Unit 4 The Number System: Rational Numbers

Transcription

1 Unit 4 The Number System: Rational Numbers Introduction In this unit, students will compare, add, and subtract decimals and fractions, including positive and negative numbers. Students will write answers to whole-number division questions as fractions and decimals, including for multi-digit numbers divided by one-digit numbers. Notation for adding integers. In Unit 2, we introduced adding and subtracting integers using gains and losses. We used the notation to mean a gain of $3 followed by a loss of $4. As students gain experience with integers, students learn to change the notation of adding integers +3 + ( 4) to this gains and losses notation, making the addition of integers easier for students. Although it can be tempting to interpret as adding 3 and subtracting 4, we teach students to think of it as adding 3 and then adding 4. Please continue this approach in this unit. Terminology. Even though mathematically the number 1 is considered a power of 10 (with exponent 0), we are not introducing it as such in this unit. There is some confusion in naming decimal fractions. We use the convention that 1/, for example, is one hundredth, not one one hundredth. When this causes confusion, as in 350/1,000 (three hundred fifty thousandths) compared to 300/50,000 (three hundred fifty thousandths), always clarify by showing the fraction you are referring to. Note that 350/1,000 would be confused with 300/51,000 if we did read the one in one thousandths, therefore, doing so would not eliminate the confusion. Do not shorten decimal point to decimal. This creates confusion between two different concepts: decimal (a number) and decimal point (the symbol separating parts of the number). Make sure students use proper terminology. When writing negative fractions, be sure to write the negative sign in front of the fraction, not in front of the numerator (this notation will be introduced in Unit 5). Like this: 1 - Not like this: NOTE: Even though fractions often appear in line with the text in our lesson plans (e.g., 1/2), remember to always either stack fractions when you show them to your students (e.g., 1 ) or to 2 introduce the non-stacked notation. Materials. We recommend that students always work in grid paper notebooks. Paper with 1/4-inch grids works well in most lessons. In this unit, grid paper will be especially useful when adding and subtracting multi-digit numbers. If students who have difficulties in visual organization will be working without grid paper, they should be taught to draw a grid before starting to work on a problem. Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-1

2 NS7-18 Decimal Fractions Pages Standards: preparation for 7.NS.A.1 Goals: Students will add tenths, hundredths, and thousandths when written as fractions. Prior Knowledge Required: Recognizes increasing and decreasing patterns Can use grids to represent tenths and hundredths Can write equivalent fractions Can add fractions with the same denominator and fractions with different denominators Can multiply whole numbers by 10 Vocabulary: decimal fraction, denominator, equivalent fraction, hundredth, numerator, power of 10, represent, tenth, thousandth Introduce powers of 10. Write on the board: 10 = = = = Have volunteers fill in the blanks. SAY: These numbers are called powers of 10. We will learn about powers of other numbers later in the year. Review multiplying powers of 10. SAY: Multiplying powers of 10 is easy because you just write more zeros at the end of the number you are multiplying by. Exercises: Multiply. a) 10 b) c) 1, d) Answers: a) 1,000; b) ; c) 10,000; d) 10,000 Exercises: What do you multiply by? a) 10 = 1,000 b) = 1,000 c) 10 = Bonus: d) = 10,000 e) 1,000 = 10,000,000,000 Answers: a) ; b) 10; c) 10; Bonus: d) ; e) 10,000,000 Students who are struggling can write the number of zeros under each power of 10. For example, in part a), write 1 under 10 and 3 under 1,000. E-2 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

3 (MP.7) Introduce decimal fractions. Write on the board: Decimal fractions Not decimal fractions , ,000 Have volunteers suggest fractions. Have the rest of the class point their thumbs toward the correct group to signal where each fraction should be placed. Have students guess the rule for putting the fractions in each group. Explain that a decimal fraction is a fraction whose denominator is a power of 10. Decimal fractions are important because powers of 10 are easy to work with. Point out that while some of the denominators in the not decimal fractions group are multiples of 10, they are not powers of 10. Also, some fractions (such as 1/2 and 2/5) are equivalent to decimal fractions but are not decimal fractions. Review equivalent tenths and hundredths. Draw on the board: SAY: The picture shows why three tenths equals 30 hundredths. The second square has 10 times as many shaded parts and 10 times as many parts altogether. Write on the board: Exercises: Write an equivalent fraction with the denominator. Answers: a) 70/, b) 40/, c) 90/ Equivalent tenths, hundredths, and thousandths. Write on the board: SAY: Now you have to decide what to multiply the numerator by to get an equivalent fraction. You have to figure out what the denominator was multiplied by and then multiply the numerator by the same thing. Have volunteers tell you what to multiply by; then have other volunteers fill in Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-3

4 the numerators. (30, 700, 50) SAY: To make an equivalent fraction, you just have to add the same number of zeros at the end of the numerator and denominator. Exercises: Write the missing numerator in the equivalent fraction. Answers: a) 80; b) 300; c) 900; Bonus: 300,000 Adding tenths and hundredths. Draw on the board: SAY: If you can add hundredths, and if you can change tenths to hundredths, then you can add tenths and hundredths. Three tenths is 30 hundredths, and six more hundredths is 36 hundredths. Remind students that they can change tenths to hundredths without using a picture: Exercises: Add. a) b) c) Bonus: Answers: a) 47/, b) 74/, c) 58/, Bonus: 93/ Adding tenths, hundredths, and thousandths. Write on the board: , ASK: How can you change the fractions to make them easier to add? (change all denominators to 1,000) Write underneath: + + = 1,000 1,000 1,000 1,000 Have volunteers complete the equation: 300/1, /1, /1,000 = 396/1,000. Point out how adding fractions with denominators 10,, and 1,000 is easy because when you make all the denominators 1,000, the numerators are in expanded form. E-4 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

5 Exercises: Add. a) b) Bonus: 10 1, ,000 Answers: a) 439/1,000; b) 521/1,000; Bonus: 438/1, , Adding decimal fractions with missing tenths or hundredths. Write on the board: = 10 1,000 ASK: How many thousandths are in 4/10? (400) So how many thousandths are there altogether? ( = 409) Write the answer: = 10 1,000 1,000 SAY: 4 tenths, 0 hundredths, and 9 thousandths add to 409 thousandths. Have students add more tenths and thousandths. Exercises: Add. a) b) c) , , ,000 Answers: a) 307/1,000; b) 901/1,000; c) 206/1,000 Repeat the process with 4/ + 9/1,000. Then write on the board: 0 tenths + 4 hundredths + 9 thousandths = 49 thousandths SAY: We might be tempted to write this as 049/1,000, but we do not write the zero at the beginning of a number. Exercises: Predict the sum. Then check by adding a) + b) + 1,000 1,000 c) d) , ,000 Bonus: e) + f) + 1, ,000 Answers: a) 37/1,000; b) 82/1,000; c) 802/1,000; d) 508/1,000; Bonus: e) 208/1,000; f) 802/10,000 SAY: You might need to add thousandths or just hundredths. Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-5

6 Exercises: Add the decimal fractions. Write the answer as a decimal fraction. a) b) + c) d) , , ,000 Bonus: e) + + f) + 1, ,000 Answers: a) 892/1,000; b) 806/1,000; c) 96/; d) 35/1,000; Bonus: e) 268/1,000; f) 50,005/,000 Extensions 1. Write 1 as a decimal fraction. Sample answers: 10/10, / 2. (MP.1) a) Is there a largest power of 10? (MP.3) b) Is there a smallest decimal fraction? How do you know? Answers: a) No, because you can multiply any power of 10 by 10 to get an even larger one. b) No, because you can make the fraction smaller by making the denominator a larger power of Find the missing number. 4 9 a) + = b) = c) + = Bonus: d) = e) + = f) + = Answers: a) 5, b) 8, c) 4, Bonus: d) 7/, e) 27/, f) 63/ E-6 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

7 NS7-19 Place Value and Decimals Pages Standards: preparation for 7.NS.A.1 Goals: Students will identify the place value and the actual value of digits in whole numbers and decimals. Prior Knowledge Required: Knows the definition of a decimal fraction Understands place value for whole numbers and the use of zero as a placeholder Can write expanded form for whole numbers Can write equivalent fractions Can add fractions with the same denominator and fractions with different denominators Vocabulary: decimal, decimal fraction, decimal point, denominator, hundredth, place value, placeholder, power of 10, represent, tenth, thousandth The place value system. Write on the board: 5,834 = 5, SAY: We use place value to write numbers. That means that where a digit is placed in the number tells you its value. Because the 5 is in the thousands place, it is worth 5,000. Exercises: What does the 7 represent? a) 6,742 b) 9,017 c) 6,572 d) 7,904 Answers: a) 700; b) 7; c) 70; d) 7,000 Exercises: Find the actual value of the digit 6 in the number. a) 632 b) 5,632 c) 75,632 d) 875,632 Answers: a) 600, b) 600, c) 600, d) 600 Large place values ten thousands, hundred thousands, and millions. SAY: The next three place values after thousands are ten thousands, hundred thousands, and millions. Write on the board: 7, 9 0 2, millions hundred thousands ten thousands Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-7

8 Point out that we write commas to separate groups of three digits, between hundreds and thousands, hundred thousands and millions, and so on. Exercises: Write the place value of the digit 3 in the number. a) 312,607 b) 453,207 c) 3,762,906 d) 7,401,235 e) 8,435,241 f) 8,004,312 Answers: a) hundred thousands, b) thousands, c) millions, d) tens, e) ten thousands, f) hundreds The place value system extends to include tenths. Write on the board: thousands hundreds tens ones SAY: 10 hundreds fit into a thousand, 10 tens fit into a hundred, and 10 ones fit into a ten. Tell students that you want to continue the place value system so that you can use place value for fractions too. ASK: Ten of what make one whole? (tenths) To guide students, draw pictures of 10 equal parts fitting into one whole and shade 1/10: ASK: Does anyone remember how to show 1/10 using place value? (0.1) Write on the board: = 3 = = 4 27 = SAY: We call these numbers decimals. The dot between the whole numbers and the number of tenths is called a decimal point. Decimals are similar to mixed numbers. There s a wholenumber part to the left of the decimal point and a fractional part to the right. But when the number is less than 1 whole, we write 0 as the whole-number part. Exercises: Write the decimal for the number. a) 5 8 b) 3 c) Answers: a) 0.5, b) 3.8, c) 74.6, Bonus: Bonus: Extending the place value system beyond tenths. Write on the board: hundreds tens ones tenths ASK: What should the next place value be? (hundredths) PROMPT: Ten of what fit into a tenth? Point out that there is symmetry in the place value names, with the ones as the center of reflection: hundreds tens ones tenths hundredths E-8 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

9 SAY: There is also symmetry in the values. Write on the board: Ask volunteers to continue the place values in both directions. Show students how to write decimals for one-digit hundredths and thousandths: = 8 = , 000 SAY: The next place value after tenths is hundredths. The next place value after that is thousandths. Exercises: Write the decimal a) b) c) 1,000 d) 9 1,000 Answers: a) 0.07, b) 0.004, c) 0.05, d) 0.009, e) e) 6 1, 000 Tell students that there are two ways to read 0.03 out loud: zero point zero three or three hundredths. SAY: We write 0.03 as three hundredths when we use words to write it on paper. Exercises: Write the decimal in words. a) 0.04 b) 0.8 c) d) 0.07 e) Answers: a) four hundredths, b) eight tenths, c) nine thousandths, d) seven hundredths, e) three thousandths More than one non-zero digit in decimals. Write on the board: = Read the place values in the decimal to show how they correspond to the expanded form: 9 ones, 6 tenths, and 7 hundredths. Tell students that they can read 9.67 out loud as nine point six seven. NOTE: Reading 9.67 as nine point sixty-seven is incorrect and should be discouraged. It may create the misconception that 9.67 is greater than 9.8, since 67 > 8. Exercises: Write the decimal a) b) c) Answers: a) 3.49, b) 8.53, c) ,000 Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-9

10 Using 0 as a placeholder. Write on the board: 3 = = = = Ask a volunteer to write the last decimal. (5.03) Point out that because there are no tenths, the tenths place has a zero. Write on the board: ,000 Ask volunteers to write the decimals. (0.53, 0.503) Point out how the denominator tells you how many places the digit goes after the decimal point: tenths go one place after the decimal point, hundredths go two places, and thousandths go three places. SAY: You have to be careful because some place values might be missing. You will have to write zeros in those positions. Exercises: Write the decimal. a) b) , 000 Answers: a) 2.03, b) 8.008, c) 5.007, d) c) , d) ,000 Exercises: Write the value of the 6 as a fraction or whole number. a) b) c) d) e) Bonus: What places do the zeros hold in ? in ? Answers: a) 6/10; b) 6/; c) 6/10; d) 6; e) 6/1,000; Bonus: ones place and thousandths place in , ones place and ten thousandths place in Extensions (MP.1) 1. Write the correct decimal: $800,000 + $ = $ Answer: $800, (MP.1, MP.8) 2. a) How many times as much as the second 3 is the first 3 worth? i) 28,331 ii) 24,303 iii) 320,135 iv) 3,789,453 b) Which is worth more in each number, the 3 or the 6? How many times more? i) 63 ii) 623 iii) 6,342 iv) 36 v) 376 vi) 3,006 Hint: Pretend the 6 is a 3, compare the first 3 to the second 3 and then solve the harder problem. Example: In part ii), look at the number 323. The first 3 is worth times as much as the second 3. Since 6 is 2 times as much as 3, the 6 in 623 is worth 200 times as much as the 3. c) How many times as much as the 5 is the 2 worth? i) 25 ii) 253 iii) 2,534 iv) 342,580 v) 3,472,508 vi) 2,345 vii) 23,457 viii) 234,576 ix) 2,345,768 d) How much more is the 2 worth than the 5 in the decimal ? E-10 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

11 Answers: a) i) 10, ii), iii) 10,000, iv) 1,000,000; b) i) 6, 20 times; ii) 6, 200 times; iii) 6, 20 times; iv) 3, 5 times; v) 3, 50 times; vi) 3, 500 times; c) As long as the numbers are immediately next to each other, the 2 is always worth 4 times as much as the 5; as long as the numbers are 3 digits apart, the 2 is always worth 400 times as much as the 5. i) 4, ii) 4, iii) 4, iv) 4, v) 4, vi) 400, vii) 400, viii) 400, ix) 400; d) 400,000 Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-11

12 NS7-20 Positive and Negative Decimals Pages Standards: preparation for 7.NS.A.1 Goals: Students will write mixed numbers as decimals and decimals as mixed numbers, including negative numbers. Prior Knowledge Required: Can write mixed numbers and decimals Can add fractions with different denominators Vocabulary: decimal, decimal fraction, decimal point, hundredth, mixed number, negative, positive, tenth, thousandth Writing decimals as proper fractions. Write on the board: = 2 tenths + 6 thousandths = + = 10 1,000 1,000 Point out that the numerator is the decimal without the zero point in front. The number of zeros in the denominator is equal to the number of digits after the decimal point. Write on the board: = 206 1, = 37 1, 000 SAY: There are three digits after the decimal point, so there are three zeros in the denominator. Exercises: Hold up the correct number of fingers to signal how many zeros you would put in the denominator. a) 0.3 b) c) d) Answers: a) 1, b) 3, c) 3, d) 6 Exercises: Write the fraction for each decimal in the previous exercise. Answers: a) 3/10; b) 56/1,000; c) 801/1,000; d) 437/1,000,000 Writing proper fractions as decimals. Write on the board: 34 1,000 Tell students that you want to write this fraction as a decimal. SAY: The numerator tells you what digits to write. The denominator has three zeros, so you have to put three digits after the E-12 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

13 decimal point. Write the digits 34 on the board, and point out that there are only two digits. So, to make three digits after the decimal point, students need to add a 0 before the 3. Write on the board: Point out that 34 thousandths is the same as 3 hundredths and 4 thousandths, so 0 ones and 0 tenths makes sense. Write on the board: 3 Have students raise the correct number of fingers to signal the answer as you ASK: How many digits would you put after the decimal point in the decimal? (2) How many digits are in the numerator? (1) How many zeros do you need to write after the decimal point? (1) How do you know? (The number of zeros you need to add is the number of zeros in the denominator minus the number of digits in the numerator.) Finally, write the decimal on the board beside the fraction: 3 = 0.03 In the exercises below, have students hold up the correct number of fingers to signal the number of zeros. Exercises: How many zeros do you need to write after the decimal point in the fraction? a) ,405 b) c) d) 1, ,000,000 Answers: a) 0 (closed fist), b) 2, c) 2, d) 1 Exercise: Write the fractions from the previous exercise as decimals. Answers: a) 0.34, b) 0.007, c) , d) Tell students that some people don t write the 0 in front of the decimal point. So, for example, instead of writing 0.4, they will write.4. SAY: Be careful not to miss the decimal point; you don t want to mistake.4 for 4. Comparing tenths and hundredths. Write on the board: ASK: How many hundredths are in 0.63? (63) How many hundredths are in 0.9? (90) Which is greater, 0.63 or 0.9? (0.9) Remind students that comparing fractions is easier when both fractions have the same denominator. So it s convenient to change 9 tenths to 90 hundredths. Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-13

14 Exercises: Write both decimals as hundredths. Which one is greater? a) 0.5 and 0.42 b) 0.6 and 0.78 c) 0.3 and 0.05 Answers: a) 0.50 > 0.42, b) 0.60 < 0.78, c) 0.30 > 0.05 Reading decimals. Remind students that decimals can be used to represent mixed numbers. The whole-number part of the mixed number goes to the left of the decimal point, and the fractional part goes to the right. Write on the board: Tell students that we read the decimal the same way we read the mixed number, as 5 and 28 hundredths. Point out that the decimal point is read as and. NOTE: Another correct way to read 5.28 is five point two eight. Write on the board: three six hundredths three and six hundredths Ask a volunteer to write the missing decimal. (3.06) Exercises: Write the decimal. a) five and eight hundredths b) thirty-five thousandths c) seven and twelve thousandths d) twenty and two thousandths Answers: a) 5.08, b) 0.035, c) 7.012, d) SAY: Remember look at the number of digits after the decimal point to tell you whether the decimal is tenths, hundredths, or thousandths. Exercises: Write tenths, hundredths, or thousandths. a) 4.7 = four and seven b) 8.03 = eight and three c) 5.13 = five and thirteen d) = three and twenty-eight Answers: a) tenths, b) hundredths, c) hundredths, d) thousandths Emphasize how important the and can be when reading decimals. Write on the board: SAY: is said twenty and four hundredths while 0.24 is said twenty-four hundredths. E-14 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

15 Writing decimals as mixed numbers. Write on the board: Remind students that the decimal point separates the whole-number part from the fractional part. ASK: What is the whole-number part? (45) What is the fractional part? (891 thousandths) Write the equivalent mixed number on the board: ,000 Exercises: Write the decimal as a mixed number. a) 25.4 b) 1.73 c) d) Bonus: 123,456.9 Answers: a) 25 4/10, b) 1 73/, c) 20 7/, d) 5 303/1,000, Bonus: 123,456 9/10 Review converting improper fractions into mixed numbers using division. Remind students that they can use division with remainders to convert improper fractions to mixed numbers. For example, = 3 R 7, so 37/10 = 3 7/10. Point out that this makes sense, because 37 tenths = 3 ones and 7 tenths. Write on the board: = R, 608 = R, so =. so =. 10 Have volunteers fill in the blanks. (7 R 9, so 7 9/10; 6 R 8, so 6 8/) Exercises: Write the improper fraction as a mixed number. a) 43 b) 780 c) 3, ,000 Answers: a) 4 3/10, b) 7 80/, c) 3 524/1,000, d) 12 34/ Converting improper fractions to decimals. Write on the board: d) 1, = 3 = SAY: Once you can change an improper fraction to a mixed number, you can change it to a decimal. Exercises: Write the improper fraction as a mixed number, then as a decimal. a) 28 b) 728 c) 793 d) 7,845 Bonus: 63, , 000 Answers: a) 2 8/10 = 2.8, b) 72 8/10 = 72.8, c) 7 93/ = 7.93, d) 78 45/ = 78.45, Bonus: /1,000 = Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-15

16 (MP.8) Ask students to compare the improper fractions and the decimals. ASK: Is there a shorter way to find the answer without converting the improper fraction to a mixed number? PROMPTS: Where do you see the numerator in the answer? (the number without the decimal point) Where do you see the denominator in the answer? (the number of zeros in the denominator is the number of digits after the decimal point) Point out that the decimal can be obtained by writing the numerator and then making sure that the number of digits after the decimal point is the same as the number of zeros in the denominator. (MP.7) Exercises: Convert into a decimal without writing the mixed number. a) 654 b) 43,654 c) 3,756 d) 53,094 Bonus: 845,036, ,000,000 Answers: a) 65.4, b) , c) 375.6, d) , Bonus: 8, NOTE: When writing decimals, the convention is to separate groups of three place values with commas before the decimal point, but not after. Negative decimals. Tell students that, just as fractions can be negative, decimals can be negative too. When two numbers are equal, their opposites are equal too. Write on the board: 734 = , so - =-7.34 (MP.7) Exercises: Write the decimal for the negative number. a) b) c) d) ,000 Answers: a) 61.2, b) 3.82, c) 4.07, d) Extensions 1. Teach students to interpret whole numbers written in decimal format. Examples: 5.1 million is 5,,000; 3.7 thousand is 3, Have students look for decimals in the media (e.g., news stories, billboards, advertisements, posters) and write the decimals as mixed numbers. 3. Find the mistakes = 0.05 = 0.47 = 10 Answers: = 3/1,000, not 3/; 0.47 = 47/, not 47/ = 0, E-16 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

17 NS7-21 Comparing Fractions and Decimals Pages Standards: preparation for 7.NS.A.1 Goals: Students will place simple fractions and decimals on number lines. Students will compare positive and negative decimals and simple fractions. Prior Knowledge Required: Can order and compare fractions Can order and compare decimals Can write equivalent fractions and decimals Understands that opposite numbers are the same distance from 0, on the opposite side of 0 Is familiar with < and > signs Is familiar with number lines, including negative decimals and fractions Vocabulary: decimal, decimal fraction, decimal point, equivalent decimals, equivalent fractions, hundredth, negative, positive, tenth, thousandth Materials: BLM Number Lines from 2 to 2 and 0.2 to 0.2 (p. E-45) BLM Hundredths Number Lines (p. E-46) Decimals on number lines. Draw on the board: A B C D E A = C = Have volunteers write point A as a decimal (0.3) and a fraction (3/10). Then SAY: Point C is 6/10 more than 1. Demonstrate counting the increments after the 1 to verify this. SAY: So it is one whole and six tenths. Have volunteers write the number as a mixed number (1 6/10) and as a decimal (1.6). Point out how the wholes and the tenths are shown in both ways of writing the number. Exercises: Write a decimal and a fraction or mixed number for points B, D, and E. Answers: B. 0.8 and 8/10, D. 2.4 and 2 4/10, E. 2.7 and 2 7/10 Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-17

18 Positive and negative mixed numbers and decimals on a number line. Display the number line below: Point to the point A. ASK: How far from 0 is this? (3 tenths) Is it positive or negative? (negative) Write.3 under A. Ask a volunteer to write the decimal for B. ( 0.5 or.5) Now point to C, and ASK: How far from 0 is this? (1 and 4 tenths) Is it positive or negative? (negative) Write 1.4 under C. Ask a volunteer to write the decimal for D. ( 1.8) Exercises: Write a decimal and a fraction or mixed number for each point marked. A B C D E Answers: A. 1.9, 1 9/10; B. 1.3, 1 3/10; C. 0.5, 5/10; D. 0.3, 3/10; E. 0.9, 9/10 Remind students that, on a number line, numbers on the left are less than numbers on the right. Provide students with a strip from BLM Number Lines from 2 to 2 and 0.2 to 0.2. Exercises: Write < or >. a) b) c) Answers: a) <, b) >, c) < Partial number lines divided into hundredths. Project on the board a number line from BLM Hundredths Number Lines. Circle the interval from 0.20 to 0.30 and tell students you want to enlarge this section. Draw on the board the number line shown, but without the two points marked: Point out that we can label.20 and.30 as tenths,.2 and.3. The length of this part of the number line is 1 tenth and it is divided into 10 equal parts, so each part is a hundredth. Now mark the two points and, pointing to each in turn, ASK: What number is this? (.21 and.25) Now draw on the board: E-18 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

19 Have students point to which side of the number line 0 will be. (point right) ASK: How do you know? (because the numbers are negative) Which decimal, 0.5 or 0.4, is farther from 0? ( 0.5) Point out that as you go left from 0, the numbers without the minus signs get bigger 0.5 is bigger than 0.4 but the numbers themselves get smaller. Exercise: Write the points marked from least to greatest. Answers: 0.49, 0.47, 0.44, 0.42 Exercises: Use the number lines from BLM Number Lines from 2 to 2 and 0.2 to 0.2. Write < or >. a) b) c).13.2 d) Answers: a) <, b) <, c) >, d) < (MP.4) Word problems practice. Before giving part b) below, remind students that integers are used to describe elevation, with 0 being sea level, anything higher being positive, and anything lower being negative. a) Which temperature, 3.6 F or 2.58 F, is warmer? ( 2.58 F) b) Which elevation, 14.2 m or 14.7 m, is higher? ( 14.2 m) Comparing decimal tenths to 1/2. Draw on the board: Invite volunteers to mark the missing fraction on the top number line (1/2) and the decimal increments on the bottom number line. ASK: What decimal does one half represent? (0.5) What is the decimal fraction for 0.5? (5/10) Remind students that 1/2 is equivalent to 5/10, so it makes sense that they are at the same place on the number line. Exercises: Is the decimal more than half or less than half? (Students can signal thumbs up for more than half and thumbs down for less than half.) a) 0.3 b) 0.6 c) 0.8 d) 0.4 e) 0.2 Answers: a) less, b) more, c) more, d) less, e) less Comparing decimal hundredths to one half. Remind students how to compare decimal tenths to decimal hundredths. For example, 0.4 is greater than 0.37 because 4 tenths is 40 hundredths and 0.37 is only 37 hundredths. ASK: Which is greater, 0.56 or 0.5? (0.56) PROMPTS: How many hundredths are in 0.56? (56) And in 0.5? (50) ASK: Which is greater, 0.38 or 0.5? (0.5) Students can signal the answers to the exercises below. Exercises: Is the decimal greater than or less than one half? a) 0.42 b) 0.87 c) 0.39 d) 0.51 Answers: a) less, b) greater, c) less, d) greater Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-19

20 Comparing decimal tenths to quarters. Add the marks for quarters (1/4 and 3/4) to the top number line on the board and invite volunteers to label them. Exercises: Write > or <. 1 3 a) 0.4 b) Answers: a) >, b) <, c) >, d) > c) d) Comparing negative fractions and decimals. Draw a double number line from 1 to 0, with markings for decimal tenths and fourths. Invite volunteers to label all the increments. Remind students that the order in the negative numbers is the opposite of the order in the positive numbers, so if we know that 1/2 > 0.2, then we also know that 1/2 < 0.2. ASK: How does the number line show that? ( 0.2 is to the right of 1/2 on the number line) (MP.7) Exercises: Write > or <. a) b) c) d) e) f) Answers: a) <, b) >, c) <, d) >, e) <, f) < Comparing decimal hundredths to quarters. Write on the board: 1 = = 4 ASK: Can you multiply 4 by a whole number to get 10? (no) Cross out the first equation. ASK: Can you multiply 4 by a whole number to get? (yes, 25) Show this on the board: How can you write 1/4 as a decimal? (0.25) Repeat for writing 3/4 as a decimal (0.75); then summarize on the board as follows: 1 = = 0.5 = = SAY: You can use this to compare any number of hundredths to 1/4, 1/2, or 3/4. Exercises: Which number is greater? a) 0.12 or 1/4 b).73 or 1/2 c) 0.73 or 3/4 d).29 or 1/4 e) 0.45 or 3/4 f).89 or 1/2 g) 0.87 or 3/4 h).36 or 1/4 Answers: a) 1/4, b).73, c) 3/4, d).29, e) 3/4, f).89, g) 0.87, h).36 E-20 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

21 Remind students that the order is reversed when you compare negative fractions. For example, 0.12 < 1/4, so 0.12 > 1/4. (MP.7) Exercises: Which number is greater? a).19 or 1/4 b).63 or 1/2 c) 0.67 or 3/4 d).59 or 1/2 e) 0.85 or 3/4 f).28 or 1/4 Answers: a).19, b) 1/2, c) 0.67, d) 1/2, e) 3/4, f) 1/4 Converting fractions to decimal fractions and decimals. Write on the board: 1 = = = 20 Ask volunteers to write the missing numerators. (2, 4, 15) ASK: How can you write 1/5 as a decimal? (0.2) Repeat with 2/5 (0.4) and 3/20 (0.15). Exercises: Write the fraction as a decimal hundredth. a) 1 b) 1 c) d) 7 e) 23 Bonus: Answers: a) 0.05, b) 0.04, c) 0.02, d) 0.14, e) 0.92, Bonus: 0.08 SAY: Writing the fractions as decimal hundredths is like converting them to the common denominator, so ordering fractions becomes easy. Exercise: Order the fractions from the preceding exercises from greatest to least. Bonus: Write the fractions with denominator 1,000; then put them in order from least to greatest ,000 Answers: 0.92, 0.14, 0.08, 0.05, 0.04, 0.02, so the fractions from greatest to least are 23/25, 7/50, 24/300, 1/20, 1/25, 1/50; Bonus: 35/500 = 70/1,000, 16/200 = 80/1,000, 28/7,000 = 4/1,000, so from least to greatest, the fractions are 28/7,000, 35/500, 16/200. Ordering positive and negative decimals and fractions. Remind students that if they can order positive numbers, then they can order negative numbers too. Since 2 < 3 < 5, then 2 > 3 > 5. Exercises: Write the numbers from least to greatest. a) b) Answers: a) 6/10, 0.48, 35/; b) 7/10, 0.5, 7/ Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-21

22 SAY: Positive numbers are always greater than negative numbers. Exercises: Write the numbers from least to greatest. a) b) Answers: a).45, 1/4, 3/5; b) 3 1/, 2.14, 89/, 89/10 89 (MP.4) Word problems practice. a) Which elevation, 2.8 m or m, is higher? ( m) b) Which temperature, C or 3.76 C, is warmer? ( 3.76 C) Extensions (MP.7) 1. a) Write the missing number. Hint: Write the decimal as a decimal fraction. i) = iv) = v) = ii) = 0.13 iii) = , vi) = b) Use equivalent fractions to find the missing number. i) = 0.4 ii) = 0.3 1,000 iii) = 3.27 iv) = 0.6 v) = 0.24 vi) = Sample solutions: a) i) 0.7 = 7/10, so 7 is the missing number; b) i) 0.4 = 4/10 = 40/, so the missing number is 40. Answers: a) ii) 13, iii) 327, iv), v) 10, vi) 10,000; b) ii) 300, iii) 327, iv) 3, v) 6, vi) 5 2. Write any number that works. 3 4 a) - >- > b) - <- < c) - <- 1, Answers: a) any number between 31 and 39, b) 5, c) 1 (MP.1) 3. Use 10,, and 1,000 once each to make the statement true a) - >- and - >- 1 b) = Answers: a) 27/1,000 > 3/, 4/ > 4/10; b) 1/10 = /1,000 or 1/ = 10/1,000 E-22 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

23 (MP.1, MP.2) 4. Ray saw four fish at different elevations: km, 0.18 km, 0.9 km, 1.8 km. Use the information below to decide which fish was seen at which elevation. The coelacanth lives between 150 m and 400 m below sea level. The football fish lives between 200 m and 1 km below sea level. The deep sea angler lives between 250 m and 2 km below sea level. The rattail lives between 22 m and 2.2 km below sea level. Answers: coelacanth: 0.18 km, football fish: 0.9 km, deep sea angler: 1.8 km, rattail: km Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-23

24 NS7-22 Adding and Subtracting Multi-Digit Decimals Pages Standards: 7.NS.A.1c, 7.NS.A.1d Goals: Students will add and subtract positive and negative decimals. Prior Knowledge Required: Understands place value Can represent decimals using base ten materials Can add fractions with the same denominator Can tell how many hundredths are in a number with two decimal places Can add and subtract multi-digit numbers with or without regrouping Uses addition to check subtraction Vocabulary: absolute value, algorithm, decimal point, hundredth, regrouping, tenth, thousandth Adding decimals. Write on the board: = 35 + = = Write the first equation in vertical format; then ask volunteers to write the other two equations as decimals in vertical format: Explain that you can add and subtract decimals the same way you add whole numbers line up the place values but, instead of adding or subtracting ones and tens, you re adding or subtracting tenths and ones or hundredths and tenths. Exercises: Add or subtract by lining up the place values. Use grid paper. a) b) c) Bonus: Answers: a) 4.9, b) 2.5, c) 9.79, Bonus: SAY: You might need to regroup the same way you do with whole numbers. Exercises: Add or subtract. Use grid paper. a) b) c) d) e) f) g) h) E-24 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

25 Answers: a) 25.3, b) 6.32, c) 208.2, d) or 5.42, e).36, f) 0.08, g) or 34.2, h) SAY: The answer to d) is or just ASK: What other answer can be written shorter? (part g) can be written as 34.2) Adding decimals with different numbers of digits to the right of the decimal point. Write on the board: SAY: It s the place values that need to be lined up, not the last digits. You can make sure the place values are lined up by lining up the decimal points, because the decimal point is always between the ones and tenths. Have a volunteer add (25.85) Exercises: Add. a) b) c) d) Answers: a) 1.18, b) 0.865, c) 99.95, d) Adding whole numbers and decimals. Write on the board: ASK: How can you line up the decimal points when 32 has no decimal point? PROMPT: Where would the decimal point go in 32? (after the 2) SAY: You can look at 32 as 32.0, or 32 and 0 tenths. Now you can line up the decimal points and add. Have a volunteer do so: Exercises: Add. a) b) c) Answers: a) 17.7, b) 18.3, c) SAY: You can check your answers by adding the fractions. Do this for the example on the board: = = 36 = (MP.1) Exercise: Check your answers to the exercises above by adding the fractions. Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-25

26 Subtracting decimals. SAY: You can subtract by lining up the decimal points too. You might have to write zeros to make both decimals have the same number of digits after the decimal point. Demonstrate as shown: NOTE: Writing zeros at the end of the bottom number is optional, since it is easy to subtract, say, 4 0, even when the zero isn t written. But writing zeroes at the end of the top number is necessary because you will have to regroup, and it s easier to regroup when you can write what you re doing. Exercises: Subtract. Use grid paper. a) b) c) d) e) f) g) h) Answers: a) 5.3, b) 2.41, c) 4.69, d) 3.5, e) 1.28, f) , g) 1.48, h) Students can check their answers using addition. (MP.4) Word problems practice. a) Mona made 0.6 L of milkshake by adding ice cream to 0.48 L of milk. How much ice cream did she add? (0.6 L 0.48 L = 0.12 L) (MP.3) b) Len placed a table 1.23 m long along a wall 3 m long. If his bed is 2.13 m long, will it fit along the same wall? Explain. (1.23 m m = 3.36 m; the bed will not fit) c) Sara cut 0.86 m of wood board to make a shelf. The leftover piece is 1.45 m long. How long was the board before she cut off the shelf? (2.31 m) Adding multi-digit negative decimals. Remind students that if they can add two positive numbers, then they can add two negative numbers too. Write on the board: SAY: If you lost $3.50, then lost $4.10, how much did you lose overall? ($7.60) Finish the equation on the board: = 7.60 SAY: You just had to add as though they were positive and then put the negative sign in front. Exercises: Add. a) b) c) Answers: a) 9.57, b) c) E-26 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

27 SAY: In these exercises, you will have to regroup. Exercises: Add. a) b) c) Answers: a) 11.36, b) 8.63, c) Adding multi-digit positive and negative decimals. Remind students that if they know how to subtract positive numbers, then they can add positive and negative numbers. Write on the board: ASK: If you lost 3.4 dollars and gained 2.1 dollars, did you gain more or lose more? (lose) SAY: So the answer will be negative. How much did you lose? ( = 1.3, so you lost 1.3 dollars) Write the answer on the board: = 1.3 SAY: The answer is negative because the number with the greater absolute value is negative. Remind students of the notation for absolute value. Write on the board: 3.4 = = 2.1 SAY: The number part is 1.3 because you had to subtract the absolute values of the two numbers. Exercises: Add the positive and negative numbers. a) b) c) Answers: a) 12.2, b) or 3.11, c) 1.37 SAY: The next problems will need regrouping. Exercises: Add the positive and negative numbers. a) b) c) Answers: a) 1.69, b) or 1.85, c) or 8.58 Remind students that if they know how to add positive and negative numbers, then they automatically know how to subtract positive and negative numbers, because they can always change a subtraction to an addition. Write on the board: +(+) = + +( ) = (+) = ( ) = + Exercises: Add or subtract. a) 3.4 (+6.87) b) (+6.87) c) (+15.8) d) 11.5 ( 15.8) e) +2.8 ( 3.41) f) (+11.4) Answers: a) 10.27, b) 3.47, c) 1.43, d) 4.3, e) 6.21, f) 2.39 Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-27

28 Debits and credits. Remind students that the bank records a debit when you take money out of your account and a credit when you put money in. Draw on the board: Debit ( ) Credit (+) Balance $8.31 $3.45 $7.12 Have volunteers calculate the balance after each transaction. ($3.45, $4.86, $2.26) Tell students that a negative balance means that they owe the bank. It is not a good idea, but it can happen sometimes. Now add a row to the chart, and tell students that the debit of $8.31 was actually a mistake by the bank. ASK: How can the bank correct its mistake? (add a credit of $8.31) ASK: What is the new balance? ($10.57) Write on the board: $2.26 ( $8.31) = $ $8.31 = $10.57 SAY: You can think of correcting the mistake as taking away the negative amount or as adding the positive amount, because they both do the same thing. (MP.4) Exercises: Write an addition and a subtraction to find the corrected balance. a) John s bank account balance was $ John noticed an incorrect debit to his account for $15.81, so the bank now has to correct its mistake. What is the corrected balance? b) Nancy s bank account balance was $ Nancy noticed an incorrect debit to her account for $ What is the corrected balance? c) Tony s bank account balance was $ Tony noticed an incorrect debit to his account for $ What is the corrected balance? d) Wendy s bank account balance was $7.50. Wendy noticed an incorrect debit to her account for $5.11. What is the corrected balance? Answers: a) $14.00 ( $15.81) = $ $15.81 = $29.81, b) $37.00 ( $13.29) = $ $13.29 = $50.29, c) $14.23 ( $103.29) = $ $ = $89.06, d) $7.50 ( $5.11) = $ $5.11 = $2.39 Extensions 1. a) Add mentally. i) ii) iii) b) Add the two numbers that are easiest to add first, then find the total: (MP.5) c) Would you use pencil and paper to add, or would you add mentally? i) ii) iii) Answers: a) i) 6, ii) 20, iii) 11; b) = 10 and = 17.9; c) i) mentally, ii) pencil and paper, iii) mentally 2. Make up two decimals that add to Check your answer by adding them. $0 E-28 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

29 (MP.1) 3. Subtract using the number line. Do you get the same answer by lining up the decimal points? (MP.3) 4. Abdul says that is the largest number less than 1 that can be added to without needing to regroup. Is that correct? Hint: = Answer: No, also does not need regrouping, but is larger than In fact, any other decimal produced by adding digits to the right of the 4 is larger than and can be added to without needing to regroup. (MP.1) 5. Place decimal points in each number to make the statement correct. a) = 1807 b) = 4528 Sample answers: a) = 18.07, b) = Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-29

30 NS7-23 Division with Fractional and Decimal Answers Pages Standards: preparation for 7.NS.A.2 Goals: Students will divide a whole number by a whole number and will write the answer as a fraction and as a decimal. Prior Knowledge Required: Can divide whole numbers by whole numbers with remainder Can convert an improper fraction to a mixed number Can multiply a fraction by a whole number Can draw pictures representing proper fractions, improper fractions, and mixed numbers Can create equivalent fractions with a given denominator Can write decimal fractions as decimals Understands that multiplying both terms of a division by the same number does not change the answer Materials: a calculator for each student (see Extension 5) BLM Division, Fractions, and Decimals (Advanced) (p. E-47, see Extension 8) Vocabulary: decimal, decimal point, denominator, division, fraction, improper fraction, mixed number, numerator, remainder Drawing pictures to show equal sharing. Write on the board: 4 people share 7 pies Point out that you are drawing circles for pies. ASK: How many circles should I draw? (7) Draw the 7 circles on the board and then SAY: In case some pies taste better than others, they decide to share each pie equally. ASK: How many pieces should I divide each circle into? (4) Divide the pies; then have a volunteer shade the amount that one person gets: Write the multiplication below the picture: = E-30 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

31 Exercises: Draw a picture to show how much one person gets. Then write the multiplication. a) 4 people share 2 pies b) 4 people share 5 pies c) 2 people share 3 pies d) 2 people share 5 pies Selected solution: Answers: b) 5 1/4 = 5/4, c) 3 1/2 = 3/2, d) 5 1/2 = 5/2 Using division for equal sharing when the answer is a fraction. Remind students that division is used for equal sharing. ASK: If 2 people share 6 pies, how much does each person get? (3 pies) Write on the board: 2 people share 6 pies 4 people share 5 pies 6 2 = 3 = Ask a volunteer to fill in the blanks (5 4 = 5/4) and have students signal whether they agree (thumbs up) or disagree (thumbs down). SAY: 4 people sharing 5 pies is another way of saying that 5 pies are divided equally between 4 people, so the answer is 5 4. Point out that the number of objects being divided goes first, and the number of people sharing goes second. The answer is how much each person gets. Students may be used to seeing 5 4 = 1 R 1, but now the answer includes a fraction: 5/4 or 1 1/4. Have another volunteer write the division equation to show 5 people sharing 2 pies. (2 5 = 2/5) SAY: When 2 pies are shared equally among 5 people, the answer is 2 5. Exercises: Write the division equation. Answers: a) 3 5 = 3/5, b) 4 5 = 4/5, c) 3 6 = 3/6, d) 4 6 = 4/6 Dividing whole numbers without a picture. Write on the board: 5 7 = 4 9 = 3 8 = Challenge students to predict the answers to these questions without using a picture. Point out that the first number in the division is the numerator (number on top of the fraction) and the second number in the division is the denominator (number on the bottom of the fraction). Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-31

32 Exercises: Divide. Write your answer as a fraction. a) 3 10 b) 4 7 c) 8 9 d) 7 8 Bonus: 13 1,000 Answers: a) 3/10, b) 4/7, c) 8/9, d) 7/8, Bonus: 13/1,000 Have students check their answers by multiplication. For example, since 3 10 = 3/10, they should check that 3/10 10 = 3. Writing the answer as a mixed number. Now tell students that 3 people are sharing 5 pies. ASK: How much does each person get? (5/3 pies) Remind students that we can write an improper fraction as a mixed number. Ask a volunteer to shade 5/3 to find the mixed number. Exercises: Divide. Write the answer as an improper fraction and as a mixed number. Show your answer with a picture. a) 9 4 b) 7 2 c) 6 4 Bonus: 15 8 Answers: a) 9/4 = 2 1/4, b) 7/2 = 3 1/2, c) 6/4 = 1 2/4 = 1 1/2, Bonus: 15/8 = 1 7/8 Point out that another way to get the answer to 9 4 is to not break the pies into parts until you have to. If you start by giving each person 2 whole pies, then you just have to divide the remaining pie into fourths. So each person gets 2 1/4 pies. Have students solve the other exercises the same way to verify their answers. Review writing decimal fractions as decimals. Write on the board: ,572 Remind students that the number of zeros in the denominator of the fraction is the number of digits after the decimal point. SAY: Each of these fractions has denominator, so each of the decimals will have two digits after the decimal point. Write on the board: Ask volunteers to finish writing the decimal for each fraction by writing the decimal point in the correct place. (0.82 or.82, 0.07 or.07, 7.04, 65.72) SAY: You may need to write a 0 after the decimal point. E-32 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

33 Exercises: Write the fraction as a decimal. a) 38 b) e) 4 10 f) c) 1,000 g) 3 10 d) h) 1, 000 4,503 Bonus: 1,000,000 Answers: a) 0.38, b) 1.5, c) 0.007, d) 4.76, e) 0.4, f) 54.8, g) 0.3, h) 0.091, Bonus: Division with decimal answers. Write on the board: 82 = 82 SAY: This can also be written as a decimal. Have a volunteer write the decimal answer. (0.82) Exercises: Divide. Write the answer as a fraction and a decimal. a) 432 b) 3 1,000 c) d) 4 e) 98 f) 37 1,000 Answers: a) 432/ = 4.32, b) 3/1,000 = or.003, c) 18/10 = 1.8, d) 4/ = 0.04 or.04, e) 98/ = 0.98 or.98, f) 37/1,000 = or.037 Connecting two ways of dividing. Tell students that when dividing by 10,, or 1,000, they are just moving the decimal point one, two, or three places to the left. Write on the board: So 82 = SAY: There are two zeros in, so move the decimal point two places to the left. Exercises: Divide by moving the decimal point the correct number of places. Make sure you get the same answers as above. a) 432 b) 3 1,000 c) d) 4 e) 98 f) 37 1,000 Answers: a) 4.32, b) or.003, c) 1.8, d) 0.04 or.04, e) 0.98 or.98, f) or.037 Dividing by a number that is not a power of 10. Write on the board: 4 5 Ask a volunteer to write the answer as a fraction. (4/5) Then ASK: Is there an equivalent fraction with denominator 10? Write on the board: 4 = 5 10 Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-33

34 Ask a volunteer to fill in the missing numerator. Then show the multiplicative relationship between the numerators and denominators. SAY: So 4 fifths = 8 tenths. Write on the board: 4 5 = 0.8 Exercises: Divide. Write your answer as a fraction and as a decimal. a) 2 5 b) 3 2 c) 6 5 d) 7 2 (MP.1) Bonus: Divide. Then divide your answer by 10. For each question, do you get the same answer as you got in the exercises above? e) 20 5 f) 30 2 g) 60 5 h) 70 2 Answers: a) 2/5 = 0.4; b) 3/2 = 1.5; c) 6/5 = 1.2; d) 7/2 = 3.5; Bonus: e) 4 10 = 0.4, yes; f) = 1.5, yes; g) = 1.2, yes; h) = 3.5, yes Write on the board: 1 4 = 1 4 ASK: Is there an equivalent fraction with denominator 10? (no) How do you know? (4 does not divide into 10) Is there an equivalent fraction with denominator? (yes) How do you know? (4 divides into ) Write on the board: 1 = 4 Ask a volunteer to write the missing numerator (25); then ask another volunteer to write the decimal answer (0.25). Exercises: Divide by making an equivalent fraction with denominator 10 or. a) 3 2 b) 2 5 c) 9 20 d) e) 3 4 f) 7 5 g) 9 25 h) 7 4 Bonus: Reduce the fraction if you can before making the denominator a power of 10. i) 6 8 j) 9 60 k) 6 15 l) m) n) 21, ,000 Answers: a) 3/2 = 15/10 = 1.5, b) 4/10 = 0.4, c) 45/ = 0.45, d) 62/ = 0.62, e) 0.75, f) 1.4, g) 0.36, h) 1.75, Bonus: i) 3/4 = 0.75, j) 3/20 = 0.15, k) 2/5 = 0.4, l) 0.555, m) 0.628, n) When doing Question 5 on AP Book 7.1 p. 114, students might write their answers with different numbers of digits after the decimal point. For example, in part e), students could write 3/4 = 750/1,000 instead of 75/ and get instead of Both answers are correct. E-34 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

35 Extensions 1. Write the fact family for 3/5 5 = 3. Answers: 3/5 5 = 3, 5 3/5 = 3, 3 3/5 = 5, 3 5 = 3/5 (MP.1, MP.2) 2. To divide 3 5, replace 3 with Do you get the same answer? Hint: Use the fact that = This is true because they are both equal to 30 (5 10). Answer: Yes: (30 10) 5 = (30 5) 10 = 6 10 = 6/10 = 0.6. (MP.1) 3. Write 11 2 as a decimal in two ways: first write the answer as an improper fraction and convert the improper fraction to a decimal; then write the answer as a mixed number and convert the mixed number to a decimal. Make sure you get the same answer. Answer: 11 2 = 11/2 = 55/10 = 5.5 or 11 2 = 5 1/2 = 5 5/10 = 5.5 (MP.1) 4. (1 + 2) 3 + (4 + 5) 6 = (7 + 8)? Answer: 6 (MP.1, MP.3) 5. a) Write the decimal answers to 1/4 and 5/4. Make sure that the decimal for 5/4 is exactly 1 greater than the decimal for 1/4. Why does this make sense? b) How do you expect the decimals for 1/3 and 4/3 to compare? Check on a calculator. Answer: a) The decimals are 0.25 and 1.25, and indeed = This makes sense because the fraction 5/4 is 1 1/4; b) I expect the decimal for 4/3 to look exactly like the decimal for 1/3, but with 1 in front. My calculator says 1/3 = and 4/3 = (MP.1) 6. a) Predict which divisions will have an answer greater than 1; then check by writing the answer as a decimal. i) 9 10 ii) iii) 3 4 iv) 5 4 v) 6 5 vi) 5 8 b) Estimate where B A is on the number line below. A B Answers: a) ii), iv), and v) have answers greater than 1. The answers are i) 0.9, ii) 1.1, iii) 0.75, iv) 1.25, v) 1.2, vi) 0.625; b) slightly greater than 1 (MP.7) 7. a) Write the missing numerators; then change the fractions to decimals = = = 6 4 = = = b) Are the answers equivalent decimals? Explain why this makes sense. c) What property of division does this show? Answers: a) 15 and 1.5, 150 and 1.50; b) The answers are equivalent decimals. This makes sense because the fractions they are made from, 3/2 and 6/4, are equivalent fractions. c) This shows the property that multiplying both numbers in a division by the same number keeps the answer the same. Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-35

36 (MP.1) 8. After completing Extension 7, provide students with BLM Division, Fractions, and Decimals (Advanced). As students work on the questions, they will make connections between properties of division and properties of fractions. Answers: 1. b) 4/4 = 1, c) 7/7 = 1, d) a/a = 1; 2. a) = 7 and 14 2 = 7, so (8 2) + (6 2) = (8 + 6) 2; b) = 32 and 48 8 = 6, so (48 2) + (48 6) 48 (2 + 6); a) and (8 + 6) 48 48, b) and 48 ; 4. yes, a); 5. no, b); 6. a) 6 3 = 2, (2 + 6) b) (3 5) 5 = 3, c) (7 4) 7 = 4, d) 3 4 = (3 2) (4 2), e) = 5 4, f) 12 8 = (12 4) (8 4) (MP.3) 9. Mandy says 2 R 1 = 2 1/4 because 9 4 = 2 R 1 and 9 4 = 2 1/4. Will says 2 R 1 = 2 1/3 because 7 3 = 2 R 1 and 7 3 = 2 1/3. Explain why their reasoning is incorrect. Answer: 2 R 1 is not a number. The division sign is being used in two different ways. NOTE: The two ways of using the division sign are essentially different. One way is an operation using whole numbers that can only have whole-number answers (with remainder). The other way is an operation using any numbers, including fractions, that can have any kind of answer. E-36 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

37 NS7-24 Long Division Pages Standards: preparation for 7.NS.A.2 Goals: Students will use the standard algorithm for long division to divide 2-digit, 3-digit, and 4-digit whole numbers by 1-digit whole numbers. Prior Knowledge Required: Understands remainders Understands division as finding the number in each group Can use base ten materials to represent numbers Vocabulary: dividend, divisor, quotient, remainder, standard algorithm Materials: Base ten materials (optional) BLM Hundreds Charts (p. E-48, see Extension 2) Long division: three-digit by one-digit. Using pictures of base ten materials, explain why the standard algorithm for long division works. SAY: Let s divide 726 into 3 equal groups. Step 1. Make a model of 726 units. 7 hundreds blocks 2 tens blocks 6 ones blocks Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-37

38 Step 2. Divide the hundreds blocks into 3 equal groups. Keep track of the number of units in each of the 3 groups, and the number remaining, by using the long-division algorithm hundreds blocks, or 200 units, have been divided into each group. 600 units (200 3) have been divided. 1 hundred and 2 tens still need to be divided. Point out how the long-division algorithm keeps track of the place values because hundreds are put in the hundreds column, tens in the tens column, and ones in the ones column. Exercises: Carry out Steps 1 and 2 of long division. a) b) 3822 c) d) Students should show their work using actual base ten materials or a model sketched on paper for at least one problem (but some students will need more practice). Step 3. Divide the remaining hundreds block and the 2 remaining tens blocks equally among the 3 groups. To do this, you need to regroup the hundreds block as 10 tens blocks. There are 120 units in total, so 40 units can be added to each group from Step 2. E-38 Teacher s Guide for AP Book 7.1 Unit 4 The Number System

39 SAY: You can keep track of Step 3 using the standard long-division algorithm. Write on the board, using a different color for the new step: tens (or 40 new units) have been placed into each group. 12 tens (4 tens 3) have been divided altogether. 6 units still need to be divided. Emphasize that the 1 is in the hundreds column and the 2 is in the tens column because together they represent 120. Exercises: Carry out Step 3 of long-division on the problems you started above. a) b) 3822 c) d) Exercises: Do Steps 1 3 on these problems. a) 3612 b) 4792 c) d) Students should show their work using either base ten materials or a model drawn on paper for at least one problem (but some students will need more practice). Step 4. Divide the 6 remaining blocks among the 3 groups equally. SAY: There are now 242 units in each group, so = 242. Teacher s Guide for AP Book 7.1 Unit 4 The Number System E-39