Patterns in Decimals You encounter decimals every day. Prices displayed in stores and adver- tisements, statistics in the sports section of the newspaper, and call num- bers on books in the library are often given as decimals. In this lesson, you will review the meaning of decimals, and you will practice working with decimals in a variety of situations. MATERIALS Explore • Spare Change cards (1 set per Read the rules of the Spare Change game, and then play two rounds group) with your group. • dollar charts Spare Change Game Rules (1 per player) •Place four Spare Change cards face up on the table. Place the rest of the deck face down in a pile. •To take a turn, a player chooses one of the four cards showing and shades that fraction of a dollar on his or her dollar chart. The player then places the card on the bottom of the deck, and replaces it with the top card from the deck. ① Possible answer: Ialways chose the greatest value I could, or I would try to choose a value that would not allow my opponent to get $1.00 perfectly. When I was closer to a dollar than •Play continues until one player has shaded the entire card or is my partner, I could unable to shade any of the four amounts showing. sometimes choose a •The player who has shaded the amount closest to a dollar at the card so that all the remaining cards would end of the game is the winner. put my partner over a dollar. Describe some strategies you used while playing the game. See ①. ② Possible answer: ② I learned to look Discuss what you learned about decimals. See . carefully at the decimal places—$.5 is 50 cents, not 5 cents! 112 CHAPTER 2 All about Numbers Investigation 1 Understanding Decimals Decimals are equivalent to fractions whose denominators are 10, 100, 1,000, 10,000, and so on. Each decimal place has a name based on the fraction it represents. Just Decimal Equivalent Fraction In Words the 0.1 ᎏ1ᎏ one tenth facts 10 ᎏ1ᎏ The prefix “deci” comes 0.01 100 one hundredth from the Latin word 1 0.001 ᎏᎏ one thousandth decem, meaning “ten.” 1,000 ᎏ1ᎏ 0.0001 10,000 one ten-thousandth EXAMPLE How is 9.057 different from 9.57? These decimals look similar, but they represent different numbers. You can see this by looking at the place values of the digits. 9 . 057 9 . 57 ϩ ᎏ0ᎏ ϩ ᎏ5ᎏ ϩ ᎏ7ᎏ ᎏ5ᎏ7 ᎏ9,0ᎏ57 9.057 means 9 10 100 1,000,or 91,000, or 1,000. ϩ ᎏ5ᎏ ϩ ᎏ7ᎏ ᎏ5ᎏ7 ᎏ95ᎏ7 9.57 means 9 10 100,or 9100, or 100. The number 9.057 is read “nine and fifty-seven thousandths.” The number 9.57 is read “nine and fifty-seven hundredths.” In Problem Set A, you will see how the ideas discussed above relate to the Spare Change game. LESSON 2.3 Patterns in Decimals 113 MATERIALS Problem Set A dollar charts 1. Consider the values $.3 and $.03. a. Are these values the same? no ϭ ᎏ3ᎏ 1b. $.3 10 dollar, b. Explain your answer to Part a by writing both amounts as fractions. ϭ ᎏ3ᎏ $.03 100 dollar c. Illustrate your answer to Part a by shading both amounts on a dollar chart. See below. 2. Do $.3 and $.30 represent the same value? Explain your answer by writing both amounts as fractions and by shading both amounts on a dollar chart. See below. Just 3. In her first four turns of the Spare Change game, Rosita chose $.45, $.1, $.33, and $.05. the facts a. Complete a dollar chart showing the amount Rosita should have In many countries, a shaded after the first four turns. See below. decimal comma is used instead of a decimal b. What part of a dollar is shaded on Rosita’s chart? Express your answer as a fraction and as a decimal. ᎏ9ᎏ3 point, and a space is 100, 0.93 used to separate c. How much more does Rosita need to have $1.00? Express your groups of three digits. ᎏ7ᎏ answer as a decimal and as a fraction. 100 dollar, $.07 4. Shade 12 whole 4. How could you shade a dollar chart to represent $.125? ᎏ1ᎏ rectangles and 2 of another. Now you will explore how multiplying or dividing a number by 10, 100, 1,000, and so on changes the position of the decimal point. 1c. Possible chart: ϭ ᎏ3ᎏ 2. yes; $.3 10 dollar, ϭ ᎏ3ᎏ0 ϭ ᎏ3ᎏ $.30 100 dollar 10 dollar Possible chart: 3a. 114 CHAPTER 2 All about Numbers Problem Set B 1. Copy this table: Calculation Result ؒ ϭ Just 81.07 81.07 1 81.07 ϭ 81.07 ؒ 10 810.7 10 ؒ 81.07 the facts 81.07 ؒ 10 ؒ 10 ϭ 81.07 ؒ 100 8,107 The numbers 10, 100, 81.07 ؒ 10 ؒ 10 ؒ 10 ϭ 81.07 ؒ 1,000 81,070 and so 81.07 ؒ 10 ؒ 10 ؒ 10 ؒ 10 ϭ 81.07 ؒ 10,000 810,700 ,10,000 ,1,000 ؒ on are called powers of ؒ ؒ ؒ ؒ ؒ ϭ ten because they can 81.07 10 10 10 10 10 81.07 100,000 8,107,000 be written as 10 raised to a power. For example: a. Enter the number 81.07 on your calculator. Multiply it by 10, and record the result in the second row of the table. •10 ϭ 101 ؒ • 1,000 ϭ 103 b. Find 81.07 100 by multiplying your result from Part a by 10. Record the result in the table. 2a. 78,010 ;c. Continue to multiply each result by 10 to find 81.07 ؒ 1,000 .2b. 0.3 81.07 ؒ 10,000; and 81.07 ؒ 100,000. Record your results 2c. 9,832,000 d. Describe how the position of the decimal point changed each time 2d. Possible answer: you multiplied by 10. It moved one place to the right. I moved the deci- mal point one 2. In Parts a–c, predict the value of each product without doing any place to the right calculations. Check your prediction by using your calculator. for each 0. a. 7.801 ؒ 10,000 b. 0.003 ؒ 100 c. 9,832 ؒ 1,000 3a. It is multiplied by 10. It is multiplied d. When you predicted the results of Parts a–c, how did you deter- by 100. It is multi- mine where to put the decimal point? plied by 1,000. 3. Think about how the value of a number changes as you move the 3b. Possible answer: decimal point to the right. The value of a. How does the value of a number change when you move the deci- the number is multiplied by 10 mal point one place to the right? Two places to the right? Three for each decimal places to the right? (Hint: Look at your completed table from place moved. Problem 1, or test a few numbers to see what happens.) 4a. 100; Possible b. Challenge In general, what is the relationship between the explanation: To get number of places a decimal is moved to the right and the change from 2.4 to 240, in the value of the number? the decimal point is moved two 4. Tell what number you must multiply the given number by to get places to the right. 240. Explain how you found your answer. 4b. 1,000; Possible a. 2.4 b. 0.24 c. 0.00024 explanation: To get from 0.24 to 240, 4c. 1,000,000; Possible explanation: the decimal point To get from 0.00024 to 240, the is moved three decimal point is moved six places places to the right. to the right. LESSON 2.3 Patterns in Decimals 115 Problem Set C 1. Copy this table: Calculation Result 81.07 81.07 Ϭ ϭ ᎏ1ᎏ 81.07 10 10 of 81.07 8.107 Ϭ Ϭ ϭ ᎏ1ᎏ 81.07 10 10 100 of 81.07 0.8107 81.07 Ϭ 10 Ϭ 10 Ϭ 10 ϭ ᎏ1ᎏ of 81.07 1d. It moved one place 1,000 0.08107 to the left. Ϭ Ϭ Ϭ Ϭ ϭ ᎏ1ᎏ 81.07 10 10 10 10 10,000 of 81.07 0.008107 2a. 0.001414 Ϭ Ϭ Ϭ Ϭ Ϭ ϭ ᎏ1ᎏ 81.07 10 10 10 10 10 100,000 of 81.07 0.0008107 2b. 343.72 ᎏ1ᎏ 2c. 0.877 a. Find 10 of 81.07 by entering 81.07 on your calculator and dividing by 10. Record the result in the second row of the table. 2d. Possible answer: I ᎏ1ᎏ moved the decimal b. Find 100 of 81.07 by dividing your result from Part a by 10. point one place to Record the result in the table. the left for each 0. ᎏ1ᎏ ᎏ1ᎏ c. Continue to divide each result by 10 to find 1,000 of 81.07; 10,000 3a. It is divided by 10. of 81.07; and ᎏ1ᎏ of 81.07. Record your results. It is divided by 100,000 100. It is divided d. Describe how the position of the decimal point changed each time ᎏ1ᎏ by 1,000. you divided by 10 (that is, each time you found 10). 3b. Possible answer: 2. In Parts a–c, predict each result without doing any calculations.