S OLUTIONS M ANUAL M ICHAEL M ATTHEWS University of Nebraska at Omaha M ATHEMATICS FOR E LEMENTARY T EACHERS WITH A CTIVITIES F IFTH E DITION Sybilla Beckmann University of Georgia iii TABLE OF CONTENTS Mathematics for Elementary Teachers with Activities, 5e Chapter 1 Numbers and the Base-Ten System ........................................................................ 1-1 Chapter 2 Fractions and Problem Solving ............................................................................... 2-1 Chapter 3 Addition and Subtraction ........................................................................................ 3-1 Chapter 4 Multiplication ......................................................................................................... 4-1 Chapter 5 Multiplication of Fractions, Decimals and Negative Numbers .............................. 5-1 Chapter 6 Division................................................................................................................... 6-1 Chapter 7 Ratio and Proportional Relationships ..................................................................... 7-1 Chapter 8 Number Theory ....................................................................................................... 8-1 Chapter 9 Algebra ................................................................................................................... 9-1 Chapter 10 Geometry .............................................................................................................. 10-1 Chapter 11 Measurement......................................................................................................... 11-1 Chapter 12 Area of Shapes ...................................................................................................... 12-1 Chapter 13 Solid Shapes and Their Volume and Surface Area .............................................. 13-1 Chapter 14 Geometry of Motion and Change ......................................................................... 14-1 Chapter 15 Statistics ................................................................................................................ 15-1 Chapter 16 Probability............................................................................................................. 16-1 1.1 The Counting Numbers 1-1 Chapter 1 Numbers and the Base-Ten System 1.1 The Counting Numbers 1. Answers will vary. For example, when connecting the counting numbers as a list view of numbers with the number of objects in a set view of numbers, a child must learn to associate each number in the list in a one to one correspondence with each object in the set, starting with one. Also, the child must be able to learn that the last number from the list, used to connect with the last object in the set, is the number of objects in the set. 2. Yes, there is a better way to respond. For instance, you could group the beads into sets of 10 beads in each group. Then you would have 3 groups of 10 beads in each group and there would be 5 left over beads. This grouping would facilitate a discussion about place value and allow the conversation to focus on 3 tens. 3. a. You could group the beads into sets of 10 beads in each group. Then you would have 4 groups of 10 beads in each group and there would be 7 left over beads. Using the place value system of representing numbers, 4 tens and 7 ones is 47. Figure 1.1 shows a simple math drawing that could be drawn. Figure 1.1: Representation of 47 b. You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band, 10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then you would have 3 bundles of 100 toothpicks in each bundle (or 3 hundreds) and you would have 2 bags of 10 toothpicks in each bag (or 2 tens) and there would be 8 left over toothpicks. Using the place value system of representing numbers, 3 hundreds, 2 tens, and 8 ones is 328. Figure 1.2 shows a simple math drawing that could be drawn. 1-2 Chapter 1: Numbers and the Base-Ten System Figure 1.2: Representation of 328 c. You could bag the toothpicks into sets of 10 toothpicks in each bag. Then when you get 10 bags of 10 toothpicks in each, you could bundle, with a rubber band, 10 bags of 10 toothpicks to make sets of 100 toothpicks in each bundle. Then when you have 10 bundles of 100 toothpicks, you could get a giant gallon sized plastic bag and put them into it, and group these 10 sets of 100 toothpicks into 1 set of 1000 toothpicks. Using the place value system of representing numbers, 1 thousand is represented as 1000. Figure 1.3 shows a simple math drawing that could be drawn. Figure 1.3: Representation of 1000 1.1 The Counting Numbers 1-3 4. For instance, if you we are using Popsicle sticks, you could get a large collection (say three-hundred fifty-six sticks) in an unorganized pile. Then you could ask the students that are learning about place value to start counting them. After a bit, someone (you or some of the students) will likely start to group the Popsicle sticks into piles of equal size. You could then count the Popsicle sticks faster by bundling groups of 10 together. Perhaps you might bundle these groups together physically by tying a twist tie around each set of 10 Popsicle sticks. After a while of doing this, you will have lots of bundled sets of 10 Popsicle sticks. At this stage, you could take 10 sets of twist-tied sets of 10 Popsicle sticks and put them in to a gallon sized plastic bag to make groups of 100 Popsicle sticks (consisting of 10 sets of 10 bundled Popsicle sticks). As you continue making twist-tied bundles of 10 and baggies of 100, you eventually will use up all of the Popsicle sticks. When this is done, you would end up with 3 baggies (or 3 hundreds) and 5 twist-tied bundles (or 5 tens) and 6 left over Popsicle sticks. Now you could point out that your baggies, bundles, and individual Popsicle sticks correspond directly with the base-ten representation of three hundred fifty six Popsicle sticks (or 356). Since the digit in each place value is representing a count of objects that consist of 10 bundles of objects that are represented in the digit to the right, we see that each place value represents a number that is 10 times greater than the place value to its immediate right. For example, when we count the farthest left place value in this number (365), we count groups of hundred (3 baggies). Since each baggie is made up of 10 bundles, we note that the place value immediately to right of the hundreds place is the place value where we are counting the bundles (or tens). 5. Young children must learn several key ideas about place value and overcome some linguistic hurdles to learn how to count in the base-ten system. They must understand the key role that 10 plays in our base-ten system or in representing numbers. They must understand how the location affects the value of the number or the unit of the digit in that particular location. They must overcome linguistic difficulties inherit to how we say numbers in English, especially the anomalies like eleven or the inconsistent order of twenty-two (2 tens and two ones) and nineteen (one nine and one ten). 6. The number 1001 looks like 100 and 1 put together. Calling it “one hundred one” makes sense w/o understanding the structure of our number system. See Figure 1.4. Each small block represents 1. Figure 1.4: Base-Ten Representation of 1001 1-4 Chapter 1: Numbers and the Base-Ten System 7. If we count what we’ve got in the math drawing, we see 17 individual toothpicks and 15 bags of 10 toothpicks in each bag. Naively, we might write this as 1517 toothpicks, which would be misleading since as written it would represent one-thousand five-hundred seventeen toothpicks. Since our place value system can only represent up to 9 of any particular place value, we have to regroup when we have more than 9 of a particular place value (or unit). In terms of toothpicks, this means that since 17 is greater than 9, we have enough individual toothpicks to regroup into one more baggie of 10. This gives us 7 left over toothpicks but now 16 bags. Similarly, we also have enough bags to regroup (or bundle) them with a rubber band into one group of 100 toothpicks. This gives us 1 bundle of 100 toothpicks (or 1 hundred), 6 bags of ten toothpicks (or 6 tens) and 7 individual toothpicks. See Figure 1.5 for what Figure 1.10 (in the regular text) would look like once you’ve regrouped the numbers in a way that corresponds to the structure of the base-ten system. Figure 1.5:Representation of 167 8. Answers will vary. In the base-ten system the digits represent different values of objects. The place value is integrally related to the value that any particular digit represents. The number ten plays a vital role in the system and is the basis of the value of each place. The base-ten system is much easier to represent large numbers than more primitive ways of representing numbers. However, the base-ten system is not as intuitive and is harder to learn than more primitive systems, such as a simple tally mark system. 9. a. See Figure 1.6. In the first number line I first used the larger tick marks to represent 400 each. Then I counted up to 800 and then 1200 using these sized tick marks. Realizing that 900 wasn’t going to fall perfectly on a 400 tick mark, I divided the 400 tick marks spaces into 4 smaller spaces and then each shorter tick mark represented 100. I then counted one more shorter tick mark past 800 to get to 900. b. See Figure 1.6. For the second number line, we divided the space between 0 and 300 into 3 equal spaces so between taller tick marks represents 100. Then we divided each of these spaces representing 100 into 2 small spaces representing 50. I counted up to 200 with large tick marks and then over one more smaller space to 250. 1.2 Decimal and Negative Numbers 1-5 c. See Figure 1.6. For the last number line, I let the spaces between taller tick marks represent 2000. I went up to 6,000. Next, since it takes ten 200s to make 2,000 and the gap between 6,000 and 8,000 is 2,000, then I made 10 spaces between 6,000 and 8,000 and plotted 6,200 one of these spaces above 6,000. Figure 1.6: Number lines 10. Answers will vary. For example, they could use a photocopier and shrink the poster to a small size (but where the dots are still distinguishable hopefully), then they could make 999 copies of this small sized poster. This would work since each of the 1000 shrunk down posters contain 1 million dots and 1,000,000 x 1,000 is 1 billion. It would probably be doable; however, one might need to find a special photocopier (like one that photocopies large maps) if the poster is large enough. Or for a more green approach, one could try to have each student try to draw a portion of the dots themselves. For example if there were 25 students in the class then each student would need to represent 40 million dots. This would likely be not very feasible because if you worked everyday on this for 4 weeks (for a total of 5x4 or 20 days) each student would still need to draw 40 20  million or 2 million dots a day. If they worked for 50 minutes on it, that would be 2, 000, 000 50  or 40,000 dots every minute. Even if they could do this, the class would likely revolt after two or three days of drawing dots day after day. 1.2 Decimals and Negative Numbers 1. As in Practice Exercise 3, one toothpick could represent any number of quantities. If one toothpick represents 1, then the given collection represents 346. Other possibilities are shown in the table. If 1 toothpick represents: then the collection represents: 100 34,600 10 3,460 1 346 1 10 34.6 1 100 3.46 1 1000 0.346 1-6 Chapter 1: Numbers and the Base-Ten System 2. Base-ten block sketches will be shown here. Similar sketches can be used for the bundled objects model. a. Figure 1.7 represents 0.26 if we consider each block to represent 0.01. Then by counting we see that we have 6 blocks (or 6 hundredths) and 2 bundles of 10 blocks (or 2 tenths). If we decided to have each block represent 0.1 instead, then Figure 1.7 would represent 2.6; whereas if each block represented 100 then Figure 1.7 would represent 2,600. Figure 1.7: A representation of 0.26 b. Figure 1.8 represents 13.4 if we consider each block to represent 0.1. Then by counting we see that we have 4 blocks (or 4 tenths) and 3 bundles of 10 blocks (or 3 ones) and 1 bundle of 100 blocks (or 1 ten). If we decided to have each block represent 0.001 instead, then Figure 1.8 would represent 0.134; whereas if each block stood for 1 then Figure 1.8 would represent 134. Figure 1.8: A representation of 13.4 c. Figure 1.9 represents 1.28 if we consider each block to represent 0.01. Then by counting we see that we have 8 blocks (or 8 hundredths) and 2 bundles of 10 blocks (or 2 tenths) and 1 bundle of 100 blocks (1 one). If we decided to have each block represent 0.00001 instead, then Figure 1.9 would represent 0.00128; whereas if each block stood for 10 then Figure 1.9 would represent 1,280. Figure 1.9: A representation of 1.28 1.3 Reasoning to Compare Numbers in Base Ten 1-7 d. Figure 1.10 represents 0.000032 if we consider each block to represent 0.000001. Then by counting we see that we have 2 blocks (or 2 millionths) and 3 bundles of 10 blocks (or 3 hundred-thousandths). If we decided to have each block represent 0.1 instead, then Figure 1.10 would represent 3.2; whereas if each block stood for 0.001 then Figure 1.10 would represent 0.032. Figure 1.10: A representation of 0.000032 3. See Figure 1.11. In the figure below the darkened strip next to the phrase “each 0.001”is meant to be one 0.01 strip broken into 10 smaller strips. To represent 1.438 as a length, make a long strip by laying a 1 unit long strip next to four 0.1 unit long strips, three 0.01 unit long strips and eight 0.001 unit long strips. To represent 0.804, lay eight 0.1 unit long strips next to four 0.001 unit long strips. Figure 1.11: Representing Base-Ten Numbers as Lengths 4. Jerome is trying to follow the pattern shown on the number line by thinking that the number seven and ten tenths comes after seven and nine tenths. In fact, that thinking is correct. What is incorrect is to write seven and ten tenths as 7.10. Pointing out to Jerome that 7.10 is already on the number line (7.1) might help him understand his error. Base- ten blocks could also help by allowing him to see that seven units and ten tenths of a unit is properly rearranged as eight units. 5. Answers will vary. Examples in which zeros can be dropped include such numbers as 01, 1.0, and 00.20100. Examples in which zeros cannot be dropped include such numbers as 1.01, 2.00301, and 0.00002. The issue of whether one can drop the zero directly in front of the decimal point (e.g., .1 vs. 0.1) is not a mathematical issue but one of style. Sometimes zeros which could be dropped mathematically are kept because of tolerance levels. This is discussed in the “What Is the Significance of Rounding When Working with Numbers That Represent Actual Quantities”, Section 1.4. 1-8 Chapter 1: Numbers and the Base-Ten System 6. Answers will vary. For example, see Figure 1.12. Figure 1.12: Zooming in on a number line. 7 See Figure 1.13. Figure 1.13: Zooming in on 7.0028 8. See Figure 1.14. 1.3 Reasoning to Compare Numbers in Base Ten 1-9 Figure 1.14: Locating numbers on various scales 9. Yes, Cierral may label the tick mark that way. Starting from the left, the tick marks should then be labeled 7.0001, 7.0002, 7.0003, …, 7.0009. 10. Yes, Juan may plot 9.999 where he did. Starting from the left, the intervening tick marks should then be labeled 9.9991, 9.9992, …, 9.9999. 11. a. This number line can be labeled in many different ways. Three examples are shown in Figure 1.15. Figure 1.15: Three Ways to Label the Tick Marks b. This number line can be labeled in many different ways. Three examples are shown in Figure 1.16. 1-10 Chapter 1: Numbers and the Base-Ten System Figure 1.16: Three Ways to Label the Tick Marks c. This number line can be labeled in many different ways. Three examples are shown in Figure 1.17. Figure 1.17: Three Ways to Label the Tick Marks d. This number line can be labeled in many different ways. Three examples are shown in Figure 1.18. Figure 1.18: Three Ways to Label the Tick Marks 12. The distance between 0 and 1 is the unit. We place -1 one unit to the left of the 0. We place -2 two units to the left of the 0. For -1.68, we start at 0 and move 1 unit to the left. Then from this spot, we move 6 tenths of a unit to the left from this spot. Finally, we move 8 hundredths of a unit to the left from this spot to arrive at -1.68. 13. Figure 1.19 shows the decimal numbers: -1, -0.92, -0.3, -0.03, 0, 0.07, 0.1, 0.3, 0.9, and 1. These choices allow students to distinguish between numbers such as 0.07 and 0.1, which allows students to consider place value when plotting each number. These choices also 1.3 Reasoning to Compare Numbers in Base Ten 1-11 allow a comparison of the decimals -0.03, -0.3, and 0.3, which helps students consider the place value meaning of -0.03 compared to -0.3 and also to consider that, like the numbers on the positive part of the number line, smaller decimals (-0.3 for instance) are farther to the left from larger decimals like -0.03. Figure 1.19: Comparing decimals on a number line 14. a. See Figure 1.20. Figure 1.20: -4.3 on a number line b. See Figure 1.21. Figure 1.21: -0.28 on a number line c. See Figure 1.22. Figure 1.22: -0.28 on a number line, zoomed in d. See Figure 1.23. Figure 1.23: -6.193 on a number line e. See Figure 1.24. Figure 1.24: -6.193 on a number line, zoomed in 15. Thinking of –N as the opposite of N, then if N were a negative number, then –N would be the opposite of a negative number. In other words, -N would be a positive number. N would also be a positive number. So –N and N would be the same number. 1.3 Reasoning to Compare Numbers in Base Ten 1. We compare numbers in the base-ten system in the way that we do because base-ten places of larger value count more than the largest combined value made with lower 1-12 Chapter 1: Numbers and the Base-Ten System places. For instance, when comparing 234 to 219, we see that both numbers have the same value in the hundreds place but that the 234 has a greater value in the tens place. Since the tens place counts more than the largest amount you could have in the ones place, we can essentially ignore the values in the ones place. 2. See Figure 1.25. Figure 1.25: Using bundled objects to compare decimals 3. a. See the solution to practice exercise #3. b. See Figure 1.26 Since 1.1 is farther to the right of 0.999 so 1.1 is greater than 0.999. Figure 1.26 Comparing 1.1 and 0.999 with a number line. c. See Figure 1.27. 1.1 is greater because it has more overall toothpicks. Key = 1 toothpick = 0.001 1.3 Reasoning to Compare Numbers in Base Ten 1-13 , Figure 1.27: Using bundled objects to compare 1.1 and 0.999. 4. See Figure 1.28. a. 0 is to the left of 0.6 in the number line so 0<0.6. b. 0.00 = 0, which is to the left of 0.7, so 0.00<0.7. c. 3.00, 3.0, and 3 are all the same point. d. 3.7777 is to the right of 3.77 so 3.7777 > 3.77. Figure 1.28: A number line to compare 0 and 0.6. 1.1 0.999