a situation are related to each other determines the mathematical operations that make sense. Understanding the quantitative relationships is essential to being able to answer questions reliably involving the values of quantities in the given situation. Without such an understanding one has no recourse but to guess what mathematical operations are needed, as was illustrated in the last chapter. In the absence of real understanding it is the unreliability of the guessing games that students often play that makes story problems difficult for many of them. SEE INSTRUCTOR NOTE 3.1B. It Activity 2 It’s Just a Game contains instructional Practice doing a quantitative analysis to solve this problem: information on the problem in Activity 2, Team A played a basketball game against Opponent A. Team B played a including a drawing representing the basketball game against Opponent B. The captains of Team A and Team B situation. argued about which team beat its opponent by more. Team B won by 8 more points than Team A won by. Team A scored 79 points. Opponent A scored 48 points. Team B scored 73 points. How many points did Opponent B score? Students may have difficulty interpreting the statement, ―The captain 1. What quantities are involved in this problem? (Hint: there are of Team B won the more than four.) argument by 8 points.‖ Make sure that this is 2. What does the 8 points refer to? clearly understood. 3. There are several differences (results of additive comparisons) in this situation. Sketch a diagram to show the relevant differences in After students have had a chance to work on these, the problem. someone should put a 4. Solve the problem. diagram on the board for discussion showing all of What arithmetic operations did you use to solve the problem? 5.the quantities and their 6. For many students this is a difficult problem. Why do you suppose relationships. See the this is the case? Is the computation difficult? What is difficult instructor‘s note above about this problem? for one possible diagram. It is good to share 7. Suppose you are the teacher in a fifth-grade class. Do you think diagrams that, although telling your students that all they need to solve this problem is different, describe the subtraction and addition, will help them solve it? Explain. same relationship. Take-Away Message…The problems undertaken in this section involved additive combinations (which can be expressed with an addition equation) and additive comparisons (which can be expressed with a subtraction equation). What makes these problems difficult is the complexity of the quantitative structures of the problems, not the arithmetic. The problems involve only addition and subtraction. Understanding the quantitative relationships in a problem is what’s crucial. Once again, these problems illustrate how undertaking a quantitative analysis by listing quantities and using diagrams to explore the relationships of the quantities can help one understand a complex problem situation. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 52 Learning Exercises for Section 3.1 LE 3.1. Students do not To develop skill in analyzing the quantitative structure of situations, work have answers for 1, 2, the following tasks by first identifying the quantities and their and 5b,d. They have a relationships in each situation. Feel free to draw diagrams to represent the complete solution for 9 relationships between relevant quantities. and answers only for 6 1. a. Bob is taller than Laura. Suppose you were told Bob‘s height and and 10. You may want them to show how they Laura‘s height. How would you calculate the difference arrive at the answers for between their heights? 6 and 10. b. Bob is taller than Laura. Suppose you were told the difference SEE INSTRUCTOR between Bob‘s height and Laura‘s height. Suppose you were also NOTE 3.1C about told Laura‘s height. How would you calculate Bob‘s height? advice to give students about these exercises. 2. Kelly‘s mom timed her as she swam a 3-lap race in 1 minute 43 seconds. Her swimming coach timed Kelly only on her last two laps. Describe how Kelly might calculate how long it took her to swim the Don‘t be shy about first lap using the information from her mom and the coach. assigning these problems. They provide valuable 3. Metcalf School has two third-grade rooms (A and B) and two fourth-practice in the kind of grade rooms (C and D). Together, rooms C and D have 46 students. reasoning that we want students to develop and Room A has 6 more students than room D. Room B has 2 fewer become adept at using. students than room C. Room D has 22 students. How many students are Unless they practice this there altogether in rooms A and B? type of reasoning it is unlikely that they will 4. In Exercise 3, what quantities are combined? Which quantities are become good at it. compared? Assigning all of the problems is not a bad idea, 5. Following are two variations of the activity ―It‘s Just a Game,‖ from but you may wish to space the last Activity, but the captain of Team B still wins the argument by 8 out the assignment. Have points in each case. them work on some problems in class (in pairs a. Variation #1: Team A scored 79 points. Opponent A scored 53 or small groups) and others points. Fill in the blanks with scores for Team B and Opponent at home. B so that the specified conditions are met. Be sure to assign 11, which provides the type of Team B ______ Opponent B ______ practice necessary to develop effective b. Variation #2: Team B scored 75 points. Opponent B scored 69 quantitative reasoning. You points. Give possible scores for Team A and Opponent A so that may want to have students the specified conditions are met. undertake 11 together with previous exercises. Team A _____ Opponent A _____ c. Compare your response to Variation #1 with those of classmates. Are they correct? How many possible responses are there? d. What did you find out after thinking through Variation #2? State your conclusion and explain why it is the case. 6. Connie bought several types of candy for Halloween: Milky Ways, Tootsie Rolls, Reese‘s Cups, and Hershey Bars. Milky Ways and Tootsie Rolls together were 15 more than the Reese‘s Cups. There were 4 fewer Reese‘s Cups than Hershey Bars. There were 12 Milky Ways and 14 Hershey Bars. How many Tootsie Rolls did Connie buy? Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 53 7. a. One day Annie weighed 24 ounces more than Benjie, and Benjie 1 pounds less than Carmen. How did Annie's and weighed 34 Carmen's weights compare on that day? b. Why can't you tell how much each person weighed? 8. You have two recipes which together use a pound of butter. One recipe 1takes pound more than the other one. How much butter does each 4 recipe use? 9. A hospital needs a supply of an expensive medicine. Company A has the most, 1.3 milligrams, which is twice the difference between the weight of Company B's supply and Company C's, and 0.9 mg more than Company C's supply. How many milligrams can the hospital get from these three companies? 110. A city spent about of its budget on buildings and maintenance, 41which was about of the total budget less than it spent on 8 administrative personnel. The amount spent on administrative 1personnel was about of the budget more than was spent on public 8 safety. About what part of the budget was left for other expenses? 11. All of the problems in the previous exercises involve the quantitative operations of combining and comparing quantities. Choose two of them and for each state one additive combination or one additive comparison that is involved. SEE APPENDIX B on using the video clip 3.2 Ways of Thinking About Addition and Subtraction Strategies (of first graders), on the IMAP CD. Although the idea of combining two or more quantities additively seems One possible way to handle quite simple, situations involving additive combinations can vary in this section is to assign it difficulty for young children. Situations involving subtraction are quite for reading and have students write a page or varied, and when only one type of situation is taught (for example, take two reflecting on the away), children can have difficulty with other situations (for example, significance it has for comparing) that also call for subtraction. In this chapter we consider two teaching. Class discussions types of situations calling for addition and three calling for subtraction. are also a good way for students to share ideas A problem situation that calls for addition often describes one quantity about the possible being physically, or actively, put with another quantity, as in Example 3. implications for teaching. One such implication that Example 3 is important to bring up is that teachers need to really Four girls were in the car. Two more got in. How many girls were listen to their students if there then? they want to understand how their students are Another type of problem situation that calls for addition involves making sense of conceptually (rather than physically or literally) placing quantities mathematics. SEE INSTRUCTOR together, thus leading to another view of what addition means. NOTE 3.2A on what should be in content Example 4 courses versus methods Only 4 cars and 2 trucks were in the lot. How many vehicles were courses. there altogether? Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 54 Even though both problems involve the additive combination of two quantities, and they both call for the mathematical operation of addition, many research studies have shown that the second problem is more difficult for young children. They may not yet understand concept relations as ―cars and trucks are simply two different kinds of vehicles.‖ Addition can describe situations that involve an additive combination of quantities, either literally (as in the first example) or conceptually (as in the second example). Numbers added together are called addends. The number that is the result of an addition is called the sum. In either of the problems given in Examples 3 and 4, the numbers being added (2 and 4) are addends, and the result is called the sum (6). An important distinction calling for awareness by the teachers is this: The calculation one does to solve a problem may be different from that suggested by the action described in the problem or by the problem‘s underlying structure. The following illustrates this important point. Example 5 Josie needs to make 15 tacos for lunch. She has made 7 already. How many more tacos does she have to make? You would probably regard this as a subtraction problem, because one can calculate the solution from the number sentence 15 – 7 = n. Yet, when young children have solved similar problems, perhaps with the aid of concrete materials, they often count out 7 blocks, and then put out additional blocks, one at a time, until they have 15. Their actions are a reenactment of the problem‘s story. The action of the problem suggests addition of an unknown number of tacos to the 7 tacos, to give a total of 15. Mathematically this problem can also be described by the sentence 7 + n = 15, where the sum and one addend are known, but the other addend is missing. If the numbers were larger, say 1678 + n = 4132, you would probably want to subtract to find n. Such missing-addend (or missing-part) situations, then, can be solved by subtraction, and missing-addend story problems may be in a subtraction section of an elementary textbook. Note the potential confusion for students due to the mismatch of the joining action suggesting addition and the calculation which uses subtraction. The classification of such problems as ―missing addend‘ problems serves to point out this connection. A problem situation that can be represented as a +? = b is called a missing addend problem. Although the action suggests addition, the missing value is b - a. It is therefore classified as a subtraction problem. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 55 …Why is this problem a missing-addend problem calling for Think About subtraction? Ana has to practice the piano for 15 minutes every day. Today she has practiced 7 minutes. How many more minutes does she have to practice today? In the typical elementary school curriculum, missing-addend situations are not the first ones students encounter when dealing with subtraction. Usually take-away situations are the first introduced under subtraction. In fact, the minus sign (–) is often read as ―take away‖ instead of as ―minus,‖ a reading that can limit one‘s understanding of subtraction. Contrast the following story problem with the missing-addend problem given in the above Think About . Josie made 15 tacos for her friends at lunch. They ate 7 of them. How many tacos did they still have? Think About… How do you suppose children would act out this story problem using counters? In situations like the problem above, children typically start with 15 counters, and then they take-away or separate 7 counters. The remaining counters stand for the uneaten tacos. This enables them to answer the question. In a take-away situation, it is natural to call the quantity that is left after the taking away is done, the remainder. A situation in which one quantity is removed or separated from a larger quantity is called a take-away subtraction situation. What is left of the larger quantity is called the remainder. As in the case of addition, one can distinguish between two types of situations: one in which there is a physical taking-away, and ones in which there is no ―taking-away‖ action as such, but there is a missing part, given a whole. Think About... How would you classify the following problem situation? Of Josie's 15 tacos, 7 had chicken and the rest had beef. How many of the tacos had beef? Because the discussion of missing addends is fresh in your mind, you may be saying to yourself, ―I think I can see both of the last two taco problems as missing-addend problems.‖ [number of tacos eaten (7)] + [number still to eat (?)] = total number (15) Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 56 [number of chicken tacos (7)] + [number of beef tacos (?)] = total number (15) This similarity of structure is one reason that some textbooks emphasize the missing-addend view when dealing with subtraction. Example 6 One can think about 7 – 3 = ? as 3 + ? = 7. Familiarity with 3 + 4 = 7 enables one to see 7 – 3 = 4. Some curricula build considerably on this view and may at some point teach families of facts. The family of (addition and subtraction) facts for 3, 4, and 7 would include all of these: 3 + 4 = 7, 4 + 3 = 7, 7 – 3 = 4, and 7 – 4 = 3. Do you see what might be some advantages to families of facts? A third type of subtraction comes from the additive comparison idea. This type of situation is also found in elementary school curriculum. Consider the following situation: Josie made 15 tacos and 7 enchiladas. How many more tacos than enchiladas did Josie make? Two separate quantities, the number of tacos and the number of enchiladas, are being compared in a how-many-more (or how-many-less) sense. A problem situation involving an additive comparison is referred to as Recall that addition was a comparison subtraction situation. defined as an additive Subtraction can tell how many more, or less, there are of one than of the combination, to be other. The result of the comparison is the difference between the values of distinguished now the two quantities being compared. Comparison subtraction usually from additive appears in the curriculum after take-away subtraction. comparison. When one number is subtracted from another, the result is called the difference or remainder of the two numbers. The number from which the other is subtracted is called the minuend and the number being subtracted is called the subtrahend. Example 7 In 50 - 15 = 35, 50 is the minuend, 15 is the subtrahend, and 35 is the difference (or remainder). In all of the previous examples, the objects under consideration were separate, disconnected objects, like tacos. But there are other situations in Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 57 which we combine or compare things such as distances, areas, volumes, and so forth. Quantities being considered are called discrete if they are separate, non-touching, objects that can be counted. Quantities are continuous when they can be measured only by length, area, and so on. Continuous quantities are measured, not counted. Example 8 The following problem situation is also a comparison subtraction problem. Height is a continuous quantity. The heights given are only approximations of the real height of each. One cannot count an exact number of inches. Johann is 66 inches tall. Jacqui is 57 inches tall. How much taller is Johann than Jacqui? Example 9 The following problem situation is a comparison problem calling for subtraction of discrete quantities. One can count siblings; one cannot have parts of siblings. Johann has 5 siblings and Jacqui has 4 siblings. How many more siblings does Johann have than Jacqui? Activity 3 Which Is Which? Pair up with someone in your class and together classify each of the Activity: Which Is Which? following problem situations. 1. Take-away subtraction 1. Velma received four new sweaters for her birthday. Two of them 2. Missing addend were duplicates, so she took one back. How many sweaters does subtraction she have now? 3. Addition—literally adding 2. Velma also received cash for her birthday, $36 in all. She has an 4. Comparison eye on some software she wants, which costs $49.95. Her dad subtraction, then offered to pay her to wax and polish his car. How much does she addition—implicit because one must need to earn to buy the software? recognize that both 3. Velma also received two mystery novels and three romance novels girls and boys are for her birthday. How many novels did she receive? friends. 4. There were six boys and eight girls at her birthday party. How many more girls than boys from class were at the party? How many friends were at the party? Activity 4 Writing Story Problems Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 58 1. Pair up with someone in your class. Write problem situations (not necessarily in the order given) that illustrate these different views of addition and subtraction: a. Addition that involves putting two quantities together b. Addition that involves thinking about two quantities as one quantity c. Take-away subtraction d. Comparison subtraction e. Missing addend subtraction 2. Share your problems with another group. Each group should identify the types of problems illustrated by the other group, and discuss whether each group can correctly identify the problem situations with the description. In each problem, identify whether the quantities are discrete or continuous. Discussion 1 Identifying Types of Problem Situations If there are any disagreements in the story problems written for the last activity, discuss these as a class. Take-Away Message…Addition is called for in problems of two different generic types: combining actively or implicitly. Subtraction is called for in three different generic types: taking away, comparing, and finding the missing addend. Missing addend problems are sometimes called additive comparison problems, but they are solved by subtracting. If a teacher illustrates subtraction only with take-away situations, then that teacher should not be surprised if his or her students cannot recognize other types of subtraction situations. Also, if students are presented only with discrete objects when they add or subtract, they will not necessarily be able to transfer this knowledge to situations in which the quantities are continuous. One might ask how children cope with the diversity of situations that call for addition or subtraction. The manner in which they act out the story usually shows how they have conceived the situation. The physical situations can be quite different even though the symbolic mathematics may be the same. Learning Exercises for Section 3.2 1. Illustrate the computation 8 – 5 with both continuous and discrete LE 3.2. drawings for for each of the three generic situations for subtraction: a. Students do not take-away, b. comparison, and c. missing-addend. Notice how the have answers drawings differ. for 3, 4b, 5a,b, 6, and 7. Be 2. a. The narrative in this section has neglected story problems like this sure to assign 3, one: ―Jay had 60 pieces of clean paper when she started her 8, and 9. homework. When she finished, she had 14 clean pieces. How many pieces of paper did she use?‖ What mathematical sentence would one write for this problem? Notice that the action here is take-away, but the correct sentence for the action would be 60 – n = 14, not 60 – 14 = n (one could calculate 60 – 14 to answer the question though). Such complications are usually avoided in current K-6 treatments in the United States, but several other countries give considerable attention to these types! Are we ―babying‖ our children intellectually? Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 59 b. Make up a story problem for this ―missing-minuend‖ number sentence: n – 17 = 24. 3. Teachers sometimes emphasize ―key words‖ to help children with story 3. Be sure to discuss the problems. For example, ―altogether‖ suggests addition, ―left‖ suggests serious limitations of subtraction. The intent is good: Have the children think about the this very common situation. But unfortunately children often abuse the key words. They strategy, key words, that many teachers use to skim, looking just for the key words, or they trust them too much, ―help‖ students know taking what is intended as a rough guide as a rule. In the following, tell which operation to use. how the italicized key words could mislead a child who does not read the whole problem or who does not think about the situation. a. Dale spent $1.25. Then Dale had 55 cents. How much did Dale have at the start? b. Each classroom at one school has 32 children. The school has 12 classrooms. How many children are at the school altogether? c. Ben divided up his pieces of candy evenly with Jose and Cleveland. Each of the three boys got 15 pieces of candy. How many pieces did Ben start with? d. Flo has 3 times as much money as Lacy does. Flo has 84 cents. How much does Lacy have? e. Manny's mother bought some things at the grocery store. She gave the clerk $10 and got $1.27 in change. In all, how much did she spend at the store? f. Each package of stickers has 6 pages, and each page has 12 stickers on it. What is the total number of stickers in 4 packages? 4. Consider this comparison subtraction situation: ―Ann has $12.85 and Bea has $6.43. How much less money does Bea have than Ann?‖ Note that the question is awkward. This is believed to be one explanation of why young children have more trouble with comparison story problems than with take-away problems. When story problems are phrased less awkwardly, children tend to do much better. For example, ―There are 12 bowls and 8 spoons. How many bowls do not have a spoon?‖ is easier for the children than, ―There are 12 bowls and 8 spoons. How many more bowls are there than spoons?‖ This may be due to the fact that the phrasing in the former case is suggestive of an action (matching bowls with spoons) that enables students to obtain an answer. Re-phrase the questions to make these problems easier. a. The baseball team has 12 players but only 9 gloves. How many more players are there than gloves? b. There are 28 children in the teacher‘s class. The teacher has 20 suckers. How many fewer suckers are there than children? 5. Give the families of facts for these: a. 2, 6, 8 b. 12, 49, 61 Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 60 c. x, m, and p, where m – p = x 7. Responses here should assume that the 6. Choose a family of facts in 5 above; write one addition word child already views problem and three subtraction word problems, one of each type, subtraction as the for that family of facts. operation needed in ―take away‖ situations. ii 7. In a first-grade class the teacher gave the following problem:Building on this understanding would be There are 12 boys and 8 girls. How many more boys than girls? pedagogically sound. A majority of the children answered ―four more boys‖ correctly, but Thus, helping the child five children chose to add and gave ―20 children‖ as their answer. One conceive the comparison situation as a take-away of these five children insisted that subtraction could not be used situation would because it is impossible to take away 8 girls from 12 boys. None of probably help. One way the other children could respond to this child‘s claim. In light of what to do this is to do a you have learned in this section, describe how you would handle this matching between the situation if you were the teacher, in order to help this child understand girls and the boys. There are 8 boys that are that subtraction is the appropriate operation in this situation. Note that matched with girls. If simply telling a student ―the right way‖ does not necessarily help we ―take these away‖ him/her understand. the remaining 4 are not matched, and this shows 8. Tell which type of subtraction is indicated in each of these story how many more boys problems, with an explanation of your choice. than girls there are. a. Villi is running a 10-kilometer marathon. He has run 4.6 8. These are kilometers. How far does he have yet to go? admittedly trivial word problems. b. Diego is saving up for a car. He needs $2000 to buy one from Students are asked his uncle. Thus far he has $862. How much more does he need? not to answer them but to c. Laresa just got paid for the week. She received $200. She owes understand their her mother $185 for some clothes. Once she pays it, what will structure. she have left? d. Laresa‘s friend works at a different place and is paid $230 a week. How much more does she make per week than Laresa? e. Bo just bought gas for $2.79 per gallon. Last week he bought gas for $2.72. How much more did he pay for gas this week than he did last week? 9. Make up a word problem for each of the two different types of addition situations; make one for the three different types of subtraction situations. 10. Mr. Lewis teaches second grade. He has been using ―take-away‖ problems to illustrate subtraction. A district text contained the following problem: ―Vanessa‘s mother lost 23 pounds on a diet last year. Vanessa herself lost 9 pounds on the diet. How many more pounds did Vanessa's mother lose?‖ Most of Mr. Lewis‘s students were unable to work the problem, and he was very discouraged because he had spent so much time teaching subtraction. Do you have any advice for him? 11. For each of the following problems, name the quantities, their values, note the relationships among the values, make a drawing for the problem, then compute the answer. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 61 Example: JJ earned $2.50 an hour for helping a neighbor. JJ worked 5 hours. Then JJ bought a T-shirt for $7.35. How much money did JJ have left? Solution: Quantities with values are: Amount JJ earns per hour: $2.50. Number of hours JJ works: 5. Cost of T-shirt: $7.35. Amount of money left after purchase of T-shirt: unknown. Amount earned in one hour Amount earned in 5 hours Amount spent on shirt Amount remaining To find the amount earned, I would multiply her wage per hour by the number of hours she worked or add five $2.50s. Then I would subtract the amount she spent on the shirt because that cost is taken away from the total earned. $2.50 5 , $7.35 = $5.15 ， a. A post 12 feet long is pounded into the bottom of a river. 2.25 feet of the post are in the ground under the river. 1.5 feet stick out of the water. How deep is the river at that point? 32b. At one school of all the eighth graders went to one game. of 35 those who went to the game traveled by car. What part of all the eighth graders traveled by car to the game? c. A small computer piece is shaped like a rectangle which is 2.5 centimeters long. Its area is 15 square centimeters. How wide is the piece? d. Maria spends two-thirds of her allowance on school lunches and one-sixth for other food. What fractional part of her allowance is left? 5e. On one necklace, of the beads are wooden. There are 40 beads in 8 all on the necklace. How many beads are wooden? f. A painter mixes a color by using 3.2 times as much red as yellow. How much red should he use with 4.8 pints of yellow? 3.3 Children’s Ways of Adding and Subtracting Children who understand place value and have not yet learned a SEE INSTRUCTOR standard way to subtract often have unique ways of undertaking NOTE 3.3A on ivsubtraction. Prepare to be surprised! preparing students for this activity. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 62 Activity 5 Children's Ways 1. Consider the work of nine second graders, all solving 364 – 79 (in written form, without calculators or base ten blocks). Identify: The children‘s ways of subtracting shown in this a. which students clearly understand what they are doing; section usually come as a complete, sometimes b. which students might understand what they are doing; and humbling, experience to both prospective and c. which students do not understand what they are doing. practicing teachers, because they do not immediately understand the thinking used by each of the students. We advise that you not skip this section. When students realize that they must understand methods other than the ones they use, they are motivated to learn to attend to students‘ reasoning. 2. Pick out places where errors were made, and try to explain why you think the errors occurred. In Activity 5 the solutions of students 2, 3, and 5 are all based on the standard algorithms that are taught in school. Student 3 correctly uses the ―regrouping‖ algorithm taught in most U.S. schools. Describe in writing the steps followed when one uses the standard regrouping algorithm for subtraction. Activity 6 Making Sense of Students’ Reasoning SEE INSTRUCTOR 1. Justify, using your knowledge of place value and the meanings of NOTE 3.3B on the addition and subtraction, the procedure used in Solution 3 of the equal additions method used in activity called Children‘s Ways. Make up some other problems, many other countries. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 63 and talk them through, using the language of ―ones, tens, hundreds‖ etc. 2. Discuss what you think happened in the solutions of Students 2 and 5. 3. The procedure used by the fourth student is called the ―equal additions‖ method. Figure out how this algorithm works, and the mathematical basis for it. Hint: Because 9 cannot be taken from 4, ten was added to both numbers, but in different places. Sometimes teachers dismiss solutions that do not follow a standard procedure. Yet sometimes these solutions show a great deal of insight and understanding of our base ten numeration system and how numbers can be decomposed in many helpful ways. But there are times that a solution that is ―different,‖ such as the equal additions method of subtraction, may not demonstrate any insight. Rather, any procedure can be learned as a sequence of rules. In the United States, subtraction is usually taught by a ―regrouping method‖ (see Student 3‘s method in Activity 5) but in some other countries children are taught the ―equal additions‖ method (see the work of Student 4). Either can be taught rotely or meaningfully. Activity 7 Give It a Try Go back over the solutions of students 1, 6, 7, 8, and 9. Make sure you understand the thinking of each student. You can do this by thinking through each method as you try 438 – 159. The other students appear to have used ―nonstandard algorithms‖ or ―invented algorithms,‖ that is, procedures they have developed on their own, or perhaps as a class for solving subtraction problems. In Activity 7, you should have thought about place value understanding for each of the Discussion 2. Some methods here. students will favor the standard algorithm Think About... If you were (or are) a parent, what would you say to your because it is familiar. child if he or she subtracted like Student 1? or Student 9? What about This is fine, if it is Student 4? Have you ever seen this subtraction method before? If so, tell taught in a meaningful your instructor so that your method might be shared with others. way. Some methods would not be Discussion 2 Is One Way Better? appropriate to teach if students have never Is any one method of subtraction (from the ones illustrated at the worked with negative beginning of this lesson) better than others? Why or why not? numbers. Which methods do you think could be more easily understood? Are there any methods that should not be taught (rather than invented by a student)? Why or why not? Take-Away Message…Children do not think about mathematics in the same way that adults do. Part of the reason for this is that adults are more mature and have a wider variety of experiences with numbers. But also part of the reason is that we have had limited opportunities to explore numbers and the meaning of operations on numbers. When children enter school they are often inquisitive and willing to explore in ways that they often lose in later grades, unfortunately. But when Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 64 children are given numerous opportunities to think about numbers and come to understand the place value system we use, and then apply that understanding in a variety of situations, they often come up with novel ways of approaching problems. The ways they invent for undertaking addition and subtraction are useful to them, and often can be generalized. This knowledge can also help them understand and remember any traditional algorithms they are taught. This has the advantage that they are unlikely to make the common errors that result when place value is not well understood. Learning Exercises for Section 3.3 LE 3.3. Students have 1. Make up some new subtraction problems and try solving them by each Cases C and D for 2, 3, and of the methods illustrated in the activity, Children‘s Ways. Try 4 only, not for Cases A, B, inventing some other ways you think a child might approach your E, F, or G. They also do subtraction problems. not have answers for 1 or 5. 2. Here are several cases of addition and subtraction problems solved by Students will profit by other first and second graders in classrooms where the standard doing all of these exercises. algorithms were not taught. Some procedures are more sophisticated Although there are than others, based on the individual student's understanding of numerically few problems, numeration. Some were done mentally; others were written down. they take quite a bit of Study each method until you understand the thinking of the student, time, so this is actually a and then do the problem given using the same strategy. rather long assignment. Case A. Teacher: What is 39 + 37? Student 1: 30 and 30 is 60, then 9 more is 69, then 7 more is 70, 71, 72, 73, 74, 75, 76. You do: 48 + 59. Case B. Teacher: What is 39 + 37? Student 2: 40 and 40 is 80, but then you need to take away. First 1, and get 79, then 3, and get 76. You do: 48 + 59. Case C. Teacher: What is 39 + 37? Student 3: 40 and 37 is, let's see, 40 and 30 is 70, and 7 more is 77, but you need to take away 1, so it's 76. D. This answer was You do: 48 + 59. actually given by a first-grade child who Case D. A student talks aloud as he solves 246 + 178: was not considered to be special, but who Student 4: Well, 2 plus 1 is 3, so I know it's 200 and 100, so now it's had had a great many somewhere in the three hundreds. And then you have to add the tens opportunities to on. And the tens are 4 and 7. . . .Well, um. If you started at 70; 80, explore number 90, 100. Right? And that's four hundreds. So now you're already in structure. the three hundreds because of this [100 + 200], but now you're in the four hundreds because of that [40 + 70]. But you've still got one more ten. So if you're doing it 300 plus 40 plus 70, you'd have 410. But you're not doing that. So what you need to do then is add 6 more onto 10, which is 16. And then 8 more: 17, 18, 19, 20, 21, 22, 23, 24. iiiAnd that's 124. I mean 424. You do: 254 + 367. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 65 Case E. Teacher: What is 65 – 7? Student 5: 65 take away 5 is 60, and take away 2 more is 58. You do: 58 – 9. Student 6: 65 – 7. 7 and 8 is 15, and 10 is 25, and 10 is 35 and 10 is 45 and 10 is 55 and 10 is 65. Five tens is 50, and 8 is 58. (Uses fingers to count tens.) You do: 58 – 9. Case F. Written solution for 654 - 339. Student 7: 6 take away 3 is 3 and you make it into hundreds so 300. Then you add 50—>350–30 and it comes to 320, +4 is 324, –9 is first –4 is 320 then –5 is 315. You do: 368 – 132. Case G: Oral solution for 500 - 268. Student 8: 2 to get to 270, then 30 more to get to 300, then 200 more, so 232. You do: 800 – 452. 3. Discuss the procedures used in Exercise 2 in terms of how well you think the student understands the procedure, how long you think it took for the student to develop and understand the process being used; and how robust the procedure is, that is, can it be used on other problems? 4. Discuss the procedures used in Exercise 2 in terms of whether the procedure would be difficult for students to remember. Do you think the student, when faced with another subtraction problem, will be able to use his or her procedure? Why or why not? 5. In many European countries and in Australia, addition and subtraction are taught using an ―empty‖ number line that is not marked with 0 and 1. Here are two examples for addition, with 48 + 39: + 30 + 2 + 7 48 78 80 87 + 30 or + 7 + 2 48 50 80 87 Subtraction is done similarly. Consider 364 – 79, a problem you saw earlier. – 50 – 20 – 5 – 4 Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 66 285 305 355 360 364 or – 60 – 5 – 4 – 10 285 290 294 304 364 Try the following computations using the empty number line method. a. 62 + 49 b. 304 – 284 c. 72 – 38 d. 253 – 140 3.4 Ways of Thinking About Multiplication Just as there are different ways of thinking about addition and subtraction, depending upon the quantities involved and the type of situation involved, there are different ways of thinking about multiplication and division. Many children learn to think of multiplication only as repeated addition, which limits the type of situations in which they know to multiply. Discussion 3 When Do We Multiply? How would you solve these two problems? 1. One kind of cheese costs $2.19 a pound. How much will a SEE APPENDIX package weighing 3 pounds cost? B on using the 2. One kind of cheese costs $2.19 a pound. How much will a video clip Javier, which fits very package weighing 0.73 pound cost? well with this lesson. The clip is If you even hesitated in deciding to multiply to solve the second problem, on the IMAP CD. you are not alone. Researchers across the world have noticed that success on the second problem is usually 35 to 40% less than success on the first problem, even among adults! Many solvers think they should divide or subtract on the second problem. Why does this happen? This section will offer a possible reason: Each of multiplication and division, like addition and subtraction, can represent quite different situations, but solvers may not be aware of this fact. Just as there are different ways of thinking about subtraction, there are several ways to think about multiplication and division. We begin with different meanings of multiplication. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 67 When a whole number n of quantities, each with value q, are combined, ， q. the resulting quantity has value q + q + q +… .(n addends), or n This is called the repeated addition view of multiplication. Example 10 Karla invited 4 friends to her birthday party. Instead of receiving gifts, she gave each friend a dozen roses. How many roses did she give away? 4 ， 12 = 12 + 12 + 12 + 12. Some critics say that repeated addition receives attention for so long (grade 2 or 3 on) that it restricts children's attention when other situations that use multiplication appear. Nonetheless, this view does build on the children's experience with addition, and it does fit many situations. Using the notion of repeated addition, 3 ， 4 means the sum of 3 addends, each of which is 4: 3 ， 4 = 4 + 4 + 4. Notice the order; 3 ， 4 means 3 fours, not 4 threes. At least that is the case in every U.S. text series; the reverse is the case in some other countries (British or British-influenced countries, for example), which can cause confusion. Of course, you know from commutativity of multiplication that 3 ， 4 = 4 ， 3, but as a teacher, you would want to model the standard meaning, even after the children have had experience with commutativity. In contrast with commutativity of addition, commutativity of multiplication is much less intuitive: It is not obvious ahead of time that 3 eights (3 ， 8) will give the same sum as 8 threes (8 ， 3). The numbers being multiplied are called factors, and the result is called the product (and sometimes a multiple of any factor). The first factor is sometimes referred to as the multiplier, and the second the multiplicand. In some contexts, divisor is used as a synonym for factor, except divisors are not allowed to be 0 (but factors may be). Note that under the repeated-addition view of multiplication, the first factor must be a whole number. Under this view, 2.3 ， 6 cannot be interpreted; it is meaningless to speak of the sum of 2.3 addends. However, the commutative property of multiplication (which children should know by the time they are solving problems with decimal numbers) tells us that 2.3 ， 6 is the same as 6 ， 2.3. Overall, any story situation in which a quantity of any sort is repeatedly combined could be described by multiplication. 6 ， 2.3 could be used, for example, for ―How much would First Think About. 6 boxes of that dog treat weigh? Each box weighs 2.3 pounds.‖ Multiplication as repeated addition does Think About...Read again the cheese problems in Discussion 3. Does the not fit the second idea of multiplication as repeated addition fit both problems? problem, because it doesn‘t make sense to add 2.19, 0.73 times. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 68 This example illustrates the difficulty many people have with the second type of problem. An emphasis on multiplication only as repeated addition to the exclusion of other interpretations of multiplication appears to lead many children into the unstated and dangerous over-generalization that the product is always larger than the factors, expressed as: Multiplication always makes bigger. ...When does multiplication NOT ―make bigger‖? Think About A second type of situation calling for multiplication is a rectangular array with n items across and m items down, for a total of m ， n items. Or we could generate a rectangle of n squares across and m squares down for a total of m ， n squares. (This is, of course, what we use to find the area of a rectangle.) The array (or area) model of multiplication occurs in cases that can be modeled as a rectangle n units across and m units down. The We begin here to product is m ， n. use property names, which Note: When the factors are represented by letters, we often use a dot or should be known drop the multiplication symbol: m ， n = mn = mn. ,by students. However, we If n and m are whole numbers, this could be considered a special case of advise that they be repeated addition. One attraction of this model is that for continuous pointed out as quantities the n and the m do not have to be whole numbers. Another they occur. There will be many feature of this model is that commutativity of multiplication is easily seen: reminders along Length times width (or number of rows times number in each column) is the way. just reversed if the array or rectangle is turned sideways. 11Think About... Draw a rectangle that is inches across and inches 3224 high. What is the total number of square inches shown? Find the answer SEE INSTRUCTOR first by counting square inches and parts of square inches, and then by NOTE 3.4A for a multiplying. Did you get the same answer? drawing for this Think About. There is a third way to think about multiplication. The fractional part of a quantity model of multiplication occurs when we need to find a fractional part of one of the two quantities. This is sometimes referred to as the operator view of multiplication. With this model we can attach a meaning to products such as 2，17(pounds, say), meaning two-thirds of 17 pounds, or to a product of 3 0.35 ， 8.2 (kilograms), meaning thirty-five hundredths of eight and two- tenths kilograms. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 69 ... Suppose you are dealing with cookies. How could you Think About SEE INSTRUCTOR 11―act out‖ ? How could you ―act out‖ ? How are these different? 6，，622NOTE 3.4B for a Do you get the same answer for both? What is the ―unit‖ or the ―whole‖ drawing for this 1Think About. for the in each case? 2 aYou should have said that the ―unit‖ or ―whole‖ for the in the second b 1situation, , is the amount represented by the second factor, 6 cookies. ，62 We want one-half of six cookies. The reason this is referred to as the aoperator view is because the acts on, or operates on, the amount b represented by the second factor. The operator view will be treated in more detail later, but you now have the background to consider the following question. Think About …Using the part-of-an-amount way of thinking about multiplication by a fraction, tell what 0.4 x 8 or 0.7 x 0.9 might mean. Does that help you locate the decimal point in the 0.4 x 8 (= 3.2) and 0.7 x 0.9 (= 0.63) products? Think About...If you buy 5.3 pounds of cheese costing $4.20 per pound, could this be represented as a repeated addition problem? SEE INSTRUCTOR Think About... What is meant by 5.3 ， 4.20, as in finding the cost of NOTE 3.4C for the buying 5.3 pounds of meat at $4.20 a pound? third Think About. There are other situations in which ordered ―combinations‖ of objects, rather than sums of objects themselves, are being counted and which can be described by multiplication. This fourth type of situation uses what is commonly called the fundamental counting principle. These situations show that, mathematically at least, multiplication need not refer to addition at all. The next activity leads to a statement of that principle. Activity 8 Finding All Orders with No Repeats Allowed You hear that Ed, Fred, Guy, Ham, Ira, and Jose ran a race. You know there were no ties, but you do not know who was first, second, and third. In how many ways could the first three places in the race have turned out? If you were successful in identifying all 120 possible orders for the first three finishers, you no doubt used some sort of systematic method in keeping track of the different possible outcomes. One method which you may encounter in elementary school books is the use of a tree diagram. Here is an example, for a setting that is often used in elementary books: Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 70 You have 3 blouses and 2 pairs of pants in your suitcase, all color-compatible. How many different blouse-pants outfits do you have? At the start, you have two choices to make—a choice of blouse, and a choice of pants. A tree diagram records these choices this way: first choicesecond choice pants 1 blouse 1 pants 2 You will recognize this pants 1counting problem as a Cartesian-product blouse 2(start)problem. pants 2 pants 1 blouse 3 pants 2 Notice that after a blouse is chosen, both possibilities for the choice of pants make the next ―branches‖ of the tree. You make a choice by going through the tree. For example, blouse 1 and pants 2 would be one choice. You make all possible choices by going through the tree in all possible ways. Do you find 6 different outfits? What would the tree diagram look like if you chose pants first? Would there be 6 choices then also? With a situation as complicated as the 6-racers problem in Activity 8, making a tree diagram is quite laborious (try it). Although it is good experience to make tree diagrams, there is an alternative. Notice that in the outfits example, the first act, choosing a blouse, could be done in 3 ways and then the second act, choosing pants, could be done in 2 ways, no matter what the choice of blouse was. And, 3 ， 2 = 6, the number of possible outfits. Situations which can be thought of as a sequence of acts, with a number of ways for each act to occur, can be counted efficiently by this principle: In a case where two acts can be performed, if Act 1 can be performed in m ways, and Act 2 can be performed in n ways no matter how Act 1 .turns out, then the sequence Act 1-Act 2 can be performed in mn ways. This is the fundamental counting principle view of multiplication. The principle can be extended to any number of acts. For example, the racers problem has three ―acts‖: finishing first (6 possibilities), finishing second (5 possibilities after someone finished first), and finishing third (now 4 possibilities). The number of possible first-second-third outcomes to the race can then be calculated by the fundamental counting principle: 6 Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 71 ， 5 ， 4, or 120, without having to make a tree diagram. Notice that here the unit for the end result is different from any unit represented in the factors. (As an aside, notice also that the tree diagram here is not the same as the ―factor tree‖ that you may remember from elementary school.) ... At an ice cream shop, a double-decker ice cream cone can Think About be ordered from two choices of cones, 12 choices for the first dip, 12 for the second dip, and 8 kinds of toppings. How many choices of double-decker ice cream cones are possible? Does your answer include two dips of the same flavor? A later section on multiplicative comparisons (not additive comparisons) will give a fifth important way of thinking about multiplication. Until then, when you use multiplication for a problem, identify which way is involved: repeated addition, array, part-of-a-quantity, or fundamental counting principle. Activity 9 Merrily We Multiply Devise a multiplication problem for each of the types described thus far: repeated addition, array, part-of-a-quantity, or fundamental counting principle. If you do this in pairs or groups, compare your problem situations with those of a neighboring group. Earlier we referred to the commutative property of multiplication, that is, for any numbers m and n, mn = nm. This property tells us we can treat either m or n as the multiplier. You should also remember the associative property of multiplication, that is, for numbers p, q, and r, (pq)r = p(qr). This property tells us that when multiplying, we do not need parentheses, because it does not matter which product we find first. Example 11 8) or 6 ， 8 = 3 ， 16. Thus we could write (3 ， 2) ， 8 = 3 ， (2 ， the product as 3 ， 2 ， 8. Parentheses tell us which operation to do first, but the associative property tells us that when multiplying, it does not matter which multiplication we do first. The distributive property of multiplication over addition (often just shortened to the distributive property) is a particularly useful property: for numbers c, d, and e, c(d + e) = cd + ce. Example 12 Here is an illustration of this property, showing that 2 ， (4 + 3) = (2 ， 4) + (2 ， 3). (The right side could have been written without parentheses because in a string of operations, all multiplication must be done before addition.) Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 72 +4433 222 x2+7=2x( 4 + 3) ( 23)x4)( 2x= Learners often wonder, in the symbolic form, where the other 2 came from. The drawing shows it was there all the time. Take-Away Message…Problem situations that call for multiplication can be categorized in different ways, including the following: 1. Finding the sum when a whole number of like quantities is combined is called the repeated addition view of multiplication. In the United States, finding mn means n + n + n … with n as an addend m times. Using this form consistently makes understanding easier at first. Later, when commutativity is understood, students recognize that the addend and the number of addends could be reversed. 2. Finding the area of a rectangle or the number associated with a rectangular array, is called the array (or area) model of multiplication. This view highlights commutativity of multiplication and the distributive property of multiplication over addition. 3. Finding the fractional part of a quantity is sometimes called the operator view of multiplication because it appears that the multiplier is “operating” on the multiplicand. This third type of situation can involve a product that is smaller than one or both of the factors, which causes difficulty for students who have only thought about multiplication as repeated addition. The misconception that “multiplication makes bigger” is common and needs to be countered during instruction. 4. In situations where finding the number of ways in which one act follows another, the fundamental counting principle view of multiplication applies. Tree diagrams help to illustrate this principle. LE 3.4 Students do Learning Exercises for Section 3.4 not have answers for 2, 1. a. Make sketches for 6 ， 2 and for 2 ， 6 and contrast them. 4, 8, 10c, 10d, 12 and 115，，5b. Make sketches for and for and contrast them. 14. Be sure to assign 2, 223, and 4. c. Make a systematic list of possible names for this problem: Janice is ordering ice cream. There are 5 kinds available: vanilla, strawberry, chocolate, mocha, and butter pecan. There are also five kinds of sprinkles to put on top: M&Ms, coconut, Heath bar pieces, chocolate chips, and walnut bits. If she orders one scoop of ice cream and one kind of sprinkles, how many choices does she have? 2. A teacher quite often has to make up a story problem on the spot. For each view of multiplication, make up TWO story problems that you think children would find interesting. Label them for later reference (e.g., array model). Use a variety of sizes of numbers. You might want to share yours with others. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 73 3. Will multiplying a whole number (>1) by any fraction always result in a product smaller than the whole number we started with? When does multiplication make bigger? 2– – 4.Does repeated-addition make sense for 4 ， 2? For 2 ， 4? For 8 ， ? 3 2For ， 8? Explain your thinking for each case. 3 15. In terms of repeated addition and/or part-of-an-amount, what does 621， 12 mean? (Here is a context that may be helpful: dozen eggs.) 62 6. For each, make up a story problem which could be solved by the given calculation. a. 32 ， 29 b. 4 ， 6.98 c. 0.07 ， 19.95 32d. 12 ， , about pizza. e. ， 6, about pizza. f. 40% ， 29.95 83 7. Here are the ingredients in a recipe that serves 6: 6 skinned chicken breast halves 1 tablespoon margarine 11 cup sliced onion 1 cups apple juice 2 11 tablespoon olive oil 2 tablespoons honey 2 12 cups sliced tart apples teaspoon salt 2 a. You want enough for 8 people. What amounts should you use? b. You want enough for 4 people (and no leftovers). What amounts should you use? 8. The fact that m ， n and n ， m are equal is a very useful idea for, say, learning the basic multiplication facts. For example, if you know that 6 ， 9 = 54, then you automatically know that 9 ， 6 = 54 ... if you believe that m ， n = n ， m. Text series may use the array model to illustrate this idea. An array for 3 ， 5 would have 3 rows with 5 in each row (rows go sideways in almost every text series, and in advanced mathematics). With a quarter-turn, A 3 x 5 arrayof square regionsbecomes a 5 x 3 array Since the two arrays are made up of the same squares, even without counting you know that 3 ， 5 = 5 ， 3. a. Make an array for 4 ， 7 and turn the paper it is on 90?. What does the new array show? b. Use a piece of graph paper to show that 12 ， 15 = 15 ， 12. 11c. Show how this idea works in verifying that 6 ， = ， 6. 22 Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 74 9. a. Show with a drawing that 5 ， ，, ， ,， , , ， , ， , ， ,, What is the name of this property? b. How could you illustrate, without a drawing, the associative property of multiplication? Hint: (a b) c = a (b c). ，，，， c. Show with a drawing that 3 6.5 = 6.5 3. ，， 10. Make a tree diagram or a systematic list for three of these, and check that the fundamental counting principle gives the same answer. Tell how many you found. a. You flip a spinner that has four differently-colored regions (red, white, blue, green) all equal in area, and toss one die and count the dots on top (1 through 6 possible). How many color-dot outcomes are possible? b. A couple is thinking of a name for their baby girl. They have thought of 3 acceptable first names and 4 acceptable middle names (all different). How many baby girls could they have without repeating the whole name!? c. In a sixth-grade election, Raoul, Silvia, Tien, Vena, and Wally are running for president; Angela and Ben are running for vice-president; and Cara and Von are running for treasurer. In how many ways could the election come out? d. In a game you toss a red die and a green die, and count the number of dots on top of each one, for example, R2, G3. (The numbers of dots are not added in this game.) In how many different ways can a toss of the dice turn out? 11. A hamburger chain advertised that they made 256 different kinds of hamburgers. Explain how this claim is possible. Hint: One can choose 1 patty or 2, mustard or not, etc. 12. An ice-cream store has 31 kinds of ice cream and 2 kinds of cones. a. How many different kinds of single-scoop ice-cream cones can be ordered at the store? b. How many different kinds of double-scoop ice-cream cones are there? (Decision: Is vanilla on top of chocolate the same as chocolate on top of vanilla?) 13. You have become a car dealer! One kind of car you will sell comes in 3 body styles, 4 colors, and 3 interior-accessory ―packages,‖ and costs you, on average, $12,486. If you wanted to keep on your lot an example of each type of car (style with color with package) that a person could buy, how much money would this part of your inventory represent? 14. License plate numbers come in a variety of styles. a. Why might a style using 2 letters followed by 4 digits be better than a 6-digits style? b. How many more are there with a style using 2 letters followed by 4 digits than the 6-digits style? How many times as large is the number of license plates possible with a style using 2 letters Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 75 followed by 4 digits as is the number possible with a 6-digits style? (Ignore the fact that some choices of letters could not be used because they would suggest inappropriate words or phrases.) c. Design a style for license-plate numbering for your state. Explain why you chose that style. 15. Locate the decimal points in the products by thinking about the part-of-an-amount meaning for multiplication. Insert zeros in the product, if needed. x 128.5 = 7 7 7 (Where does the decimal point go?) a. 0.6 b. 0.8 x 0.95 = 7 6 c. 0.75 x 23.8 = 1 7 8 5 d. 0.04 x 36.5 = 1 4 6 e. 0.328 x 0.455 = 1 4 9 2 4 f. 0.65 x 0.1388 = 9 0 2 2 3.5 Ways of Thinking about Division Next we turn next to division. There are two common types of situations that call for division, and they both appear in virtually every curriculum. A third view, which you may recognize from your earlier coursework, is more encompassing and mathematically includes the first two views. Under any of these views, this vocabulary applies: In a division situation that can be described by a ? b = q, the a is called the dividend, b is called the divisor, and q is called the quotient. In a division situation in which b is not a factor of a, the situation can be rrdescribed as a ? b = q + , the quotient is q + , and the quantity r is bb called the remainder. This situation is also written as a ? b = q Remainder r. Note that the divisor can never be 0. Example 13 In 28 ? 4 = 7, 28 is the dividend, 4 is the divisor, and 7 is the quotient. In the usual U.S. calculation form, divisordividend, the quotient is written on top of the dividend. 1There may also be a remainder, as in 13 ? 4 = 3 R 1, or 3. 4 Division calculations that have a remainder of 0 are informally said to ―come out even.‖ One basic view of the division a ? b is called the measurement view. The measurement view of division as repeated This follows from ―How many measures of b are in a?‖ Because the subtraction is the usual answer can often be found by repeatedly subtracting b from a, this view basis for the standard algorithm for whole of division is also sometimes referred to as the repeated-subtraction number division. If I set out to find 354 ? 26, I am asking how many 26s are Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 76 in 354. The first step is to write the problem in the long division format and then place a 1 over 5, indicating that there are at least 10 26s in 354, which view of division. This view is also called the quotitive view of division. You will find that different textbooks use different terms. Example 14 When asked what 12 ’ 4 means, one would say it means ―How many 4s are there in 12‖ or ―How many measures or counts of 4 are in 12‖ and the answer can be obtained by repeatedly subtracting 4 until the 12 are gone, resulting in a quotient of 3. When asked what 13 ? 4 means, we can repeatedly subtract 4 three times again, resulting in a partial quotient of 3, but this time there is a remainder, 1, because it was not possible to again subtract 4. 31...What is meant by ? , using the measurement view? Think About42 How can one think of using repeated subtraction to find the quotient? What is meant by 0.6 ? 0.05? Using the measurement view of division one could ask, ―How many 13measures of are in ?‖ 24 Think About 2... Suppose a child has this problem situation: ―Danesse has 3 friends. She has 12 cookies to share with them. How many cookies does each of the four people get?‖ How do you suppose a child who does not yet understand division might model this situation? Another basic view of division is sharing equally. When one quantity (the dividend) is shared by a number of objects (the divisor), the quantity associated with each object is the quotient. This sharing notion of division is also called partitive division. Example 15 Danesse probably shared her cookies by saying one for you, one for you, etc. Thus the 12 cookies are shared equally by four people, and each gets three cookies. Or, you might say that the 12 cookies have been partitioned into four sets of the same size. Activity: Highway Repair Activity 10 Highway Repair 1. Measurement: Are these two problems the same? Which view of division is Repeatedly subtract 2.5 miles until there is no represented in each? Would you make the same drawing for each? highway left. The number of times 2.5 is Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 77 subtracted is the number of days. 2. Partitive: Partition the highway into 2.5 parts (probably by first cutting 1. The highway crew has several miles of highway to repair. Past experience suggests that they can repair 2.5 miles in a day. How many days will it take them to repair the stretch of highway? 2. The highway crew has the same stretch of highway to repair. But they must do it in 2.5 days. How many miles will they have to repair each day to finish the job on time? Discussion 4 Practicing the Meanings Tell what each could mean with each view of division. 1. 85 ? 7 2. 22.5 ? 2 71 (repeated subtraction view only) 3. ,88 4. 1.08 ? 0.4 (repeated subtraction view only) A more general way of thinking about division is the missing-factor view of division in which a question is asked about a missing factor that, when multiplied by the divisor, would result in the dividend. That factor is the quotient. Example 16 12 ? 4 = n would be associated with n ， 4 = 12 (with repeated subtraction in mind) or 4 ， n = 12 (with sharing in mind). Indeed, the common way to check a division calculation is to calculate divisor ， quotient, and see whether the product is the dividend. A missing-factor view could replace the earlier two views of division, once they are understood. Because our emphasis is on mathematics for the elementary school, our attention will center on the repeated subtraction Discussion: and sharing views. Keep them well in mind. Dividing by Zero It is extremely Discussion 5 Dividing by Zero important to have this discussion in ―You can‘t divide by 0.‖ An elementary school teacher should class. The fact that know why this is not the case. Perhaps the easiest way to think one cannot divide by about the issue is to think of checking a division calculation; the 0 is a mystery to related multiplication, divisor ， quotient, should equal the many students. dividend. So 12 ? 4 = 3 is true because 3 ， 4 = 12. 1. ―Check‖ 5 ’ 0 = n. What number, if any, checks? 2. ―Check‖ 0 ’ 0 = n. What number, if any, checks? 3. A child says, ―Wait! Last year my teacher said, ?Any number divided by itself is 1.‘ So isn‘t 0 ’ 0 = 1?‖ What do you reply? Thus, either no number checks (as in 5 ? 0) or every number checks (as in 0 ? 0) when dividing by zero. Either is clearly a good reason to say that division by zero is undefined, or ―You can‘t divide by zero.‖ Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 78 Take-Away Message…Just as with other operations, there are different contexts that can be described by division: 1. Finding the number of times a particular value can be repeatedly subtracted from another value. This view of division has different labels: repeated subtraction division, measurement meaning of division, and quotitive meaning of division. 2. Finding the number in each share if a quantity is shared equally by a number of objects. This view of division is called sharing-equally division, or partitive division. 3. Finding the missing factor that, when multiplied by another known factor, would result in a known product. This view of division is called the missing- factor view of division. Mathematically, this view can encompass the first two views, but we depend on the first two views when helping children to understand division. 4. When one value (the dividend) is divided by another (the divisor, which cannot be 0) the result is a value called the quotient. In situations where the quotient must be expressed as a whole number, there is sometimes a value remaining, called the remainder. 5. For good reasons, division by zero is undefined. That is, one cannot divide by zero. Learning Exercises for Section 3.5 1. For the repeated-subtraction (also called measurement or quotitive) LE 3.5 Students do view of division, make up TWO story problems that you think children not have answers for 2, 3, 4, 5a, 5c, 5d, 7, or 8. would find interesting. Do the same for the sharing (partitive) view of Be sure to assign 1, 2, division. and 9. 2. Illustrate 12 ? 3 with drawings of counters using the repeated-subtraction view and then the sharing view. Notice how the drawings differ. Write story problems that could lead to the following equations. 3. a. 150 ? 12 = n (partitive) b. 150 ? 12 = n (repeated-subtraction) c. 3 ? 4 = n (partitive) d. 3 ? 4 = n (repeated-subtraction) 4. After years of working with division involving whole numbers, children often form the erroneous generalization, division makes smaller. Give a division problem which shows that division does not always make smaller. 5. Consider the following situation: ―Mount Azteca is 1.3 km high. There are 3 different trails from the starting place at the bottom to the top of the mountain. One trail is 4.5 km long, another is 3.5 km long, and the most difficult is 2.4 km.‖ For each computation indicated below, finish the story problem started above so that your problem could lead to the computation described. As Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 79 always, it may be helpful to make a drawing of the situation and think about the quantities involved. a. 3 ， 2.4 b. (4.5 + 3.5) – (2 ， 2.4 c. 4.5 ? 3 (partitive) d. 4.5 ? 3 (repeated subtraction) e. 10 ? 2.4 6. a. A national test once used a question like ―The bus company knows that after a ball game, 1500 people will want to catch a bus back to the main bus terminal. Each bus can hold 36 people. How many buses should the company have ready?‖ What is your answer to the question? b. There will be 300 children singing in a district song fest. Each singer gets a sash. Sashes come in boxes of 8. Calculate 300 ? 8, and then answer the following from your calculation. b1. How many boxes of sashes should the district order? b2. If the district orders only 37 boxes, how many children would not get a sash? 7. Analyze the quantities in this setting to generate some questions you can answer: One company plans to open 505 acres around an old gold-mining site and to process the ground chemically. The company hopes to extract 636,000 1-year ounces of gold from 48 million tons of ore and waste rock, over a 62 period. The company expects to get about $200 million for the gold. 8. The six problems below were taken from elementary school mathematics textbooks. Discuss the meaning or interpretation of division in each problem and explain how you would help students visualize the action in the problem and connect it to a meaning of division. Identify the quantities in each and tell how to use them to draw a diagram for each problem. a. The park bought 900 kg of seal food. The food comes in boxes of 12 kg each. How many boxes did the park buy? (fourth grade) b. The dolphins ate 525 kg of food in 3 weeks. What is the average number of kilograms of food they ate each day? (fourth grade) c. A salesperson drove about 34,500 km in 46 weeks. What was the average distance the salesperson drove each week? (fifth grade) d. The art teacher has 128 straws in the supply room. Each student needs 16 straws for an art project. How many students can do the art project? (fifth grade) e. A strange coincidence. Six hens each weighs the same. Their total weight is 35 lbs. How much does each hen weigh? (sixth grade) f. It takes 1 minute to make a gadget. How many gadgets can be made in 14 minutes? (sixth grade) 9. What is 24 ? 3? 240 ? 30? 2400 ? 300? 24000 ? 3000? 2.4 ? 0.3? Formulate a rule for finding quotients such as these. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 80 10. Reflect on: If you were not taught by a teacher who clarified the meanings of operations, how did this affect your ability to solve, and develop an attitude toward, story problems? 3.6 Children Find Products and Quotients When students can analyze situations that involve finding a product or a quotient, they might use novel methods for computing, perhaps not even Discussion 6: Indian... realizing that they are multiplying or dividing. In this section you will see The work of Student 1 is a variety of ways that children think about multiplication and division ―like‖ the standard algorithm problems. except the work is left to right. Student 2 wanted to Discussion 6 Indian River Oranges find twenty-four 16s, so it made sense to write 16 In a third-grade classroom the focus had been on understanding twenty-four times. She problem situations and the children knew that multiplication can be combined pairs of 16s, v thought of as repeated addition. They were working on this problemgetting twelve 32s, then combined pairs of 32s to get ―Every Christmas my father gets big boxes of fruit, like Indian River six 64s. When combining oranges, from his company. We get so many pieces of fruit that we pairs of 64s, she first have to give some away. This year we ended up making 24 grocery combined pairs of 60 to get bags of fruit with 16 pieces in each bag. How many pieces did we bag three 120s then added on the altogether?‖ Discuss what you think each child was thinking about in six 4s. While this is one of the less efficient ways of each case below. finding the answer, it does 1. work—the student can at 2. 20 x 10 = 200 16least work the problem, based 3220 x 6 =1201664on understanding. A more 64120320 1664efficient way should come 322401664about as she views the work 4 x 10 = 40 120166432of others. 360 4 x 6 =24166464120Look for opportunities for 6416643232016good questions like this one 24384for #3: What if there were 14 (continued 384pieces of fruit in each of the 24 times)th case, the 24 bags? In the 4 student used the equal sign improperly, but did appear to 3.4.understand what the problem 2440816 x 12 = 10 x 12 = 120required. However, 24801672prospective teachers also 6 x 12 = 2416840192often use the equal sign in 2496192this fashion and should not be 96allowed to do so. It can be 38496 very confusing to students 192192and models incorrect writing 384of mathematics. (Note: Student 2 actually wrote the number 16 twenty-four times.) Discussion 7 What Were They Thinking? 1. Justify, using your knowledge of place value and multiplication, each of the methods used in Discussion 6. 2. Discuss the relative clarity and the relative efficiency of the four methods. 3. Another child (Student 5) was asked to find 54 ， 62 and wrote: ―First I did 50 ， 60 wich (sic) came to 3000. Then I did 60 ， 4 and then I added it on to 3000 Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 81 wich (sic) came to 3240. Then I did 50 ， 2 which came to 100 and this I added it on to the 3240 which came to 3340 then I did 4 ， 2 which came to eight and that's how it came to 3348." Work through this thinking to see whether it is correct. Discussion 8 Thinking About Division Discussion 8: 1. 10 is far too viHere is some children's work on what we typically regard as few, 14 is too many, so use 13. The remainder is less than 160 division problems. None of these children had learned the standard so 13 works. The reminder is division algorithm but coped quite well without it. You are to study 120 ml. each method and tell what you think each child is thinking. The first 2. Ask whether the solution child was about 7 years old and had not learned about division, but he shows a partitive (sharing knew how to multiply using the calculator. The second child, also equally) or repeated subtraction way of thinking about division. about 7, knew simple division facts and used them to his advantage. A story context, as in #1, is Note the way he deals with remainders. often helpful in thinking about a solution. There are three 20s in 78, with 18 left over. 18 = 10 + Student 1 Student 2 8. Divide both 8 and 10 by 3. How many cups containing What is 78 divided by 3? There are 2 threes in 8 and 3 160 ml are in 2200 ml? threes in 10. We now have 20 + 20 + 20 + 20 = 60 2 + 3 = 25 threes. Turning to 10 ， 160 = 1600 8 ? 3 = 2 rem 2 the remainders, we find another 13 ， 160 = 2080 10 ? 3 = 3 rem 1 three for a total of 26 threes. 3. 35 twice is 70. So subtracting 2 + 1 = 3 14 ， 160 = 2240 successive 70s is subtracting 3 ? 3 = 1 So 13 cups, 120 ml left two 35s as a time. over. So 78 ? 3 = 26 4. Also repeated subtractions. The Greenwood or ―scaffold‖ algorithm was revived during The next two solutions were from children in fourth grade who had New Math days in the 1960s. been solving problems requiring division. The teacher had not yet 5. One way: Think of sharing; 42 objects are to be placed in 7 taught them a particular method for dividing. However, they knew piles. First put 40 of the objects that division could be thought of as repeated subtraction. 12 ? 3 could into 10 piles (because the child be found by asking how many times 3 could be subtracted from 12; can do this division). I have 4 the answer is 4 times. things in each pile, and I have 3 piles too many. Put those objects together; there are 12 Student 3 Student 4 objects. Add the 2 that were left 280 ? 35 273247from using only 40 objects ，rather than 42, giving 14 280 2700 100 objects. This allows 2 more in - 70 547 each of the seven remaining 270 10 210 piles. There are now 4 + 2 or 6 277 - 70 in each pile. 270 10 140 7 120 - 70 70 So 3247 ? 27 = 120 remainder 7. - 70 The fourth way is the Greenwood or 0 “scaffold” algorithm, dating back to the So four 70s is eight 35s. seventeenth century. The next child was in first grade and knew a few simple division and Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 82 multiplication facts. He also knew what division meant, and in his head he made up a story that he explained after he had given the vii method that appears here. Student 5 What's 42 ? 7? Well, 40 divided by 10 is 4, and 3 times 4 is 12, and 12 and 2 is 14, and 14 divided by 7 is 2, and 2 plus 4 is 6, so it's 6. Do you understand the thinking of each student? (The last one may be difficult.) What knowledge of place value did these students have? Take-Away Message… When young children can work problems such as “Amelia has 6 vases and 24 roses. She wants to put the same number of roses in each vase. The pagination from How many roses will she put in each vase?” or “Amelia has 24 roses and some vases. this page to the end of the chapter is not the If she wants to place 4 roses in each vase, how many vases will she need?,” they are same as in the student doing division. They may find the quotient in each case by acting it out. As the text. numbers in each situation become larger, the methods they use may seem more complex because we find it more difficult to follow their reasoning. In actuality, they are using one of the three different views of division to solve each problem. They deal with the situation at hand, solving the problem in a sense-making manner. Student-devised methods may help them to later develop and understand algorithms for finding quotients. LE 3.6 Students do not Learning Exercises for Section 3.6 have answers to 2 and 3. Assign all six exercises. Exercises 1 through 5 refer to the children‘s work in Discussion 8. 1. How is the first student's method related to estimating? 4. Scaffolding may take 2. The second student shows remarkable facility in dealing with longer because it involves remainders. Try this method yourself on 56 divided by 4. writing more down. (Is this important?) But the 3. Student 3 solved the division problem by repeatedly subtracting. What scaffolding method is more are advantages and disadvantages of this method? easily understood and remembered and takes Student 4's method of dividing is sometimes called the scaffolding 4. much less time to learn. method, and in some schools it is taught as a first algorithm for doing division. Compare it to the standard division algorithm in terms of advantages and disadvantages. 5. Student 5's method for dividing is perhaps the most difficult one to understand. Suppose you start by telling yourself you have 42 candies to share among 7 people, and since you don't know how many each person should get, you begin with 10 piles. When you have figured this method out, try it on 63 ? 9. 6. The following problem was written by a third grader who was challenged by her father to make up some story problems for him to solve. How much of the information given is used to solve the problem? What would you ask this child about this problem (in addition to: Where did you learn the word ―heedless?‖)? Alicia was a heedless breaker, and she broke 24 lamps in one week. Her parents paid $7.00 for every light-bulb she cracked to replace Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 83 it. Her parents only paid for 5 bulbs in a week, like Monday- Friday. Answer the question below: How much money did her parents pay for one week, not counting the week-end? 3.7 Issues for Learning: Developing Number Sense The idea of developing number sense in the elementary grades is a crucial one. Too often, when focusing just on answers rather than on both reasoning and answers, children do not try to make sense of mathematics as shown by contrasting the two elementary classrooms described by viiiHowden. She asked students in one first grade classroom in a very transient neighborhood to tell what came to mind when she said twenty-four. The children immediately gave a variety of answers: two dimes and four pennies, two dozen eggs, four nickels and four pennies, take a penny away from a quarter, the day before Christmas, my mother was 24 last year, when the hand (on a grocery scale hanging in the room) is almost in the middle of twenty and thirty. When she asked this question in a third-grade classroom in a professional community, 24 was just a number that is written in a certain way, that appears on a calendar or on a digital watch. Howden claimed the students in the first grade class had more number sense. She said: Number sense can be described as a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms. Since textbooks are limited to paper-and-pencil orientation, they can only suggest ideas to be investigated, they cannot replace the ―doing of mathematics‖ that is essential for the development of number sense. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels (p. 11). There has been a great deal written about number sense in the past two decades. Educators have come to realize that developing good number sense is an important goal in the elementary school. In fact, an important ixdocument from the National Research Council, called Everybody Counts, contains the statement: ―The major objective of elementary school mathematics should be to develop number sense. Like common sense, number sense produces good and useful results with the least amount of effort‖ (p. 46). Research has shown that children come to school with a good deal of intuitive understanding, but that for many this understanding is eroded by instruction that focuses on symbolism and does not build on intuitive knowledge. Many educators now recognize that we can introduce symbols too soon and that students need to build an understanding of a mathematical phenomenon before we attempt to symbolize it. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 84 Unfortunately, many teachers believe that while an emphasis on understanding is good for some students, others, particularly those in inner-city schools and in remedial programs, need a more rigid approach x to mathematics. Yet a very large research studyof academic instruction for disadvantaged elementary school students showed that mathematics instruction that focused on understanding was highly beneficial to the students. ―By comparison with conventional programs, instruction that emphasizes meaning and understanding is more effective at inculcating advanced skills, is at least as effective at teaching basic skills, and engages children more extensively in academic learning‖ (p. i ). Number sense is a way of thinking about numbers and their uses, and it has to permeate all of mathematics teaching if mathematics is to make sense to students. It develops gradually, over time. All of the instruction in these pages focuses on helping you develop number sense and recognize it when it occurs. For example, mental computation and computational estimation both build on number sense and continue to develop it. The examples of children's methods of operating on numbers demonstrate, in most cases, a good deal of number sense on the part of the students. Take-Away Message…Students exhibit good number sense when they can find ways to tackle problems, such as subtraction, multiplication, and division, even though they have not been taught procedures to solve these types of problems. Teachers must have good number sense if they are going to be successful in helping their students develop it. Do you have good number sense? Do you think you are developing better number sense? Students do not have Learning Exercises for Section 3.7 answers for 1a, c, d; The first five Learning Exercises will, like the earlier children's work, give 2; 4, 5b, c, d; 5; 6 e, f; 7b, 9 you a better idea of what number sense means. 1. In each pair, choose the larger. Explain your reasoning (and use number sense rather than calculating). a. 135 + 98, or 114 + 92 b. 46 – 19, or 46 – 17 131c. +, or 1 d. 0.0358, or 0.0016 + 0.313 242 2. Is 46 ， 91 more, or less, than 5000? Is it more, or less, than 3600? Explain. 3. Suppose you may round only one of the numbers in 32 ， 83. Which one would you choose, 32 or 83, to get closer to the exact answer? Explain with a drawing. 4. Without computing exact answers, explain why each of the following is incorrect. a. 310 b. 119 c. 27 ， 3 = 621 Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 85 520 46 630 137 d. 36 ? 0.5 = 18 150 940 + 470 + 300 2081 602 5. Suppose that the sum of 5 two-digit whole numbers is less than 100. Decide whether each of the following must be true, false, or may be true. Explain your answers. a. Each number is less than 20. b. One number is greater than 60. c. Four of the numbers are greater than 20, and one is less than 20. d. If two are less than 20, at least one is greater than 20. e. If all five are different, then their sum is greater than or equal to 60. 6. Look for compatible numbers to estimate these: a. 36 + 47 + 52 + 18 + 69 b. 39 ， 42 c. 1268 – 927 d. 34,678 ? 49 e. 19 + 26 + 79 + 12 + 74 f. 4367 ? 73 7. a. A quick estimate of 56 ? 9.35 is 5.6, from 56 ? 10. A better estimate would adjust the 5.6 up. Explain why, using a meaning for division. b. A quick estimate of 715 ? 10.2146 is 71.5. A better estimate would adjust the 71.5 down. Explain why, using a meaning for division. 8. Consider 150.68 ， 5.34. In estimating the product by calculating 150 ， 5, one has ignored the decimal part of each number. In refining the estimate, which decimal part (the 0.68 or the 0.34) should be the focus? Explain. 9. In multiplying decimals by the usual algorithm, the decimal points do not need to be aligned (in contrast to the usual ways of adding and subtracting decimals). Why not? Section 3.8 Check Yourself This very full chapter focused on how children can come to understand the four arithmetic operations on whole numbers. As teachers, either now or in the future, you need to be able to reason through a child‘s solution that is different from one you may have seen before. Of course, this cannot always be done quickly, but children should not be told their work is incorrect when it is not. Sometimes a student has a unique and wonderful way of using knowledge of number structure to find a solution--such a child has good number sense. At other times the answers are wrong because of a lack of understanding of the underlying place value structure of the numbers. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 86 You should be able to work problems like those assigned and to meet the following objectives. 1. Identify the operations and the view of that operation that fits a problem situation involving any of the four operations. 2. Write story problems that correctly illustrate any specified view of an operation; for example, write a story problem with the missing addend view of subtraction. 3. Provide reasons why some views of operations are more difficult than others. 4. Explain why using ―key words‖ is not a good strategy for answering story problems (see Section 3.2 Learning Exercises). 5. Study students‘ arithmetic procedures and explain how the students are reasoning. 6. Describe why, in many countries, the empty number line is a popular way of beginning addition and subtraction of multiple digit numbers. 7. Describe how you would explain to someone that ―multiplication makes bigger‖ is not always true. 8. Describe what is meant by ―number sense‖ and how you would recognize it. References for Chapter 3 iThompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, B. Greer, P. Nesher, & G. Goldin (Eds.), Theories of learning mathematics (pp. 267–283). Hillsdale, NJ: Erlbaum. iiHatano, G., & Inagaki, K. (1996). Cultural contexts of schooling revisited: A review of The Learning Gap from a cultural psychology perspective. Paper presented at the Conference on Global Prospects for Education: Development, Culture and Schooling, East Lansing, MI. iiiCarpenter, T. P. (1989, August). Number sense and other nonsense. Paper presented at the Establishing Foundations for Research on Number Sense and Related Topics Conference, San Diego, CA. ivShuard, H., Walsh, A., Goodwin, J., & Worcester, V. (1991). Primary initiatives in mathematics education: Calculators, children and mathematics. London: Simon and Schuster. vKamii, C., & Livingston, S. J. (1994). Classroom activities. In L. Williams (Ed.), Young children continue to reinvent arithmetic in 3rd grade: Implications of Piaget's theory (pp. 81–146). New York: Teachers College Press. viShuard, H., Walsh, A., Goodwin, J., & Worcester, V. (1991). Primary initiatives in mathematics education: Calculators, children and mathematics. London: Simon and Schuster. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 87 viiHarel, Guershon. Personal conversation with a child he interviewed. viiiHowden, H. (1989). Teaching number sense. Arithmetic Teacher, 36(6), 6–11. ixNational Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press. xKnapp, M. S., Shields, P. M., & Turnbull, B. J. (1992). Academic challenge for the children of poverty (Summary LC88054001). Washington, DC: U.S. Department of Education. Reasoning About Numbers and Quantities Chapter 3 Instructor‘s Version p. 88 Chapter 4 Some Conventional Ways of Computing The previous chapter introduced you to some of the ways children invent SEE INSTRUCTOR NOTE 4 for an overview for carrying out arithmetic operations. The procedures you yourself use and. Introduction to this are probably the standard ones you learned in elementary school. Contrary section. to what some believe, these algorithms, or procedures, took centuries to evolve, and other algorithms are standard elsewhere. In this section we will look at those procedures for carrying out arithmetic operations more carefully, and come to understand how and why they work. 4.1 Operating on Whole Numbers and Decimal Numbers The arithmetic operations on whole numbers or on decimal numbers are similar since the same base-ten structure underlies both of these forms. The computations are highly dependent on this base-ten structure. Developing sensible step-by-step procedures, or algorithms, to carry out the arithmetic operations of addition, subtraction, multiplication, and division demands understanding our base ten system. When students have difficulty with computational algorithms, many errors they make can be The addition, traced back to a lack of understanding of our base-ten system. subtraction, and multiplication Think About... Are the algorithms you use for the four arithmetic algorithms are right-operations based on right-to-left or left-to-right procedures? Can all be to-left procedures, but undertaken working from left-to-right? need not be. And the division algorithm is Some children understand algorithms better when they are illustrated with left-to-right. Mental base ten materials or drawing. computation and estimation are often Example 1 Use base ten drawings to illustrate 174 + 36. undertaken left-to- right. Some of the Solution Step 1: Represent the problem, using the small cube as 1. student-developed procedures in the last section illustrate this. 174 + 36 Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 89 Step 2. Place all cubes together; replace 10 small cubes with one long. 1 174 +36 0 Step 3. Place all longs together; replace 10 longs with one flat 11 174 +36 10 Step 4. Put like blocks together to represent the sum. 11 174 + 36 210 Activity 1 Act out the following calculations using base ten blocks or drawings, First model a problem as whichever is appropriate. Then re-do each and record numerically at indicated in the each step. It is important that one writes down the steps as the introduction to this procedures are undertaken, with the numerical work linked to the chapter. Point out that the old term ―carry‖ does work with the blocks, software, or drawings. When you use base ten correspond to work with blocks or drawings, be sure you specify which block or drawing is the blocks, but the term being used to represent one whole. ―borrow‖ for subtraction 1. 312 – 124 does not work as well. Terms like ―rename‖ or 2. 3 x 21 ―regroup‖ are commonly 3. 123 + 88 used nowadays rather than the older terms. See 4. 12.3 + 88 Instructor Note 2.4A for 5. 12.3 + 8.8 an example of how to model a subtraction 6. 12.3 + 0.88 problem with drawings. 7. Describe what new blocks would be needed to compute 235.42 + 6.345. Multiplication and division algorithms are more difficult to understand than addition and subtraction algorithms. In the last section you saw some multiplication algorithms that could easily lead to the standard algorithm that you use. If children understand place value and use that to multiply, they will understand the process. The first example on the next page shows all six partial products. The second example is a shortened way to find the product using just two partial products. If children first learn to write all partial products, they will understand where the partial products come from in the condensed algorithm. Note the use of decomposing 348 as 8 + 40 + 300 and then using the distributive property. Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 90 348 348 x 26 x 26 48 = 6 x 8 2088 = 6 x 348 240 = 6 x 40 6960 = 20 x 348 1800 = 6 x 300 9048 160 = 20 x 8 800 = 20 x 40 (Note that 348 = 300 + 40 + 8.) = 20 x 300 6000 9048 We can also use all partial products to make sense of multiplying by decimal numbers. 3.48 3.48 Students may need x 2.6 x 2.6 practice on part-of-an- .048 = .6 x .08 2.088 = .6 x 3.48 amount thinking for .240 = .6 x .4 6.960 = 2 x 3.48 products involving 0.6 1.800 = .6 x 3 9.048 .160 = 2 x .08 .800 = 2 x .4 6.000 = 2 x 3 9.048 Activity 2 Your Turn Multiply 1.3 x 2.4, showing all partial products. When prospective teachers Here is an example of a series of division algorithms for 472 ? 37 that have learned how easy the division algorithm is to leads to the standard algorithm that you probably use. In the first example, understand, and also realize 37s are subtracted 10 at a time or 1 at a time. In the second, we subtract that they could use any one other multiples of 37 (in this case 2). In the third, we simply move the 10 of the algorithms given and 2 to the top. In the last, we do not write 10, but only the first digit of below to do long division, 10, aligned with the ten‘s place in the dividend, and thus leaving room for they sometimes react in anger, saying that they a digit in the ones place so that the 2 can be filled in. Notice that the should have been taught standard algorithm you learned is just a condensed version of subtracting this in elementary school 37s. rather than being made to suffer over years of learning how to divide. 12 2 10 12 37472374723747237472 ，，，， 370 10 370 10 370 37 102 102 102 102 37 1 74 2 74 74 65 28 12 28 28 37 1 28 12 Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 91 Thus 472 divided by 37 is 12 with a remainder of 28. Notice that each successive time the division problem is worked, it becomes a little more condensed, so that the final illustration matches the way most of us learned to divide. However, the previous three illustrations show that the problem can be thought of as how many 37s are in 472, and subtracting multiples of 37 until no more can be subtracted. While this is happening, SEE we keep count at the side or on top of the number of 37s that have been INSTRUCTOR subtracted and record them. Studying this series of algorithms will help NOTE 4.1 for another example you understand the last one, which is the one most widely used in the of moving from a United States but often never understood. nonstandard to a standard algorithm Activity 3 Why Move the Decimal Point? for division. As you know, the first step in using the usual algorithm for calculating, say, 56.906 ? 3.7 is to move the decimal point in the divisor to make a whole number (37) and to move the decimal point in the dividend that number of places (giving 569.06). Use a calculator in parts a.-d. to Find answers to the following (using a calculator if you wish): a. 56.906 ? 3.7 10) ? (3.7 10) b. (56.906 ，， c. 1.728 ? 1.44 100) ? (1.44 100) d. (1.728 ，， e. What insights into the ―move the decimal point in the divisor‖ rule does this activity suggest? Activity 4 Now I Know How to Divide Decimal Numbers Try dividing 0.18 by 1.5 using each of the techniques demonstrated for 472 ? 37. Use knowledge gained from the previous activity to account for ―moving the decimal point.‖ Take-Away Message…Undertaking the fundamental arithmetic operations using the standard procedures is easier when you can first see them “unpacked,” with each step along the way clarified by using properties and thinking about the meaning of the operation involved. This work can lead to an understanding of where all the numbers come from and how the algorithm can then be condensed into the procedures we all know and probably all use. Learning Exercises for Section 4.1 LE 4.1 Students 1. Work through the following problems, thinking about each step you do not have perform, and why. If you have used base ten blocks or sketches of base answers to 1b and ten blocks, you can use them to help you with these problems. In each 2b case, specify what you decide to use as your unit. For example, if the Be sure to assign 1, 2, 3, and 5 flat is used to represent 1, then 34.52 can be represented as follows: .. . Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 92 a. 56.2 + 34.52 b. 4 x 0.39 c. 345.6 – 21.21 d. 2912 ? 8 (Think of this as a sharing problem.) 2. Work through the following problems in the base indicated. Write down your procedure, and specify what you use as your unit in each case. a. 231 + 342 in base five b. 1000 – 555 in base six If you feel at all uncertain about these problems, make up and try some of your own. 3. Show how you could use a series of algorithms, such as those in this section, to teach another person to understand the standard algorithms for multiplication and division: 35 x 426; 14910 ? 426. 4. There have been, in the past, many different algorithms developed for Check out students' carrying out computation. Here are two such algorithms for computing understanding of the steps in this algorithm; 36 x 342. The first is called the lattice method for multiplication, used they often have by the Arabs in the 1600s and carried to Europe. The method depends difficulty divorcing on knowing the multiplication facts, but not much on place value. The their thinking from the factors are written across the top and right, and the answer, 12,312, is algorithm they know. read off going down on the left and around the bottom. Compare this Students most often, unfortunately, do not ethod to showing all partial products. m want to know why these work. The right-342to-left slants in the lattice give entries with 01031the same place value. 962 21162482 312Note that the right 0column is 342 x 2, The second is called the Russian peasant algorithm. One number is 1342 x 2, 342 x successively halved until 1 is reached (if it is odd, 1 is subtracted 252, . . . 342 x 2. 36 before halving) and the other number is doubled the same number of is 100100 in base 2, 5times the first is halved. Numbers in the second column are crossed that is, 36 = 1 x 2 + 43out when the corresponding number in the first column is even, and 0 x 2 + 0 x 2 + 1 x 210the remaining numbers are added. Reread the preceding steps as you 2 + 0 x 2 + 0 x 2. 5study this calculation of 36 x 342. So 36 x 342 = (2 + 22) x 342. But this is 1368 + 10944. Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 93 36 342 18 684 9 1368 4 2736 2 5472 1 10,944 12,312 Try these algorithms on 57 x 623. They are not magic! Each can be This algorithm is on justified mathematically. page 91 of the student text. The 5. Use your understanding of division to complete each equation. remaining pages in this chapter do not If 4000 ? 16 = 250, then match the pages in a. 8000 ? 16 = ___________ b. 16,000 ? 16 = _________ the student text. c. 2000 ? 16 = ___________ d. 4000 ? 32 = ___________ e. 4000 ? 64 = ___________ f. 4000 ? 8 = _________ g. 4000 ? 4 = _________ h. 4000 ? 0.4 = _________ 4.2 Issues for Learning: The Role of Algorithms This is an extremely important discussion. As stated earlier, In Section 4.1 we focused on the methods used in carrying out the many prospective teachers (and arithmetic operations on whole numbers and decimal numbers, methods some practicing teachers) believe that learning the that are likely to be already familiar to you. By learning well one standard algorithms in the same algorithm for each operation, traditional long division for example, that way they did is the most skill becomes automatized and allows one to think about other things, for important component of example, the problem at hand that led to the division. elementary school mathematics, and that calculators in Discussion 1 What Do You Remember? elementary school are harmful and if used will lead to inability What do you remember about learning algorithmic procedures in to calculate in other ways. This school? Did you understand these procedures? Were they easy or section will help them reflect on painful to learn? Were they easy or difficult to remember? Do you these issues. There is an use them often now? How important is it to be able to carry out excellent discussion of using algorithms in Adding It Up, operations rapidly with paper and pencil? published by the National The standard algorithms are called "standard" because children are taught to Research Council. do them the same way. The evolution of these algorithms began in the 1600s, with the demise of the use of the Roman numeral system due to the hard-won acceptance of the Hindu-Arabic decimal system of numeration (our place- value system). The algorithms came into existence as people attempted to become as efficient and speedy as possible in their calculations. Even so, some of the algorithms we use here are not standard in other parts of the world. For example, our subtraction algorithm was selected for teaching in U.S. ischools because a research study in 1941 showed it to be ―slightly better‖ Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 94 than the equal-additions method of subtraction you saw in Chapter 3, a method that is taught in many other countries. Those who learned the equal-additions algorithm find the algorithm used in the United States to be strange. Many mathematics educators say that there should be less emphasis on teaching pencil-and-paper algorithms because students can use calculators to compute. But they do not argue that we should therefore not teach methods of calculating that are independent of calculator use. One reason for teaching calculation methods is because calculation is a useful tool. Another reason is that it may lead to a deeper understanding of number and number operations. Students make better sense of standard algorithms when they have had opportunities to explore arithmetic operations and find their own ways of operating on numbers, exemplified by those children whose work you have studied in the last chapter. Students can then learn standard algorithms as required, because in most states and school districts they are required. If they learn standard algorithms with an understanding of why the procedures lead to correct answers they are more likely to remember them and use them correctly. ... Do you have a better understanding of the multiplication and Think About division algorithms after seeing, in the previous section, how longer algorithms can be used to show why standard algorithms work? What, then, is the role of calculators in elementary school? Calculators are ubiquitous, and are much faster than any human calculator. Will calculators someday completely replace paper-and-pencil calculations? Not likely. But efficiency and automaticity should be reconsidered as valid reasons for teaching arithmetic skills. This does not mean that people should not have these skills, but rather that finding an answer using paper and pencil may no longer need to be rapid (except in some testing situations). There are times when pencil-and-paper computing should be preferred, and times when calculators should be used. Discussion 2 What Is the Role of Calculators? What is the role of calculators in school mathematics? A newspaper cartoon showed parents and grandparents sitting at desks using calculators for computing, while the 12-year-old son was sitting at his desk doing worksheets of long-division practice. What do you think was the message? Contrary to what is believed by many, research has shown that calculators do not hinder the learning of basic skills and can, in fact, enhance learning and ii,iii,iv skill development when used appropriately in the classroom.The National Assessment of Educational Progress (NAEP) from years 1990 to 2003 shows no difference in scores for fourth grade students in classrooms where vcalculators were frequently used and where calculators were not used. (The NAEP test in mathematics is administered nationally at grades 4, 8, and 12 every 2-4 years.) In eighth grade, students who frequently used calculators scored significantly higher than students who did not. Keep in mind that Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 95 students often do not understand enough about arithmetic operations to be able to choose appropriate operations to solve real problems, and in such cases a calculator is useless. If understanding algorithms leads to less need for practice, more time can be devoted to learning which operations are appropriate and when. In the next chapter you will be given opportunities to consider a third option using standard algorithms or calculators. In many cases an exact answer is not needed. What should be done then? 4.3 Check Yourself By now you should be able to explain the multiplication and division algorithms to someone else who does not understand the processes. You should know the pros and cons of the standard algorithms and student-invented algorithms, and provide a cogent argument defending your view of what is important for children to learn about the procedures for carrying out arithmetic procedures. You should also be able to do problems like those assigned and to meet the following objectives. 1. Explain each step of the addition (e.g., for 378 + 49) and subtraction algorithms (e.g., for 378 – 49) and show how renaming is used. 2. Explain each step of whole number multiplication (e.g., 375 x 213) using partial products. 3. Explain division of whole numbers (e.g., 764 ? 46) using a series of algorithms that leads to the standard algorithm. 4. Recognize the role of place value in the algorithms. 5. Discuss the role of both nonstandard and standard algorithms. References for Chapter 4 iBrownell, W. A. (1941). Arithmetic in grades I and II: A critical summary of new and previously reported research. Durham, NC: Duke University Press.iiShuard, H., Walsh, A., Goodwin, J., & Worcester, V. (1991). Primary initiatives in mathematics education: Calculators, children and mathematics. London: Simon and Schuster. iiiKloosterman, P., & Lester, F. K., Jr. (Eds.) (2004). Results and interpretations of the 1990 through 2000 mathematics assessments of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics. ivKilpatrick, J., Swafford, J., & Findell, B. (Eds). (2001). Adding it up. Washington, DC: National Research Council, National Academy Press. vKloosterman, P., Sowder, J., & Kehle, P. (2004). Clearing the Air About School Mathematics Achievement: What Do NAEP Data Tell Us? Presentation at the April 2004 American Educational Research Association, April, 2004, San Diego. Reasoning About Numbers and Quantities Chapter 4 Instructor‘s Version p. 96 Chapter 5 Using Numbers in Sensible Ways SEE INSTRUCTOR People who are comfortable working with numbers have many ways to NOTE 5 for an overview think about numbers and number operations. For example, they use of this section. ―benchmarks,‖ that is, numbers close to particular numbers that are easier to use in calculating. Thinking about numbers in various ways allows them to estimate answers, do mental arithmetic, and recognize when answers are wrong. This chapter will help you acquire a better number sense, that is, the ability to recognize when and how to use numbers in sensible ways. 5.1 Mental Computation When people hear the word ―computation‖ they usually think of paper- and-pencil computation using the methods they learned in school. But computation can take many forms, and an individual with a good understanding of numbers will choose the most appropriate form of computation for the problem at hand. Today, this often means using a calculator because it is fast and efficient. Often, though, one can mentally compute an answer faster than one can compute with paper and pencil or even a calculator. At other times, one might wish to compute mentally simply because it is more convenient than finding pencil and paper or a calculator. To become skilled at mental computation takes some practice. Number properties must be understood and used, although you may not even be aware of the properties because they are so natural for you to use. One important advantage of practicing mental computation, for children and for adults, is that it helps develop flexibility of use of numbers and These three ways properties of operations and gives them a sense of control over numbers. of finding 12 ， 50 provide an Consider some of the following ways of mentally computing 12 ， 50: opportunity to review the 1. 12 ， 50 can be thought of as 12 ， 5 ， 10 = 60 ， 10 = 600. This commutative, required decomposing 50 into 5 ， 10, and then using the associative, and associative property of multiplication: 12 ， (5 ， 10) = (12 ， 5) ， distributive 10 = 60 ， 10. properties. 2. 12 ， 50 can be thought of as (10 + 2) ， 50 = (10 ， 50) + (2 ， 50) (using the distributive property of multiplication over addition), which is 500 + 100 = 600. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 97 3. 12 ， 50 =(6 ， 2) ， 50 = 6 ， (2 ， 50) = 6 ， 100 = 600 (using the associative property). Remember that addition is also commutative and associative, giving a great deal of flexibility in adding numbers. There are also several ways of thinking about how to mentally compute a difference. For example: 1. 147 , 66 could be found by adding 4 to each number: 151 , 70 is 151 – 50 – 20 = 101 – 20 = 81. 2. 147 , 66: Count up 4 to 70, then 30 to 100, then 40 to 140, then 7 to 147. This is 4 + 30 + 40 + 7 which is 81. 3. 147 , 66 is 140 – 60 which is 80, plus 7 is 87, minus 6, is 81. And of course there are other ways… Activity: I Can Do It In My Head Activity 1 I Can Do It in My Head! Discuss some of these Before reading ahead, do the following calculations in as many ways in class. There are many different ways to as possible. mentally calculate, 24 1. 152 – 47 2. 1000 – 729 3. 25 ，some easier than 1others. Students should 4. 38.6 + 27.2 + 4.8 – 38.6 5. 12 ? 6. 6 ， 43 + 6 ， 7 4be expected to justify their answers, and to Many times the ease of mental computation depends on the numbers used, discuss different and on familiarity with properties of operations. The computation should methods in terms of not begin before looking over the numbers and thinking of different ways their ease. See also SEE INSTRUCTOR of reorganizing or rewriting the numbers. Consider Problem 4 in Activity NOTE 5.1A for a 1. Many people would begin by trying to add 38.6 and 27.2 mentally, a discussion of challenge but certainly possible. But as one becomes more adept at mental strategies. computation, the knowledge that the numbers could be reordered, allowing the 38.6 – 38.6 to be the first operation, makes this a much easier problem to compute mentally: 27.2 + 4.8 is 31 + 1 is 32. Now consider Problem 3, which is 25 ， 24. One way to think about this problem is as (5 ， 5) ， (4 ， 6) = (5 ， 4) ， (5 ， 6) = 20 ， 30 = 600. Notice that you have really done the following, but without writing it all: 25 ， 24 = (5 ， 5) ， (4 ， 6) = 5 ， (5 ， 4) ， 6 using the associate property of multiplication, = 5 ， (4 ， 5) ， 6 using the commutative property of multiplication, = (5 ， 4) ， (5 ， 6) using the associative property again. 100When 25 is a factor, it could also be thought of as. Now 25 ， 24 4 100100，24,，4，6becomes = 100 ， 6 = 600. 44 Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 98 For Problem 6, the arithmetic is easy if we use the distributive property: (6 ， 43) + (6 ， 7) = 6 ， (43 + 7) = 6 ， 50 = 300. For Problem 2, which is 1000 – 729, the mental computation by counting up is actually easier than using the standard algorithm, which, because of the zeros, often leads to errors. Add 1 to 730, 70 to 800, then 200 to 1000, 1 + 70 + 200 is 271. Notice that this is based on the missing-addend view of subtraction. In Problem 5 think of how many quarters are in $12. Too often, people give the answer 3. Why do you think this happens? Discussion 1 Becoming Good at Mental Computation Discussion 1. Becoming... Try these individually, then together discuss different ways of SEE INSTRUCTOR undertaking these mental computations. NOTE 5.1B about discussing these problems 1. Mentally compute 50 ， 35 ， 2 ? 5 in class so that students can be exposed to a variety of 2. Mentally compute 19 ， 21 (Is this the same as 20 ， 20?) ways of thinking. 3. Mentally compute 5,400,000 ? 600,000 4. Mentally compute 3476 + 2456 – 1476 5. Mentally compute 50 ， 340 6. Mentally compute 24 ， 38 + 24 ， 12 7. Mentally compute 8 ， 32 ， 4 (Hint: You can use powers of 2.) Students may need to 8. Mentally compute 35 ， 14 + 35 ， 6 review changing percents to decimals or fractions. Mental calculation with some percents is not difficult. Here are some A review is found in the Appendix: A Review of hints about calculating with percents. Some "Rules." 11. 10% of a number is or 0.1 times the number. You may prefer to omit 10these percent problems now and return to them 10% of 50 is 5. 10% of 34.5 is 3.45. in Chapter 9 if your students have difficulty 2. Multiples of 10% can be found using the 10% calculation: later working with 20% of 50 is twice 5, or 10. percents. 20% of 34.5 is 3.45 twice, or 6.90. 70% of 20 is 10% seven times, or 14. (Or think: 10% of 20 is 2, and 2 ， 7 = 14 is 70% of 20.) 3. 15% is 10% plus half that again. 15% of 640 is 64 + 32 or 96. 4. 25% is a quarter of an amount, and 50% is half an amount. Thus 25% of $360 is $90 because one quarter of 360 is 90. Or, you could say 10% is 36, 10% more is 36 more, then 5% is half of 10%, and half of 36 is 18. Finally, 36 + 36 + 18 is 90. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 99 Activity 2 I Can Do Percents in My Head Activity 2. a. 10 Determine the following mentally. Does a mental drawing like the b. 6 is 10% of 60, so 12 one here help in some cases? is 20% of 60 c. 26 a. What is 25% of 40? d. 100 e. 10% is 5.5; 30% is b. Twelve is 20% of what number? 15 + 1.5 = 16.5 % f. 33.3 c. Thirteen is 50% of what number? 1Estimating with or 3 d. Ten percent of what number is 10? % was not 33.3 covered above. e. What is 30% of 55? f. What percent of 66 is 22? Take-Away Message…There are many ways to mentally compute if one knows the properties of operations on numbers. As people become more comfortable using numbers, they are more likely to mentally compute answers when this is easy to do. Computing mentally leads to greater facility with numbers and more confidence in one’s ability to work with numbers. Learning Exercises for 5.1 Students do not have 1. How can the following computations be done mentally using the answers for 1a, 1c, 2b, 2c, distributive property? Write out your solutions. 2e, 2f, 3b, 3c, 3e 3f, 4b, 4d, 5, 6 second row. a. 43 ， 9 b. 23 ， 98 + 23 ， 2 c. 72 ， 30 Be sure to assign 2 and 3. 2. Tell how you could mentally compute each. 28 c. 2391 + 431 –1391 a. 365 + 40 + 35 b. 756 – d. 499 – 49 e. 124 ， 25 f. 42 – 29 g. 44 ， 25 h. 75 ， 88 i. 8 ， 32 3. Compute each of the following mentally. a. 25% of 60 b. 10% of 78 c. 30% of 15 d. 80% of 710 e. 15% of $40 f. 100% of 57 g. 125% of 40 h. 20 is 20% of ? i. 5% of 64 4. Compute each of the following mentally. a. 24 is 20% of what number? b. 38 is 10% of what number? c. 89 is 100% of what number? d. 50 is 125% of what number? Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 100 5. The figure shown below is 120% of a smaller figure. Can you show #5 is easier than it first 20% of the smaller figure? Shade 100% of the smaller figure. seems. Draw in diagonals so that the hexagon shows six triangles. 120% is 6/5, and 5/5 would be five of the triangles. 6. If you wish, shade a grid to help you find the fractional equivalent in simplest form for each of the following percents. 1% = _____ 75% = _____ 50% = _____ 333 266% = _____ 12.5% = _____ 20% = _____ 3 5.2 Computational Estimation Many times an estimate rather than an exact answer to a calculation may be sufficient. Estimates are used not only when an exact answer could be found but is not needed, but also when an exact answer is not possible, such as when you make a budget and estimate how much your total utilities bills will be each month. Mental computation is used in computational estimation, but when you estimate, you are looking only for an approximate answer. However, estimation is sometimes considered more difficult because it involves both rounding and mental computation. Think About… Think back on the times you have needed a numerical answer to a computation in the past week. How many times did you need an exact answer? How many times did you need just an estimate? Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 101 Discussion 2 Is One Way Better Than Another? 1. Estimate 36 ， 55 alone before discussing this problem. Discussion 2. The methods used by the 2. Carefully read these students' solutions to the first exercise. Be different children deserve sure you understand the thinking behind each one. Then discuss a class discussion. them in terms of whether each way is a good way to estimate. Did Students often don't appreciate good you use one of these ways? estimation skills and need to reflect on what is going Shawn: Round to 40 and 60. 40 ， 60 = 2400. on in each example. Jack: First round down: 30 ， 50 = 1500. Then round up: 40 ， 60 = Discuss with students the 2400. So it's about in the middle, maybe a little past. So I'd say relative sophistication of these methods. For 2000. example, Shawn rounds Maria: Rounding both up would make it too big, so I'll round 36 to 40 both numbers up—and 55 to 50. 40 ， 50 = 2000. forcing the estimate to be Jimmy: A little more than 36 ， 50, which is 36 ， 100 ? 2 and that's too high. There is no compensation made, such 18 ， 100 = 1800. It's about 5 ， 36 more, or about 180 more, so I'll as saying ―It will be less say 1980. than 2400.‖ Deb: Rounding both up gives 40 ， 60 = 2400. Since that's too big, I'll say it's about 2200. See INSTRUCTOR NOTE 5.2 for additional Sam: A little more than 6 ， 6 ， 50, which is 6 ， 300 = 1800. So I'll information on these say 1900. children‘s methods. Each of the above ways provide good estimates, but some are better than others because they are efficient and/or because they are more accurate. For example, Shawn rounded both numbers up to the closest multiple of 10. Rounding up gives a high estimate. Jack rounded both up and down then takes something in the middle. His way takes slightly more work but gives a closer estimate. But depending on the context, you might want a high estimate. Suppose you are at the hardware store buying drawer pulls that cost 55? each. You need 36 of them and wonder if you have enough cash with you. According to Shawn you need $24, and according to Jack, Activity 3. Some of your students may make the you need $20. Before you reach for your wallet, which estimate would same errors in 3 and 4 as you use? Why? those described in the Think Abouts below, Activity 3 Now I’ll Try Some particularly the fourth problem. Help them 1. Try each of the strategies from the above discussion (if appropriate) understand why the to estimate 16 ， 48. Which estimates do you think are better? Why? solution containing a 2. If a city's property tax is $29.87 per $1000 of assessed value, what decimal portion is would be an estimate of the tax on property assessed at $38,600? incorrect. You may or may not want to get into 3. Estimate 789 ， 0.52. a discussion of 4. Estimate 148.52 + 49.341. significant digits here. iThink About: In a research study many middle school students were asked to estimate 789 ， 0.52. Some students rounded 0.52 to 1, rounded 789 to 800, and said their answer was 800 ， 1 = 800. Is 1 a good substitute for 0.52? Why or why not? Suppose something cost $0.52, and you bought 789 of these objects. Would $800 be a good estimate of the price? Why do you think students rounded 0.52 to 1? Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 102 In the same study the students were asked to estimate Think About: 148.52 + 49.341. Some students said, ―148 is almost 150 and 49 is almost 50. And 0.52 is almost 0.6 , and 0.341 is almost 0.3. And 0.6 + 0.3 is 0. 9. So the answer is 200.9.‖ Find the flaw in this argument. (Hint: Is the 148 being rounded up to 150, or is the 148.52 being rounded up to 150?) Research has shown that good estimators use a variety of strategies and demonstrate a deep understanding of numbers and of operations. They are flexible in their thinking and are disposed to make sense of numbers. A good estimator uses the rounding of numbers a great deal, but seldom uses iithe formal rounding procedures often taught in schools. Proficiency in flexible rounding requires that an individual has an intuitive notion of the magnitude of the number in question, and this intuition is acquired through practice in comparing and ordering numbers and using benchmarks, especially for fractions and decimals. For example, to Activity 4. 1. Round 71 to 72 and estimate 257 + 394 + 2 + 49, a good strategy (but not the only one) would cancel with the 8, or be to round 257 to 250, 394 to 400, 49 to 50, and drop the 2 since it is round 89 to 88 and insignificant here (or think of rounding it to 0). School students who are cancel with the 8. This is inflexible in their rounding will insist that 257 must be rounded to 300, or a good example of using perhaps 260 if rounding to the nearest 10, but not to 250. In another ―compatible‖ numbers. Discuss why rounding 71 example, if you want to estimate the quotient of 6217 ? 87, you might find to 70 is not helpful, it more convenient to round to 6300 and 90 rather than 6000 and 90, since 2. Rounding both up 70 ， 90 is 6300. The activity below is intended to provide you with would call for some practice in estimating. compensation. Rounding to 40 and 35 still leads to Activity 4 Finding “Just About” an easy computation. Be sure to check your solutions and methods with others. You may be 3. About 6 ? 12 which is ?. surprised to find many good ways of doing these estimations. 4. 344 is a large error if the 71，89exact product is 2924, but 1. Estimate 8is a relatively small error if the exact product is 42,656. 2. Estimate 42 ， 34 5. 20 ， 86 introduces an 3. Estimate 5.8 ? 12 error of 2 ， 86, whereas 18 ， 90 introduces an error 4. If 34 ， 86 is estimated as 30 ， 86, then the estimate is 2580. The of 4 ， 19, which is smaller exact answer is 2924. The difference between these two numbers than 2 ， 86. is 344. If 496 ， 86 is estimated as 500 ， 86, then the exact answer 6. Many students will claim is 42,656 and the estimate is 43,000. Again the difference is 344. this is the same as 53 ， 27. Which of these would you choose: (a) the first estimate is better; (b) However, they are the same; (c) the second estimate is better. Why? (50+3)(27-3), 50 ， 30. In fact, the estimate is 50 ， 3–5. 18 ， 86 can be estimated as (a) 20 ， 90; or (b) 20 ， 86; or (c) 18 3 ， 27 off, so the actual ， 90. Which of these three ways gives the closest estimate? product is less than the estimated product of 1500. 6. If you estimate 53 ， 27 by saying 50 ， 30 is 1500, is the exact answer less than, equal to, or more than your estimate? By now you know that when multiplying two numbers, rounding both up gives a high estimate, and rounding both down gives a low estimate. Thus a more accurate estimate of a product could be found by rounding one number up and one number down. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 103 Discussion 3 Rounding Up and Down If you are adding two numbers is it usually best to round both Discussion 3. When estimating a sum numbers up? If not, what is the better thing to do? What about rounding one number up subtraction? What about division? and the other down is usually better. When Estimating with percents is very common. When you shop and see a sale subtracting or dividing, at ―30% off‖ you need to know what you have to pay. Here are some hints rounding both numbers about estimating with percents, building on the earlier mental computation up or down is better, but with percents. it depends a great deal on the numbers. For 1. Estimating 10% of 34.5 could be 10% of 35 which is 3.5. example in estimating 24354 ? 13, rounding 2. If you estimate 20% of 34.5, it is 3.5 twice, or 7. An estimate of 30% of both down to 24000 and 5 = 15. 50.3 is 10% of 50 three times: 3 ，12 would provide a good estimate. 3. 15% is 10% plus half that again. 15% of 642 is 64.2 + 32.1 or 96.3 But if you are estimating, such as for a tip, you could say 15% of 640 is 64 + 32, or 96, or even 100. Once again, we remind 4. 25% is a quarter of an amount. 25% off of $350 is about 25% of $360 you that we assume some which is $90. (Note that 1/4 of 360 is 90.) Or, you could say 10% is 35, basic knowledge of 10% more is 70, half of 35 is about 18, and 35 + 35 + 18 is 88. percent here. Students may need to review the 5. When estimating a discount on price, it may be easier to estimate the sale knowledge of percents value. Thus, if a vase is regularly priced at $140 and is 20% off, then the we assume. See the appendix: A Review of actual price is 80% of $140. 50% of 140 is 70. 10% of 140 is 14 and so Some ―Rules.‖ 30% of 140 is 3 ， 14, or 42. So the actual price is $70 + $42 = $112. To estimate, 80% of $150 would be 8 $15 or 4 $30 or $120. In this case, ，，There are additional finding the actual value was probably as easy as estimating it. lessons in later sections that include practice with Activity 5 What Does It Cost? Estimate. percents. 1. A coat is 30% off. It was originally $150. How much is it now? Activity 5. 2. A dress, marked as $90 before the sale, is 40% off. What percent a. 30% of 150 is 3 x 15 of the original price must one pay to buy it? What is the discounted = 45, and $150 - $45 = cost? $105. 3. A jacket cost $45 after the 50% discount. How much was it b. 60%; 60% of $90, 6 originally? x 9 = $54 An eighth-grade teacher of a low-ability class told one of the authors that c. $90 she spent the last 8 weeks of math classes on mental computation and estimation because her students were so weak in this area, and she wanted them to better understand how to operate with numbers. Her students did extremely well. She said that they came to feel very confident, mathematically, and were no longer afraid of numbers. Take-Away Message…Computational estimation builds a greater facility with numbers but at the same time reinforces facility. Estimation plays a major role in our daily lives. Learning how to make better estimates can be a valuable skill, one we want to pass on to students. You should be estimating calculations throughout this course. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 104 Learning Exercises for Section 5.2 62. 1. Estimate 0.76 ， Students do not have 2. You buy 62 tablets at 76? each. You want to estimate the price before answers to 4; 5; 6a, 6c, you get to the cash register to make certain you have enough money 6d, 6e; 7b, 7c, 7d, and 8. with you. Do you estimate differently than in Exercise 1? Discuss times Be sure to assign 1, 2, 6, 7, 8. when an estimate should be an overestimate, and times when an underestimate is preferable. 3. Which is greater (without calculating): 0.21 84.63 or 84.63 ? 0.21? ， Explain your answer. 4. Your restaurant bill is $27.89. What should you leave as a tip if you want to tip 15%? 5. Reflect in writing on the six ways of estimating the product of 36 and 55 used in the first discussion in this section. Can all be done mentally? Are some better than others? Why or why not? Are some easier than others? Which ones show a better ―feel‖ for numbers? 6. Estimate the following products using whatever strategies you prefer. 6. should be a. 49 890 b. 25 76 c. 16 650 ，，，discussed in class. d. 341 6121 e. 3 532 ，， 7. Explain how you would estimate the cost of the following items with the discounts indicated. a. a dress marked $49.99 at a 25% discount b. a CD marked $16.99 at 15% off c. a blouse at 75% off the $18.99 clearance price d. a suit at 60% off the already marked-down price of $109.99 8. Use your ―close to‖ knowledge to estimate the following. The first one is done for you. a. 0.49 ， 102 is about of half of 100, or 50 b. 32% of 12 c. 94% of 500 d. 0.52 ， 789 e. 23% of 81 f. 35% of 22 g. 76% of $210 9. Return to the activities in this section and once again work though each part on your own. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 105 5.3 Estimating Values of Quantities Many times when people talk about estimation, they mean estimating numerosity rather than estimating calculations, for example, the crowd at some big event. Perhaps you have at some time entered a contest where you had to guess the number of jelly beans in a large jar and found it very difficult to do. Numerosity refers not just to guessing, but to making intelligent guesses. If you went home and found a similar jar that you filled with jelly beans, then counted them, you might make a better guess than had you not done so. For a long time you will be able to make pretty good judgments concerning the number of jelly beans in jars. Although that may not be a skill worth developing, there are other contexts where estimating skills can be useful. Skill in estimating values of quantities, whether discrete counts (like jelly beans) or continuous measures (like the height of a tree), require practice. Money is certainly one area in which understanding estimates can be There is a video Powers useful. A recent news report recommended that those speaking about of Ten (8 minutes) social security legislation speak only about numbers in the millions, commercially available that would be appropriate because people don't know what a trillion means. Perhaps large numbers to use with the material will make more sense to you after this section. in this section. It may also still be available at Activity 6 Developing a “Feel” for the Size of Quantities du/primer/java/scienceop1. How big is a crowd of 100 people? Of 1000 people? Of 10,000 people? ticsu/powersoften/index.Do you have a sense for how large each of these groups would be? html a. Would 100 people fit in a typical classroom? If so, how crowded would the room be? b. Would 1000 people fit in a typical classroom? Inside your home? Where would it be natural to find a group this size? Are there enough seats in a standard movie theater for 1000 people? c. Where would you expect to see 10,000 people? Do you have a feel for how large a group this would be? If exactly this many people were in attendance at your local stadium, would it be completely full? Half full? How full? 2. How long is 10 feet? 100 ft? 1000 ft? 10,000 ft? Do you have a sense for how long each of these lengths is? a. Name at least five objects whose typical length or height is approximately 10 feet. (Think of the length of your bedroom. Is it more or less than 10 feet?) b. Do the same for 100 feet. Think about a point or object in the environment surrounding you that is approximately 100 feet away from you, 100 feet tall, or 100 feet long. If necessary, go outside and find something that will help you make sense of 100 feet. c. Name some things that you could use as a reference for a length, height, or distance of 1000 ft. (One way--a mile is 5280 feet, so 1000 feet is about....?) Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 106 Having good number sense requires having some sense not only of the 11 and) relative size of a number (e.g., the relative size of 3 and 30, or of 32 but also of the absolute size of a number. (For example, how much is a million? Could you reasonably ask a third-grade class to bring in their pennies until they reached a million?) To develop a ―feel‖ for the size of numerical amounts it is helpful to compare them to some reference amount with which you are familiar. Such a reference amount is often called a benchmark. A benchmark is a personally meaningful and recognizable amount that can be used to make size estimates. Benchmarks are particularly useful when we want to have a feel for very large or very small numerical amounts. Benchmarks can involve money, population, time, distance, height, weight, a collection of objects, or any other physical or non-physical attributes. Example 1 It is 250,000 miles from the earth to the moon. How far is that? If you know that it is approximately 25,000 miles around the earth along the equator, you could think that traveling 10 times around the earth would be comparable in distance to traveling to the moon. Or. . . You could think of a place that is roughly 100 miles away from where you live and imagine traveling there back and forth 1250 times! Or . . .You could think of a place that is roughly 10 miles away from where you are and imagine covering that distance 25,000 times! Think About… Does the example above give you a ―sense‖ for how far away the moon is? Is it farther away than you thought? Or closer? How long would it take you to walk to the moon? Having one‘s own benchmarks can be very helpful. If these referents are to be useful to you they must be personalized. This means that you are the one who needs to be familiar with the amount. For example, if you attend games at the local stadium and you know its capacity, you can use that as a benchmark for that many people. In one city, the football stadium holds more than 60,000 people. Such a reference can give you a feel for that many people. Having a feel for 60,000 people, however, does not necessarily help you have a feel for how far 60,000 miles is, or how much $60,000 can buy. Therefore, the nature of the quantity, not just its numerical value, is important in deciding the usefulness of a reference amount. Take-Away Message… Many people have little “feel” for the size of numbers unless they practice, within some context. For example, a builder can easily estimate the height of a building and its square footage. Having some benchmarks for numbers as they are used in different contexts is an important aspect of living with numbers every day. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 107 Learning Exercises for Section 5.3 1. Think of a personal referent that will give you a feel for each of the following distances: a. 3 miles b. 30 miles c. 300 miles LE 5.3 Students do not have answers for any of d. 3000 miles e. 30,000 miles these exercises. 2. How long would it take you to travel each of the distances in Exercise 1, if you were to drive your car? It is very useful to discuss and share their 3. Find personal benchmarks for at least five large numbers (groups) of benchmarks in class. people. For example, you might say that 1000 people is about the Allow sufficient time for students to think of their number of students that attended your high school; your university has own. about 25,000 students; the city you live in has a population of about half a million people, and so on. (These are estimates of numerosity.) Do the same thing for amounts of money. For example, my car is worth about $10,000; the salary of a beginning teacher in my city is about $35,000; the average price of a home in my city is $250,000, and so on. But find your own! 4. A congressman once said, ―I don‘t care how much it costs—a billion, a trillion. We need to solve the AIDS problem.‖ Can the United States afford to spend one million dollars to fund AIDS research? One billion dollars? One trillion dollars? Why or why not? The population of the United States is now about 300 million. What would each of these amounts mean for each person in this country, in terms of a person‘s share of the national debt? 5. Write a children's story (not a school lesson) that uses at least five quantities with large values in ways that will help children understand If you worry about having how big the values really are. Include references to places, things, and sufficient time to events that will make sense to them. The story should have between complete the entire 500 and 1000 words. (Do you have a feel for how many pages that is?) course, you may want to skip Section 5.4 and As you develop your own personal referents, make a list of them so that return to it later if you have time. The point of you can refer to them at any time. In a later study of measurement, you this section is simply to will again be asked to prepare a list of personal referents. show that there are ways of writing values for very large and very small quantities. On the practical side, scientific 5.4 Using Scientific Notation for Estimating Values of Very notation is required in Large and Very Small Quantities some credentialing tests. Most very large and very small numbers are actually estimates. For a 12- digit number, the digit in the ones place is not going to matter. Indeed, for a large number, only a few digits to the far left really matter. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 108 Many people do not even know the names of numbers past one trillion, even though uses for very large numbers exist. Very small numbers also need to be expressed in some standard way that can be shared. We can express very large and very small numbers using scientific notation. A number is written in scientific notation when it has the form a ， b10 where a is a number between 1 and 10, and b is an integer. Example 2 7a. 50 million is 50,000,000, and in scientific notation this is 5 10. ， b. For small numbers, scientific notation requires the use of 1-3 negative integers. 10 means . We formalize this by This is the first 310appearance of ―integers.‖ ,d1c,saying that for any non-zero c and any d, You may want to check dcyour students‘ interpretations. They 6c. 6 ten-thousandths is .0006 or and in scientific notation this may be very rusty 10,000working with negative -4exponents. If so, skip, for is 6 10. Sometimes we write .0006 as 0.0006 so that the ， now, the operations on decimal point is more obvious. numbers in scientific notation. A future section d. 350,000,000 = 3.5 100,000,000 and in scientific notation this is ， covers negative numbers. 83.5 10. ， e. 0.00052 = 5.2 0.0001 and in scientific notation this is 5.2 ，， -4.10 Note that both the 3.5 and the 5.2 are between 1 and 10, and that the second part of the numerals have ten to a power. Operating on numbers in scientific notation requires knowing how to use 3 4exponents to your advantage. Recall that 55 means (5 5 5 ) ，，，，The appendix ―A 7 (5 5 5 5) which is 5 5 5 5 5 5 5 or 5, and that ，，，，，，，，，Review of Some 1?Rules‘ has a section 5，5，534 -15 ? 5 means which is or 5. In general we have: 5，5，5，55on exponents. Two laws of exponents useful in scientific notation: Example 3: The solution could be m n m+nm n m-n a a= a and a? a= a，used to illustrate the ―canceling‖ of units, if you wish to point 8 -4 8-4)Example 3 (3.5 10) (5.2 10) = (3.5 5.2) (10 10 ，，，，，，this out. We do not 4= 18.2 10. But this is not in scientific notation because 18.2 is ，pursue this shortcut 4 5(even though it larger than 10. 18.2 10= 1.82 10. ，， might be useful) 8 -4 8-4because it by-passes Example 4 (3.5 10)? (5.2 10) = (3.5 ? 5.2) (10 ? 10) ，，， thinking about the 120.67 10. But this is not in scientific notation because 0.67 is ,，multiplications. 12 11smaller than one. 0.67 10= 6.7 10. ，， Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 109 Example 5 Light travels 186,000 miles in a second. A light-year is the distance that light travels in one year, even though it might sound like a time unit. About how many miles are in a light-year? Solution days512mismin.hmi so 1.86，10，60，60，24，365,5.87，10smin.hdayyearyear12a light-year is about miles. (Is this a meaningful 5.87，10 number?) Activity 7 Do You Agree? Write out two very large numbers and two very small numbers and give them to a partner to write in scientific notation. Your partner should do the same with you. Check the numbers to see if you agree with the ways in which these numbers are written in scientific notation. Make certain that you are using correct notation. Then write other numbers in scientific notation and give them to one another to write in full notation. If you have questions, seek clarification. Take-Away Message…Scientific notation provides an efficient way to express and operate on very large and very small numbers, using powers of ten and laws of exponents. Learning Exercises for Section 5.4 4 21. a. What is (6.12 10) (3 10 )? ，，，Students do not have answers 4 2 b. What is (6.12 10) ? (3 10 )? ，，for 1a, 4, 6, 8, and 9. 2 4 c. What is (6.12 10) ? (3 10 )? ，， 4 6 102. Why is (3 10) (4 10) not 12 10 in scientific notation? ，，，， 3. a. Write 45,000,000 220,000,000,000 in scientific notation, then ， compute and be sure the answer is in scientific notation. b. Do the same for 6,900,000 ? 23,000,000,000. 4. The word billion c. Do the same for 0.0000000000056 ? 70,000. means different things in different countries: d. Do the same for 0.0084 ? 0.000004. In the U.S it means 4. a. Write 1.5 billion in scientific notation. one thousand million, b. Write 4.27 trillion in scientific notation. but in Britain and in some other countries it 6 45. Find (3.14 10)+ (2.315 10). Write a sentence about the role of ，，means one million scientific notation in addition and subtraction. million. 6. Experiment with scientific notation on a scientific calculator. Where does the power of ten appear on the calculator? Do the computations in Exercises 1, 3, and 5 using a scientific calculator. Think about how to enter and read the numbers. n2n7. Choose some number to be n, and then find 2n; 2; n; 10. As n gets larger, describe how the results change in each case. Does the result grow faster when n is an exponent, or when it is a base? 8. How long is 1 billion seconds in hours? In days? In years? Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 110 100 . Have you ever heard this term used? If so, where? 9. A googol is 10 10. Not all large numbers are commonly expressed in scientific notation. 10Computer memory is expressed in powers of two. A kilobyte is 2 102030(kilo means 1000, and 2 = 1024); a megabyte is 2; a gigabyte is 2. Which of these is close to 1 billion? 5.5 Issues For Learning: Mental Computation iiiIn a research study fourth- and sixth-grade students received almost daily instruction on mental computation that always allowed students to discuss a variety of strategies for mental computation problems. At the beginning, these students wanted to perform the mental analogue of the paper-and-pencil procedure, that is, they would say (when asked to add 38 and 45 mentally) ―I put the 45 under the 38. Then I add 5 and 8 and that‘s 13 and I write down a 3 and carry the 1. Then I add 1 and 3 and 4 and I get 8, so the answer is 83.‖ When asked to add 345 and 738 these students could not mentally remember the steps and made frequent errors. By the end of the research study, students would have other methods of mental computation that did not depend on trying to remember all the numbers from the standard algorithms. For example, they would add 38 and 45 by saying ―Thirty and 40 is 70, and 8 and 5 is 13, and 70 and 13 is 83.‖ Now they were using place value. Instead of adding 3 and 4, they were adding 30 and 40. They could now mentally compute with skill and accuracy. Mental computation is used in many cultures by people who need to do ivfrequent calculations. For example, in studies of the Diola people of the Ivory Coast, unschooled children who worked with their parents in the marketplace developed excellent mental computation skills that showed deep insight into the structure and properties of the whole number system. Mental computation can play an important role in developing number sense through explorations that force students to use numbers and number relations in novel ways that are likely to increase awareness of the structure of the number system. Unfortunately some teachers view mental computation as a skill to be practiced very rapidly. In such cases instruction tends to focus on drill using chain calculations (e.g., 4 + 15; – 9; 2; 3, –15, ? 5 is 9) and on learning "tricks" such as those for ，， multiplying by 9 or by 11. Although such skills are valuable and could have a place in the curriculum, it is questionable whether they should be emphasized at the expense of instruction that could lead children to better number sense. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 111 5.6 Check Yourself The power of mental computation and computational estimation skills is that they both cause and result from number flexibility. As you develop these skills you will feel more comfortable with numbers. This is true for elementary school children as well. By providing them opportunities to work with numbers in this personal way, you will allow them to develop a comfort with numbers that can permeate their study of numbers and operations. You should be able to work problems like those assigned and to meet the following objectives. 1. Perform mental calculations when appropriate, depending on the numbers involved, and do so in a variety of different ways. 2. Identify the associative and commutative properties of addition and multiplication, and the distributive property of multiplication over addition when they are used in a mental computation. 3. Estimate calculations in a variety of ways, for a variety of forms of numbers. Evaluate computational estimations in terms of what they reveal about 4. the number understanding of the person doing the calculations. 5. Use personal benchmarks to estimate, at the least, numerosity and distance. 6. Determine when a number is in scientific notation, and what it means. 7. Discuss why scientific notation makes it easier to express large and small numbers. 8. Express given numbers or do calculations involving scientific notation. References for Chapter 5 iThreadgill-Sowder, J. (1984). Computational estimation procedures of school children. Journal of Educational Research, 77, 332-336. iiSowder, J. T. (1992). Making sense of numbers in school mathematics. In G. Leinhardt, R. Putnam, & R. Hattrup (Eds.), Analysis of arithmetic for mathematics education. Hillsdale, NJ: Erlbaum. iiiMarkovits, Z., & Sowder, J. T. (1994). Developing number sense: An intervention study in grade seven. Journal for Research in Mathematics Education, 25, 4-29. ivGinsburg, H. P., Posner, J. K., & Russell, R. L. (1981). The development of mental addition as a function of schooling and culture. Journal of Cross-Cultural Psychology, 12(2), 163-178. Reasoning About Numbers and Quantities Chapter 5 Instructor‘s Version p. 112