Thomas' Calculus: Early Transcendentals , 14th Edition Solution Manual

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SOLUTIONSMANUALMICHAELMATTHEWSUniversity of Nebraska at OmahaMATHEMATICSFORELEMENTARYTEACHERSWITHACTIVITIESFIFTHEDITIONSybilla BeckmannUniversity of Georgia

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iiiTABLE OF CONTENTSMathematics for Elementary Teachers with Activities, 5eChapter 1Numbers and the Base-Ten System........................................................................ 1-1Chapter 2Fractions and Problem Solving............................................................................... 2-1Chapter 3Addition and Subtraction........................................................................................ 3-1Chapter 4Multiplication ......................................................................................................... 4-1Chapter 5Multiplication of Fractions, Decimals and Negative Numbers .............................. 5-1Chapter 6Division................................................................................................................... 6-1Chapter 7Ratio and Proportional Relationships ..................................................................... 7-1Chapter 8Number Theory....................................................................................................... 8-1Chapter 9Algebra ................................................................................................................... 9-1Chapter 10Geometry .............................................................................................................. 10-1Chapter 11Measurement......................................................................................................... 11-1Chapter 12Area of Shapes...................................................................................................... 12-1Chapter 13Solid Shapes and Their Volume and Surface Area .............................................. 13-1Chapter 14Geometry of Motion and Change ......................................................................... 14-1Chapter 15Statistics................................................................................................................ 15-1Chapter 16Probability............................................................................................................. 16-1

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1.1 The Counting Numbers 1-1Chapter 1Numbers and the Base-Ten System1.1The Counting Numbers1.Answers will vary. For example, when connecting the counting numbers as a list view ofnumbers with the number of objects in a set view of numbers, a child must learn toassociate each number in the list in a one to one correspondence with each object in theset, starting with one. Also, the child must be able to learn that the last number from thelist, 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 of10 beads in each group. Then you would have 3 groups of 10 beads in each group andthere would be 5 left over beads. This grouping would facilitate a discussion about placevalue 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 wouldhave 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 47b.You could bag the toothpicks into sets of 10 toothpicks in each bag. Then whenyou 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 youwould have 3 bundles of 100 toothpicks in each bundle (or 3 hundreds) and youwould have 2 bags of 10 toothpicks in each bag (or 2 tens) and there would be 8left over toothpicks. Using the place value system of representing numbers, 3hundreds, 2 tens, and 8 ones is 328. Figure 1.2 shows a simple math drawing thatcould be drawn.

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1-2Chapter 1: Numbers and the Base-Ten SystemFigure 1.2: Representation of 328c.You could bag the toothpicks into sets of 10 toothpicks in each bag. Then whenyou 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. Thenwhen you have 10 bundles of 100 toothpicks, you could get a giant gallon sizedplastic bag and put them into it, and group these 10 sets of 100 toothpicks into 1set of 1000 toothpicks. Using the place value system of representing numbers, 1thousand is represented as 1000. Figure 1.3 shows a simple math drawing thatcould be drawn.Figure 1.3: Representation of 1000

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1.1 The Counting Numbers 1-34.For instance, if you we are using Popsicle sticks, you could get a large collection (saythree-hundred fifty-six sticks) in an unorganized pile. Then you could ask the studentsthat are learning about place value to start counting them. After a bit, someone (you orsome 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 aroundeach set of 10 Popsicle sticks.After a while of doing this, you will have lots of bundled sets of 10 Popsicle sticks. Atthis stage, you could take 10 sets of twist-tied sets of 10 Popsicle sticks and put them into a gallon sized plastic bag to make groups of 100 Popsiclesticks (consisting of 10 setsof 10 bundled Popsicle sticks). As you continue making twist-tied bundles of 10 andbaggies 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) and6 left over Popsicle sticks. Now you could point out that your baggies, bundles, andindividual Popsicle sticks correspond directly with the base-ten representation of threehundred fifty six Popsicle sticks (or 356). Since the digit in each place value isrepresenting a count of objects that consist of 10 bundles of objects that are representedin the digit to the right, we see that each place value represents a number that is 10 timesgreater than the place value to its immediate right. For example, when we count thefarthest 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 toright 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 somelinguistic hurdles to learn how to count in the base-ten system. They must understand thekey role that 10 plays in our base-ten system or in representing numbers. They mustunderstand how the location affects the value of the number or the unit of the digit in thatparticular location. They must overcome linguistic difficulties inherit to how we saynumbers in English, especially the anomalies like eleven or the inconsistent order oftwenty-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” makessense w/o understanding the structure of our number system. See Figure 1.4. Each smallblock represents 1.Figure 1.4: Base-Ten Representation of 1001

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1-4Chapter 1: Numbers and the Base-Ten System7.If we count what we’ve got in the math drawing, we see 17 individual toothpicks and 15bags of 10 toothpicks in each bag. Naively, we might write this as 1517 toothpicks, whichwould be misleading since as written it would represent one-thousand five-hundredseventeen toothpicks.Since our place value system can only represent up to 9 of any particular place value, wehave to regroup when we have more than 9 of a particular place value (or unit). In termsof toothpicks, this means that since 17 is greater than 9, we have enough individualtoothpicks to regroup into one more baggie of 10. This gives us 7 left over toothpicks butnow 16 bags. Similarly, we also have enough bags to regroup (or bundle) them with arubber 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. SeeFigure 1.5 for what Figure 1.10 (in the regular text) would look like once you’veregrouped the numbers in a way that corresponds to the structure of the base-ten system.Figure 1.5:Representation of 1678.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. Thenumber 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 waysof representing numbers. However, the base-ten system is not as intuitive and is harderto 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 torepresent 400 each. Then I counted up to 800 and then 1200 using these sized tickmarks. Realizing that 900 wasn’t going to fall perfectly on a 400 tick mark, Idivided the 400 tick marks spaces into 4 smaller spaces and then each shorter tickmark represented 100. I then counted one more shorter tick mark past 800 to getto 900.b.See Figure 1.6. For the second number line, we divided the space between 0 and300 into 3 equal spaces so between taller tick marks represents 100. Then wedivided 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 to250.

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1.2 Decimal and Negative Numbers 1-5c.See Figure 1.6. For the last number line, I let the spaces between taller tick marksrepresent 2000. I went up to 6,000. Next, since it takes ten 200s to make 2,000and the gap between 6,000 and 8,000 is 2,000, then I made 10 spaces between6,000 and 8,000 and plotted 6,200 one of these spaces above 6,000.Figure 1.6: Number lines10.Answers will vary. For example, they could use a photocopier and shrink the poster to asmall size (but where the dots are still distinguishable hopefully), then they could make999 copies of this small sized poster. This would work since each of the 1000 shrunkdown posters contain 1 million dots and 1,000,000 x 1,000 is 1 billion. It would probablybe doable; however, one might need to find a special photocopier (like one thatphotocopies large maps) if the poster is large enough. Or for a more green approach, onecould try to have each student try to draw a portion of the dots themselves. For example ifthere were 25 students in the class then each student would need to represent 40 milliondots. This would likely be not very feasible because if you worked everyday on this for 4weeks (for a total of 5x4 or 20 days) each student would still need to draw4020million or 2 million dots a day. If they worked for 50 minutes on it, that would be2, 000, 00050or 40,000 dots every minute. Even if they could do this, the class wouldlikely revolt after two or three days of drawing dots day after day.1.2 Decimals and Negative Numbers1.As in Practice Exercise 3, one toothpick could represent any number of quantities. If onetoothpick represents 1, then the given collection represents 346. Other possibilities areshown in the table.If 1 toothpick represents:then the collection represents:10034,600103,460134611034.611003.46110000.346

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1-6Chapter 1: Numbers and the Base-Ten System2.Base-ten block sketches will be shown here. Similar sketches can be used for the bundledobjects model.a.Figure 1.7 represents 0.26 if we consider each block to represent 0.01. Then bycounting we see that we have 6 blocks (or 6 hundredths) and 2 bundles of 10blocks (or 2 tenths). If we decided to have each block represent 0.1 instead, thenFigure 1.7 would represent 2.6; whereas if each block represented 100 then Figure1.7 would represent 2,600.Figure 1.7: A representation of 0.26b.Figure 1.8 represents 13.4 if we consider each block to represent 0.1. Then bycounting we see that we have 4 blocks (or 4 tenths) and 3 bundles of 10 blocks (or3 ones) and 1 bundle of 100 blocks (or 1 ten). If we decided to have each blockrepresent 0.001 instead, then Figure 1.8 would represent 0.134; whereas if eachblock stood for 1 then Figure 1.8 would represent 134.Figure 1.8: A representation of 13.4c.Figure 1.9 represents 1.28 if we consider each block to represent 0.01. Then bycounting we see that we have 8 blocks (or 8 hundredths) and 2 bundles of 10blocks (or 2 tenths) and 1 bundle of 100 blocks (1 one). If we decided to haveeach 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

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1.3 Reasoning to Compare Numbers in Base Ten 1-7d.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 of10 blocks (or 3 hundred-thousandths). If we decided to have each block represent0.1 instead, then Figure 1.10 would represent 3.2; whereas if each block stood for0.001 then Figure 1.10 would represent 0.032.Figure 1.10: A representation of 0.0000323.See Figure 1.11. In the figure below the darkened strip next to the phrase “each 0.001”ismeant 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 0.01unit long strips and eight 0.001 unit long strips. To represent 0.804, lay eight 0.1 unitlong strips next to four 0.001 unit long strips.Figure 1.11: Representing Base-Ten Numbers as Lengths4.Jerome is trying to follow the pattern shown on the number line by thinking that thenumber seven and ten tenths comes after seven and nine tenths. In fact, that thinking iscorrect. What is incorrect is to write seven and ten tenths as 7.10. Pointing out to Jeromethat 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 unitis 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 as1.01, 2.00301, and 0.00002. The issue of whether one can drop the zero directly in frontof 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 tolerancelevels. This is discussed in the “What Is the Significance of Rounding When Workingwith Numbers That Represent Actual Quantities”, Section 1.4.

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1-8Chapter 1: Numbers and the Base-Ten System6.Answers will vary. For example, see Figure 1.12.Figure 1.12: Zooming in on a number line.7See Figure 1.13.Figure 1.13: Zooming in on 7.00288.See Figure 1.14.

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1.3 Reasoning to Compare Numbers in Base Ten 1-9Figure 1.14: Locating numbers on various scales9.Yes, Cierral may label the tick mark that way. Starting from the left, the tick marksshould 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 marksshould then be labeled 9.9991, 9.9992, …, 9.9999.11.a.This number line can be labeled in many different ways. Three examples areshown in Figure 1.15.Figure 1.15: Three Ways to Label the Tick Marksb.This number line can be labeled in many different ways. Three examples areshown in Figure 1.16.

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1-10Chapter 1: Numbers and the Base-Ten SystemFigure 1.16: Three Ways to Label the Tick Marksc.This number line can be labeled in many different ways. Three examples areshown in Figure 1.17.Figure 1.17: Three Ways to Label the Tick Marksd.This number line can be labeled in many different ways. Three examples areshown in Figure 1.18.Figure 1.18: Three Ways to Label the Tick Marks12.The distance between 0 and 1 is the unit. We place -1 one unit to the left of the 0. Weplace -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, wemove 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, whichallows students to consider place value when plotting each number. These choices also

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1.3 Reasoning to Compare Numbers in Base Ten 1-11allow a comparison of the decimals -0.03, -0.3, and 0.3, which helps students considerthe place value meaning of -0.03 compared to -0.3 and also to consider that, like thenumbers on the positive part of the number line, smaller decimals (-0.3 for instance) arefarther to the left from larger decimals like -0.03.Figure 1.19: Comparing decimals on a number line14.a.See Figure 1.20.Figure 1.20: -4.3 on a number lineb.See Figure 1.21.Figure 1.21: -0.28 on a number linec.See Figure 1.22.Figure 1.22: -0.28 on a number line, zoomed ind.See Figure 1.23.Figure 1.23: -6.193 on a number linee.See Figure 1.24.Figure 1.24: -6.193 on a number line, zoomed in15.Thinking of –N as the opposite of N, then if N were a negative number, then –N would bethe opposite of a negative number. In other words, -N would be a positive number.Nwould also be a positive number. So –N andNwould be the same number.1.3 Reasoning to Compare Numbers in Base Ten1.We compare numbers in the base-ten system in the way that we do because base-tenplaces of larger value count more than the largest combined value made with lower

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1-12Chapter 1: Numbers and the Base-Ten Systemplaces. For instance, when comparing 234 to 219, we see that both numbers have thesame 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 onesplace, we can essentially ignore the values in the ones place.2.See Figure 1.25.Figure 1.25: Using bundled objects to compare decimals3.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 than0.999.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

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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 0.6.1.10.999

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