Unit 6 Number and Operations in Base Ten: Decimals


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1 Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation, and they will extend their knowledge to comparing, adding, and subtracting multidigit numbers. Students will also multiply fractions and decimals by whole numbers. There is some confusion in naming decimal fractions. We use the convention that 1/, for example, is one hundredth, not one onehundredth. 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 onethousandths, therefore doing so would not eliminate the confusion. Another way to avoid confusion is to not shorten decimal point to decimal. Shortening to decimal 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. Number and Operations in Base Ten G1
2 NBT538 Introduction to Decimals Pages STANDARDS preparation for 5.NBT.A.1, 5.NBT.A.3 Goals Students will use decimal notation for fractions with denominators and, place decimal hundredths on number lines, and order decimal hundredths using a number line. Vocabulary decimal notation decimal point equivalent decimals hundreds hundredths ones place value tens tenths PRIOR KNOWLEDGE REQUIRED Knows that, on number lines, greater whole numbers appear to the right of lesser whole numbers Can name fractions from area models and number lines MATERIALS BLM Hundredths Number Lines (p. G62) Introduce decimal tenths. Tell students that the fraction 1/ can be represented in various ways. Show three ways on the board: Point out that each way means 1 part out of equal parts. Tell students that mathematicians have invented an even easier way to write one tenth, called decimal notation. Show this on the board: 1 = 0.1 SAY: The dot is called a decimal point. Mathematicians use decimal notation because it takes up less space on the page and is easier to write. Ask volunteers to show how they would write 2 tenths as a decimal (0.2), 3 tenths (0.3), and other numbers up to 9 tenths (0.9). SAY: The 0 before the decimal point tells you that the number is less than 1. Representing decimal tenths on a number line. Draw a number line from 0 to 1, and ask students to place various decimal tenths on the number line. (0.8, 0.5, 0.2, 0.7) Exercises: Write the decimal for each marked point. 0 1 Answers: 0.3, 0.4, 0.9 Representing decimal tenths using pictures. Draw various shapes, such as circles, squares, or rectangles, and have volunteers represent various numbers given in decimal notation by shading the pictures. G2 Teacher s Guide for AP Book 5.1
3 a) 0.2 b) 0.3 c) 0.5 d) 0.6 Writing decimal notation for pictures. Now ask students to do the reverse. Exercises: Write the decimal for the picture. a) b) c) d) Answers: a) 0.8, b) 0.7, c) 0.5, d) 0.8 Introduce decimal hundredths. Tell students that the fraction 1/ can also be represented in various ways. Show four ways on the board: 1 one hundredth 0.01 Point out how a hundredth is written differently from a tenth: there are two digits after the decimal point instead of only one. Ask a volunteer to show how they would write two hundredths as a decimal (0.02) and then read it as zero point zero two. ASK: How would you write three hundredths as a decimal? (0.03). Exercises: Write the fraction as a decimal. a) 9 b) 4 c) 8 Answers: a) 0.09, b) 0.04, c) 0.08, d) 0.07, e) 0.05 d) 7 e) 5 Writing equivalent tenths and hundredths as fractions and decimals. Draw on the board: = ASK: How many tenths are shaded? (3) Fill in the first numerator. SAY: Each column is one tenth, and three of them are shaded. ASK: How many hundredths are shaded? (30) PROMPT: How many hundredths are in each column? () So there are, 20, 30 hundredths shaded. Fill in the second numerator. ASK: How would you write 3 tenths as a decimal? (0.3) How would you write 30 hundredths as a decimal? (0.30) Write on the board: Number and Operations in Base Ten 538 G3
4 0.3 = 0.30 SAY: These decimals are called equivalent decimals because one decimal equals the other decimal, just as equivalent fractions equal each other. Exercises: Write two equivalent fractions and two equivalent decimals for the amount shaded. a) b) c) d) Answers: a) 5/ = 50/ = 0.5 = 0.50, b) 2/ = 20/ = 0.2 = 0.20, c) 7/ = 70/ = 0.7 = 0.70, d) 4/ = 40/ = 0.4 = 0.40 Using a picture to show a combination of tenths and hundredths. Draw the first picture below on the board. ASK: How many hundredths are shaded? (30) How many tenths are shaded? (3) Then shade two more hundredths. ASK: Now how many hundredths are shaded? (32) Summarize by saying that 32 hundredths equals 3 tenths and 2 more hundredths. Write on the board: 32 hundredths = 3 tenths + 2 hundredths Exercises: Describe the fraction shaded as hundredths, and as tenths and hundredths. Bonus a) b) c) d) Answers: a) 64 hundredths = 6 tenths + 4 hundredths, b) 47 hundredths = 4 tenths + 7 hundredths, Bonus: c) 85 hundredths = 8 tenths + 5 hundredths, d) 86 hundredths = 8 tenths + 6 hundredths Comparing dimes and pennies to tenths and hundredths. Write the decimals 0.2 and 0.17 on the board and ask students to say which decimal is greater and why they think so. Some students may think that 0.17 is greater than 0.2 because 17 is greater than 2. Tell students that they may find it easier to compare decimals that represent tenths and hundredths if they think of the decimals as representing amounts of money. Write on the board: $0.56 = 56 cents = 5 dimes and 6 pennies G4 Teacher s Guide for AP Book 5.1
5 SAY: We use decimal notation for money because a dime is a tenth of a dollar and a penny is a hundredth of a dollar. Write on the board: 0.56 = 56 hundredths = 5 tenths + 6 hundredths ASK: How many hundredths are in 0.7? (70) Write on the board: 0.7 = 0.70 = 70 hundredths = 7 tenths 0 hundredths Exercises: Write the amount as hundredths and as mixed units (tenths and hundredths). a) 0.34 b) 0.68 c) 0.90 d) 0.5 Answers: a) 34 hundredths = 3 tenths + 4 hundredths, b) 68 hundredths = 6 tenths + 8 hundredths, c) 90 hundredths = 9 tenths + 0 hundredths, d) 50 hundredths = 5 tenths + 0 hundredths Now write 0.73 on the board and challenge students to write the decimal in four different ways using the terms tenths, hundredths, dimes, and pennies. Answer: 7 dimes 3 pennies, 7 tenths 3 hundredths, 73 pennies, 73 hundredths Exercises: Write the amount in three more ways. a) 8 dimes 5 pennies b) 0 dimes 6 pennies c) 9 dimes 0 pennies Answers: a) 8 tenths 5 hundredths, 85 pennies, 85 hundredths; b) 0 tenths 6 hundredths, 6 pennies, 6 hundredths; c) 9 tenths 0 hundredths, 90 pennies, 90 hundredths Now ask students to compare the decimals 0.2 and Students should see that 0.2 is greater than 0.17 because 2 dimes is greater than 1 dime and 7 pennies. Another way to compare decimals such as 0.2 and 0.17 or 0.3 and 0.25 is to think about how many hundredths are in the decimals. ASK: How many hundredths are in 0.3? (30, because 3/ is the same fraction as 30/) How many hundredths are in 0.25? (25) So 0.3 is greater than 0.25, because 30 hundredths is more than 25 hundredths. Tell students that they can change a decimal written in tenths to one written in hundredths by adding a zero to the decimal: Examples: 0.4 = 0.40 Four tenths is the same as forty hundredths. 0.9 = 0.90 Nine tenths is the same as ninety hundredths. 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 Relate tenths and hundredths to place value. Tell students that just like there is a ones place and a tens place in whole numbers, there is a tenths place and a hundredths place in decimals. Show this on the board: Number and Operations in Base Ten 538 G5
6 68 hundredths = 6 tenths 8 hundredths 68 = SAY: The first digit to the right of the decimal point is the number of tenths, and the second digit is the number of hundredths. Write on the board: tenths hundredths 0. Exercises: Describe the hundredths using the three ways shown above. a) 54 hundredths b) 8 hundredths c) 37 hundredths Answers: a) 54/, 5 tenths 4 hundredths, 0.54; b) 8/, 0 tenths 8 hundredths, 0.08; c) 37/, 3 tenths 7 hundredths, 0.37 Relate tenths and hundredths to number lines. Project onto the board BLM Hundredths Number Line. Label the tenths as shown Demonstrate counting 4 tenths and then 3 more hundredths. Then demonstrate counting 43 hundredths (count by hundredths until 40, then 1 hundredths until 43). Mark 0.43 on the number line. Exercises: Use BLM Hundredths Number Line to display these exercises. Write the fraction of the distance from 0 to 1 as hundredths, and as tenths and hundredths. a) b) c) d) Answers: a) 9 hundredths = 0 tenths 9 hundredths, b) 28 hundredths = 2 tenths 8 hundredths, c) 52 hundredths = 5 tenths 2 hundredths, d) 70 hundredths = 7 tenths 0 hundredths Extensions 1. Write the fraction as tenths and then as a decimal. a) 1 2 b) Write the fraction as hundredths and then as a decimal. a) 3 20 b) 7 50 c) c) Explain how you know that 0.7 = Answers: 1. a) 5/ = 0.5, b) 2/ = 0.2, c) 4/ = 0.4, d) 6/ = 0.6, e) 8/, 0.8, 2. a) 15/ = 0.15, b) 14/ = 0.14, c) 24/ = 0.24, d) 32/ = 0.32, e) 45/ = 0.45, = 7/ = 70/ = 0.70 d) d) e) e) G6 Teacher s Guide for AP Book 5.1
7 NBT539 Decimal Fractions Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will represent decimal fractions in expanded form. Vocabulary decimal fraction denominator equivalent fraction expanded form hundredth numerator powers of tenth thousandth PRIOR KNOWLEDGE REQUIRED Recognizes increasing and decreasing patterns Can use grids to represent tenths and hundredths Can write equivalent fractions Can add fractions with like and unlike denominators Review powers of. Write this pattern on the board: = = = = Have volunteers fill in the blanks. Tell students that these numbers are called powers of. Review multiplying by powers of. SAY: Multiplying by powers of is easy because you just write zeros at the end of the number. Exercises: Multiply. a) b) c) 1,000 d) Answers: a) 1,000, b), c),000, d),000 Exercises: What do you multiply by? a) = 1,000 b) = 1,000 c) = Bonus: d) =,000 e) 1,000 =,000,000,000 (MP.7) Students who are struggling can write the number of zeros under each power of. For example, in part a), write 1 under and 3 under 1,000. Answers: a), b), c), Bonus: d), e),000,000 Introduce decimal fractions. Display a table with the headings decimal fractions (with examples such as 5/, 4/, 3/, 425/1,000) and not decimal fractions (with examples such as 1/2, 2/5, 4/17, 9/20, 289/3,000). Have volunteers suggest additional fractions for the table, and have the rest of the class signal in which group each fraction should be placed. Have students guess the rule for putting the fractions in each group. NOTE: Do not use decimal notation until the next lesson. Focus on the concept of a decimal fraction. Number and Operations in Base Ten 539 G7
8 Explain that a decimal fraction is a fraction whose denominator is a power of. Decimal fractions are important because powers of are easy to work with. Point out that while some of the denominators in the not decimal fractions group are multiples of, they are not powers of. Also, some fractions (such as 1/2 and 2/5) are equivalent to decimal fractions but they are not decimal fractions. Review equivalent tenths and hundredths. Draw the two squares on the board. SAY: The picture shows why 3/ equals 3/. The second square has times as many shaded parts and times as many parts altogether. Write on the board: 3 = 30 Exercises: Write an equivalent fraction with denominator. Show your work. a) 7 = b) 4 = c) Answers: The numerators are: a) 70, b) 40, c) 90 9 = Equivalent tenths, hundredths, and thousandths. Write on the board: 3 7 = 5 = 1,000 = 1,000 SAY: Now you have to decide what to multiply the numerator by to get an equivalent fraction. You have to decide 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 the numerators (30, 700, 50). SAY: To make an equivalent decimal fraction, you just have to add the same number of zeros to the numerator and the denominator. Exercises: Write the missing numerator in the equivalent fraction. Bonus 8 a) = 1,000 b) 3 = 9 c) 1,000 = 3 1,000 =,000 Answers: a) 80, b) 300, c) 900, d) 30, Bonus: 30,000 Adding tenths and hundredths. Draw the picture below on the board: 3 + = + 6 = 36 G8 Teacher s Guide for AP Book 5.1
9 SAY: If you can add hundredths, and if you can change tenths to hundredths, then you can add tenths and hundredths. For example, 3 tenths is 30 hundredths, and 6 more hundredths is 36 hundredths. Remind students that they can change the tenths to hundredths without using a picture: = = 36 Exercises: Add. Show your work. a) b) c) Answers: a) 47/, b) 74/, c) 58/, Bonus: 93/ Bonus Adding tenths, hundredths, and thousandths. Write on the board: = 6 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 Have volunteers complete the equation: 300/1, /1, /1,000 = 396/1,000. Point out how adding fractions with denominators,, and 1,000 is easy because it s just using expanded form. Exercises: Add. 4 a) b) c) Bonus: For the bonus, students will need to carefully look at the denominators. Answers: a) 439/1,000, b) 521/1,000, c) 978/1,000, Bonus: 438/1,000 Adding decimal fractions with missing tenths or hundredths. Write the equation on the board but without the answer shown in italics = 1,000 ASK: How many thousandths are in 4/? (400) So, how many thousandths are there altogether? ( = 409) Write the answer, then SAY: 4 tenths, 0 hundredths, and 9 thousandths add to 409 thousandths. Have students add more tenths and thousandths. Exercises: Add. 3 a) b) + 1 c) Answers: a) 307/1,000, b) 901/1,000, c) 206/1, Number and Operations in Base Ten 539 G9
10 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 answer, then check by adding. 3 a) b) c) + 6 SAY: Be careful to watch for which denominator is missing. 8 d) e) f) Bonus: g) h) + 2, 000 Answers: a) 37/1,000, b) 82/1,000, c) 56/1,000, d) 802/1,000, e) 37/1,000, f) 508/1,000, Bonus: g) 208/1,000, h) 802/,000 SAY: You might need to add thousandths or just hundredths. Exercises: Add the decimal fractions. Write the answer as a decimal fraction. 3 a) b) c) d) e) f) Bonus: g) h) 5 + 5, 000 Answers: a) 333/1,000, b) 806/1,000, c) 96/, d) 35/1,000, e) 892/1,000, f) 51/, Bonus: g) 268/1,000, h) 50,005/,000 Extensions 1. Write 1 as a decimal fraction. Sample answers: /, / (MP.3) (MP.3) (MP.1, MP.3) 2. a) Is there a largest power of? (no, because you can multiply any power of by to get an even larger one) b) Is there a smallest decimal fraction? How do you know? (no, because you can make the fraction smaller by making the denominator a larger power of ) 3. Find the value of x. Explain how you can find your answer in b) and f). 4 a) + x = 9 3 b) + x = 38 x c) + 5 = 45 Bonus 6 d) + x = 67 4 e) + x = 67 7 f) + x = 4 Answers: a) 5, b) 8, c) 4, Bonus: d) 7/, e) 27/, f) 33/ G Teacher s Guide for AP Book 5.1
11 NBT540 Place Value and Decimals Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will represent decimals in expanded form. Vocabulary decimal decimal fraction decimal point denominator hundredth place value placeholder tenth thousandth 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 like and unlike denominators Review 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 digit 5 is in the thousands place, it is worth 5,000. Exercises: What does the digit 7 represent? a) 6,742 b) 9,017 c) 6,572 d) 7,904 Answers: a) 700, b) 7, c) 70, d) 7,000 Point out that the place value system extends to include tenths. Write on the board: thousands hundreds tens ones SAY: The place values get times as big. For example, tens are times as big as ones, hundreds are times as big as tens, and so on. Tell students that you want to continue using the place value system so that you can use place value for fractions, too. ASK: What is 1 ten times as big as? PROMPT: Ten of what make one whole? (a tenth) To guide students, draw pictures of equal parts fitting into 1 whole. Tell students that there is way to show 1/ that uses place value. Write on the board: 1 = = = = 27.4 SAY: We call these numbers decimals. The dot between the whole numbers and the number of tenths is called a decimal point. Decimals are like mixed numbers. Point to 3/ = 0.3 on the board and SAY: There s a wholenumber part to the left of the decimal point and a fractional part to the Number and Operations in Base Ten 540 G11
12 right. However, when the number is less than one whole, we write 0 as the wholenumber part. Exercises: Write the decimal for the number. a) 5 b) 3 8 c) 74 6 Answers: a) 0.5, b) 3.8, c) 74.6, Bonus: Bonus Extending the place value system beyond tenths. Write the following sequence 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: to some extent, they mirror each other on either side of the ones : hundreds tens ones tenths hundredths SAY: There is also symmetry in the values; in other words, what each place is worth is mirrored on either side of the ones. Draw on the board the picture below. Ask volunteers to continue the place values in both directions Show students how to write decimals for onedigit hundredths and onedigit thousandths: 3 = = SAY: The next place value after tenths is for hundredths. The one after that is for thousandths. Exercises: Write the decimal for the fraction. a) 7 b) 4 c) 5 d) 8 Answers: a) 0.07, b) 0.004, c) 0.05, d) 0.008, e) e) 6 Expressing decimal fractions in different ways. Tell students that there are two ways to say 0.03 out loud: zero point zero three and three hundredths. We use the second way to spell the number out in words, 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 G12 Teacher s Guide for AP Book 5.1
13 Writing a decimal with more than one nonzero digit. Write on the board: = 9.67 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 say 9.67 out loud as nine point six seven. NOTE: Reading and saying 9.67 as nine point sixtyseven is incorrect and should be discouraged because it can create the misconception that 9.67 is greater than 9.8 (since 67 > 8). Exercises: Write the decimal. a) d) f) b) c) e) Answers: a) 3.49, b) 8.53, c) 6.21, d) 2.759, e) 4.381, f) 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: Ask volunteers to write the decimals. (0.53, 0.503) Point out how the denominator tells you how many places after the decimal point the digit goes: tenths go one place after the decimal, hundredths go two places, and thousandths go three places. SAY: You have to be careful because some place values might be missing. You ll have to write zeros in those positions as placeholders. Exercises: Write the decimal. 3 a) 2 + b) d) e) c) 5 + f) Answers: a) 2.03, b) 3.502, c) 5.007, d) 8.039, e) 0.05, f) Exercises: Write the value of the digit 6 as a fraction or a whole number. a) b) c) d) e) Bonus: What place are the zeros holding in ? In ? Number and Operations in Base Ten 540 G13
14 Answers: a) 6/, b) 6/, c) 6/, d) 6, e) 6/1,000, Bonus: ones place and thousandths place in , ones place and tenthousandths place in Extensions (MP.8) 1. How much more is the 2 worth than the 5 in the decimal ? Answer: Because each place value is times the next value to the right, the relative values between the 2 and the 5 are the same as for 240,675. In 240,675, the 2 is worth 200,000 and the 5 is worth 5. How many times as much as 5 is 200,000 worth? Draw the table in the on the board: Number How many times as much as 5? , ,000 4, ,000 40,000 So the 2 is worth 40,000 times as much as the 5. (MP.1) 2. Write the correct decimal: $700 + $ = $. Answer: $ G14 Teacher s Guide for AP Book 5.1
15 NBT541 Decimals Greater Than 1 Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will write mixed numbers as decimals and decimals greater than one as mixed numbers, to thousandths. Vocabulary decimal decimal point digit fraction improper fraction mixed number PRIOR KNOWLEDGE REQUIRED Understands place value for whole numbers and the use of zero as a placeholder Can write an improper fraction as a mixed number Can write expanded form for whole numbers Can write mixed numbers and decimals Writing decimals in words. Tell students that just as they can write whole numbers and fractions in words, they can also write decimals in words. SAY: Just as a decimal point separates the wholenumber part and the fractional part, we can use the word and to separate the whole number from the fractional part. Write on the board: three and seventeen hundredths Exercises: Write the missing words. a) 4.08 = and eight hundredths b) 17.6 = and six tenths c) 16.5 = sixteen and tenths d) 3.07 = three and hundredths e) = thirtyeight and hundredths f) 30.8 = thirty and eight g) 3.08 = three and eight h) 4.17 = four seventeen Answers: a) four; b) seventeen; c) five; d) seven; e) fourteen; f) tenths; g) hundredths; h) and, hundredths Exercises: Write the decimal in words. a) 3.8 b) c) d) 41.5 Bonus: 3,000, Answers: a) three and eight tenths, b) twentysix and nine hundredths, c) thirty and forty hundredths, d) fortyone and five tenths, Bonus: three million and fortyfive hundredths Writing the decimal for the word. Refer to the picture on the board: three and seventeen hundredths Number and Operations in Base Ten 541 G15
16 SAY: The and tells you where to put the decimal point, and the fraction word tells you how many digits to put after the decimal point. Exercises: Write the decimal. a) twelve and thirteen hundredths b) fifty and three tenths c) two and fiftythree hundredths d) two hundred and five hundredths e) two hundred five and six tenths Hint: Some students might find it helpful to circle or underline the word and. Answers: a) 12.13, b) 50.3, c) 2.53, d) , e) Writing decimals as proper fractions. Write on the board: = 2 tenths + 6 thousandths = = 206 Point out that the same digits appear in both the decimal and the fraction s numerator. Also point out that the number of digits after the decimal point is the same as the number of zeros in the fraction s denominator. Write on the board: = = 37 There are three digits after the decimal point. So there are three zeros in the denominator. three digits after the decimal point so three zeros in the denominator Write on the board: a) 0.3 b) c) d) ASK: How many zeros would you put in the denominator? Students can signal their answers. (1, 3, 3, 6) Exercises: Write the fraction for each decimal above. Answers: a) 3/, b) 56/1,000, c) 801/1,000, d) 437/1,000,000. Writing decimals as mixed numbers. Write on the board: 3.14 ASK: What is the wholenumber part? (3) What is the fractional part? (14 hundredths) Write on the board: 3.14 = 3 14 Exercises: Write the decimal as a mixed number. Bonus a) 2.7 b) 3.07 c) 4.80 d) e) Answers: a) 2 7/, b) 3 7/, c) 4 80/, d) 235 6/, e) 17 9/, Bonus: 3 801/1,000 G16 Teacher s Guide for AP Book 5.1
17 Writing improper decimal fractions as decimals. Write on the board: 382/. Ask a volunteer to change it to a mixed number (3 82/), then have another volunteer change the mixed number to a decimal (3.82). Write more improper fractions on the board with denominators and (Examples: 45/, 402/, 637/), and ask volunteers to do both steps of converting to a decimal (4.5, 4.02, 6.37). Exercises: Write the improper fraction as a decimal. a) 28 b) 154 c) 769 Answers: a) 2.8, b) 1.54, c) 7.69, d) 6.1, e) 3.2 d) 61 e) 32 (MP.8) Introduce a shortcut for converting improper fractions to decimals. Write the answers to the exercises above on the board: 28 = = = = = 3.2 SAY: The numerator tells you what number to write, and the denominator tells you how many digits go after the decimal point. Write on the board: 384 = = = 4 3 Have volunteers show where to put the decimal point. (3.84, 38.4, 4.3) SAY: A denominator of means there must be 1 digit after the decimal point, and a denominator of means there must be 2 digits after the decimal point. Exercises: Write the decimal. a) 497 b) 84 c) 604 d) 307 Answers: a) 4.97, b) 8.4, c) 6.04, d) 30.7, Bonus: 78,523.4 Bonus 785, 234 Extensions 1. Have students list at least five decimal numbers that each take exactly six words to say. Make it clear to students that two words joined by a hyphen count as one word (e.g., fiftyeight is one word). This exercise gives students practice writing number words for decimals, being organized, and looking for patterns. Sample answers: 600, six hundred thousand and fortythree hundredths 9,000, nine million eighty and nine hundredths 2. Teach students to interpret large whole numbers written in decimal format and words. For example, 5.1 million is 5,,000; 3.7 thousand is 3,700. Number and Operations in Base Ten 541 G17
18 3. Point out the symmetry in the place values around the ones position. For example, in the number 27, , some of the place values are as follows: hundreds tens ones tenths hundredths Challenge students to name the place values in the number by using this symmetry. (ones, tenths, hundredths, thousandths, tenthousandths, hundredthousandths, millionths) NOTE: Some students might look for a oneths position, from thinking of the decimal point as the center of symmetry. This might seem natural because the decimal point is the only part of the number that looks different. However, ones are the basic units and in fact are the basis for the symmetry. 4. Add whole numbers, tenths, and hundredths. Example: = Have students look for decimals greater than 1 in the media and write the decimals as mixed numbers. Students can clip out examples of decimals in headlines and article text from newspapers and discuss how and where big numbers are used. G18 Teacher s Guide for AP Book 5.1
19 NBT542 Expanded Form and Place Value Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will write numbers in many different, but equivalent, forms. Vocabulary decimal decimal fraction equivalent decimal equivalent fraction expanded form lowest terms PRIOR KNOWLEDGE REQUIRED Understands place value for whole numbers and the use of zero as a placeholder Can write the expanded form for whole numbers Can write mixed numbers and decimals Can write the expanded form for whole numbers and decimals Introduce equivalent decimals. Write on the board: 3 = 30 = 1,000 SAY: Three tenths is the same as thirty hundredths. ASK: How many thousandths is that? (300) Fill in the missing numerator, then ask volunteers to write the decimal for each fraction on the board (0.3, 0.30, 0.300). Explain that these are called equivalent decimals, because the fractions they are equal to are equivalent. Exercises: Write the equivalent decimals from the equivalent fractions. a) d) 4 = 40 2 = 20 b) = = 70 c) e) 6 = 60 8 = 80 = 800 Answers: a) 0.4 = 0.40, b) 0.7 = 0.70, c) 0.6 = 0.60, d) 0.2 = 0.20 = 0.200, e) 0.8 = 0.80 = Tell students that saying 0.3 = 0.30 is the same as saying 3 tenths is equal to 30 hundredths or 3 tenths and 0 hundredths. Point out that 3 tenths and 0 hundredths or 0 thousandths and so on is the same as 3 tenths. ASK: How many hundredths is 8 tenths equal to? (80) Have a volunteer write the equivalent decimals on the board. (0.8 = 0.80) Write five ways to write seven tenths on the board: seven tenths seventy hundredths Ask students to write more ways. Then ask volunteers to show ways on the board. Sample answers: 0.7, 0.70, 0.700, 70/, 700/1,000, seven hundred thousandths. Exercises: Some of these are wrong. Which ones? How do you know? a) 0.37 = 37 b) 0.05 = 50 c) = 47 d) 0.62 = 62 Number and Operations in Base Ten 542 G19
20 Answers: Parts b) and d) are incorrect: in part b), the numerator should be 5 or the decimal should be 0.50; in part d), the denominator should be or the decimal should be Review decimal fractions in lowest terms. Remind students what it means for a fraction to be in lowest terms. ASK: Is 6/9 in lowest terms? (no, because 6 and 9 both divide by 3, so you can reduce the fraction): 6 = Is 2/3 in lowest terms? (yes, because the only whole number that divides both 2 and 3 is 1) Remind students that when you divide a power of by a smaller power of, you remove as many zeros from the dividend as there are in the divisor, for example,,000 = 1,000. Exercises: Divide. a) 1,000 b),000 c),000 Bonus: 1,000 1,000 Answers: a), b), c),000, Bonus: 1 SAY: Suppose I want to reduce the fraction /1,000 to lowest terms. ASK: Will my answer be in lowest terms if I divide the numerator and denominator by? Demonstrate what happens when you divide by : = ASK: Is the result in lowest terms? (no, because you can divide the numerator and denominator by again) 1 = What number should we have divided by, rather than? () Ask students to say what number they would divide by to reduce the following fractions to lowest terms:, 000 Students should notice the following: if the numerator and denominator of a fraction are both powers of, then when you divide both by the smaller power of, the resulting fraction is in lowest terms. Ask students to show how they would reduce the fractions above: 1 = = = 1 and so on. 1 NOTE: Many people reduce decimal fractions by crossing out an equal number of zeros in the numerator and denominator: = = 1 G20 Teacher s Guide for AP Book 5.1
21 Students can use this method, but they should be able to explain why it works. In the example, you cross out two zeros in the numerator and two zeros in the denominator because to reduce the fraction you divide both by. Multiplying by powers of. Remind students that when you multiply a fraction by a whole number, you multiply the numerator by the whole number and leave the denominator the same: = = 3 2 = Exercises: Find the product and reduce your answer to lowest terms. a) 1 b) 1,000 1 c) 1 d),000 1 Answers: a), b) 1, c) 1, d) 1 1, Ask students say how they could find the products above mentally. They should be able to articulate the following kind of strategy: 1 There are 2 zeros in the whole number. There are 3 zeros in the fraction. 1 There will be 3 2 = 1 zero left in the 2 zeros 3 zeros denominator, so the answer is 1. Make sure that students know that if there are more zeros in the whole number than in the fraction, the answer will be a whole number. Also note for students that any number divided by 1 is the same number. Exercises: Solve the puzzle. 1 1 a) = b) 1 d) 1, 000 = 1 Answers: a), b), c) 1,000, d) = 1 c) 1, 000 = 1 Comparing place values. Students are now prepared to compare place values. Use the following steps to show them how. ASK: How many times as large as is? Step 1: If any of the numbers you are comparing are decimals, write them as fractions: = Step 2: ASK: What number do you need to multiply the smaller number by to produce the larger number? Solve the problem by writing an expression as shown below: 1 1? = so,000 = Number and Operations in Base Ten 542 G21
22 Exercises: How many times as large as the smaller number is the larger number? Show your work. a) 0.1 and 0.01 b) 0.01 and c) and 0.01 d) and Answers: a), b), c) 1,000, d),000 Remind students that in a multidigit number, a digit in one place represents times as much as it represents in the place to its immediate right and 1/ of what it represents in the place to its immediate left. Students can use this knowledge to compare digits that are farther apart. Example: In the number , how many times as much as the 3 on the right is the 3 on the left? Step 1: Write the value of each 3. The first (left) 3 has value 30 and the second (right) 3 has value. Step 2: Write an expression and find the missing number in the box.? 3 = 30 so 3 = 30 Students can also compare place values by using the following diagram: hundreds tens ones tenths hundredths thousandths Exercise: In the number , how many times as large as the value of the 7 on the right is the value of the 7 on the left? Answer: The first 7 is 4 place values to the left of the second 7: so the value of the 7 on the left is =,000 times as large as the value of the 7 on the right. Review writing decimals in expanded form. Write on the board: = 8 tenths + 3 hundredths + 7 thousandths = Tell students that writing like this as a sum of decimal fractions is writing it in expanded form. Exercises: Write the decimal in expanded form. Bonus a) b) 0.45 c) d) Answers: a) 6/ + 7/ +2/1,000, b) 4/ + 5/, c) 2/ + 3/1,000, d) 8/ + 9/1,000, Bonus 2/ + 2/1,000,000 G22 Teacher s Guide for AP Book 5.1
23 Students can use their knowledge of multiplication to write a decimal in expanded form in two ways. For example, 0.7 can be written as 0.7 = 7 Extension or Decimals can be written in three different ways using fractions. For example, 0.03 can be written as: = or 0.03 = 3 or 0.03 = 3 2 Ask students to write the decimal in three different ways. a) 0.05 b) c) d) e) Answers: a), 5 1, ; b), 7, 7 ; c), 6, 6 ; d), 17, 17 ;, 000, e), 323, 000 1, 000, Number and Operations in Base Ten 542 G23
24 NBT543 Comparing Decimal Fractions and Decimals Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will place decimal numbers and mixed fractions on number lines and compare them. Vocabulary decimal equivalent decimal improper fraction mixed number PRIOR KNOWLEDGE REQUIRED Understands decimals and fractions on number lines Understands decimal numbers with up to two decimal places and their equivalent fractions (proper or mixed) Can use number lines Review number lines with fractions. Draw on the board: Have students count out loud with you from 0 to 1 by tenths: zero, one tenth, two tenths,, nine tenths, one. Then have a volunteer write the equivalent decimal for 1/ on top of the number line: Continue in random order until all the equivalent decimals have been added to the number line. Then have students write in their notebooks the equivalent decimals and fractions for the spots marked on these number lines: Answers: a) 0.7, b) 0.2, c) 0.5, d) 0.6, 1.2, 1.9, 2.6 Draw the number line below on the board and ask students to draw it in their notebooks. Have volunteers mark the location of the following numbers on the number line with an and the corresponding letter. A. 0.7 B. 2 7 C D. 8 E Invite any students who don t volunteer to participate. Help them with prompts and questions such as: Is the number more than 1 or less than 1? How do you know? Is the number between 1 and 2 or between 2 and 3? How do you know? Review writing improper fractions as mixed numbers. Then ask students to write the following improper fractions as mixed numbers and then place them on a number line from 0 to 3: a) 17/ b) 23/ c) 14/ d) 28/ e) 11/ Answers: a) 1 7/, b) 2 3/, c) 1 4/, d) 2 8/, e) 1 1/ G24 Teacher s Guide for AP Book 5.1
25 When students are done, ASK: When the denominator is, what is an easy way to tell whether the improper fraction is between 1 and 2 or between 2 and 3? (look at the number of tens in the numerator; it tells you how many ones are in the number) ASK: How many tens are in 34? (3) 78? (7) 123? (12) 345? (34) How many ones are in 34/? (3) 78/? (7) 123/? (12) 345/? (34) Exercises: Which two whole numbers is each fraction between? a) 29/ b) 24/ c) 127/ d) 81/ e) 143/ f) 318/ Answers: a) 2 and 3, b) 2 and 3, c) 12 and 13, d) 8 and 9, e) 14 and 15, f) 31 and 32 Invite volunteers to answer a) and b) on the board, then have students do the rest in their notebooks. Tell students that there are two different ways of saying the number We can say twelve decimal four or twelve and four tenths. Both are correct. (Note that twelve point four is also commonly used. However, we recommend students use the word decimal point. ) Point out the word and between the ones and the tenths, and tell students that we always include it when a number has both a wholenumber part and a fraction part. Have students place the following fractions on a number line from 0 to 3: a) three tenths b) two and five tenths c) one and seven tenths d) one decimal point two e) two decimal point eight Draw a number line from 0 to 3 on the board. Mark the following points with an no numbers and have students write the number words for the points you marked: Draw a line on the board with endpoints 0 and 1 marked: 0 1 Ask volunteers to mark the approximate position of each number with an : a) 0.4 b) 6/ c) 0.9 Then draw a number line from 0 to 3 with whole numbers marked: Ask volunteers to mark the approximate positions of these numbers with an : a) 2.1 b) 13/ c) 29/ d) 0.4 e) 22/ Number and Operations in Base Ten 543 G25
26 Continue with more numbers and number lines until all students have had a chance to participate. (Example: Draw a number line from 0 to 2 with whole numbers marked. Ask students to mark the approximate positions of 0.5, 1.25, and others.) Bonus: Use larger whole numbers and/or longer number lines. Extensions 1. Fill in the missing numbers Show 1.5: G26 Teacher s Guide for AP Book 5.1
27 NBT544 Comparing Fractions and Decimals Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will use fractions (one half, one quarter, and three quarters) as benchmarks for decimals. Vocabulary decimal equivalent fraction PRIOR KNOWLEDGE REQUIRED Can place decimals and fractions on number lines Understands decimal numbers with up to two decimal places and their equivalent fractions (proper or mixed) MATERIALS string colored thread clothespins Review number lines with decimals. Draw on the board: NOTE: Invite students, in small groups if necessary, to gather near the board as you go through the first part of the lesson. Have a volunteer show where 1/2 is on the number line. Draw another number line the same length and divided into two equal parts, and superimpose that second number line over the first number line so that students see that 1/2 is exactly at the 0.5 mark. ASK: Which decimal is equal to 1/2? Is 0.2 between 0 and 1/2 or between 1/2 and 1? Is 0.7 between 0 and 1/2 or between 1/2 and 1? What about 0.6? 0.4? 0.3? 0.9? Go back to the decimal 0.2 and ASK: We know that 0.2 is between 0 and 1/2, but is it closer to 0 or to 1/2? Draw on the board: ASK: Is 0.6 between 0 and 1/2 or between 1/2 and 1? (1/2 and 1) Which number is it closer to, 1/2 or 1? (1/2) Have a volunteer show the distance to each number with arrows. Which arrow is shorter? Which number is 0.4 closest to, 0, 1/2, or 1? (1/2) Which number is 0.8 closest to?(1) Repeat the questioning for all of the remaining decimal tenths between 0 and 1. Creating number lines. On grid paper, have students draw a line squares long. Then have them cut out the line leaving space above and below for writing and fold it in half so that the two endpoints meet. They Number and Operations in Base Ten 544 G27
28 should mark the points 0, 1/2, and 1 on their line. Now have students refold the line, and then fold it in half a second time. Have them unfold the line and look at the folds. ASK: What fraction is exactly halfway between 0 and 1/2? (1/4) How do you know? (the sheet is folded into 4 equal parts, so the first fold must be 1/4 the distance from 0 to 1) SAY: So every half is two quarters. ASK: What fraction is halfway between 1/2 and 1? How do you know? (3/4, because the sheet is folded into 4 equal parts, so the third fold must be 3/4 of the distance from 0 to 1) Have students mark the fractions 1/4 and 3/4 on their number line. Then have students write the decimal numbers from 0.1 to 0.9 in the correct places on their number line using the squares on the grid paper to help them. Comparing numbers on number lines. Tell students to look at the number line they ve created and to fill in the blanks in the following questions by writing less than or greater than in their notebooks. Exercises: Write less than or greater than. a) 0.4 is 1/4. b) 0.4 is 1/2. c) 0.8 is 3/4. d) 0.2 is 1/4. e) 0.3 is 1/2. f) 0.7 is 3/4. Answers: a) greater than, b) less than, c) greater than, d) less than, e) less than, f) less than Have students rewrite each statement using the greater than and less than symbols: > and <. Exercises: Which whole number is each decimal, mixed fraction, or improper fraction closest to? a) 0.7 b) c) 2.3 d) e) 2 6 f) g) 17.2 h) 16.8 i) 16 3 j) 174 e) 15.9 f) 15.3 Answers: a) 1, b) 1, c) 2, d) 2, e) 3, f) 1, g) 17, h) 17, i) 16, j) 17, k) 16, l) 15 Exercises: Which decimal is halfway between a) 1 and 2? b) 17 and 18? c) 31 and 32? d) 0 and 3? e) 15 and 18? f) 30 and 33? g) 25 and 28? Bonus: Which whole number is each decimal closest to? h) 23.4 i) 39.8 j) k) l) m) Answers: a) 1.5, b) 17.5, c) 31.5, d) 1.5, e) 16.5, f) 31.5, g) 26.5, Bonus: h) 23, i) 40, j) 314, k) 236, l) 981, m) 1,000 G28 Teacher s Guide for AP Book 5.1
29 Draw on the board: Ask volunteers to write two different fractions for the amount shaded in the pictures. Have other volunteers change the fractions to decimals. ASK: Do these four numbers all have the same value? How do you know? What symbol do we use to show that different numbers have the same value? (the equal sign) Write on the board: 0.9 = 0.90 = 9/ = 90/. Comparing numbers. Have students change more decimals to fractions with denominator. Exercises: Write as a fraction with denominator. a) 0.6 b) 0.1 c) 0.4 d) 0.8 e) 0.35 f) 0.42 Answers: a) 60/, b) /, c) 40/, d) 80/, e) 35/, f) 42/ Exercises: Put the group of numbers in order from smallest to largest by first changing all numbers to fractions with denominator. a) 0.3, 0.7, 0.48 b) 38/, 4/, 0.39 c) 2 17/, 2 3/, 2.2 Answers: a) 30/, 48/, 70/; b) 38/, 39/, 40/; c) 217/, 220/, 230/ Now show a hundreds block with half the squares shaded: SAY: This hundreds block has equal squares. How many of the squares are shaded? (50) So what fraction of the block is shaded? (50/) Challenge students to give equivalent answers with different denominators, namely and 2. PROMPTS: If we want a fraction with denominator, how many equal parts do we have to divide the block into? () What are the equal parts in this case and how many of them are shaded? (the rows; 5) What fraction of the block is shaded? (5/) What are the equal parts if we divide the block up into 2? (blocks of 50) What fraction of the block is shaded now? (1/2) Ask students to identify which fraction of the following blocks is shaded: Number and Operations in Base Ten 544 G29
30 Challenge students to find a suitable denominator by asking themselves: How many equal parts the size of the shaded area will make up the whole? For example, is the shaded part equal to 1/5 of the whole? 1/2? Have students convert their fractions into equivalent fractions with denominator. (1/5 = 20/, 1/4 = 25/, 1/20 = 5/) Finding equivalent fractions. Write on the board: 2 4 =, 3 4 = and have volunteers fill in the blanks. Then have students copy the following questions into their notebooks and fill in the blanks. a) b) 2 5 = 2 20 = 3 5 = 3 20 = 4 5 = 4 20 = Bonus: = Answers: a) 40, 60, 80,, b), 15, 20, 25, Bonus: = 5 20 = Exercises: Circle the larger number. First change the numbers to fractions with denominator. a) 1/2 or 0.43 b) 3/2 or 1.6 c) 3.7 or 3 1/2 d) 1/2 or.57 e) 1/4 or 0.23 f) 3/5 or 0.7 Answers: a) 1/2, b) 1.6, c) 3.7, d) 0.57, e) 1/4, f) 0.7 Give students groups of fractions and decimals to order from least to greatest by first changing all numbers to fractions with denominator. Include mixed, proper, and improper fractions. Start with groups of only three numbers and then move to groups of more numbers. ACTIVITY Make a number line divided into five equal parts from string that is any length. Students could mark the whole numbers with thread. Have students use clothespins to mark the following numbers in the most appropriate location: 13/ / Extensions 1. Have students rearrange the following words to create different numbers. What are the smallest and largest numbers that can be made using all the words? hundredths hundred thousand two five nine thirtyseven and G30 Teacher s Guide for AP Book 5.1
31 Answers: Greatest number: Thirtyseven thousand nine hundred five and two hundredths, Least number: Two thousand five hundred nine and thirtyseven hundredths 2. Write a decimal for each fraction by first changing the fraction to an equivalent fraction with denominator. a) 2/5 b) 1/2 c) 1/4 d) 3/5 e) 11/25 f) 47/50 Answers: a) 0.4, b) 0.5, c) 0.25, d) 0.6, e) 0.44, f) If you can run 12 km in an hour, how many kilometers can you run in a minute? Write your answer as a decimal number. (Hint: Reduce the fraction and then change to a fraction with denominator.) Answer: 0.2 km 4. Compare without using pictures, and determine which number is larger in the pair: a) 2 3/4 and 17 sevenths b) 1.7 and 17 elevenths c) 1.5 and 15 ninths d) 2.9 and 26 ninths NOTE: Students will have to convert all of the numbers into fractions in order to compare them. Is it best to use improper or mixed fractions? You can invite students to try both and see which types of fractions are easier to work with in this case (Mixed numbers work better for parts a) and d); improper fractions work better for parts b) and c).) Answers: a) 2 3/4, b) 1.7, c) 15 ninths, d) Write digits in the boxes that will make the statement true..5 <.3 Answer: any numbers a and b so that a < b (Example: 2 and 3) Number and Operations in Base Ten 544 G31
32 NBT545 Ordering Decimals Pages STANDARDS 5.NBT.A.1, 5.NBT.A.3 Goals Students will use fractions (one half, one quarter, and three quarters) as benchmarks for decimals. Vocabulary decimal equivalent fractions PRIOR KNOWLEDGE REQUIRED Can order whole numbers Can write equivalent fractions and decimals Can order proper and improper fractions with the same denominator Understands decimal place values Can translate between fractions with denominator,, or 1,000 and decimals MATERIALS play money (dimes and pennies) Comparing decimals by comparing their equivalent fractions. Write on the board: ASK: Which fraction is greater? (7/) So which decimal is greater? (0.7) SAY: You can compare decimals by comparing the fractions they are equivalent to. Exercises: Write the decimals as fractions. In the pair, which decimal is greater? a) 0.4 or 0.3 b) 0.35 or 0.27 c) 0.8 or 0.9 d) 0.76 or 0.84 Answers: a) 0.4 b) 0.35, c) 0.9, d) 0.84 Comparing different place values. Write on the board: Have students signal which number is the larger in each pair, starting from the bottom pair. SAY: You can compare decimals by writing them as fractions with the same denominator. Exercises: Write the decimals as fractions with the same denominator. Then decide which decimal is greater. NOTE: Students who struggle comparing decimals with one and two decimal places (e.g., students who are saying that 0.17 > 0.2) can use play dimes and pennies. G32 Teacher s Guide for AP Book 5.1
33 a) 0.3 and 0.24 b) 0.57 and Bonus: and Answers: a) 0.3, b) 0.614, Bonus: Ordering decimals by rewriting them to the smallest place value. Write on the board: 0.7 = SAY: I want to compare 0.7 to Writing them both as hundredths makes comparing them easy 70 is more than 64. Write the > sign between the decimals: 0.7 > 0.64 Exercises: Write both decimals in the pair as hundredths. Then compare them with the greater than or less than symbol. a) b) c) Answers: a) 0.40 < 0.51, b) 0.50 > 0.47, c) 0.30 > 0.28 Write on the board: Ask a volunteer to write 0.5 as thousandths, in decimal form. (0.500) ASK: What s greater, 500 thousandths or 487 thousandths? (500 thousandths) Write the inequality 0.5 > on the board. Exercises: Make both decimals have the same number of digits after the decimal point. Then compare them. a) 0.35 and 0.4 b) and 0.03 c) and 0.31 Bonus: 0.24 and Answers: a) 0.35 < 0.40, b) < 0.030, c) > 0.3, Bonus: > Comparing decimals with the same wholenumber part. SAY: When the wholenumber parts of a number are the same, you just have to compare the decimal parts. When the wholenumber parts are different, you only need to compare the whole numbers. Write on the board: (MP.6) 0.5 > 0.36, so 4.5 > > 3, so 5.16 > Exercises: Compare the pair of numbers. Which is larger? a) 3.4 and b) 8.56 and c) and d) 0.54 and e) 0.3 and f) 61.3 and 62.4 Bonus: and 88.4 Answers: a) 3.4, b) , c) 20.05, d) 0.54, e) 0.3, f) 62.4, Bonus: 88.4 Contrast the importance of digits after the decimal point with that of digits before the decimal point. NOTE: Many students have difficulty comparing and ordering decimals. A common mistake is regarding decimals with more digits after the decimal point as larger. Number and Operations in Base Ten 545 G33
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