Unit 1 he Number SystemT - JUMP Math for CC-Edition... · Unit 1 he Number SystemT ... They will...

$: Unit 1 he Number SystemT - JUMP Math for CC-Edition... · Unit 1 he Number SystemT ... They will multiply and divide fractions ... strip of paper with 4 dots of one color and 3 dots$
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Unit 1 The Number System

Students will divide whole numbers and interpret the answer as a fraction or decimal instead of with remainder. They will multiply and divide fractions and decimals. Students will use the correct order of operations to evaluate expressions involving fractions and decimals, including expressions that involve powers.

A note about division. Division with remainders is only used when dividing whole numbers by whole numbers. When dividing decimals by whole numbers or by decimals, students should not be asked to decide what to do when the corresponding whole number division leaves a remainder. To deal with this situation, students would need to know that they can continue the dividend by writing zeros after the decimal point. This technique is not taught in this course, as it leads to the concept of repeating decimals, which is not required in Sixth Grade. Please note in particular that it is not true that 0.7 ÷ 0.2 = 3 R 1, even though the corresponding whole number division is 7 ÷ 2 = 3 R 1. If anything, the “remainder” would be 0.1. However, the convention is that remainders are only used when dividing whole numbers by whole numbers (although decimal answers can also be used, as in 7 ÷ 2 = 3.5).

Preparation. Cut out the fraction pieces (wholes, halves, thirds, fourths, and fifths) from BLM Fraction Parts and Wholes (pp. J-84–J-88). You will need them in Lessons NS6-50 and NS6-52. Keep the pieces sorted according to their size to facilitate distribution to students.

Grid paper. Grid paper is convenient for aligning place values when adding, subtracting, multiplying, and dividing decimals. If your notebooks do not include grid paper, have lots of grid paper available for students. See BLM 1 cm Grid Paper (p. S-1).

The Number System

Teacher’s Guide for AP Book 6.2

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Introduce brackets. Remind students that addition, subtraction, multiplication, and division are called operations. Write on the board:

(8 - 5) + 2 8 - (5 + 2)

SAY: Do the operation in brackets first. (8 − 5) + 2 means 3 + 2, which is 5; 8 − (5 + 2) means 8 - 7, which is 1.

Exercises: Evaluate.

a) (8 + 4) - 3 b) 8 + (4 - 3) c) (8 - 2) × 3 d) 8 - (2 × 3)

e) 12 ÷ (2 × 3) f ) (12 ÷ 2) × 3 g) (5 + 3) × 4 h) 5 + (3 × 4)

Answers: a) 9, b) 9, c) 18, d) 2, e) 2, f ) 18, g) 32, h) 17

Point out that the operation that you do first often changes the answer, but not always.

Adding and subtracting from left to right. Tell students that mathematicians have come up with shortcuts so that they do not have to write brackets all the time. SAY: When there are no brackets, do addition and subtraction from left to right. For example, 8 - 5 + 2 means 3 + 2. If you mean 8 - 7, you have to add brackets: 8 - (5 + 2).

Exercises: Add or subtract from left to right.

a) 6 + 3 - 2 b) 6 - 3 + 2 c) 14 - 5 + 6 d) 9 + 4 - 4

Answers: a) 7, b) 5, c) 15, d) 9

Bonus: Do you need to add brackets to get the answer?

a) 9 - 4 + 2 = 3 b) 7 + 5 - 3 = 9 c) 2 + 6 - 5 = 3 d) 8 - 3 + 2 = 3

e) 7 - 3 + 2 = 6 f ) 2 + 8 + 3 = 13 g) 9 - 4 - 2 = 7 h) 9 - 4 - 3 = 2

Answers: a) yes, b) no, c) no, d) yes, e) no, f ) no, g) yes, h) no

Multiplying and dividing from left to right. SAY: When there are no brackets, do multiplication and division from left to right.

NS6-42 Order of Operations Pages 1–2

STANDARDS preparation for 6.EE.A.2

VOcABULARy brackets equation operation

GoalsStudents will understand the need for brackets in expressions and for assigning an order to the operations. Students will evaluate expressions involving more than one operation by using the correct order of operations.

PRIOR KNOWLEDGE REQUIRED

Can mentally add and subtract numbers up to 20 Knows the multiplication facts up to 6 × 6 and the related division facts

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Exercises: Multiply or divide from left to right.

a) 6 ÷ 3 × 2 b) 4 × 4 ÷ 2 c) 2 × 6 ÷ 3 d) 12 ÷ 3 × 4

Answers: a) 4, b) 8, c) 4, d) 16

Bonus: Do you need to add brackets to get the answer?

a) 3 × 6 ÷ 2 = 9 b) 12 ÷ 6 × 2 = 1 c) 5 × 8 ÷ 2 = 20

d) 8 ÷ 2 × 2 = 2 e) 2 × 3 × 4 = 24 f ) 16 ÷ 4 ÷ 2 = 2

Answers: a) no, b) yes, c) no, d) yes, e) no, f ) no

Multiply or divide before adding or subtracting. SAY: When there are no brackets, do multiplication or division before addition or subtraction.


a) 4 + 5 × 2 b) 3 × 4 + 5 c) 12 - 2 × 5 d) 3 × 8 - 5

e) 3 + 6 ÷ 3 f ) 8 ÷ 2 + 2 g) 15 ÷ 5 - 2 h) 14 - 6 ÷ 2

Bonus: i ) 34 + 50 × 2 j ) 13 × 100 - 10 k) 96 - 6 ÷ 3

Answers: a) 14, b) 17, c) 2, d) 19, e) 5, f ) 6, g) 1, h) 11, Bonus: i ) 134, j ) 1,290, k) 94

Summarizing the order of operations without brackets. Write on the board:

3 + 4 = 7 3 × 4 = 12

Have a volunteer write a subtraction equation from the first equation. (7 - 4 = 3 or 7 - 3 = 4) ASK: What kind of equation can you write from the second equation? (a division equation) Have a volunteer write the equation. (12 ÷ 3 = 4 or 12 ÷ 4 = 3) SAY: It’s easy to remember that addition and subtraction are done from left to right because you can go back and forth between addition and subtraction. That’s true for multiplication and division too. But you can’t go back and forth between multiplication and addition, so you don’t do them from left to right. You always do multiplication and division before addition and subtraction. Write on the board:

8 - 3 × 2

ASK: Which operation do I do first? (3 × 2) Circle it. Then demonstrate doing that operation only by rewriting the rest of the equation: 8 - 6. SAY: Now I’ve made it into an easier problem.

Exercises: Do the correct operation first. Then rewrite the problem as an easier one.

a) 13 - 3 × 4 b) 4 + 6 × 2 c) 9 + 7 - 3 d) 14 - 4 ÷ 2

e) 3 × 4 - 3 f ) 6 × 3 ÷ 3 g) 6 + 9 ÷ 3 h) 18 ÷ 3 × 2

Bonus: 8 - 5 + 6 ÷ 2 × 3

Answers: a) 13 - 12, b) 4 + 12, c) 16 - 3, d) 14 - 2, e) 12 - 3, f ) 18 ÷ 3, g) 6 + 3, h) 6 × 2, Bonus: 8 - 5 + 3 × 3

The Number System 6-42


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Encourage struggling students to start by circling the operation they would do first.

Exercises: Finish evaluating the expressions above.

Answers: a) 1, b) 16, c) 13, d) 12, e) 9, f ) 6, g) 9, h) 12, Bonus: 12

Evaluating long expressions. Students can evaluate longer expressions, some of which have brackets, and some of which do not.

Exercises: Do the operations one at a time, in the standard order.

a) 9 - 3 × (6 - 4) b) 5 × 3 - 5 - 3 c) 2 + 3 × 4 ÷ 6 d) 6 × 4 - (6 + 4) e) 8 ÷ 2 × 3 - 4 f ) 2 × (5 + 3) ÷ 4

Bonus: (3 + 4) × (6 - 1) + (7 + 5) ÷ (6 - 4) - (8 - 5)

Selected solution: a) 9 - 3 × 2 = 9 - 6 = 3

Answers: b) 7, c) 4, d) 14, e) 8, f ) 4, Bonus: 38

Extensions1. Use the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9, once each, to make

the equations true.

÷ ( × ) = 1

- ( + ) = 2

( + ) ÷ = 4

2. a) Evaluate expressions with more than one operation inside the brackets.

i ) 8 - (4 + 6 ÷ 2) ii ) 4 × (9 - 4 × 2) iii ) 42 ÷ (10 - 1 × 3) iv) (5 × 4 - 6) ÷ 7 v) 54 ÷ (2 × 4 + 1) vi ) 13 - (3 × 5 - 4)

b) Using exactly four 4s each time, make expressions equal to each number from 0 through 10. You may use brackets and any of the four operations. Example: (4 × 4) ÷ (4 + 4) = 16 ÷ 8 = 2 Hint: You may use the 2-digit number 44.

Answers: a) i ) 1, ii ) 4, iii ) 6, iv) 2, v) 6, vi ) 2, b) Sample answers: 0 = (4 - 4) × (4 + 4) 1 = 4 ÷ 4 × 4 ÷ 4 2 = 4 × 4 ÷ (4 + 4) 3 = (4 + 4 + 4) ÷ 4 4 = (4 - 4) × 4 + 4 5 = (4 × 4 + 4) ÷ 4 6 = (4 + 4) ÷ 4 + 4 7 = 44 ÷ 4 - 4 8 = 4 × 4 - (4 + 4) 9 = 4 ÷ 4 + 4 + 4 10 = (44 - 4) ÷ 4

How many different expressions can you come up with?

(MP.1)Sample answers

1. 6 ÷ (3 × 2) = 1 8 - (5 + 1) = 2 (7 + 9) ÷ 4 = 4

2. 8 ÷ (2 × 4) = 1 9 - (1 + 6) = 2 (5 + 7) ÷ 3 = 4

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Introduce the commutative property for addition and for multiplication. On a strip of paper, draw 7 dots in a row, with the first 3 dots shaded. Tape the paper to the board, and write underneath: 3 + 4. Now rotate the strip of paper 180° and write underneath: 3 + 4 = 4 + 3.

3 + 4 = 4 + 3

Exercises: Write the sum for each picture in two ways. Make an equation that shows that order doesn’t matter in addition.

a) b)

Bonus:

Answers: a) 2 + 5 = 5 + 2, b) 3 + 5 = 5 + 3, Bonus: 243 + 126 = 126 + 243

Point out that the equations they made are examples of the commutative property of addition. The property states that the answer will not change when they add the same numbers in any order. Write on the board the four symbols for the operations: +, -, ×, ÷. Ask students whether each operation satisfies the commutative property. Point to each symbol in turn and have students signal thumbs up for yes (+ and ×) and thumbs down for no (- and ÷). PROMPTS: Is 6 + 2 the same as 2 + 6? Is 6 - 2 the same as 2 - 6? Is 6 × 2 the same as 2 × 6? Is 6 ÷ 2 the same as 2 ÷ 6?

On a sheet of paper, draw an array with 4 rows of 3 dots. Show it to students and write 4 × 3 on the board. Rotate it so that it shows 3 rows of 4 dots and write 4 × 3 = 3 × 4 on the board.

Introduce the associative property of addition and multiplication. Exercises: Calculate both expressions and write = or ≠.

a) (3 + 4) + 5 3 + (4 + 5) b) (2 × 3) × 4 2 × (3 × 4)

c) (8 - 3) - 1 8 - (3 - 1) d) (8 ÷ 4) ÷ 2 8 ÷ (4 ÷ 2)

NS6-43 Properties of Operations Pages 3–4

STANDARDS preparation for 6.EE.A.2, 6.EE.A.3

VOcABULARy associative property brackets commutative property distributes distributive property operation

GoalsStudents will use the properties of operations to check their answers to calculations.


Knows that operations in brackets are done first Understands multiplication and division as equal groups

MATERIALS

strip of paper with 4 dots of one color and 3 dots of a different color sheet of paper with a 3 by 4 array



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Answers: a) 12 = 12, b) 24 = 24, c) 4 ≠ 6, d) 1 ≠ 4

Tell students that when the order they do the operations in doesn’t matter, the operation satisfies the associative property. As with the commutative property, the answer will not change when they do the operations with the same numbers in any order. Write on the board the four operation symbols (+, -, ×, ÷), and ask students to signal whether each operation satisfies the associative property. (+, yes; -, no; ×, yes; ÷, no)

contrasting the associative and commutative properties. Point out that the commutative property tells you that the order the numbers are in doesn’t matter. The associative property tells you that the order the operations are done in doesn’t matter.

Using the associative property to check answers. Point out that students can use the associative property to check calculations. Emphasize that everyone, even professional mathematicians, make mistakes in calculations. What is important is that they can find and correct their mistakes.

Exercises: Add or multiply in two ways. Are your answers equal? If not, find your mistake.

a) (7 + 2) + 8 and 7 + (2 + 8) b) (3 × 7) × 4 and 3 × (7 × 4)

c) (27 + 56) + 34 and 27 + (56 + 34) d) (4 × 9) × 5 and 4 × (9 × 5)

Answers: a) 9 + 8 = 17, 7 + 10 = 17; b) 21 × 4 = 84, 3 × 28 = 84; c) 83 + 34 = 117, 27 + 90 = 117; d) 36 × 5 = 180, 4 × 45 = 180

Have students circle the way of calculating the answer that they found easiest. Point out that as they practice doing the same problems two ways, they will learn which way will be faster, even before trying them.

Introduce the distributive property of addition and multiplication. Draw on the board:

Tell students that there are many ways to find out how many dots there are. Write the following expressions on the board:

6 + 8 7 + 7

ASK: Where do the 6 and 8 come from? (the black and white dots) Where do the 7 and 7 come from? (the number in each row) SAY: Because we have two different ways to write the same number, we can make an equation:

6 + 8 = 7 + 7

SAY: Now I’m going to make it harder. Write on the board:

(2 × 3) + (2 × 4) 2 × (3 + 4)

Point to each expression in turn and ASK: How does this count the dots? (the first expression counts the 2 rows of 3 black dots and the 2 rows of 4 white dots; the second expression counts the rows separately—there are 3 black dots and 4 white dots in each of the 2 rows)

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Exercise: Using only the numbers 2, 3, and 5, make an equation from the picture.

Answer: (3 × 2) + (3 × 5) = 3 × (2 + 5)

Bonus: Draw a picture to show that (3 × 2) + (3 × 4) = 3 × (2 + 4)

Tell students that this is a general property of addition and multiplication. The answers on both sides of the equal sign will be the same no matter what numbers you substitute for 2, 3, and 5 in the equation above. SAY: This property is called the distributive property and you can say that multiplication distributes over addition.

contrasting the distributive property with the associative and commutative properties. Point out that the commutative and associative properties relate to multiplication and addition separately. The distributive property shows you one way that multiplication and addition are related to each other.

Using the distributive property to check answers.

Exercises: Evaluate both expressions. Are your answers equal? If not, find your mistake.

a) 5 × (3 + 4) and (5 × 3) + (5 × 4) b) 3 × (2 + 8) and (3 × 2) + (3 × 8)

c) (2 + 3) × 7 and (2 × 7) + (3 × 7) d) (4 + 6) × 9 and (4 × 9) + (6 × 9)

e) 2 × (7 - 4) and (2 × 7) - (2 × 4) f ) (8 - 3) × 4 and (8 × 4) - (3 × 4)

Selected solution: a) 5 × 7 = 35 and 15 + 20 = 35

Answers: b) 30, c) 35, d) 90, e) 6, f ) 20

SAY: Parts e) and f) show that multiplication also distributes over subtraction. Then have students circle the way of calculating the answer that they found easiest.

Identifying properties. Write the names of the different properties on the board:

1. the commutative property 2. the associative property 3. the distributive property

Have students signal which property each equation is an example of.

a) 8 × 3 = 3 × 8 b) 2 × (7 - 4) = 2 × 7 - 2 × 4 c) 2 + 81 = 81 + 2 d) (5 × 2) × 4 = 5 × (2 × 4) e) 9 + (3 + 2) = (9 + 3) + 2 f ) (5 + 2) × 9 = 5 × 9 + 2 × 9

Answers: a) 1, b) 3, c) 1, d) 2, e) 2, f ) 3



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Exploring a property of division. Draw on the board:

8 ÷ 2 = 4

SAY: When 8 objects are shared equally between 2 people, each person gets 4. ASK: What if I wanted to share 3 times as many objects. How many would each person get? (3 times as many, or 12) What if I had 3 times as many objects, but I wanted to share them between 3 times as many people? Now how many would each person get? (4 again) Show this on the board:

(8 × 3) ÷ (2 × 3) = 8 ÷ 2

SAY: I’ve drawn 3 times as many dots, and I’ve made 3 times as many groups. I didn’t change the number in each group, so both divisions have the same answer.

Exercises: Evaluate all expressions. Make sure your answers are equal.

a) 20 ÷ 5 b) 30 ÷ 3 (20 × 2) ÷ (5 × 2) (30 × 2) ÷ (3 × 2) (20 × 3) ÷ (5 × 3) (30 × 3) ÷ (3 × 3)

Answers: a) 4, 4, 4; b) 10, 10, 10

Exercises: Investigate: Does dividing each number in a division question by the same number get the same answer?

a) 60 ÷ 12 b) 90 ÷ 30 (60 ÷ 2) ÷ (12 ÷ 2) (90 ÷ 2) ÷ (30 ÷ 2) (60 ÷ 3) ÷ (12 ÷ 3) (90 ÷ 3) ÷ (30 ÷ 3)

Answers: a) 5, 5, 5; b) 3, 3, 3; yes, they all have the same answer

ExtensionSolve the equation. Justify your answers.

a) 2 × 5 = w × 2 b) w × 6 = 6 × 3

c) 2 × (3 × w) = (2 × 3) × 7 d) 3 × (w × 6) = 3 × 24

e) 10 × 3 = (2 × w) × 3 f ) (3 × w) × 4 = 3 × 20

g) (2 × 3) × w = 2 × 12 Bonus: 14 × w = 2 × 35

Selected solutions: a) Using the commutative property, 2 × 5 = 5 × 2, so w = 5, d) w × 6 = 24, so w = 4, g) By the associative property, (2 × 3) × w = 2 (3 × w), so 3 × w = 12 and w = 4.

Answers: b) 3, c) 7, d) 4, e) 5, f ) 5, g) 4, Bonus: 5

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NOTE: We do not recommend using the word “squared” for numbers with the exponent 2, unless you first explain where the word comes from.

Introduce powers as repeated multiplication. First, remind students that multiplication is a short form for repeated addition. Then point out that just as we can add the same number repeatedly, we can multiply the same number repeatedly. Write on the board:

3 × 5 = 3 + 3 + 3 + 3 + 3 35 = 3 × 3 × 3 × 3 × 3

Point to 3 × 5 and SAY: The 3 tells you what number to add, and the 5 tells you how many times to write it in the sum. Point to 35 and SAY: The 3 tells you what number to multiply, and the 5 tells you how many times to write it. Have volunteers demonstrate the first two exercises below.

Exercises: Write the repeated multiplication.

a) 23 b) 32 c) 24 d) 53

e) 102 f ) 74 g) 45 h) 04

Bonus: 1,0004

Answers: a) 2 × 2 × 2, b) 3 × 3, c) 2 × 2 × 2 × 2, d) 5 × 5 × 5, e) 10 × 10, f ) 7 × 7 × 7 × 7, g) 4 × 4 × 4 × 4 × 4, h) 0 × 0 × 0 × 0, Bonus: 1,000 × 1,000 × 1,000 × 1,000

Introduce the word “power.” Tell students that the short form notation for repeated multiplication is called a power.

Exercises: Write the power.

a) 2 × 2 × 2 × 2 × 2 b) 3 × 3 × 3 × 3

c) 10 × 10 × 10 d) 1 × 1

Bonus: 1,000,000 × 1,000,000 × 1,000,000 × 1,000,000 × 1,000,000

Answers: a) 25, b) 34, c) 103, d) 12, Bonus: 1,000,0005

NS6-44 Powers Pages 5–6

STANDARDS 6.EE.A.1

VOcABULARy base exponent power

GoalsStudents will investigate properties of powers, and evaluate powers by using repeated multiplication.


Can multiply a sequence of more than two whole numbers (e.g., 2 × 3 × 5) Can use a calculator to multiply numbers Can add, subtract, multiply, and divide two whole numbers Knows that multiplication commutes (e.g., 2 × 4 = 4 × 2)



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Introduce the words “base” and “exponent.” Write on the board: 35. SAY: 3 is called the base and 5 is called the exponent. Remind students that, in English, the word “base” means the bottom of something. Here, the word “base” is the bottom part of a power. Have students signal the base in each expression below by holding up the correct number of fingers:

a) 23 b) 72 c) 45 d) 64 e) 108 f ) 89 g) 91

Answers: a) 2, b) 7, c) 4, d) 6, e) 10, f ) 8, g) 9

Then have students signal the exponent in each expression above. Answers: a) 3, b) 2, c) 5, d) 4, e) 8, f ) 9, g) 1

Evaluating powers. Students can evaluate powers by keeping track of the partial products. Write on the board:

25 = 2 × 2 × 2 × 2 × 2 =

Have volunteers dictate the partial answers as you write them in the boxes. Write the final product as the answer.


a) 33 b) 24 c) 103 d) 53 e) 14 f ) 05

Bonus: g) 017 h) 132 i ) 108

Answers: a) 27, b) 16, c) 1,000, d) 125, e) 1, f ) 0, Bonus: g) 0, h) 1, i ) 100,000,000

comparing powers. Tell students that powers stand for numbers. Once you know what numbers they are, you can compare them. Write on the board:

25 = 32 52 =

Have a volunteer write the number that 52 equals. (25) Prompt students if necessary by saying “5 times 5.” Then ASK: Which is greater—2 to the exponent 5, or 5 to the exponent 2? (2 to the exponent 5)

Exercises: Evaluate the powers. Then write <, >, or =.

a) 34 43 b) 23 32 c) 24 42 d) 101 110 e) 102 210

Answers: a) 81 > 64, b) 8 < 9, c) 16 = 16, d) 10 > 1, e) 100 < 1,024

Adding, subtracting, multiplying, and dividing powers. Tell students that since powers just stand for numbers, we can add, subtract, multiply, and divide them. But we have to evaluate what numbers they stand for before adding, subtracting, multiplying, or dividing them. Write on the board:

32 + 23 = + 42 ÷ 22 = ÷ = =

Have volunteers tell you how to fill in the blanks. (9 + 8 = 17, 16 ÷ 4 = 4)

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a) 102 + 52 b) 42 - 22 c) 102 × 23 d) 102 - 23 e) 102 ÷ 52 f ) 32 × 23

Answers: a) 100 + 25 = 125, b) 16 – 4 = 12, c) 100 × 8 = 800, d) 100 - 8 = 92, e) 100 ÷ 25 = 4, f ) 9 × 8 = 72

Using the order of operations with powers. Remind students that when expressions involve more than one operation, there is a standard order to do them in. ASK: Which operation do you perform first, addition or multiplication? (multiplication) SAY: Multiplication is repeated addition, and multiplication is done before addition. ASK: Using this pattern, which operation would you do first, multiplication or powers? (powers) Write on the board:

3 × 25 = ? 3 × 32 or 65

ASK: Would you evaluate 2 to the exponent 5, then multiply by 3, or would you multiply 3 by 2, then take the result to the exponent 5? (evaluate the power, then multiply by 3) Have a volunteer circle the correct way to evaluate the expression. (3 × 32) SAY: If you meant 65, you would have to add brackets: (3 × 2)5 = 65.


a) 5 × 23 b) (5 × 2)3 c) 6 × 22 d) (6 × 2)2 e) 3 × 32 f ) (3 × 3)2 g) 4 × 32 h) (4 × 3)2

Answers: a) 40, b) 1,000, c) 24, d) 144, e) 27, f ) 81, g) 36, h) 144

SAY: Evaluate powers before multiplying or dividing, and multiply or divide before adding or subtracting.

Exercises: Evaluate using the correct order of operations.

a) 3 + 42 b) (3 + 4)2 c) 8 ÷ 23 d) (8 ÷ 2)3 e) 42 – 3 × 2 f ) 7 + 22 – 3 g) 9 – 42 ÷ 4 h) 18 ÷ 32 × 5

Bonus: 40 ÷ (7 – 5)3 + 32 – (2 – 1)5

Answers: a) 19, b) 49, c) 1, d) 64, e) 10, f ) 8, g) 5, h) 10, Bonus: 13

Extensions1. Create the sequence of numbers with base 2 and exponents

1, 2, 3, 4, and 5. Find the difference between terms in the sequence. What do you notice?

Answers: 2, 4, 8, 16, 32 has differences of 2, 4, 8, 16—the same sequence you started with!

2. Create the sequence of numbers with bases 1, 2, 3, 4, and 5, and exponent 2. Find the difference between terms in the sequence. What do you notice?

Answer: 1, 4, 9, 16, 25 has differences of 3, 5, 7, 9—odd numbers that always increase by 2.

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3. Write the missing number in the box.

a) 2 = 128 b) 3 = 64 c) 3 = 243

Answers: a) 7, b) 4, c) 5

4. Use the numbers 1, 2, 3, 4, 5, and 6 once each to make the equation true.

+ + = 112

Answer: 61 + 52 + 34 = 112 Make up a similar problem for a partner to solve.

5. How many 8s must you add together to get a sum equal to 83?

Answer: 64 because 83 = (8 × 8) × 8 = 64 × 8

6. a) What is the greatest common factor of 56 and 57?

b) What is the lowest common multiple of 56 and 57?

Answers: a) 56, b) 57

7. Computer codes are written as sequences of 0s and 1s. Investigate how many sequences of each length there are. Students will discover that the answer doubles at each stage.

2 sequences of length 1: 0, 1 4 sequences of length 2: 00, 01, 10, 11 8 sequences of length 3: 000, 001, 010, 011, 100, 101, 110, 111

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(MP.7)

(MP.7)

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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, then SAY: In case some pies are 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 underneath the picture: 7 × 1/4 = 7/4.

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: a) , 2 × 14

= 24

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 ÷ =

NS6-45 Division with Fractional Answers Pages 7–8

STANDARDS 6.NS.A.1

VOcABULARy division fraction improper fraction mixed number remainder

GoalsStudents will recognize a fraction as division of the numerator by the denominator.


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 Understands division as equal sharing

MATERIALS

11 “granola bars” made from 4 connecting cubes each



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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). 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. Point out also that now the answer is a fraction, not a whole number with remainder.

Have another volunteer write the division equation to show 5 people sharing 2 pies. (2 ÷ 5 = 2/5)

Exercises: Write the division equation.

a) 5 people share 3 pies b) 5 people share 4 pies

c) 6 people share 3 pies d) 6 people share 4 pies

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 top number of the fraction, and the second number in the division is the bottom number of the fraction.

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

Writing the answer as a mixed number. Now tell students that 3 people are sharing 5 apple 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.

53

= 123

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 or 3/2 = 1 1/2, Bonus: 15/8 = 1 7/8

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Dividing the whole objects equally first. Prepare ahead 11 “granola bars” each made from 4 connecting cubes. Tell students that four people are sharing 11 granola bars. Ask for four volunteers. Start by giving one to each volunteer, then give one more to each volunteer. Write on the board:

11 ÷ 4 = R

Have a volunteer fill in the blanks. (2 R 3) Point out that instead of dividing each granola bar into quarters, we just have to divide the remaining 3 bars into quarters. ASK: How many quarters does each person get? (3) Demonstrate giving one quarter of each bar to each volunteer. SAY: You can use division with remainders to tell you the mixed-number answer. Write on the board:

11 ÷ 4 = 2 R 3, so 11 ÷ 4 = 234

Emphasize that these are two different ways of using the division sign. The first way has two different parts to the answer—a quotient and a remainder. The second way just has a fractional quotient.

Exercises: Write the answer as a mixed number and as an improper fraction.

a) 5 people share 12 granola bars. How many granola bars does each person get?

b) 3 people share 16 pounds of rice. How much rice does each person get?

c) 4 people share 9 pounds of flour. How much flour does each person get?

Answers: a) 12/5 = 2 2/5, b) 16/3 = 5 1/3, c) 9/4 = 2 1/4

Exercises: Use multiplication to check the answer to each division.

a) 3 ÷ 4 = 34

b) 2 ÷ 5 = 25

c) 7 ÷ 3 = 73

Selected solution: a) 4 × 3/4 = 12/4 = 3

Extensions1. Write the fact family for

35

× 5 = 3.

Answers: 3/5 × 5 = 3, 5 × 3/5 = 3, 3 ÷ 3/5 = 5, 3 ÷ 5 = 3/5

2. Ron says 2 R 1 = 2 14

because 9 ÷ 4 = 2 R 1 and 9 ÷ 4 = 214

.

Is this reasoning correct? Explain.

Answer: No. The division symbol is being used in two different ways. (In fact, by the same incorrect reasoning, 2 R 1 could also equal 2 1/3 since 7 ÷ 3 = 2 R 1 and 7 ÷ 3 = 2 1/3.)

NOTE: The two ways of using division 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 number, including fractions, that can have any kind of answer. Later, students will learn how to divide fractions by fractions.

(MP.4)

(MP.1)

(MP.3)



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3. When you divide 11 granola bars among 4 people, you divide 11 items among 4 groups. If you calculate 11 ÷ 4, the answer you get represents the number of items in each group.

total number of items ÷ number of groups = number of items in each group

11 ÷ 4 = 2 34

Number of items

We make 4 groups because there are 4 people

Each person gets 2 whole items

Plus 3/4 of an item

In general, when the divisor (in this case 4) is the number of groups, the answer is the number of items in the group or the size of the group.

However, when the divisor is the size of the group, the answer is the number of groups of that size.

Example: If you cut 11 meters of rope into 4 meter pieces, how many pieces can you make?

2 whole pieces of 4 meters in size

3 meters left over (3 meters is 3/4 of 4 meters)

4 4

As the diagram shows, the whole number (2) in the answer tells you how many groups or pieces of 4 meters in size you can make (two groups of 4 meters in size). The fraction in the answer (3/4) does not mean that you have 3/4 of a meter left over; it means that you have 3/4 of the thing you were dividing by left over. The thing you are dividing by is 4 meters long, so you have 3/4 of 4 meters left over. (3/4 of 4 meters is 3 meters, which is exactly how much rope is left over.)

Ask students to draw a picture to interpret the quotient as the number of groups. What does the fraction part of the answer mean?

a) 7 ÷ 3 = 213

b) 13 ÷ 5 = 235

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Review decimal fractions. Remind students that decimals are special fractions: tenths, hundredths, thousandths, and so on. SAY: Decimals are fractions in which the denominator is a power of 10, such as 10, 100, or 1,000. Write some fractions on the board and ask students to signal whether each one is a decimal fraction. (yes, no, yes, no)

310

100345

82100

632 000,

Review writing decimal fractions as decimals. Write on the board:82

100

7100

704100

6 572100,

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 100, so each of the decimals will have two digits after the decimal point. Ask volunteers to write the decimal for each fraction by writing the decimal point in the correct place. SAY: You may need to add a 0 after the decimal point.

8 2 7 7 0 4 6 5 7 2

Answers: 0.82 or .82, 0.07 or .07, 7.04, 65.72

Exercises: Write the fraction as a decimal.

a) 38

100 b)

1510

c) 7

1000, d) 476100

e) 4

10 f )

54810

g) 3

10 h)

911000,

Bonus: 4 503

1000 000,

, ,

Answers: a) 0.38, b) 1.5, c) 0.007, d) 4.76, e) 0.4, f ) 54.8, g) 0.03, h) 0.091, Bonus: 0.004053

NS6-46 Division, Fractions, and Decimals Page 9

STANDARDS 6.NS.B.3

VOcABULARy decimal decimal fraction decimal point denominator equivalent fraction hundredths improper fraction numerator tenths thousandths

GoalsStudents will divide whole numbers and write the answer as a decimal. Students will use equivalent fractions to make division questions with the same answer.


Can write decimal fractions as decimals Can create equivalent fractions with a given denominator Understands how multiplying both terms of a division by the same number does not change the answer

MATERIALS

calculators



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Division with decimal answers. Remind students that when dividing whole numbers, we can write the answer as an improper fraction. Write on the board:

82 ÷ 100 = 82

100SAY: 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 ÷ 100 b) 3 ÷ 1,000 c) 18 ÷ 10 d) 4 ÷ 100 e) 98 ÷ 100 f ) 37 ÷ 1,000

Answers: a) 432/100 = 4.32, b) 3/1,000 = 0.003 or .003, c) 18/10 = 1.8, d) 4/100 = 0.04 or .04, e) 98/100 = 0.98 or .98, f ) 37/1,000 = 0.037 or .037

connecting between two ways of dividing. Tell students that when dividing by 10, 100, or 1,000, they are just moving the decimal point one, two, or three places to the left. Write on the board:

8 2 . 0 So 82 ÷ 100 = 0.82

SAY: There are two 0s in 100, 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 ÷ 100 b) 3 ÷ 1,000 c) 18 ÷ 10 d) 4 ÷ 100 e) 98 ÷ 100 f ) 37 ÷ 1,000

Answers: a) 4.32, b) 0.003 or .003, c) 1.8, d) 0.04 or .04, e) 0.98 or .98, f ) 0.037 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:

45

= 10

Ask a volunteer to fill in the missing numerator. Then show the multiplicative relationship between the numerators and denominators.

45

× 2

× 2 =

810

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

Answers: a) 2/5 = 0.4, b) 3/2 = 1.5, c) 6/5 = 1.2, d) 7/2 = 3.5

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Bonus: Divide.

a) 20 ÷ 5 b) 30 ÷ 2 c) 60 ÷ 5 d) 70 ÷ 2

Then divide your answer by 10. For each question, do you get the same answer as you got in the exercises above?

Answers: a) 4 ÷ 10 = 0.4, yes, b) 15 ÷ 10 = 1.5, yes, c) 12 ÷ 10 = 1.2, yes, d) 35 ÷ 10 = 3.5; yes

Write on the board:

1 ÷ 4 = 14

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 100? (yes) How do you know? (4 divides into 100) Write on the board:

14

= 100

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 100.

a) 3 ÷ 2 b) 2 ÷ 5 c) 9 ÷ 20 d) 31 ÷ 50

e) 3 ÷ 4 f ) 7 ÷ 5 g) 9 ÷ 25 h) 7 ÷ 4

Bonus: Reduce the fraction before making the denominator 10 or 100.

i ) 6 ÷ 8 j ) 9 ÷ 60 k) 6 ÷ 15

Make the denominator 1,000.

l ) 111 ÷ 200 m) 314 ÷ 500

Make the denominator 1,000,000.

n) 21,403 ÷ 500,000

Answers: a) 3/2 = 15/10 = 1.5, b) 4/10 = 0.4, c) 45/100 = 0.45, d) 62/100 = 0.62, e) 0.75, f ) 1.4, g) 0.36, h) 1.75, Bonus: i ) 3/4 = 0.75, j ) 3/20 = 15/100 = 0.15, k) 2/5 = 0.4, l ) 0.555, m) 0.628, n) 0.042806

Exploring how equivalent divisions give equivalent fractional and decimal answers.

Exercises: Write the missing numerators, then change the fractions to decimals.

a) 3 ÷ 2 = 32

= 10

= b) 6 ÷ 4 = 64

= 100

Bonus: 12 ÷ 8 = 128

= 1,000

=

Answers: a) 15, 1.5; b) 150, 1.50; c) 1,500, 1.500



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ASK: Are all the decimals the same? (yes) How could you have predicted that? (because the fractions are equivalent) Tell students that equivalent fractions will give equivalent decimal answers. Write on the board:

3 ÷ 2 = 6 ÷ 4 32

= 64

1.5 = 1.50

SAY: 6 is twice as much as 3 and 4 is twice as much as 2. Remind students that multiplying both numbers in a division by the same number gives the same answer. Multiplying the numerator and denominator of a fraction by the same number gives an equivalent fraction.

Exercises: Predict whether the answers will be the same. Then check your prediction on a calculator.

a) 7 ÷ 2 and 14 ÷ 4 b) 9 ÷ 2 and 27 ÷ 4 c) 8 ÷ 5 and 32 ÷ 20

Bonus: d) 30 ÷ 8 and 90 ÷ 24 e) 5 ÷ 2 and 45 ÷ 6

Answers: a) same, b) different, c) same, d) same, e) different

Extensions1. Use a calculator to check your answers to the questions on

AP Book 6.2 p. 9.

2. Compare the decimal answers to 14

,54

, and 94

. Explain why this makes sense.

Answer: The decimals are 0.25, 1.25, and 2.25; each decimal is 1 greater than the previous decimal, which makes sense because 5/4 = 1 1/4 and 9/4 = 2 1/4.

3. Provide students with BLM Division, Fractions, and Decimals (Advanced) (p. J-89). As students work on the questions, they will make connections between properties of division and properties of fractions.

(MP.1, MP.7)

(MP.1)

(MP.3)

(MP.7)

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Half of a number. Remind students that they can find half of a number by finding half of a set. For example, to find half of 6, make a set with 6 objects and take half of it. ASK: How many are in half the set of 6? To illustrate, draw this on the board:

Ask a volunteer to circle half of the dots. SAY: So half of 6 is 3.

Exercises: Find half of the number.

a) 12

of 4 b) 12

of 8 c) 12

of 10 Bonus: 12

of 200

Answers: a) 2, b) 4, c) 5, Bonus: 100

“Of” can mean multiply. Remind students that “of” can mean multiply. SAY: 3 groups of 4 means 3 × 4. Half of a group of 4 means 1/2 × 4.

Exercises: Multiply.

a) 12

× 4 b) 12

× 8 c) 12

× 10 Bonus: 12

× 200

Answers: a) 2, b) 4, c) 5, Bonus: 100

Half of a fraction. Explain to students that just as we can talk about half of a whole number, we can also talk about half of a fraction. Demonstrate finding half of 3/5 by dividing an area model of the fraction into a top half and a bottom half:

35

12

of 35

= 3

10

Exercises: Find half of the fraction.

a) 29

b) 57

c) 311

d) 25

e) 56

f ) 47

Answers: a) 2/18 = 1/9, b) 5/14, c) 3/22, d) 2/10 = 1/5, e) 5/12, f ) 4/14 = 2/7

NS6-47 Multiplying Fractions Pages 10–12

STANDARDS preparation for 6.NS.A.1, 6.NS.B.3

VOcABULARy denominator improper fraction mixed number numerator unit fraction

GoalsStudents will develop and apply the formula for multiplying all types of fractions by other fractions.


Can multiply fractions by whole numbers Understands that “of” can mean multiply



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Show students another way of dividing the fraction 4/7 in half. Instead of dividing in half from top to bottom, divide in half from left to right (you can do this because 4 is even). Notice that the two methods seem to give different answers:

12

of 47

= 414

12

of 47

= 27

ASK: Are these answers the same? (yes, they are equivalent) Explain that no matter how you take half of 4/7, the answer should always be the same.

A unit fraction of a unit fraction. Tell students you want to find 1/3 of 1/2. Draw on the board:

12

13

of 12

Point to the first picture and SAY: Here is half of a rectangle. Point to the second picture and SAY: Here is one third of half the rectangle. ASK: What fraction of the rectangle is one third of half of it? (one sixth) Write on the board:

13

of 12

= 16

Exercises: Draw pictures to find the fraction of the fraction.

a) 12

of 14

b) 13

of 14

c) 14

of 12

d) 15

of 12

e) 15

of 13

Answers: a) 1/8, b) 1/12, c) 1/8, d) 1/10, e) 1/15

Have volunteers share their pictures, and point out that the total number of pieces in the rectangle is the product of the denominators. That is because one of the denominators tells you the number of rows, and the other denominator tells you the number of columns. So the answer is always the unit fraction with denominator equal to the product of the denominators.

Exercises: Find the fraction of the fraction without using a picture.

a) 13

of 18

b) 12

of 16

c) 14

of 15

Bonus: 12

of 1

2 134,

Answers: a) 1/24, b) 1/12, c) 1/20, Bonus: 1/4,268

Multiplying unit fractions. Remind students that “of” can mean multiply. SAY: To multiply 1/2 times 1/3, you can find 1/2 of 1/3.


a) 12

× 13

b) 13

× 15

c) 18

× 15

Bonus: 1

1000, ×

123

Answers: a) 1/6, b) 1/15, c) 1/40, Bonus: 1/23,000

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Multiplying fractions in general. Tell students you want to find 4/5 of 2/3. Draw on the board:

23

SAY: Here is two thirds of a rectangle. Then complete the picture to show 4/5 of 2/3:

45

of 23

SAY: Here is four fifths of two thirds of the rectangle. ASK: What fraction of the rectangle is four fifths of two thirds? (8/15) PROMPT: How many parts are in the rectangle with the thick outline? (8) How many parts are in the whole rectangle? (15) Write on the board:

45

× 23

= 45

of 23

= 8

15

Exercises: Draw pictures to find the fraction of the fraction.

a) 12

× 23

b) 23

× 14

c) 34

× 12

d) 25

× 34

e) 25

× 23

Answers: a) 2/6 or 1/3, b) 2/12 or 1/6, c) 3/8, d) 6/20 or 3/10, e) 4/15

ASK: How can you get the total number of pieces in the whole rectangle from the two fractions? (multiply the denominators) Pointing to the rectangle with the thick outline, ASK: How can you get the number of pieces in this rectangle from the two fractions? (multiply the numerators) Write on the board:

45

× 23

= 8

15 5 × 34 × 2

SAY: Multiply the numerators to get the numerator and multiply the denominators to get the denominator.

Exercises: Multiply without using a picture.

a) 35

× 34

b) 37

× 45

c) 45

× 23

d) 35

× 67

e) 38

× 78

Answers: a) 9/20, b) 12/35, c) 8/15, d) 18/35, e) 21/64

Multiplying improper fractions. SAY: You can also multiply fractions greater than 1. Draw on the board:

32

=

Tell students that you want to find out 4/5 of 3/2. Draw on the board:

45

of 32



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ASK: How many pieces are in one whole? (10) So then how many pieces are in one half? (5) Now how many pieces are in four fifths of three halves? (12) Write on the board:

45

× 32

= 1210 5 × 2

4 × 3


a) 35

× 72

b) 53

× 6

25 c)

73

× 85

Answers: a) 21/10, b) 30/75 or 2/5, c) 56/15

Multiplying mixed numbers. Write on the board:

123

× 214

= 3

× 4

215

× 234

= 5

× 4

Have volunteers write the missing numerators, then multiply the improper fractions. Tell students that when a question gives the numbers as mixed numbers, the answer is usually a mixed number too. Have volunteers change the answers to mixed numbers.

Exercises: Multiply by changing the numbers to improper fractions. Write your answer as a mixed number.

a) 135

× 213

b) 312

× 215

c) 234

× 112

Bonus: 41325

× 112

Answers: a) 8/5 × 7/3 = 56/15 = 3 11/15, b) 77/10 = 7 7/10, c) 33/8 = 4 1/8, Bonus: 339/50 = 6 39/50

Real-world problems. Tell students that you are making 3/4 of a recipe that calls for 3 1/2 cups of flour. Tell students that you want to calculate how much flour to use. Have a volunteer show what expression you need to evaluate. (3/4 × 3 1/2) Have students evaluate the product, then ask a volunteer to tell you the answer. (21/8 or 2 5/8) Now tell students that you have 2 1/2 cups of flour. ASK: Is that enough? (no) What if I used all my flour? Will the recipe turn out? PROMPT: How close to 2 5/8 is 2 1/2? (it is quite close, so the recipe will likely turn out) Draw on the board:

2 12

cups

2 58

cups3 cups

2 cups

1 cup

SAY: 2 1/2 cups of flour is very nearly as much as 2 5/8 cups of flour.

Exercises: Will the recipe turn out?

a) I’m making 5 1/2 batches of gravy. Each batch needs 3/8 cup of flour. I use 2 cups of flours.

(MP.4)

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b) I’m making 3/8 of a recipe for cupcakes. The recipe calls for 2 1/2 cups of flour. I use 1 cup of flour.

c) I’m making 1 1/2 batches of cookies. Each batch needs 1 1/2 cups of flour. I use 3 cups of flour.

d) I’m making 2 1/2 batches of cookies. Each batch needs 1 2/3 cups of flour. I use 4 cups of flour.

Answers: a) yes, I need 33/16 = 2 1/16 cups; b) yes, I need 15/16 cups; c), no, I need 9/4 = 2 1/4 cups; d) yes, I need 25/6 = 4 1/6 cups

Extensions1. Multiply. Reduce your answers to lowest terms. What do you notice?

Why does this make sense?

a) 25

× 22

b) 25

×33

c) 25

× 44

d) 25

× 55

Answer: The answer is always 2/5, because 2/5 is always being multiplied by 1.

2. John adds 25

+ 53

= 78

. What mistake did he make? How can you tell

by estimating that the answer is incorrect? Hint: Look for any fractions

in the equation that are more than 1.

Answer: He added the numerators and denominators to add the fractions. You can’t do that when adding fractions—you have to change them to the same denominator before you add. He was probably thinking about the method for multiplying fractions. To estimate the answer, note that 5/3 is more than 1. When you add something to a number more than 1, you should get a number more than 1, but 7/8 is less than 1, so the answer is incorrect.

3. Jan bought 5/2 cups of sugar. She used 3/4 of the sugar to bake a cake. Each person eats 1/6 of the cake. How much sugar does each person eat? Is that more or less than 1/3 cup of sugar?

Answer: Each person eats 5/16 cup of sugar, which is less than 1/3 cup.

4. Multiply 34

× 29

and 43

× 29

, then compare your answer to 2/9 to

check your answer. Multiplying 2/9 by a number greater than 1 should

produce an answer greater than 2/9 because you are taking more

than one 2/9. If you are taking less than one 2/9, the answer should

be less than 2/9.

Answer: 6/36 = 1/6 < 2/9 because 3/18 < 4/18, and 8/27 > 6/27 = 2/9

(MP.1)

(MP.3)

(MP.1)


should Answers be flushed to a)


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4. Anna needs these ingredients to make 12 muffins.

134

cups flour 12

cup sugar

14

teaspoon salt 113

teaspoons cinnamon

1 cup milk 7 tablespoons butter

2 teaspoons baking powder 1 egg

2 tablespoons brown sugar

Anna has an 8-muffin pan.

a) What fraction of the ingredients should she use?

b) Anna needs to use a whole egg. Her egg has a volume of about 3/8 cup. How much extra liquid does this create in her muffin mix?

c) Anna needs to reduce the milk by the amount of extra liquid she used for the egg, to keep the total amount of liquid ingredients the same. How much milk should she use?

Answers: a) 2/3; b) 2/3 of an egg would be 2/8 cup, but she used 3/8 cup, so she used 1/8 cup extra liquid; c) 2/3 cup – 1/8 cup is 13/24 cup, or just over 1/2 cup

(MP.4)

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The connection between digits after the decimal point and the denominator of a decimal fraction. Write on the board:

3 tenths 3 hundredths 3 thousandths

Have volunteers write the fractions represented, then write the decimals:

310

3100

31000,

0.3 0.03 0.003

Remind students that the notation for decimals means that, for every 0 in the denominator of a decimal fraction, there is another digit after the decimal point. SAY: The number of thousandths is given by the third place value after the decimal point. So if the denominator of the decimal fraction is 1,000 (which has three 0s), then the decimal has three digits after the decimal point. For example, fifty-two thousandths is 52/1,000 or 0.052.

Exercises: Change the fraction to a decimal.

a) 15

10 000, b)

201000,

c) 3

100 d)

4210

e) 32

100 000,

Answers: a) 0.0015, b) 0.020 = 0.02, c) 0.03, d) 4.2, e) 0.00032

Exercises: Write the decimal as a decimal fraction.

a) 0.3 b) 0.0045 c) 32.54 d) 0.0000234

Answers: a) 3/10, b) 45/10,000, c) 3,254/100, d) 234/10,000,000

Review multiplying by 10, 100, and 1,000.


a) 10 × 10 b) 10 × 100 c) 1,000 × 10 d) 10 × 1,000 e) 100 × 1,000 f ) 1,000 × 1,000 g) 1,000 × 100 h) 100 × 10 i ) 100 × 100

NS6-48 Multiplying Decimals by Decimals Pages 13–14

STANDARDS 6.NS.B.3

VOcABULARy decimal decimal fraction decimal fraction names (tenths, hundredths,

and thousandths)decimal point denominator fraction numerator

GoalsStudents will multiply decimals by decimals.


Can multiply fractions Can translate between decimal fractions and decimals Can multiply and divide by powers of ten (10, 100, 1,000, and so on) Can name fractions from a picture

MATERIALS

calculators



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Answers: a) 100, b) 1,000, c) 10,000, d) 10,000, e) 100,000, f ) 1,000,000, g) 100,000, h) 1,000, i ) 10,000

ASK: What is a short way to find the answer? (count the number of 0s in total and put that many 0s in the answer) To guide students who don’t see the pattern, suggest that they write the number of 0s in each factor and in the answer.

Once all students see the pattern, explain why it makes sense. Remind students that to multiply a whole number by 10, we add a 0 to it; so 10 × 1,000 has four 0s: 10,000. Similarly, to multiply by 100, we add two 0s, so 100 × 1,000 has five 0s: 100,000. SAY: We always add the number of 0s in the first factor to the number of 0s already there in the second factor.

Exercises: Count the 0s to multiply.

a) 100 × 10 × 1,000 b) 100 × 100 × 100 × 100 c) 1,000 × 100 × 100 × 10 × 10 d) 10 × 1,000 × 100 × 10,000

Answers: a) 1,000,000, b) 100,000,000, c) 1,000,000,000, d) 10,000,000,000

Review multiplying fractions in the special case of decimal fractions. Write on the board:

2100

× 3

1000, =

100 × 1,000 =

6100 000,

Go through the multiplication with the students.

Exercises: Multiply the decimal fractions.

a) 3

10 ×

21000,

b) 2

100 ×

41000,

c) 3

1000, ×

5100

d) 3

100 ×

21000,

× 7

100 e)

210

× 3

100 ×

3100

× 7

1000,

Answers: a) 6/10,000, b) 8/100,000, c) 15/100,000, d) 42/10,000,000, e) 126/100,000,000

Multiplying decimals. Write on the board:

0.03 × 0.004

SAY: I don’t know how to multiply decimals, but I do know how to multiply fractions. ASK: How can I change this problem into one I already know how to do? (change the decimals to fractions) Have a volunteer change the decimals to fractions, without writing the answer:

3100

× 4

1000,

Have another volunteer write the answer. (12/100,000) Then remind students that we’re not done yet. SAY: We now have an answer, but the question was given in terms of decimals, so the answer needs to be given using decimals. Ask a volunteer to write the decimal answer. (0.00012) Repeat with the product 0.004 × 0.5. This time, when students write the fraction answer (20/10,000), ASK: Are we done? (no) What do we have left to do? (change the fraction to a decimal) Ask a volunteer to write the decimal. (0.0020 or 0.002)

(MP.8)

2 × 3

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Exercises: Change the decimals to fractions, multiply the fractions, then change the answer back to a decimal. When a question is given in terms of decimals, you need to give the answer as a decimal.

a) 0.2 × 0.3 b) 0.02 × 0.3 c) 0.2 × 0.003 d) 0.02 × 0.03 Bonus: 0.0002 × 0.00003 e) 0.3 × 0.004 f) 0.006 × 0.0002 g) 0.5 × 0.4 h) 0.05 × 0.0004 i ) 3 × 0.002 j ) 4 × 0.006 k) 1.1 × 0.5 l ) 1.2 × 0.003 m) 2.1 × 0.004

Selected solution: a) 2/10 × 3/10 = 6/100 = 0.06

Answers: b) 0.006, c) 0.0006, d) 0.0006, Bonus: 0.000000006, e) 0.0012, f ) 0.0000012, g) 0.20 = 0.2, h) 0.000020 = 0.00002, i ) 0.006, j ) 0.024, k) 0.55, l ) 0.0036, m) 0.0084

connect the procedures for multiplying decimals and for multiplying decimal fractions. Write the fraction multiplication on the board, and have a volunteer write the equivalent decimals:

310

× 5

1000, =

1510 000,

0.3 × 0.005 = 0.0015

Summarize the steps that students follow to multiply the fractions, as follows:

Step 1: Multiply the fractions as though they are whole numbers by pretending the denominator doesn’t exist: 3 × 5 = 15.

Step 2: Add the 0s in the denominators to find the number of 0s in the denominator of the product: 1 + 3 = 4.

Step 3: Write the fraction with the numerator from Step 1 and the denominator from Step 2: 15/10,000.

Now apply each step to multiplying the decimals: 0.3 × 0.005.

Step 1: Multiply the decimals as though they are whole numbers: 3 × 5 = 15.

Step 2: Add the numbers of digits after the decimal points: 1 + 3 = 4.

Step 3: Starting with your answer from Step 1, shift the decimal point left the number of places given by the answer in Step 2.

Point out that these steps are just a shortcut for converting between fraction and decimal notation in which you write the numerator of the fraction and shift the decimal point the correct number of places. Write on the board:

310

×5

1000,=

1510 000,

1 place for tenths

3 places for thousandths

4 places for ten thousandths

3 . 5 . 1 5 .

So 0.3 × 0.005 = 0.0015

(MP.7)



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Exercises: Multiply the decimals using the shortcut: first multiply as though they are whole numbers, then shift the decimal point the correct number of places.

a) 2.1 × 0.04 b) 1.3 × 0.003 c) 1.02 × 0.03 d) 3.2 × 2.4 e) 0.4 × 0.75 f ) 7.6 × 0.0035

Answers: a) 0.084, b) 0.0039, c) 0.0306, d) 7.68, e) 0.300 = 0.3, f ) 0.02660 = 0.0266.

NOTE: Some students might count digits after the decimal point in the product instead of how many places the decimal point needs to be shifted in the answer. This can incorrectly lead some students to answer e) as 0.003 and f) as 0.00266.

Word problems practice. Use a calculator for these problems.

a) A garden measures 4.7 m × 3.2 m. What is the area of the garden? b) A rectangle has length 7.5 cm and width 1.2 cm. What is its area? c) A square has side length 3.4 cm. What is its area?

Answers: a) 15.04 m2, b) 9 cm2, c) 11.56 cm2

Extensions1. Since 0.5 =

12

, to multiply the number by 0.5, find 12

of the number.

a) 0.5 × 0.846 b) 0.5 × 14.008 c) 0.5 × 90.604

Answers: a) 0.423, b) 7.004, c) 45.302

2. A triangle has a base of 1.6 cm. Its height is 4 times as long as its base.

a) How tall is the triangle? b) What is its area?

Answers: a) 6.4 cm, b) 1/2 × 1.6 cm × 6.4 cm = 1/2 × 10.24 cm2 = 5.12 cm2

3. Marnie thinks that 0.4 × 0.5 = 0.02 because 4 × 5 = 20, and the number of digits after the decimal point in the product is the sum of the number of digits after the decimal points in each product. Explain her mistake.

Answer: It is true that you have to add the number of digits after the decimal points, but the answer doesn’t tell you the number of digits after the decimal point in the product. The answer tells you how many places to shift the decimal point from the whole-number product.

4. Multiply the fractions and the equivalent decimals. Do you get the same decimal answer? If not, find your mistake.

a) 52

× 25

and × b) 34

× 45

and ×

Answers: a) 10/10 = 1 and 2.5 × 0.4 = 1.00 = 1, b) 12/20 = 60/100 = 0.60 = 0.6 and 0.75 × 0.8 = 0.600 = 0.6

(MP.8)

(MP.4)

(MP.7)

(MP.3)

(MP.1)

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Review dividing whole numbers by whole numbers. Remind students that division can be used for sharing equally. Tell students that two people are sharing pizzas. Write on the board:

6 ÷ 2 = 3 ÷ 2 = 1 ÷ 2 =

Have volunteers use shading to show how much one person gets, then answer the division questions. (3, 1 1/2 = 1.5, 1/2 = 0.5)

Dividing unit fractions by whole numbers. Write on the board:

13

÷ 2 =

Have a volunteer show how much each person gets by dividing the shaded part in two, then ASK: Does this picture show a fraction? (no) PROMPT: Are the parts equal? How do we have to change the picture to show a fraction? (divide the unshaded parts into equal parts too)

ASK: What fraction of the pizza is each piece? (one sixth) Finish the equation, 1/3 ÷ 2 = 1/6, and SAY: When two people share one third of a pizza, each person gets one sixth of the pizza.

Ask a volunteer to divide one half into four equal parts, as in the picture below. Outline one piece and SAY: This is how much each person would get if four people were sharing half a square piece of cake. ASK: What fraction of the whole cake is that? (1/8) Have a volunteer draw lines to show the fraction. Have a volunteer write the division equation. (1/2 ÷ 4 = 1/8) PROMPTS: What fraction was being divided? How many equal parts was it divided into? What fraction of the whole is each equal part?

NS6-49 Dividing Fractions by Whole Numbers Page 15

STANDARDS 6.NS.A.1

VOcABULARy fraction whole number

GoalsStudents will use models to divide fractions by whole numbers and will develop the formula for this type of division through examples.


Can divide whole numbers by whole numbers Can name fractions of models when parts shown are unequal Understands division as equal sharing Understands fractions of areas



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Exercises: Use the picture to divide.

a)

12

÷ 3

b)

13

÷ 2

SAY: Now imagine the lines extended. What fraction is each part?

c)

14

÷ 2

d)

13

÷ 3

e)

12

÷ 2

f)

14

÷ 3

Answers: a) 1/6, b) 1/6, c) 1/8, d) 1/9, e) 1/4, f ) 1/12

Bonus: Divide 14

÷ 2 in three ways.

Answer: 1/8

Exercises: Write the division equation shown by the picture.

a) b)

Answers: a) 1/3 ÷ 4 = 1/12, b) 1/2 ÷ 5 = 1/10

Using a rule to divide unit fractions by whole numbers. Write on the board:

15

÷ 3

SAY: This picture shows 1/5 of the rectangle divided into three equal parts. ASK: How can we find out what fraction one of the three parts is of the whole rectangle? (extend the lines) Have a volunteer extend the lines. Then ASK: How many equal parts are there? (15) Write on the board:

15

÷ 3 = 1

15

ASK: Without using the picture, how can you get 15 from 5 and 3? (multiply) Point out that you drew 5 columns to show 1/5 and 3 rows to find 1/5 divided by 3, so there are 5 × 3 = 15 parts altogether. That means each part is one fifteenth of the rectangle.

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Exercises: Divide without drawing a picture.

a) 14

÷ 5 b) 12

÷ 8 c) 12

÷ 11 d) 1

11 ÷ 5 e)

15

÷ 7

Bonus

f ) 17

÷ 8 g) 19

÷ 6 h) 18

÷ 8 i ) 1

23 ÷ 3 j )

1200

÷ 30

Answers: a) 1/20, b) 1/16, c) 1/22, d) 1/55, e) 1/35, Bonus: f ) 1/56, g) 1/54, h) 1/64, i ) 1/69, j ) 1/6,000

Dividing any fraction by a whole number. Draw on the board:

35

÷ 4

Ask a volunteer to divide the shaded part into four equal rows. Outline one of the shaded rows and SAY: The amount in one group shows us what 3/5 divided by 4 is. ASK: How can we change the picture so that it is easy to see what fraction of the whole this is? (extend the lines) Have a volunteer extend the lines.

SAY: Now all the parts are equal. ASK: How many parts did I outline? (3) How many parts are there altogether? (20) So three twentieths are outlined. Write on the board:

35

÷ 4 = 3

20 5 × 4

Exercises: Divide without using a picture.

a) 38

÷ 2 b) 25

÷ 3 c) 34

÷ 2 d) 45

÷ 5 e) 57

÷ 2

Bonus

f ) 58

÷ 7 g) 58

÷ 9 h) 58

÷ 300 i ) 543800

÷ 2 j ) 1740

÷ 50

Answers: a) 3/16, b) 2/15, c) 3/8, d) 4/25, e) 5/14, Bonus: f ) 5/56, g) 5/72, h) 5/2,400 i ) 543/1,600, j ) 17/2,000

Word problems practice.

a) Three people share 5/8 of a cake equally. What fraction of the cake does each person get?

b) Eight people share 1/2 lb of chocolate equally. How much chocolate does each person get?

c) Four people share 3/4 of a meat pie. What fraction of the pie does each person get?

Bonus: Five people share 3/5 of a pie. Did they each have more or less than 1/8 of the pie?

Answers: a) 5/24, b) 1/16, c) 3/16, Bonus: 3/25 < 3/24 = 1/8, or 3/25 = 24/200 < 25/200 = 1/8

(MP.4)



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Extensions1. Write the fact family for

25

÷ 3 = 2

15.

Answers: 2/5 ÷ 3 = 2/15, 2/5 ÷ 2/15 = 3, 3 × 2/15 = 2/5, 2/15 × 3 = 2/5

2. Teach students a shortcut way to divide when the numerator is a multiple of the whole number being divided by:

6 sevenths ÷ 2 = 3 sevenths

the same way that:

6 apples ÷ 2 = 3 apples

Divide and write the fraction notation for the division equation.

a) 4 fifths ÷ 2 = fifths b) 8 ninths ÷ 2 = ninths

c) 8 thirds ÷ 4 = thirds d) 12 fifths ÷ 4 = fifths

Answers: a) 4/5 ÷ 2 = 2/5, b) 8/9 ÷ 2 = 4/9, c) 8/3 ÷ 4 = 2/3, d) 12/5 ÷ 4 = 3/5

3. a) You can divide improper fractions by whole numbers using the same rule that you use to divide proper fractions by whole numbers. Draw a picture to show why this works to divide 5/3 ÷ 2.

b) Divide mixed numbers by whole numbers by first changing them to improper fractions, such as 5 1/2 ÷ 3.

c) Use the distributive property to divide mixed numbers by whole numbers. For example:

512

÷ 3 = 512

+

÷ 3

= (5 ÷ 3) + 12

÷

3

= 53

+ 16

= 106

+ 16

= 116

d) Investigate with several examples to check whether you get the same answers both ways.

(MP.8)

(MP.7)

(MP.2)

(MP.1)

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Review division as fitting into. Remind students that division can be looked at as fitting into. For example, to divide 6 ÷ 2, you can ask how many objects of length 2 fit across an object of length 6:

2 2 2

SAY: Three 2s fit into 6, so 6 ÷ 2 = 3.

Dividing 1 by a unit fraction. Give students the prepared cutouts from BLM Fraction Parts and Wholes (1 whole, 2 halves, 3 thirds, 4 fourths, and 5 fifths for each student). ASK: How many 1/2s fit into 1? (2) Students should show their answer by lining up pieces. Write on the board:

1 ÷ 12

= 2

ASK: How many 1/3s should fit into 1? (3) Students should check this with their cutouts. Have a volunteer write the division equation. (1 ÷ 1/3 = 3) Repeat for how many fourths fit into 1 (1 ÷ 1/4 = 4) and how many fifths fit into 1. (1 ÷ 1/5 = 5)

Exercises: Divide.

a) 1 ÷ 16

b) 1 ÷ 17

c) 1 ÷ 1

10 d) 1 ÷

19

Bonus: 1 ÷ 1

372

Answers: a) 6, b) 7, c) 10, d) 9, Bonus: 372

Dividing a whole number by a unit fraction. Have students work in groups of four. Ask students to use their fraction pieces from BLM Fraction Parts and Wholes to determine how many 1/2s fit into a) 1, b) 2, c) 3, d) 4.

NS6-50 Dividing Whole Numbers by Unit Fractions Pages 16–17

STANDARDS 6.NS.A.1

VOcABULARy division fraction unit fraction whole number

GoalsStudents will divide whole numbers by unit fractions.


Understands division as fitting into Understands 1/n as one of n equal parts of a whole Can use number lines to represent whole numbers Understands fractions of lengths, areas, capacities, and number lines

MATERIALS

pre-cut pieces from BLM Fraction Parts and Wholes (pp. J-84–J-88) 1 die per pair of students 1/3 cup measure 1 cup measure enough counters to fill a cup



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Then show students how to write the division equations:

1 ÷ 12

= 2 2 ÷ 12

= 4 3 ÷ 12

= 6 4 ÷ 12

= 8

Repeat for thirds and fourths, but this time have students write the division equations themselves. Take up the answers on the board. Point out that no matter how many fit into 1, twice as many will fit into 2 as fit into 1, three times as many will fit into 3, and four times as many will fit into 4. ASK: How many sixths fit into 1? (6) How many sixths fit into 3? (3 × 6 = 18) Write on the board:

1 ÷ 16

= 6 so 3 ÷ 16

= 3 × 6 = 18

Exercises

a) 5 ÷ 14

b) 2 ÷ 15

c) 3 ÷ 17

d) 5 ÷ 16

e) 9 ÷ 12

f ) 10 ÷ 17

g) 8 ÷ 17

h) 9 ÷ 18

Bonus

i ) 100 ÷ 13

j ) 5 ÷ 1

1000, k) 13 ÷ 1

100 l ) 400 ÷

17 000,

Answers: a) 20, b) 10, c) 21, d) 30, e) 18, f ) 70, g) 56, h) 72, Bonus: i ) 300, j ) 5,000, k) 1,300, l ) 2,800,000

Showing division on a number line. Draw on the board:

0 1 2 3

ASK: How many steps of size 1/2 fit into 3? (6) Write on the board:

3 ÷ 12

= 6

Tell students that drawing number lines is another way to show how many halves fit into three. Ask a volunteer to extend the number line to find how many halves fit into four. (8) Then draw a number line from 0 to 2, divided into fourths. Write on the board:

2 ÷ =

ASK: How big is each step? (1/4) Fill in the first blank. How many of them fit into two? (8) Fill in the second blank.

Exercises: Write the division statement to show how many steps fit into the number line.

a) 0 1 2 3

b) 0 1 2 3 4 5

Bonus

0 1 2 3 4 5

Answers: a) 3 ÷ 1/4 = 12, b) 5 ÷ 1/2 = 10, Bonus: 5 ÷ 1/3 = 15

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Exercises: Draw a number line to determine 2 ÷ 13

.

Bonus: Draw pizzas divided into fourths to determine 3 ÷ 14

.

Answers: 0 1 2

Bonus:

2 ÷ 13

= 6 3 ÷ 14

= 12

AcTIVITy

Students play in pairs. Player 1 rolls a die and takes that many steps. Player 2 tries to take steps that are only 1/3 or 1/4 as long, and cover the same distance. Player 1 decides how many steps Player 2 should take to succeed before Player 2 tries, and records the division equation. Then players switch roles.

Different contexts for dividing fractions. Show students a 1/3 cup measure, a 1 cup measure, and enough counters to fill up the cup. Tell students that the small measure is labeled as 1/3 cup and the big measure as 1 cup. ASK: How many small cupfuls should fill up the big cup? (3) Ask a volunteer to check that this is the case. Tell students that a recipe calls for 2 cups of flour, but you only have a 1/3 cup measure. ASK: How many cupfuls do you need? (6) Have a volunteer write the division equation. (2 ÷ 1/3 = 6)

Exercises: Solve the problems.

a) Tegan needs 5 cups of sugar. She only has a 1/2 cup measure. How many cupfuls does she need?

b) Alex needs 3 cups of water for a recipe. He only has a 1/4 cup measure. How many cupfuls does he need?

c) Mary has 5 feet of ribbon. She uses 1/3 of a foot for each gift. How many gifts can she put ribbon on?

d) Rosa has 2 apples. She cuts them each into fourths. How many pieces does she have?

e) Miki has 6 muffins. He cuts them into halves.

i ) How many pieces does he have?

ii ) Four people share the muffins. How many pieces does each person get?

Answers: a) 10, b) 12, c) 15, d) 8, e) i ) 12, ii ) 3

(MP.4)



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Extensions1. How can you use:

a) a yard stick to show that 3 ÷ 1

12 = 36?

b) two hundreds blocks to show that 2 ÷ 1

100 = 200?

c) your hands and fingers to show that 2 ÷ 15

= 10?

2. Six people are sharing three oranges. Each orange is cut into eighths. How many pieces does each person get?

Answer: 4

3. Discuss why it is easier to look at division as sharing equally when dividing by whole numbers, but as fitting into when dividing by fractions.

Answer: It is hard to see how many pieces of size 3 fit into 1/2 so 1/2 ÷ 3 would be hard to find by thinking of division as fitting into. Also, 3 ÷ 1/2 would be hard to think of as sharing equally between 1/2 of a person.

(MP.2)

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Dividing unit fractions by unit fractions when the answer is a whole number. Draw on the board:

Ask a volunteer to shade half of each circle. ASK: How many fourths fit into half the circle? (2) What division equation can you write from that? (1/2 ÷ 1/4 = 2) Repeat for how many sixths fit into half (1/2 ÷ 1/6 = 3) and how many eighths fit into half (1/2 ÷ 1/8 = 4).

Exercises: Divide using the picture.

a) b) c)

13

÷ 16

14

÷ 18

13

÷ 1

12

Bonus: Write another division equation for each picture by dividing the shaded fraction by a whole number.

Answers: a) 2, b) 2, c) 4; Bonus: a) 1/3 ÷ 2 = 1/6, b) 1/4 ÷ 2 = 1/8, c) 1/3 ÷ 4 = 1/12

Exercises: Check your answers using multiplication.

Answers: a) 1/6 × 2 = 2/6 = 1/3, b) 1/8 × 2 = 2/8 = 1/4, c) 1/12 × 4 = 4/12 = 1/3, Bonus: a ) 2 × 1/6 = 2/6 = 1/3, b) 2 × 1/8 = 2/8 = 1/4, c) 4 × 1/12 = 4/12 = 1/3

Dividing fractions by unit fractions with the same denominator. ASK: How many 1/8s are in 3/8? (3) Write on the board:

38

÷ 18

= 3

ASK: How many 1/5s are in 2/5? (2) Have a volunteer write the division statement. (2/5 ÷ 1/5 = 2)

NS6-51 Dividing Fractions by Unit Fractions Pages 18–19

STANDARDS 6.NS.A.1

VOcABULARy denominator fraction fraction names (halves, thirds, fourths,...)whole number

GoalsStudents will divide fractions by unit fractions in cases where the answer is a whole number.


Can locate fractions on number lines Can name fractions from pictures Understands division as fitting into



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Exercises: Divide.

a) 47

÷ 17

b) 56

÷ 16

c) 58

÷ 18

d) 7

10 ÷

110

Bonus183245

÷ 1

245

Answers: a) 4, b) 5, c) 5, d) 7, Bonus: 183

Dividing fractions by unit fractions when the answer is a whole number. Draw on the board a circle divided into sixths as shown in the margin and have a volunteer shade 2/3 of it. ASK: How many 1/6s are in 2/3? (4) Write on the board:

23

÷ 16

= 4

Then show students how they can use a double number line to show the same equation:

Step 1: Draw and label a number line representing the fraction that is being divided.

0 1

23

Step 2: Draw a second number line below the first to represent the number you are dividing by.

0 1

23

16

SAY: It takes four 1/6s to equal 2/3, so 2/3 ÷ 1/6 = 4.

Exercises: Use the picture to divide.

0 1tenths

fifths

a) 15

÷ 1

10 b)

25

÷ 1

10 c)

35

÷ 1

10 d)

45

÷ 1

10 e)

55

÷ 1

10

Answers: a) 2, b) 4, c) 6, d) 8, e) 10

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Using multiplication to divide fractions by unit fractions. Tell students that 8 stamps make a strip that is 1 foot long. Draw on the board:

1 foot

1 ÷ 18

= 8

SAY: Each stamp is 1/8 of the strip. The picture shows that 8 eighths fit into 1, or that 1 divided by 1/8 is 8. ASK: If 8 stamps fit into 1 strip, how can we know how many stamps fit into 3/4 of a strip? Allow volunteers to articulate an answer, then SAY: A strip that is 3/4 of a foot long can only contain 3/4 as many stamps as one that is 1 foot long. So if 8 stamps fit into 1, then 3/4 of 8 stamps will fit into 3/4 of 1 strip. Write on the board:

1 ÷ 18

= 8 so 34

÷ 18

= 34

of 8 = 34

× 8 = 6

Exercises: Redo the exercises above using this method. Make sure you get the same answer.

Answers: a) 1/5 × 10 = 10/5 = 2, b) 2/5 × 10 = 20/5 = 4, c) 3/5 × 10 = 30/5 = 6, d) 4/5 × 10 = 40/5 = 8, e) 5/5 × 10 = 50/5 = 10 or 1 × 10 = 10

Remind students that this is similar to dividing whole numbers by fractions. SAY: We know that 3 times as many objects will fit into 3 as fit into 1, so 3 ÷ 1/10 is 3 × 10. In the same way, 2/5 as many objects will fit into 2/5 as fit into 1, so 2/5 ÷ 1/10 = 2/5 × 10.

Exercises: Use 1 ÷ 1

12 = 12 to divide.

a) 23

÷ 1

12 b)

34

÷ 1

12 c)

56

÷ 1

12

Bonus: Use 1 ÷ 1

100 = 100 to divide.

d) 34

÷ 1

100 e)

35

÷ 1

100 f )

710

÷ 1

100

Answers: a) 2/3 × 12 = 24/3 = 8, b) 3/4 × 12 = 36/4 = 9, c) 5/6 × 12 = 60/6 = 10, Bonus: d) 3/4 × 100 = 300/4 = 75, e) 3/5 × 100 = 300/5 = 60, f ) 7/10 × 100 = 700/10 = 70

Exercises: Divide.

a) 56

÷ 1

30 b)

53

÷ 19

c) 32

÷ 1

10 d)

73

÷ 1

15

Bonus423

1000, ÷

11000 000, ,

Answers: a) 25, b) 15, c) 15, d) 35, Bonus: 423,000



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Extensions1. Divide by finding the missing factor:

a) 1

15 × =

13

so 13

÷ 1

15 =

b) 1

628 × =

12

so 12

÷ 1

628 =

Answers: a) 5, b) 314

2. Use pattern blocks to determine 12

÷ 16

.

3. Divide by changing the first fraction to have the same denominator as the second fraction.

a) 23

÷ 1

12 b)

23

÷ 1

15 c)

34

÷ 1

20 d)

35

÷ 1

20

Answers: a) 8/12 ÷ 1/12 = 8, b) 10/15 ÷ 1/15 = 10, c) 15/20 ÷ 1/20 = 15, d) 12/20 ÷ 1/20 = 12

(MP.7)

(MP.4)

(MP.8)

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Using pictures and concrete materials to divide whole numbers by fractions. ASK: How many 1/5s fit into 1? (5) Draw on the board:

1

15

15

15

15

15

ASK: How many fit into 4? (20) SAY: Four times as many fit into 4 as fit into 1. Extend the picture to show this:

1 1 1 1

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

Tell students you want to know how many 2/5s fit into 4. Write on the board:

4 ÷ 15

= 20 4 ÷ 25

= ?

Tell students that instead of just counting 1/5s, they need to count blocks of size 2/5. Demonstrate by drawing the first block and have a volunteer draw the rest to see how many fit into 4:

1 1 1 1

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

15

SAY: 10 blocks of size 2/5 fit into 4, so 4 ÷ 2/5 = 10.

STANDARDS 6.NS.A.1

VOcABULARy phrases such as “three times as many”

or “one third as many”

GoalsStudents will divide whole numbers by fractions in cases in which the answer is a whole number.


Can divide a whole number by a unit fraction Understands division as fitting into Understands the relationship between multiplication and division

MATERIALS

BLM Fraction Parts and Wholes (pp.J-84–J-88) toothpicks

NS6-52 Dividing Whole Numbers by Fractions (Introduction)Page 20



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AcTIVITy

Give students (or groups of students) toothpicks and fraction parts and wholes from BLM Fraction Parts and Wholes.

To show 4 ÷ 2/3:

1) Place 4 wholes in a row.

2) Line up 1/3 size pieces underneath.

3) Use toothpicks to mark where the groups of size 2/3 end:

Six groups of size 2/3 fit into 4, so 4 ÷ 2/3 = 6.

Students make pictures like the one above, using toothpicks to mark where a group ends. Students then divide by counting the groups they make. Students draw pictures to divide:

a) 2 ÷ 25

b) 3 ÷ 34

c) 3 ÷ 35

d) 4 ÷ 45

Answers: a) 5, b) 4, c) 5, d) 5

Using division by a unit fraction to divide by a fraction. Refer to the pictures above that show 4 ÷ 1/5 = 20 and 4 ÷ 2/5 = 10. Point out that 2/5 is twice as long as 1/5, so half as many longer bars fit than shorter bars. Write on the board:

6 ÷ 15

= 6 ÷ 25

= 6 ÷ 35

=

ASK: How many 1/5s fit into 6? (30) Write in the answer. How many times longer is 2/5 than 1/5? (twice as long) SAY: So only half as many 2/5 will fit. ASK: What is half of 30? (15) Write in the answer again. ASK: How many 3/5s fit into 6? (10) How do you know? (3/5 is three times as long as 1/5, so only one third as many will fit)

Exercises: Use 8 ÷ 16

= 48 to divide.

a) 8 ÷ 26

b) 8 ÷ 36

c) 8 ÷ 46

Bonus: Divide and match your answers to above.

d) 8 ÷ 12

e) 8 ÷ 13

Answers: a) 24, b) 16, c) 12, Bonus: d) 16, same as 8 ÷ 3/6, e) 24, same as 8 ÷ 2/6

Write on the board:

10 ÷ 23

= (10 × 3) ÷ 2

(MP.5)

(MP.3)

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SAY: 10 × 3 = 30 one thirds fit into 10, and two thirds is twice as big as one third, so only half as many will fit. That’s why you divide by 2.

Exercises: Divide.

a) 10 ÷ 23

b) 8 ÷ 45

c) 6 ÷ 27

d) 15 ÷ 34

Answers: a) (10 × 3) ÷ 2 = 30 ÷ 2 = 15, b) (8 × 5) ÷ 4 = 40 ÷ 4 = 10, c) (6 × 7) ÷ 2 = 42 ÷ 2 = 21, d) (15 × 4) ÷ 3 = 60 ÷ 3 = 20

checking answers through multiplication. Remind students that they can check their answers using multiplication. Write on the board:

6 ÷ 2 = 3 10 ÷ 23

= 15

Have volunteers circle the two numbers you would multiply in each equation to make sure the answer is the other number. Have another volunteer do the multiplication 2/3 × 15 to check the second division.

Exercises: Check your answers to each question above, for b) to d).

Answers: b) 4/5 × 10 = 40/5 = 8, c) 2/7 × 21 = 42/7 = 6, d) 3/4 × 20 = 60/4 = 15


a) A ribbon is 4 m long. Yu needs a piece 2/3 m long for each gift. How many gifts can she wrap?

b) Sam lives 10 miles from school. Nina lives 2/5 miles from school. How many times farther from school does Sam live than Nina?

c) Ravi lives 12 miles from school and 4/3 miles from the library. How many times closer is he to the library than to school?

Answers: a) 6, b) 25, c) 9

Extensions1. To divide by unit fractions, use the division property that multiplying both

terms by the same number doesn’t change the answer. For example:

32

÷15

=32

×

5 ÷

15

×

5

= 32

×

5 ÷1=

32

×5

2. Which do you expect to be greater?

21,417,613 ÷ 12

or 21,417,613 ÷ 35

Explain.

Answer: 21,417,613 ÷ 1/2 because 1/2 = 5/10 < 3/5 = 6/10 and dividing by a smaller number gets a larger answer.

(MP.7)

(MP.1, MP.3)



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3. a) 60 pieces of size 1

10 fit into 6. How many pieces of size …

i) 2

10 will fit into 6? ii )

310

will fit into 6? iii ) 5

10 will fit into 6?

b) Find 6 ÷ 12

. Which answer from part a) is the same?

Why is this so?

Answers: a) i) 30, ii) 20, iii) 12; b) 12, iii) is the same because 5/10 = 1/2

4. Anika lives 14 miles from school and 213

miles from the library.

How many times closer to the library is she than to school?

Answer: 14 ÷ 7/3 = 6

5. To divide 3 ÷ 23

, draw a picture with thirds:

1 1 1

13

13

13

13

13

13

13

13

13

Group two thirds at a time:

1 1 1

13

13

13

13

13

13

13

13

13

4 pieces remainder

Now do the algorithm: 3 ÷ 23

= (3 × 3) ÷ 2 = 9 ÷ 2 = 92

= 412

Now look at the picture. The “4” in “412

” is the number of pieces of

size 23

, but what does the “12

” mean? (It means 1/2 of the thing you

are dividing by or 1/2 of 2/3.)

12

of 23

= 13

The remaining piece is one-half of what you are dividing by. There are four and one-half groups of size 2/3 that fit into 3, so 3 ÷ 2/3 = 4 1/2.

(MP.3)

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Finding the number of whole parts that fit into 1. Draw on the board:

1 piece of chocolate

17

17

17

17

17

17

17

This piece of chocolate has 7 pieces, so each piece is 1/7. Write on the board:

1 ÷ 27

= ?

Tell students that you want to know how many pieces of size 2/7 fit into 1. Then outline pieces of size 2/7:

17

17

17

17

17

17

17

ASK: How many whole blocks of size 2/7 fit into the whole piece? (3) SAY: There’s some leftover, but we’ll think about that part later.

Exercises: How many whole blocks fit?

a) 18

18

18

18

18

18

18

18

There are whole pieces of size 38

in 1 whole.

b) 19

19

19

19

19

19

19

19

19

There are whole pieces of size 29

in 1 whole.

Answers: a) 2, b) 4

NS6-53 Dividing 1 by a Fraction Pages 21–22

STANDARDS 6.NS.A.1

VOcABULARy denominator fraction improper fraction mixed number numerator

GoalsStudents will divide 1 by a fraction, writing the answer as a mixed number and an improper fraction.


Can convert a mixed number to an improper fraction Knows how many times a given unit fraction fits into 1 Understands division as fitting into

MATERIALS

several connecting cubes of one color for each student (at least two different colors for the class) grid paper



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Have students shade on grid paper a rectangle consisting of 1 row 9 squares across:

ASK: What fraction of the rectangle is each square? (one ninth) Have students draw blocks of size 4/9 to find out how many whole blocks of size 4/9 fit into one whole. (2) Now write on the board:

1 ÷ 25

Have students draw a rectangle on grid paper so that each part is one fifth. ASK: How many squares long is your rectangle? (5) Be sure everyone drew the correct rectangle before continuing. Then ask them to divide the rectangle into pieces of size two fifths. ASK: How many whole pieces fit into one whole? (2)

Exercises: Draw a picture to decide how many whole pieces fit into 1.

a) 1 ÷ 38

b) 1 ÷ 211

c) 1 ÷ 311

d) 1 ÷ 411

Answers: a) 2, b) 5, c) 3, d) 2

Writing the remainder as a fraction of the number you are dividing by. SAY: To divide 1 by 2/7, you need to figure out how many blocks of size 2/7 fit into 1. The answer, in this case, is not a whole number of blocks because there is a leftover part.

1

17

17

17

17

17

17

17

leftover

Point to the leftover piece, and SAY: I want to know what fraction of 2/7 the leftover piece is. SAY: Just like a whole number can be a fraction of another whole number, a fraction can be a fraction of another fraction. Draw on the board:

5

2

ASK: 2 is what fraction of 5? (2/5)

Exercises

a) 3 is what fraction of 5? b) 2 is what fraction of 9?

Bonus: 9 is what fraction of 1,000?

Answers: a) 3/5, b) 2/9, Bonus: 9/1,000

15

15

15

15

15

2 whole pieces of size 2/5 fit into 1

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Draw on the board:

17

17

17

17

17

Tell students that this is 5/7 and you want to know what fraction 2/7 is of 5/7. Ask a volunteer to shade 2/7.

17

17

17

17

17

ASK: What fraction of the 5/7 is shaded? (2/5) SAY: 2 of the 5 parts are shaded, so 2/5 is shaded. Write on the board:

27

is 25

of 57

2 sevenths is 25

of 5 sevenths

SAY: Just like 2 is 2/5 of 5, 2/7 is 2/5 of 5/7. Have students draw on grid paper two rectangles each consisting of a row 8 squares long. ASK: What fraction of the rectangle is each square? (1/8) SAY: You can use your rectangles to answer these questions.

Exercises: Shade the first fraction and draw a group showing the second fraction. Then fill in the blank.

a) 38

is of 48

? b) 38

is of 58

?

Answers: a) 3/4, b) 3/5

Exercises: Draw a picture to find the answer.

a) 25

is of 35

b) 26

is of 36

c) 27

is of 37

Answers: a) 2/3, b) 2/3, c) 2/3

SAY: You don’t even need to draw a picture. Write on the board:

2100

is of 3

100

Have a volunteer fill in the blank. (2/3) Tell students that 2 of anything is 2/3 of 3 of anything, and that’s true for thirds, fourths, fifths, hundredths, or baseballs.

Exercises: Fill in the missing numbers.

a) 59

is of 79

b) 411

is of 511

c) 38

is of 78

Bonus: 4

500 000, is of 9

500 000,

Answers: a) 5/7, b) 4/5, c) 3/7, Bonus: 4/9

Dividing 1 by a fraction. Write on the board:

1 ÷ 27

3/8 is 3/4 of 4/8

18

18

18

18

(MP.7)



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From the previous exercises, students should know that 1/7 is 1/2 of 2/7. Or students could extend the leftover piece to make a whole block of size 2/7:

1

17

17

17

17

17

17

17

Exercises: An extra block was added. What fraction of the last block is the remainder?

a) 1

18

18

18

18

18

18

18

18

remainder

b) 1

15

15

15

15

15

remainder

c) 1

19

19

19

19

19

19

19

19

19

remainder

Answers a) 1

18

18

18

18

18

18

18

18

The remainder is 2/3 of the last block.

b) 1

15

15

15

15

15


c) 1

19

19

19

19

19

19

19

19

19


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Demonstrate how to finish the division for part a) above:

1 ÷ 38

= 223

Have volunteers demonstrate completing the next two divisions:

b) 1 ÷ 25

= 212

c) 1 ÷ 49

= 214

Exercises: Divide.

a) 1

17

17

17

17

17

17

17

1 ÷ 37

=

b) 1

18

18

18

18

18

18

18

18

1 ÷ 58

=

c) 1

19

19

19

19

19

19

19

19

19

1 ÷ 59

=

Answers: a) 2 1/3, b) 1 3/5, c) 1 4/5

Exercises: Draw a picture to divide.

a) 1 ÷ 29

= b) 1 ÷ 49

= c) 1 ÷ 38

=

Answers: a) 4 1/2, b) 2 1/4, c) 2 2/3

Writing the answer as an improper fraction. Review converting mixed numbers to improper fractions.

1 ÷ 27

= 312

= 72

(3 × 2) + 1

Exercises: Write the answers above as improper fractions.

Answers: a) 7/3, b) 8/5, c) 9/5, d) 9/2

Using a shortcut way to divide 1 by a fraction. Tell students to look at their answers. ASK: How can you change the fraction you are dividing by to get the answer? (swap the numerator and denominator or turn the fraction upside down)

2 whole pieces of size 38

fit into 1

23

of another piece of size 38

fits into 1

(MP.2, MP.5)



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Exercises: Divide by turning the fraction upside down.

a) 1 ÷ 37

b) 1 ÷ 3

10 c) 1 ÷

58

d) 1 ÷ 49

Bonus: 1 ÷ 33501

Answers: a) 7/3, b) 10/3, c) 8/5, d) 9/4, Bonus: 501/33

Have students draw a picture to check the answer to part a). Discuss why it would be difficult to draw a picture to check the bonus problem. (the numbers are too large)

Extensions1. Divide 1 by a fraction by writing 1 as a fraction with the same

denominator. For example,

1 ÷ 25

= 55

÷ 25

So just like:

5 apples ÷ 2 apples in each group = 52

groups

We also have:

5 fifths ÷ 2 fifths in each group = 52

groups

So you can just divide the numerators:

1 ÷ 25

= 55

÷ 25

= 5 ÷ 2 = 52

2. Divide 1 by a fraction by multiplying both terms by the denominator.

1 ÷ 25

= 5 ÷ 2 = 52

(MP.8)

NOTE: In the next lesson, students will learn more formally why 1 ÷ a/b = b/a.

(MP.2)

(MP.2)

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Multiplying fractions with swapped numerator and denominator. Write on the board:

35

× 53

ASK: How do you multiply fractions? (multiply the numerators and multiply the denominators) ASK: What is 3 × 5? (15) What is 5 × 3? (15) Write on the board:

35

× 53

= 1515

= 1

Remind students that when the numerator and denominator are equal, the fraction is equal to 1. Ask volunteers to explain why that’s true. Then SAY: When you take all the parts in the whole, you get the whole.


a) 34

× 43

b) 27

× 72

c) 38

× 83

d) 49

× 94

Answers: a) 12/12 = 1, b) 14/14 = 1, c) 24/24 = 1, d) 36/36 = 1

Point out that a fraction multiplied by its upside down version is always going to be 1 because the same numbers you multiply to get the numerator are the numbers you multiply to get the denominator.

Understanding the rule for dividing 1 by a fraction. Write on the board:

2 × 3 = 6

ASK: What division equations can you write from this? (6 ÷ 2 = 3 and 6 ÷ 3 = 2)

Exercises: Write two division equations from the multiplication.

a) 34

× 43

= 1 b) 27

× 72

= 1 c) 38

× 83

= 1 d) 49

× 94

= 1

Answers: a) 1 ÷ 3/4 = 4/3, 1 ÷ 4/3 = 3/4; b) 1 ÷ 2/7 = 7/2; 1 ÷ 7/2 = 2/7, c) 1 ÷ 3/8 = 8/3, 1 ÷ 8/3 = 3/8; d) 1 ÷ 4/9 = 9/4, 1 ÷ 9/4 = 4/9

NS6-54 Dividing by Fractions Pages 23–25

STANDARDS 6.NS.A.1

VOcABULARy commutative property denominator fraction improper fraction mixed number numerator proper fraction whole number

GoalsStudents will divide fractions by fractions.


Can multiply fractions Can divide 1 by a fraction Understands division as fitting into Understands the relationship between multiplication and division



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Point out that when you divide 1 by a fraction, proper or improper, you always get the fraction turned upside down.

Exercises: Divide.

a) 1 ÷ 35

b) 1 ÷ 56

c) 1 ÷ 92

Bonus: 1 ÷ 76281

Answers: a) 5/3, b) 6/5, c) 2/9, Bonus: 281/76

Using pictures to understand why the fraction is turned upside down when dividing 1 by the fraction. Draw on the board:

1 ÷ 25

1

15

15

15

15

15

ASK: How does 1 ÷ 1/5 compare to 1 ÷ 2/5? Emphasize that 2/5 is twice as large as 1/5, so only half as many will fit. But 5 pieces of size 1/5 fit into 1, so 1/2 of 5, or 5/2, pieces of size 2/5 fit into 1.

Dividing any whole number by any fraction. Remind students that if you know how many of any object fit into 1, then twice as many fit into 2, 3 times as many fit into 3, and so on. Write on the board:

1 ÷ 35

= 53

so 4 ÷ 35

= 4 × 53

SAY: Four times as many will fit into 4 as will fit into 1. Have a volunteer do the multiplication. (20/3)

Exercises: Divide.

a) 1 ÷ 37

= b) 1 ÷ 54

=

so 5 ÷ 37

= so 3 ÷ 54

=

c) 8 ÷ 34

d) 9 ÷ 23

e) 7 ÷ 53

f ) 6 ÷ 45

Bonus: 30 ÷ 7

200

Answers: a) 7/3, 35/3; b) 4/5, 12/5; c) 32/3; d) 27/2; e) 21/5; f ) 30/4; Bonus: 6,000/7

Dividing any fraction by any fraction. Remind students that if you know how many of any object fit into 1, then 3/4 as many will fit into 3/4. Write on the board:

1 ÷ 25

= 52

so 34

÷ 25

= 34

× 52

Exercises: Write the missing fraction.

a) 38

÷ 27

= 38

× ? b) 43

÷ 56

= 43

× ? c) 59

÷ 38

= 59

× ?

Answers: a) 7/2, b) 6/5, c) 8/3

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Ask a volunteer to multiply the fractions in the example above: 3/4 × 5/2. (15/8) Remind students that to multiply fractions, they can multiply the numerators to get the numerator and multiply the denominators to get the denominator.

Exercises: Divide.

a) 35

÷ 74

b) 32

÷ 58

c) 32

÷ 49

d) 67

÷ 74

Bonus: 33

100 ÷

10003

,

Answers: a) 12/35, b) 24/10 = 12/5, c) 27/8, d) 24/49, Bonus: 99/100,000

Exercises: Check your answers using multiplication.

Selected solution: a) 12/35 × 7/4 = 84/140 = 12/20 = 3/5

Dividing mixed numbers. Write on the board:

53

÷ 47

123

÷ 47

ASK: How are these problems the same? PROMPT: Do you think they will have the same answer? (yes) Why? (because they are dividing the same numbers) How do you know that they are dividing the same number? (because 5/3 = 1 2/3) Have a volunteer circle the easier one to do. (5/3 ÷ 4/7) Point out that students have a way to divide improper fractions, so they can use that way to divide mixed numbers—they just have to change the mixed numbers to improper fractions.

Exercises: Change the mixed numbers to improper fractions. Then divide the improper fractions.

a) 134

÷ 25

b) 215

÷ 34

c) 57

÷ 312

d) 323

÷ 119

Answers: a) 7/4 ÷ 2/5 = 35/8, b) 11/5 ÷ 3/4 = 44/15, c) 5/7 ÷ 7/2 = 10/49, d) 11/3 ÷ 10/9 = 99/30 = 33/10

Remind students that when the question asks you to divide mixed numbers, it means the answer should be written as a mixed number as well, unless it is less than 1. Demonstrate the first exercise below for students.

Exercises: Use division with remainders to write your answers above as mixed numbers when they are greater than 1.

Answers: a) 35 ÷ 8 = 4 R 3 so 35/8 = 4 3/8, b) 2 14/15, c) 10/49, d) 1 13/15, e) 3 9/30 or 3 3/10

Exercises: Divide the mixed numbers. Write any improper fraction answers as mixed or whole numbers.

a) 234

÷ 123

b) 312

÷ 134

c) 313

÷ 214

d) 215

÷ 312

Answers: a) 1 13/20, b) 2, c) 1 13/27, d) 22/35

(MP.7)

(MP.1)

(MP.7)



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context problems. Tell students that containers of food items often tell you the size of an expected serving. For example, a single serving of yogurt might be 2/5 cup. ASK: If you have 1 cup of yogurt, how many servings do you have? (5/2) What is that as a mixed number? (2 1/2)


a) How many 3/8 cup servings are in 2/3 cup of yogurt?

b) How many 2/3 cup servings are in 1 1/2 cups of yogurt?

Answers: a) 2/3 ÷ 3/8 = 16/9 = 1 7/9 servings, b) 1 1/2 ÷ 2/3 = 3/2 ÷ 2/3 = 9/4 = 2 1/4 servings


a) A rectangle has width 1 3/7 inches and area 3 1/3 square inches. How long is the rectangle?

b) A park with area 2 1/3 square miles is 3 1/2 miles long. How wide is it?

Answers: a) 10/3 ÷ 10/7 = 70/30 = 2 10/30 or 2 1/3 inches long, b) 7/3 ÷ 7/2 = 14/21 or 2/3 miles long

For students who struggle with these exercises, you can prompt them with whole number problems: A rectangle has width 3 inches and area 12 square inches. How long is it? What did they do with the 12 and 3 to get 4? Now compare to the actual problem and have students draw the two rectangles:

3 inches 137

inches

12 squareinches

3 13

square inches

Tell students that it is sometimes easier to replace the fractions with whole numbers and reread the problems. That will help them know what to do with the numbers in the problem.

Extensions1. Divide a fraction by a fraction by writing both fractions with the

same denominator. For example:34

÷ 25

= 1520

÷ 8

20 = 15 ÷ 8 (multiply both terms by 20)

= 158

(MP.4)

(MP.4)

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2. a) Divide. Then describe a rule for dividing unit fractions by unit fractions.

i ) 18

÷ 14

ii ) 13

÷ 1

12 iii )

17

÷ 13

Answers: i ) 4/8 or 1/2, ii ) 12/3 or 4, iii ) 3/7; Rule: Divide the second denominator by the first denominator.

b) Divide. Then describe a rule for dividing fractions with the same numerator.

i ) 15

÷ 1

15 ii )

25

÷ 2

15 iii )

35

÷ 3

15 iv)

45

÷ 4

15

Answers: i ) 3, ii ) 3, iii ) 3, iv) 3; Rule: Same as part a): Divide the second denominator by the first denominator.

c) Divide 3/4 ÷ 2/5 by using a common numerator for the fractions.

Solution: 3/4 ÷ 2/5 = 6/8 ÷ 6/15 = 15 ÷ 8 = 15/8



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Interpreting mixed number answers when the answer has to be a whole number. Tell students that when they divide, some problems will have a mixed number answer even when the answer needs to be a whole number. Write on the board:

Nomi can carry 16 lb. How many 1 1/2 lb books can she carry?

Have volunteers tell you what to divide (16 ÷ 1 1/2), change the mixed number to an improper fraction (16 ÷ 3/2), do the division (16 × 2/3 = 32/3), and write the answer as a mixed number (10 2/3). Point out that she can’t carry 2/3 of a textbook, so she has to carry only 10 books. Emphasize that the answer they got by dividing is a mixed number, but the answer to the problem has to be a whole number.

Exercises: Solve the problems. Make sure the answer is a whole number.

a) Ron can carry 16 2/3 lb . How many 2 2/3 lb textbooks can he carry at once?

b) Diane can carry 15 1/2 lb. How many 1 2/3 lb textbooks can she carry at once?

c) Lina has 3/5 lb of dry pasta. Each person needs 3/16 lb. How many people can she feed?

d) Bilal has 7/8 lb of dry pasta. Each person needs 3/16 lb. How many people can he feed?

Answers: a) 50/3 ÷ 8/3 = 6 1/4, so he can carry 6 textbooks at once; b) 31/2 ÷ 5/3 = 9 3/10, so she can carry 9 textbooks at once; c) 3/5 ÷ 3/16 = 3 1/5, so she can feed 3 people; d) 7/8 ÷ 3/16 = 4 2/3, so he can feed 4 people

Extensions1. a) Write the fractions in order from least to greatest:

34

, 25

, 12

b) Divide, then write the results in order from least to greatest:

78

÷ 34

, 78

÷ 25

, 78

÷ 12

(MP.4)

(MP.1, MP.3)

NS6-55 Word Problems (Advanced) Page 26

STANDARDS 6.NS.A.1

VOcABULARy fraction improper fraction mixed number remainder whole number

GoalsStudents will solve word problems involving multiplication and division of fractions.


Can add, multiply, and divide fractions by fractions

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c) How do your answers to parts a) and b) compare? Why does this make sense?

d) Predict which quotient will be greater than 29

= 29

÷ 1, then check your prediction.29

÷ 34

or 29

÷ 43

Answers: a) 2/5, 1/2, 3/4; b) 7/8 ÷ 3/4 = 1 1/6, 7/8 ÷ 1/2 = 1 3/4, 7/8 ÷ 2/5 = 2 3/16; c) the order of the answers is reversed; it makes sense because dividing the same number by a greater number gets a lesser answer; d) 3/4 < 1, so 2/9 ÷ 3/4 should be greater than 2/9. Indeed, 2/9 ÷ 3/4 = 8/27 > 8/36 = 2/9. Also, 2/9 ÷ 4/3 should be less than 2/9. Indeed, 2/9 ÷ 4/3 = 6/36 = 1/6, and 1/6 = 3/18 < 2/9 = 4/18.

2. How do the answers in each pair compare? Which answers are greater than 1? Why does this make sense?

a) 23

÷ 35

and 35

÷ 23

b) 14

÷ 13

and 13

÷ 14

c) 92

÷ 52

and 52

÷ 92

d) 32

÷ 73

and 73

÷ 32

Answer: The second answer is the first answer turned upside down. When the first fraction is greater than the second, the answer is greater than 1. This makes sense because division is asking how many of the second fraction fit into the first fraction. The answer is more than 1 precisely when fitting a smaller object into a larger object.

3. Determine if the mixed number answer should become the whole-number part, or one more than the whole-number part. Katie needs to carry 34 1/2 pounds of groceries, but she can only carry 15 pounds on each trip. How many trips does she need to make?

Answer: 34 1/2 ÷ 15 = 2 9/30, so she needs to make three trips (two trips will only let her carry 30 pounds)

4. a) 45

÷ 13

=

b) 45

÷ 26

=

c) Compare your answers to parts a) and b). What do you notice? Why is this the case?

Answer: a) 12/5, b) 24/10, c) 12/5 = 24/10, which makes sense because you were dividing 4/5 by equivalent fractions in the first place.

(MP.1, MP.3)

(MP.4)

(MP.1, MP.3)



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Review adding and subtracting fractions with the same denominator. Write on the board:

38

+ 28

= 58

SAY: You can add and subtract eighths the same way you can add and subtract apples, so just like 3 apples + 2 apples = 5 apples, so 3 eighths + 2 eighths = 5 eighths.

Exercises: Add or subtract.

a) 47

+ 17

b) 47

- 17

c) 3

10 +

310

d) 811

- 311

Answers: a) 5/7, b) 3/7, c) 6/10, d) 5/11

Review multiplying and dividing fractions by whole numbers. Demonstrate multiplying and dividing fractions by whole numbers with one example of each:

38

× 2 = 68

38

÷ 2 = 3

16

Pointing to the multiplication, SAY: Here I multiplied the numerator by 2. That makes twice as many pieces. Pointing to the division, SAY: Here I multiplied the denominator by 2. That makes the pieces half as big.

Exercises: Multiply or divide.

a) 38

× 5 b) 38

÷ 5

c) 25

× 3 d) 25

÷ 3

Answers: a) 15/8, b) 3/40, c) 6/5, d) 2/15

Review the order of operations without powers. Write pairs of operation symbols on opposite sides of the board and have students signal which operation is done first by pointing to it. If they are done in the order they appear, students can point to the middle. Then remind students that when adding or subtracting fractions with different denominators, they can change the fractions to have the same denominator, then add or subtract them.

STANDARDS 6.EE.A.2c

VOcABULARy basedecimaldecimal pointdenominatorfractionnumeratoroperationpowerwhole number

GoalsStudents will evaluate numerical expressions involving fractions and whole-number exponents. Students will perform arithmetic operations in the conventional order.


Can use the conventional order of operations to evaluate numerical expressions involving non-zero whole-number exponents Can add, subtract, multiply, and divide fractions

NS6-56 Fractions and Order of Operations (Advanced)Pages 27–28

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Exercises: Which operation is done first? Do that operation, then rewrite the rest of the expression.

a) 23

- 13

+ 14

b) 23

13

14

− +

c) 13

+ 12

÷ 14

d) 23

- 12

× 35

Bonus: 12

+ 13

- 14

× 23

÷ 5

Answers: a) 1/3 + 1/4, b) 2/3 – 7/12, c) 1/3 + 2, d) 2/3 – 3/10, Bonus: 1/2 + 1/3 – 2/12 ÷ 5

Powers of fractions. Remind students that powers are short for repeated multiplication. Write on the board:

34 = 3 × 3 × 3 × 3 23

4

=

Ask a volunteer to write the multiplication. (2/3 × 2/3 × 2/3 × 2/3)

Exercises: Write the power as repeated multiplication.

a) 12

3

b)

35

2

c)

34

5

d)

13

4

Answers: a) 1/2 × 1/2 × 1/2, b) 3/5 × 3/5, c) 3/4 × 3/4 × 3/4 × 3/4 × 3/4, d) 1/3 × 1/3 × 1/3 × 1/3

Draw students’ attention to the power above: (2/3)4. ASK: How do you multiply fractions? (multiply the numerators to get the numerator, and multiply the denominators to get the denominator) Write on the board:

23

23

23

23

23

2 2 2 23 3 3 3

1681

4

= × × × =

× × ×× × ×

=

Exercises: Evaluate the power.

a) 13

4

b)

32

4

c)

12

8

d)

58

2

Answers: a)1/81, b) 81/16, c) 1/256, d) 25/64

Review the conventional order of operations, including powers. Remind students that evaluating powers comes before multiplication and division, which comes before addition and subtraction. One way to remember this is that the repeated form of an operation is done before any other operation. Draw on the board:

Brackets

Powers

Multiplication and division

Addition and subtraction



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Exercises: Evaluate using the correct order of operations.

a) 214

2

+

b) 2

14

2

+

c) 2

14

2

−

d) 2

14

2

−

e) 235

2

×

f ) 2

35

2

×

g)

25

103

× h)

25

103

×

i ) 45

22÷ j ) 45

22

÷

k) 2

15

3 ÷

Bonus: 235

3 51

15

32

2

− ×

× ÷

Answers: a) 81/16, b) 2 1/16, c) 49/16, d) 1 15/16, e) 36/25, f ) 18/25, g) 80/125 = 16/25, h) 8,000/125 = 64, i ) 1/5, j ) 4/25, k) 40, Bonus: 9

Review multiplying decimals. SAY: To multiply decimals, multiply as though they are whole numbers, then shift the decimal point the total number of places it is shifted in each decimal. Ask a volunteer to demonstrate multiplying 0.03 × 0.004:

3 × 4 = 12, so 0.03 × 0.004 = 0.0 0 0 1 2 .


a) 0.4 × 0.3 b) 0.6 × 0.002 c) 0.4 × 0.05 d) 0.003 × 0.02

Bonus e) 0.3 × 0.2 × 0.4 f ) 0.6 × 0.002 × 0.00003

Answers: a) 0.12, b) 0.0012, c) 0.020 = 0.02, d) 0.00006, Bonus: e) 0.024, f ) 0.000000036

Review powers of decimals. Tell students that the notation for decimal powers can sometimes be confusing. Write on the board:

0.32 = 0.3 × 0.3 = 0.23 = 0.2 × 0.2 × 0.2 =

Ask volunteers to evaluate the powers. (0.09 and 0.008) Tell students that someone you know wrote this:

0.32 = 0.9 and 0.23 = 0.8

ASK: What mistake did they make? (they thought you were supposed to evaluate 32 and put “0.” in front) SAY: This is like writing 132 = 19, because 32 is 9! Tell students to be careful to read the numbers correctly and, if they find it easier, put brackets around the entire base of the power. Show some examples:

(0.3)2 (0.2)3 (13)2

Exercises: Evaluate. Put brackets around the base if it helps.

a) 0.24 b) 0.032 c) 0.15 d) 0.00052

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Answers: a) 0.0016, b) 0.0009, c) 0.00001, d) 0.00000025

compare decimal and fraction powers. Write on the board:

0.82 45

2

Have different volunteers evaluate each expression. (0.64, 16/25) Have another volunteer change the answer for (4/5)2 to a decimal. ASK: What do you notice? (the answer is 0.64—the same as it was for 0.82) ASK: Why is this the case? (because 0.8 and 4/5 are equivalent)

Exercises: Evaluate the same power of equivalent numbers. Make sure your answers are equivalent.

a) 0.12 and 1

10

2

b) 0.52 and

12

2

c) 0.62 and

35

2

Bonus

0.53 and 12

3

Answers: a) 0.01 and 1/100, b) 0.25 and 1/4 = 25/100, c) 0.36 and 9/25 = 36/100, Bonus: 0.125 and 1/8 = 125/1,000

Extensions1. Evaluate. What do you notice?

23

32

×

23

32

2 2

×

23

32

3 3

×

23

32

4 4

×

Answer: The answers are all 1.

2. Compare.

a) 3 32 b) 0.3 0.32 c) 2 22 d) 0.2 0.22

e) 13

13

2

f )

43

43

2

g)

34

34

2

h) 1 12

Answers: a) 3 < 32, b) 0.3 > 0.32, c) 2 < 22, d) 0.2 > 0.22, e) 1/3 > (1/3)2, f ) 4/3 < (4/3)2, g) 3/4 > (3/4)2, h) 1 = 12

3. In the previous question, which positive numbers are greater than their square? Why does this make sense?

Answer: When a positive number is less than 1, multiplying it by itself results in a lesser number, so the positive number is greater than its square.

(MP.1)

(MP.7)

(MP.3, MP.7)



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Review dividing by powers of 10. Remind students that when dividing a number by 10, we move the decimal point one place left. Write the number 17 on the board. SAY: It doesn’t look like there is a decimal point, but there is one that’s understood. ASK: Where is it? (after the 7) Have a volunteer put it in. Then demonstrate moving the decimal point one place left to divide by 10:

17.0 ÷ 10 = 1.7

Exercises: Divide.

a) 45 ÷ 10 b) 3 ÷ 10 c) 0.4 ÷ 10 d) 13.4 ÷ 10

Answers: a) 4.5, b) 0.3, c) 0.04, d) 1.34

ASK: How many places would you move the decimal point to divide by 100? (2) to divide by 1,000? (3)

Exercises: Divide.

a) 342 ÷ 100 b) 84 ÷ 1,000 c) 90 ÷ 100 d) 23.45 ÷ 1,000

Answers: a) 3.42, b) 0.084, c) 0.90 = 0.9, d) 0.02345

Dividing decimals by multiples of 10. Write on the board:

543 ÷ = 54.3 45 ÷ = 0.045

For each question, ASK: How many places did the decimal point move? (1 and 3) Write 10 and 1,000 in the blanks. Point out that the number of places the decimal point moved is equal to the number of zeros in the number being divided by.

Exercises: Write 10, 100, or 1,000 in the blank.

a) 74 ÷ = 0.74 b) 8 ÷ = 0.008 c) 90 ÷ = 0.9 d) 745 ÷ = 74.5 e) 830 ÷ = 8.3 f ) 4,052 ÷ = 4.052

NS6-57 Dividing Decimals by Whole Numbers Pages 29–30

STANDARDS 6.NS.B.3

VOcABULARy decimalfractionwhole number

GoalsStudents will divide decimals by whole numbers.


Can multiply and divide by powers of 10 Understands that multiplying the dividend by a number will multiply the result of the division by the same number Knows to align place values when using the long division symbol Can use long division to divide multi-digit numbers by 1-digit numbers

MATERIALS

grid paper (optional) 32 tens blocks 32 ones blocks

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Answers: a) 100, b) 1,000, c) 100, d) 10, e) 100, f ) 1,000

Some students may find it helpful to use grid paper to see how many places the decimal point moved.

Dividing decimals by whole numbers. Show students a tens block and tell them that you are using it as the whole. Ask for four volunteers and distribute 32 tens blocks among them. ASK: How many did each person get? (8) Write on the board:

32 ÷ 4 = 8

Show students a ones block and SAY: This is one tenth. Now distribute 32 ones blocks among the 4 volunteers. ASK: How many did each person get? (8) Write on the board:

3.2 ÷ 4 = 0.8

SAY: Each person got 8 tenths of a tens block. If you divide one tenth as much among 4 equal groups, you get one tenth as much in each group.

Exercises: Divide.

a) 28 ÷ 4 = 7 so 2.8 ÷ 4 = b) 48 ÷ 6 = 8 so 4.8 ÷ 6 =

Bonus: 834 ÷ 3 = 278 so 83.4 ÷ 3 =

Answers: a) 0.7, b) 0.8, Bonus: 27.8

Write on the board:

(32 ÷ 10) ÷ 4 = (32 ÷ 4) ÷ 10

Tell students that you can change the order of the numbers you are dividing by and still get the same answer. Dividing by 10, then dividing by 4 gets the same answer as dividing by 4, then by 10—and they’re both the same as dividing by 40. Point out that the second expression divides a whole number by 10, which they already know how to do.

Exercises: Rewrite the division statement so that the final division is by 10, 100, or 1,000.

a) 0.36 ÷ 9 = (36 ÷ ) ÷ 9 b) 0.45 ÷ 5 = ( ÷ ) ÷

= (36 ÷ 9) ÷ = ( ÷ ) ÷

c) 0.042 ÷ 7 d) 3.6 ÷ 6 e) 0.21 ÷ 7 f ) 0.018 ÷ 9

Bonus: Make the final division a higher power of ten: 0.00000024 ÷ 4

Answers: a) 100, 100; b) (45 ÷ 100) ÷ 5 = (45 ÷ 5) ÷ 100; c) (42 ÷ 7) ÷ 1,000; d) (36 ÷ 6) ÷ 10; e) (21 ÷ 7) ÷ 100; f ) (18 ÷ 9) ÷ 1,000; Bonus: (24 ÷ 4) ÷ 100,000,000

Exercises: Do the divisions from the previous exercises.

Answers: a) 0.04, b) 0.09, c) 0.006, d) 0.6, e) 0.03, f ) 0.002, Bonus: 0.00000006



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Using the symbol for division. Point out that what students are really doing is dividing as though they are whole numbers, then dividing by the appropriate power of 10: either 10, 100, or 1,000. Tell students that you did the dividing as though the decimal was a whole number, and their job is to add the decimal to make the answer correct. Ask volunteers to finish writing the answers.

76 4.2

94 0.0 3 6

75 0.3 5

PROMPTS: What would you divide 42 by to get 4.2? (10) What is 7 ÷ 10? (0.7)

Answers0.7

6 4.2

0.00 94 0.0 3 6

0.0 75 0.3 5

SAY: Because 42 was divided by 10, so was 7 in the answer; because 36 was divided by 1,000, so was 9; and because 35 was divided by 100, so was 7. Have students look carefully at the answers. ASK: How can you tell where the decimal point goes in the answer? (it’s in the same place as in the number being divided) Point out how using the long division notation made it easier to see where to put the decimal point. Write on the board:

. 74 0.0 2 8

ASK: I divided as though the decimal was a whole number, then I put the decimal point in the same place as in the question. Am I done? (no) Why not? PROMPT: Is the answer “.7”? (no) Point out that you still have to put the 0s in, and do so:

.00 74 0.0 2 8

Exercises: Divide by copying the decimal point’s location. Add any 0s that you need to.

a) 7

6 4.2 b) 7

6 0.4 2 c) 7

6 0.0 4 2

Bonus7

6 0.0 0 00 4 2

Answers: a) 0.7, b) 0.07, c) 0.007, Bonus: 0.000007

Have volunteers fill in the answers to these problems:

6 4 2 6 4 2 0 6 4,2 0 0 6 42,00 0

Answers: 7, 70, 700, 7,000

Exercises: Use 84 ÷ 3 = 28 to divide.

a) 3 8 4 0 b) 3 0.8 4 c) 3 0.0 84 d) 3 8 4,00 0 e) 3 8.4

Answers: a) 280, b) 0.28, c) 0.028, d) 28,000, e) 2.8

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Using long division to divide decimals. Have volunteers guide you through the steps of the long division algorithm to divide 96 by 4 (see margin).

Exercises: Divide as though the decimal point isn’t there. Then put the decimal point in the correct place.

a) 3 4.6 2 b) 8 1.7 6 c) 5 76 .5 d) 3 85 .41

Bonus

e) 7 5.6 7 f ) 8 4.3 2

Answers: a) 1.54, b) 0.22, c) 15.3, d) 28.47, Bonus: e) 0.81, f ) 0.54

Exercises: Divide. Be sure to add any required 0s.

a) 2 0.0 51 4 b) 3 0.0 0 6 5 4 c) 4 0.0 30 5 6 d) 4 0.1 84 8

Bonus

e) 7 0.0 0 0 8 5 4 f) 9 0.0 0 0 0 0 5 3 1

Selected solution: a) 2 5 7

2 0 . 0 5 1 4- 4

1 1- 1 0

1 4- 1 4

0

0 . 0 2 5 72 0 . 0 5 1 4

- 41 1

- 1 01 4

- 1 40

Answers: b) 0.00218, c) 0.00764, d) 0.0462, Bonus: e) 0.000122, f ) 0.00000059

Exercises: Check your answers by multiplication.

Selected solution: a) 2 × 0.0257 = 0.0514


a) Six cans of juice cost $4.56. How much does each can cost?

b) A four-person relay has a total distance of 5.6 km. How far does each person run?

c) What is a better deal: three pencils for $2.34 or five pencils for $3.95?

Answers: a) 0.74 = 74¢, b) 1.4 km, c) three pencils are 78¢ each and five pencils are 79¢ each, so the three pencils are the better deal

Extensions1. To divide by 5, multiply by 2, then divide by 10.

a) 0.8 ÷ 5 b) 1.4 ÷ 5 c) 3 ÷ 5 d) 3.2 ÷ 5 e) 42 ÷ 5

Answers: a) 0.16, b) 0.28, c) 0.6, d) 0.64, e) 8.4

2. Divide 0.6 ÷ 2 and 3/5 ÷ 2. Are your answers the same? Why is this the case?

Answer: yes, they are both 0.3; it makes sense because 0.6 and 3/5 are equivalent

2 44 9 6- 8

1 6- 1 6

0

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Using number lines to divide decimals by decimals. Write on the board:

0 1Ask volunteers to write various decimals on the number line: 0.5, 0.4, 0.8, 0.9, 0.2, then fill in the remaining decimals yourself. Point out that there are 10 equal parts, so each part is a tenth, and the decimals are counting the tenths: 0.1 = 1 tenth, 0.2 = 2 tenths, and so on.

Remind students that division means fitting into, so they can use the number line to divide 0.8 ÷ 0.4 by asking how many steps of size 0.4 fit into 0.8:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 ÷ 0.4 = 2

Have students draw number lines on grid paper from 0 to 1 divided into tenths. You may need to help some students. Then have them do the following exercises.

Exercises: Divide.

a) 0.6 ÷ 0.2 b) 0.6 ÷ 0.3 c) 0.9 ÷ 0.3 d) 0.4 ÷ 0.2 e) 0.5 ÷ 0.1 f ) 1 ÷ 0.2 g) 1 ÷ 0.5 h) 0.8 ÷ 0.2

Answers: a) 3, b) 2, c) 3, d) 2, e) 5, f ) 5, g) 2, h) 4

comparing decimal division to whole-number division. Draw on the board two number lines from 0 to 10. Have volunteers show the following divisions on the number lines: 6 ÷ 2 and 6 ÷ 3.

Point out that 6 ÷ 2 is the same division as 0.6 ÷ 0.2, just on a scale ten times larger. SAY: 6 is ten times 0.6 and 2 is ten times 0.2. So the same

NS6-58 Dividing by Decimals Pages 31–32

STANDARDS 6.NS.B.3

VOcABULARy decimaldividenddivisororder of operationswhole number

GoalsStudents will divide decimals by decimals.


Can add and subtract decimals Can label decimals on a number line Understands division as fitting into Understands that multiplying both terms in a division expression will result in the same answer Can treat fraction parts such as halves or tenths as objects to be added, subtracted, multiplied, or divided

MATERIALS

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number of 0.2s fit into 0.6 as 2s fit into 6. ASK: What decimal division is the same as 6 ÷ 3? (0.6 ÷ 0.3) Write on the board:

6 apples ÷ 3 apples in each group = 2 groups 6 tenths ÷ 3 tenths in each group = 2 groups

Point out that it doesn’t matter what the objects are that you are dividing—apples or tenths—dividing 6 of them among groups of 3 of them results in 2 groups.

Exercises: Write the whole-number division that has the same answer. Then divide.

a) 0.8 ÷ 0.4 b) 0.8 ÷ 0.2 c) 0.6 ÷ 0.1 d) 0.9 ÷ 0.3

Answers: a) 8 ÷ 4 = 2, b) 8 ÷ 2 = 4, c) 6 ÷ 1 = 6, d) 9 ÷ 3 = 3

Using dividing by whole numbers to divide by decimals. Write on the board:

0.08 ÷ 0.4

Remind students that if they multiply both numbers by 10, they will get a division with the same answer. Have a volunteer write the new division. (0.8 ÷ 4) ASK: Do you know how to do this problem? (yes) SAY: As long as the divisor, in this case 4, is a whole number, then we can do the question the same way as we did problems last class. Write on the board:

0.8 ÷ 4 = 0.2

SAY: 8 ÷ 4 is 2 so 0.8 ÷ 4 is 2 ÷ 10.

Exercises: Multiply both numbers by 10 to make a division with the same answer. Then divide.

a) 0.36 ÷ 0.9 b) 36 ÷ 0.4 c) 1.8 ÷ 0.3 d) 0.006 ÷ 0.2

Bonus: 0.846 ÷ 42.3

Answers: a) 3.6 ÷ 9 = 0.4, b) 360 ÷ 4 = 90, c) 18 ÷ 3 = 6, d) 0.06 ÷ 2 = 0.03, Bonus: 8.46 ÷ 423 = 0.02

SAY: You might have to multiply both numbers by 100 or 1,000. You just have to make sure the second number, the divisor, becomes a whole number.

Exercises: Multiply both numbers by 10, 100, or 1,000 to make the divisor become a whole number. Then divide.

a) 0.8 ÷ 0.02 b) 6 ÷ 0.003 c) 0.00018 ÷ 0.6 d) 0.09 ÷ 0.3

Answers: a) 80 ÷ 2 = 40, b) 6,000 ÷ 3 = 2,000, c) 0.0018 ÷ 6 = 0.0003, d) 0.9 ÷ 3 = 0.3

Using long division to divide decimals by decimals. SAY: To divide by a decimal, make the divisor a whole number. Multiply both the dividend and the divisor by the same power of 10. Show students what this looks like when using the long division symbol.

4.0 .6 2.4

.40 .6 0. 2 4

4 0 0 .0 . 0 6 2 4.0 0



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SAY: Make sure you do the same thing to both numbers. As long as the divisor becomes a whole number, you can divide.

Exercises: Divide by determining the decimal point’s location. Add any 0s that you need to.

a) 6

0.7 4.2 b) 6

0.07 0.4 2 c) 6

0.07 0.0 4 2

d) 6

0.0 07 4.2 e) 6

0.7 0.4 2

Bonus: 6

0.0 07 0.0 0 0 0 0 0 4 2

Answers: a) 6, b) 6, c) 0.6, d) 600, e) 0.6, Bonus: 0.00006

Exercises: Make another question with the same answer so that the divisor becomes a whole number. Then divide.

a) 0.7 1.8 9 b) 0.3 2.9 1 c) 0.5 0.3 1 7

d) 0.04 1.3 2 4 e) 0.0 06 3 6 1.8 f ) 0.0 08 1 7

Answers: a) 2.7, b) 9.7, c) 0.634, d) 33.1, e) 60,300, f ) 2,125

Review the order of operations without powers.

Exercises: Evaluate. Use the correct order of operations.

a) 1.2 ÷ 3 × 2 b) 1.2 ÷ (3 × 2) c) 3 ÷ (0.5 + 0.1)

d) 3 ÷ 0.5 + 0.1 e) 0.8 + 0.2 × 3 f ) (0.8 + 0.2) × 3

Answers: a) 0.8, b) 0.2, c) 5, d) 6.1, e) 1.4, f ) 3.0 = 3

Some students may need to be reminded how to add and subtract decimals: Line up the decimal points and add or subtract as you would whole numbers.


a) Lina has 4.2 pounds of cheese. She needs 0.05 pounds of cheese for each sandwich. How many sandwiches can she make?

b) Tom has $2.73. How many $0.07 candies can he buy?

c) A shelf is 25.8 inches long. How many books 0.6 inches thick can the shelf hold?

d) Each sheet of paper is 0.008 cm thick. How many sheets are in a stack that is 2.4 cm high?

Answers: a) 84, b) 39, c) 43, d) 300

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Extensions1. Calculate 3 ÷ 0.2 and 3 ÷

15

. Are your answers the same? Why is this the case?

Answer: 30 ÷ 2 = 15, and 3 × 5 = 15. The answers are the same because we are dividing 3 by equivalent numbers.

2. Divide 8.56 ÷ 0.4 and 8.56 ÷ 0.2. How do your answers compare? Why does this make sense?

Answer: 8.56 ÷ 0.4 = 21.4 and 8.56 ÷ 0.2 = 42.8. The second answer is double the first because it is dividing by half as much.

3. Divide 7.14 ÷ 0.03 and 714 ÷ 0.03. How do your answers compare? Why does this make sense?

Answer: 7.14 ÷ 0.03 = 238 and 714 ÷ 0.03 = 23,800. The second answer is one hundred times the first answer because 714 is 100 times 7.14.

4. How can you use coins and dollar bills to show each division?

a) 1 ÷ 0.01 = 100 b) 1 ÷ 0.05 = 20

c) 1 ÷ 0.1 = 10 d) 1 ÷ 0.25 = 4

Answers: a) 100 pennies make a dollar, b) 20 nickels make a dollar, c) 10 dimes make a dollar, d) 4 quarters make a dollar

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Introduce division by 2-digit numbers. Write on the board:

4 9 3 8 4 2 1 7 11 5 8 6 11 1 0 7

Have volunteers show the long division for the first two. Discuss how they are different. If it doesn’t come up, point out that the second problem:

• requires fewer steps • doesn’t have enough hundreds to put into four groups

Remind students that we can look at division as dividing the objects among equal groups. Then SAY: We had to start by finding the first part of the number that was at least equal to the number of groups. Circle that number in the first two problems. Pointing to the last two problems, ASK: How many groups are we making for these two divisions? (11) Have volunteers circle the first part of the number being divided that is at least 11.

4 9 3 8 4 2 1 7 11 5 8 6 11 10 7

Remind students that the number of objects they are dividing is called the dividend and the number being divided by, which we are looking at as the number of groups, is the divisor.

Exercises: Write the first part of the dividend that is at least as large as the divisor.

a) 37 4 2 6 b) 37 1, 1 9 6 c) 23 2,3 4 7 d) 15 1, 4 3 9 e) 15 1 6 7

Answers: a) 42, b) 119, c) 23, d) 143, e) 16

Write on the board:

11 4 1 5

NS6-59 2-Digit Division (Introduction) Pages 33–34

STANDARDS 6.NS.B.3

VOcABULARy dividenddivisor

GoalsStudents will perform the long division algorithm, dividing by 2-digit numbers, for cases in which the relevant times table is provided for them.


Understands division as dividing objects among equal groups, with remainder Can divide multi-digit numbers by 1-digit numbers using long division Understands long division as a way to record dividing base ten blocks among equal groups Can quickly multiply 11 by any 1-digit number

MATERIALS

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Tell students to pretend that they have to divide 415 ones blocks into 11 groups. ASK: What is more convenient to use than ones blocks? (tens blocks) Would we have enough hundreds blocks to put one hundreds block in each group? (no) SAY: So let’s not bother with using four hundreds blocks at all. Let’s just start with tens blocks right away. How many tens blocks do we need? (41) Tell students to imagine dividing the tens blocks among the 11 equal groups. Write on the board:

41 ÷ 11

ASK: How many do we put in each group? (3) Show this on the board:3

11 4 1 5

Remind students that they write the 3 above the tens digit because it is the number of tens blocks in each group. Have students do the first step for these problems. Some students might find aligning the place values easier if they use grid paper.

Exercises: Write how many tens blocks are in each group.

a) 11 3 6 1 b) 11 5 0 4 c) 11 1 9 4 d) 11 8 4 6

Bonus

e) 12 3 9 2 f) 12 4 8 7 g) 12 8 9 4 h) 12 5 1 6

Answers: a) 3, b) 4, c) 1, d) 7, Bonus: e) 3, f ) 4, g) 7, h) 4

SAY: Once you decide on the first place value to divide, dividing by 2-digit numbers is the same as dividing by 1-digit numbers. It just uses harder times tables. Demonstrate completing part a), then have students complete the remaining division questions.

3 211 3 6 1

- 3 33 1

-2 29

Answers: a) 32 R 9, b) 45 R 9, c) 17 R 7, d) 76 R 10, Bonus: e) 32 R 8, f ) 40 R 7, g) 74 R 6, h) 43 R 0

Dividing when you don’t know the times table off-hand. Write on the board:

18 1, 4 7 2

ASK: What makes this division harder? (most people don’t have the 18 times table memorized) Tell students that they will learn ways to get around this later. But, for now, we are just going to write the 18 times table on the board (see margin).

Refer students to the division on the board: 18 1, 4 7 2 . ASK: What is the first part of the dividend that is at least 18? PROMPTS: Is 1 at least 18? (no) Is 14? (no) Is 147? (yes) Circle 147, then ASK: How many times does 18 go

1 8 1 8 1 8× 1 × 2 × 3

1 8 3 6 5 4

1 8 1 8 1 8× 4 × 5 × 6

7 2 9 0 1 0 8

1 8 1 8 1 8× 7 × 8 × 91 2 6 1 4 4 1 6 2



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into 147? (8) Demonstrate how to use the times table to determine that. Then write 8 above the 7. Point out that 147 is the number of tens to divide, so 7 tells us how many tens are in each group. In general, whatever place value 147 is in, we would put 7 in that same place value. Have volunteers do so for these questions:

18 1 4 7, 2 1 3 18 1 4 7 18 14,7 6 4

Exercises: Copy the questions below onto grid paper. Circle the first part of the dividend that is at least 18. In the correct place, write how many times 18 goes into the number you circled.

a) 18 1, 0 3 2 b) 18 8,5 3 6 c) 18 1, 2 8 7

d) 18 1, 1 4 6 e) 18 4,5 0 9

Bonus f ) 832,176 ÷ 18 g) 298,473 ÷ 18

Answers: a) 103, 5; b) 84, 4; c) 128, 7; d) 114, 6; e) 45, 2; Bonus: f ) 83, 4; g) 29, 1

Show students how to finish the division by referring to the 18 times table.

718 1, 2 9 6

- 1 2 63

7 2

18 1, 2 9 6- 1 2 6

3 6-3 6

0

Exercises: Divide.

a) 18 1, 4 0 4 b) 18 6,2 4 6 c) 18 9, 8 1 0 d) 18 1, 7 4 6

Bonus: 151,416 ÷ 18

Answers: a) 78, b) 347, c) 545, d) 97, Bonus: 8,412

Bonus: Explain why it makes sense that the answer to d) is close to 100.

Answer: 1,746 is close to 1,800, which is 18 × 100, so close to one hundred 18s will fit into 1,746

SAY: These next questions will sometimes have remainders.

Exercises: Divide. Write your answer with the remainder.

a) 11 4 2 5 b) 11 4,7 8 9 c) 11 2,2 5 0

d) 18 2 1 6 e) 18 1,3 3 8 f) 18 8,6 1 7

Answers: a) 38 R 7, b) 435 R 4, c) 204 R 6, d) 12 or 12 R 0, e) 74 R 6, f) 478 R 13

Exercises: Try to figure out these questions by doing the 21 times table in your head.

a) 21 7 2 9 b) 21 5 0 4 c) 21 8 8 7 d) 21 7 0 0 e) 21 4 3 8

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Bonus

f ) 21 6,5 9 6 g) 21 4,9 1 4 h) 21 8,4 6 3 i ) 42,252 ÷ 21

Answers: a) 34 R 15, b) 24 or 24 R 0, c) 42 R 5, d) 33 R 7, e) 20 R 18, Bonus: f) 314 R 2, g) 234 or 234 R 0, h) 403 or 403 R 0, i ) 2,012 or 2,012 R 0

Extensions1. Divide 4,326 ÷ 21 and 8,652 ÷ 21. Compare your answers.

Why does this make sense?

Answer: 206 and 412. The second answer is twice as much as the first because the second dividend is twice as much as the first.

2. Divide: 660,726 ÷ 213.

Answer: 3,102

3. Fill the boxes using the digits 0, 1, 2, or 3 only. Don’t start a number with a “0” digit.

×

Multiply the numbers using a calculator or long multiplication. Then give your partner the following problem:

(the product you found) ÷ (the 2-digit number)

Check that your partner gets the 5-digit number you started with. For example, a student might multiply the numbers 32 × 10,232, get 327,424, and so have a partner divide 327,424 ÷ 32 using long division. The answer should be 10,232. Alternatively, use only the digits 1, 2, 3, or 4 for the 2-digit number, and the digits 0, 1, or 2 for the 5-digit number.

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Introduce rounding to estimate the quotient using multiples of 10. Draw on the board:0 5 10 15 20 25

23 ÷ 5

0 50 100 150 200 250237 ÷ 50

Ask volunteers to place 23 approximately where it would go on the first number line and 237 on the second number line. ASK: What is the whole-number quotient, before taking remainders? (they are both 4) Point out that if they know which two multiples of 5 that 23 is between, then they know which two multiples of 50 that 237 is between. Write on the board:

29 ÷ 8 296 ÷ 80

ASK: Which two multiples of 8 is 29 between? (24 and 32) So which two multiples of 80 is 296 between? (240 and 320) Write on the board:

253 ÷ 30

ASK: What is an easier division that has the same quotient? (25 ÷ 3) What two multiples of 3 is 25 between? (24 and 27) So what two multiples of 30 is 253 between? (240 and 270) Show both long divisions on the board:

83 2 5- 2 4

1

8

30 2 5 3- 2 4 0

1 3

SAY: The remainders are different, but the quotients are the same, so you can use the easier division to help you do the harder division. SAY: We know the quotient is 8 because 3 goes into 25 eight times. Once you know the quotient, you can finish the long division.

Exercises: Divide.

a) 20 1 7 5 b) 30 1 4 3 c) 50 3 8 4 d) 40 3 5 2

Bonus

e) 80 7 1 5 f) 60 4 7 3 g) 90 6 0 8 h) 70 5 7 1

NS6-60 2-Digit Division Pages 35–36

STANDARDS 6.NS.B.3

VOcABULARy dividenddivisorestimatemultiplequotientremainder

GoalsStudents will use rounding to estimate the quotient when dividing by 2-digit numbers, for cases in which doing so gets the correct quotient.


Can divide using a number line Can divide multi-digit numbers by 1-digit numbers

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Answers: a) 8 R 15, b) 4 R 23, c) 7 R 34, d) 8 R 32, Bonus: e) 8 R 75, f ) 7 R 53, g) 6 R 68, h) 8 R 11

Rounding the divisor to estimate the quotient. Write on the board:

49 1 6 7

ASK: What makes this problem harder than the other ones we’ve done already? (49 is not a multiple of 10) ASK: Is it close to a multiple of 10? (yes) Which one? (50) SAY: 49 is close to 50 so 49 will go into 167 about the same number of times as 50 does. ASK: How many times does 50 go into 167? (3) How did you get that? (because 5 goes into 16 three times) Write 3 as the quotient, then SAY: We don’t know for sure that 3 is the right quotient, we’re just guessing because 49 is so close to 50.

Exercises: Estimate the quotient by rounding the divisor.

a) 19 1 6 4 b) 38 2 5 1 c) 41 2 5 1 d) 81 3 4 2

Bonus

e) 79 5 8 1 f) 62 5 0 2 g) 91 5 5 7 h) 78 7 2 4

Answers: a) 8, b) 6, c) 6, d) 4, Bonus: e) 7, f) 8, g) 6, h) 9

Keep these exercises on the board for later.

Multiplying the divisor by the estimated quotient. Refer to the division on the board:

349 1 6 7

ASK: Now that we have a guess for what the quotient is, what’s the next step? (multiply the quotient by the divisor) Emphasize that it is not the rounded divisor you multiply by, but the actual divisor. SAY: We just used the 50 to find the 3; now that we have the 3, we can use it to finish the division. Ask a volunteer to multiply 49 × 3 on the board. Have another volunteer show where to put the answer in the division:

24 9

× 31 4 7

3

49 1 6 7- 1 4 7

Exercises: Multiply the estimated quotient by the divisor (not the rounded divisor) for the problems you estimated above.

a) 19 1 6 4 b) 38 2 5 1 c) 41 2 5 1 d) 81 3 4 2

Bonus e) 79 5 8 1 f) 62 5 0 2 g) 91 5 5 7 h) 78 7 2 4

Answers: a) 152, b) 228, c) 246, d) 324, Bonus: e) 553, f) 496, g) 546, h) 702

Exercises: Round the divisor to estimate the quotient, then multiply the divisor by your estimate.

a) 18 1 4 3 b) 52 2 7 4 c) 48 3 6 1 d) 31 1 9 4



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Bonus

e) 78 6 5 0 f) 71 4 4 4

Answers: a) 7, 126; b) 5, 260; c) 7, 336; d) 6, 186; Bonus: e) 8, 624; f) 6, 426

Finishing the long division. Refer to the example on the board:

349 1 6 7

- 1 4 72 0

SAY: We are sharing 167 objects among 49 groups. ASK: How many are in each group? (3) How many objects have been divided so far? (147) How many have not been divided? (20) Have a volunteer show where to put the answer. Add in the subtraction sign to emphasize that they got the answer by subtracting. ASK: Are we done? (yes) How do you know? (the ones have been divided)

Exercises: Subtract to finish the long division you started above.

a) 18 1 4 3 b) 52 2 7 4 c) 48 3 6 1 d) 31 1 9 4

Bonus e) 78 6 5 0 f) 71 4 4 4

Answers: a) 7 R 17, b) 5 R 14, c) 7 R 25, d) 6 R 8, Bonus: e) 8 R 26, f) 6 R 18

SAY: Now put all the steps together.

Exercises: Divide.

a) 327 ÷ 51 b) 184 ÷ 28 c) 148 ÷ 31 d) 211 ÷ 47

Bonus: e) 583 ÷ 62 f) 642 ÷ 91

Answers: a) 6 R 21, b) 6 R 16, c) 4 R 24, d) 4 R 23, Bonus: e) 9 R 25, f) 7 R 5

Extensions1. Without solving, predict the answer to 14 ÷ 1.4. Explain your prediction.

Then check your prediction using long division.

Answer: 14 is 10 times 1.4, so 14 ÷ 1.4 = 10. Using long division, 140 ÷ 14 = 10.

2. a) Divide 99 ÷ 0.9 and 99 ÷ 1.1. Which answer is greater than 99? Why does that make sense?

b) Predict which answer will be greater than 84, then check by long division: 84 ÷ 2.1 or 84 ÷ 0.3.

Answers: a) 99 ÷ 0.9 = 110 and 99 ÷ 1.1 = 90. It makes sense that 99 ÷ 0.9 > 99 ÷ 1 = 99 > 99 ÷ 1.1, since dividing by a smaller number gets a larger answer; b) 84 ÷ 0.3 should be greater than 84; 84 ÷ 0.3 = 280 and 84 ÷ 2.1 = 40

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Review the guess, check, and revise strategy. Give each student a copy of BLM Hundreds charts. Ask a volunteer to pick a number from 1 to 100 and circle it on the first hundreds chart. Have other students try to guess the answer. The volunteer is only allowed to answer “yes” or “no.” Students can use a pencil to cross out on their chart any number that got “no” so they know not to use it again. If students guess the correct number quickly, play again until it becomes clear that the strategy is not very effective.

Then change the rules. Tell students that now the volunteer is allowed to answer “too high” or “too low.” Have a different volunteer choose a number. This time, students can cross out all the numbers that are ruled out by the volunteer’s answer. For example, if 29 is too low, then so are 1 to 28.

Tell students that when mathematicians talk about the guess, check, and revise strategy, they don’t mean to check only if the guess is wrong, but how it’s wrong. That way, they can use the information to make a better guess.

Applying the guess, check, and revise strategy to division. Tell students that different people guessed the quotient for different divisions. Challenge students to decide whether the quotient guessed is too high or too low. Write on the board:

53 1 9- 1 5

4

93 2 5- 2 7but 27 > 25

92 1 7- 1 8but 18 > 17

62 1 5- 1 2

3

Point to the first one and ASK: We have 4 left over. Can we put one more in each group? (yes) Is 5 too low or too high? (too low) Point to the second one and ASK: With 9 objects in each group, we would place 27 objects, but we only have 25 objects. Is 9 too low or too high? (too high) Point to the third one and ASK: Is 9 too low or too high? (too high) How do you know?

NS6-61 2-Digit Division—Guess and check NS6-62 cumulative Review Pages 37–40

STANDARDS 6.NS.B.3

VOcABULARy decimaldecimal pointdividenddivisorestimateguess, check, and revise strategyquotientremainderrounding

GoalsStudents will use rounding to estimate the quotient when dividing by 2-digit numbers, including for cases in which doing so requires adjusting the estimate.


Can use long division to divide by 2-digit numbers when the divisor has easy times tables Can divide by decimals

MATERIALS

BLM Hundreds charts (p. J-90)

The Number System 6-61, 6-62


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(because 18 objects is too many—we only have 17) Repeat for the fourth one. (6 is too low because we have 3 left over, so we can put one more in each group) Write on the board:

a) 7

23 1 4 9- 1 6 1

b) 6

17 1 2 3- 1 0 2

2 7

c) 8

19 1 7 4- 1 5 2

2 2

d) 9

31 2 8 4- 2 7 9

5

e) 8

33 2 5 5- 2 6 4

Have students signal thumbs up for too high, thumbs down for too low, or flat hand for just right. (a) too high, b) too low, c) too low, d) just right, e) too high)

SAY: Now do the subtraction yourself to decide whether the estimate is too low, too high, or just right.

a) 5

48 2 9 1- 2 4 0

b) 9

32 2 8 5- 2 8 8

c) 7

58 4 6 5- 4 0 6

d) 6

28 1 9 3- 1 6 8

e) 4

47 2 3 5- 1 8 8

Allow students time to subtract, then have all students signal their answers. (a) 51, too low; b) too high; c) 59, too low; d) 25, just right; e) 47, too low)

Exercises: Multiply the estimated quotient with the divisor. Is the estimate too high, too low, or just right?

a) 7

36 2 9 8 b) 6

27 1 8 3 c) 6

73 4 3 5 d) 9

24 1 9 8

Bonus

e) 7

82 5 7 3 f) 6

68 4 5 5 g) 7

86 7 0 2 h) 6

71 4 2 2

Answers: a) too low, b) just right, c) too high, d) too high, Bonus: e) too high, f) just right, g) too low, h) too high

Revising the estimate. Write on the board:

723 1 5 6

- 1 6 1too high

623 1 5 6

823 1 5 6

ASK: What is a better next guess, 6 or 8? (6) How do you know? (even 7 was too much) Erase the “8” guess. Have a volunteer multiply 23 × 6 on the board. (138) Have another volunteer show where to put the answer and how to finish the division.

Exercises: Use the first estimate to make a better estimate. Then divide.

a) 3

19 7 8- 5 7

2 121 > 19, too low!

b) 4

42 1 6 4- 1 6 8

too high!

c) 6

72 4 3 0- 4 3 2

too high!

d) 7

75 6 3 8- 5 2 5

1 1 3113 > 75, too low!

Bonus

e) 7

35 2 9 5- 2 4 5

5 0too

f ) 5

53 2 6 2- 2 6 5

too

g) 7

66 5 4 4- 4 6 2

8 2too

h) 4

58 2 9 1- 2 3 2

5 9too

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Answers: a) 4 R 2, b) 3 R 38, c) 5 R 70, d) 8 R 38, e) 8 R 15, f ) 4 R 50, g) 8 R 16, h) 5 R 1

Exercises: Divide.

a) 850 ÷ 32 b) 231 ÷ 17 c) 654 ÷ 86 d) 632 ÷ 57 e) 984 ÷ 31

Answers: a) 26 R 18, b) 13 R 10, c) 7 R 52, d) 11 R 5, e) 31 R 23

Dividing 4-digit numbers by 2-digit numbers. Write on the board:

46 9,6 3 1 46 3,8 8 7

Have volunteers circle the first part of the number that is at least as big as 46. (96, 388) Then have volunteers start the long division by dividing the circled number by 46.

Exercises: Start the long division by dividing the circled number by 46. Then finish dividing.

a) 46 7,1 4 2 b) 46 1,8 9 4 c) 46 9,3 0 4 d) 46 2,9 6 1

Answers: a) 155 R 12, b) 41 R 8, c) 202 R 12, d) 64 R 17

Exercises: Divide using long division.

a) 49 6,5 3 2 b) 38 2,8 0 7 c) 47 4,7 8 1 d) 55 5,7 3 8

Answers: a) 133 R 15, b) 73 R 33, c) 101 R 34, d) 104 R 18

Dividing by 3-digit numbers. SAY: You can divide by 3-digit numbers the same way you divide by 2-digit numbers. Start by finding the first part of the dividend that is at least as big as the divisor.

Exercises: Copy these questions in your notebook in long division notation. Circle the first part of the dividend that is at least as big as the divisor.

a) 2,136 ÷ 512 b) 842,605 ÷ 512 c) 51,284,033 ÷ 512

d) 256,813 ÷ 357 e) 3,941,078 ÷ 357 f ) 35,725,321,098 ÷ 357

Answers: a) 2,136, b) 842, c) 512, d) 2,568, e) 394, f ) 357

SAY: Then you can round the divisor to the nearest hundred, and ask how many times the rounded divisor goes into the circled number. This will give you a guess to start the long division that you might have to adjust after.

Exercises: Do the long divisions for the problems above.

Answers: a) 4 R 88, b) 1,645 R 365, c) 100,164 R 62, d) 719 R 130, e) 11,039 R 155, f ) 100,070,927 R 159

Encourage students to check their answers by multiplication. Emphasize that doing so is very important when dividing multi-digit numbers because there is so much opportunity for making mistakes.

(MP.1)



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Review dividing decimals. Remind students that as long as they can divide whole numbers by whole numbers, they can divide decimals by decimals too. Write on the board:

34 1.7 6 5 2 8 3.4 1 7 6.5 2 8

Point to the first one and ASK: Once you do the long division, how do you know where to put the decimal point? PROMPT: Where do you put the decimal point when you divide by a whole number? (in the same place it is in the dividend) Have a volunteer place the decimal point.

Now draw students’ attention to the second example. SAY: Although you are not dividing by a whole number here, you can change it to a problem in which you are. ASK: How can you do that? (multiply both the divisor and dividend by 10) Demonstrate multiplying the divisor and dividend by 10, then have a volunteer place the decimal point in the quotient.

3 .4 1 7 6 .5 2 8 34 1,7 6 5 .2 8

.34 1, 7 6 5 .2 8

Have a volunteer demonstrate how to change this problem:

0.0 0 34 1 7 6 .5 2 8 34 1,7 6 5,2 8 0

Exercises: Create an equivalent problem that divides by a whole number. Then use the guess, check, and revise strategy to divide.

a) 8.4 9 8 1 .1 2 b) 0.0 0 27 3 7 .8 c) 5.7 7 6.3 8

Answers: a) 116.8, b) 14,000, c) 13.4


a) Lina has 4.2 pounds of cheese. She needs 0.12 pounds of cheese for each sandwich. How many sandwiches can she make?

b) Tom has $12.60. How many $0.45 candies can he buy?

c) A shelf is 41.4 inches long. How many 2.3 inch thick books can the shelf hold?

Bonus A quarter is about 0.18 cm thick. About how much is a stack worth that is 6.3 cm high?

Answers: a) 35, b) 28, c) 18, Bonus: 35 quarters are worth $8.75

Extensions1. Investigate: Are the estimates more likely to be correct when the

divisor is closer to the rounded number you used to make your estimate? For example, when the divisor is 31 rounded to 30, is your estimate more likely to be correct than when the divisor is 34 rounded to 30? Try these examples:

(MP.4)

(MP.1)

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31 2 4 3 31 2 4 9 31 2 5 7 31 2 6 5 31 2 7 4

34 2 4 3 34 2 4 9 34 2 5 7 34 2 6 5 34 2 7 4

Answer: Rounding 31 to 30 gives the right answer in all cases, except the first and last cases. Rounding 34 to 30 doesn’t give the right answer in any case. So rounding 31 to 30 gives the correct quotient more often than rounding 34 to 30.

2. Fill the boxes using the digits 3 to 9 once each.

×

Multiply the numbers using a calculator or long multiplication. Then give your partner the following problem:

(the product you found) ÷ (the 2-digit number)

Check that your partner gets the same 5-digit number you started with.

3. Decide how the first triangle was made. Then finish the second triangle using the same rule.

a) 20

13 7

9 4 3

6

4.2

2.52

b) 48

6 8

3 2 4

7.65

1.5

0.5

Answers: a) 6

4.2 1.8

2.52 1.68 0.12

b) 7.65

1.5 5.1

0.5 3.0 1.7

(MP.1)

(MP.7)


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