In the previous lesson, you learned what dividing by fractions means. In this lesson you will divide with fractions to solve problems. Take a look at this problem.
Charlie is growing vegetables in planters. He has 4 bags of soil and uses 2 ··
3 of a bag
of soil to fill each planter. How many planters can he fill?
Explore It
Use the math you already know to solve the problem.
Think of the number of planters that Charlie can fill as how many 2 ·· 3 s are in 4. Will that
number be greater than or less than 4? Explain your reasoning.
The model below represents the 4 bags of soil. Draw horizontal lines to divide each bag into thirds.
Circle and count groups of 2 ·· 3 in the model. How many did you circle?
Why do you circle groups of 2 ·· 3 to represent this problem?
How many planters can Charlie fill?
Explain how the model helped you solve the problem.
When you found the number of 2 ·· 3 s that are in 4, you were dividing. You are solving the
problem 4 4 2 ·· 3 . You can solve this problem by multiplying.
You know that multiplication and division are related. 4 divided by 2 is the same as 1 ·· 2 of 4,
or multiplying 4 by 1 ·· 2 .
4 4 2 2
4 1 ·· 2 2
When dividing with unit fractions, you learned that dividing 4 by 1 ·· 3 is the same as multiplying 4 by 3.
4 4 1 ·· 3 12
4 3 12
Dividing with any fraction works the same way. Dividing 4 by 2 ·· 3 is the same as
multiplying 4 by 3 ·· 2 .
4 4 2 ·· 3 6
4 3 ·· 2 12 ·· 2
6
You can solve any division problem using multiplication. To divide by any number, you can multiply by its multiplicative inverse, which is also known as the reciprocal.
Reflect
1 Explain how you can solve this division problem by using multiplication.
• Solve word problems using division of fractions.
• Write an equation to solve a problem using division of fractions.
• Write a story problem that will use division of fractions.
PReRequisite skiLLs
• Know that multiplication and division are inverse operations.
• Know that division is either fair sharing (partitive) or repeated subtraction (quotative).
• Divide with whole numbers.
• Divide a whole number by a fraction.
• Model division with manipulatives, diagrams, and story contexts.
vocabuLaRy
reciprocal: the multiplicative inverse of a number; with fractions, the numerator and denominator are switched
the LeaRning PRogRession
In Grade 5, students learn to understand fractions as division and to divide whole numbers by unit fractions.
In Lesson 6, students built upon the understanding from Grade 5 using models to show division of fractions. In this lesson, students continue to build upon their knowledge by using visual models and equations to divide whole numbers by fractions, fractions by fractions, and mixed numbers by fractions to solve word problems.
In Grade 7, students will continue their work with fractions to include all rational number operations.
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Prerequisite Skills 6.NS.1
Ready Lessons
Tools for Instruction
Interactive Tutorials ✓ ✓
ccLs Focus
6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by
using visual fraction models and equations to represent the problem. For example, create a story context for 2 ··
3 4 3
··
4 and use a
visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2 ··
3 4 3
··
4 5 8
··
9
because 3 ··
4 of 8
··
9 is 2
··
3 . 1 In general, a
·
b 4 c
·
d 5 ad
··
bc . 2 How much chocolate will each person get if 3 people share 1
··
2 lb of chocolate equally?
How many 3 ··
4 -cup servings are in 2
··
3 of a cup of yogurt? How wide is a rectangular strip of land with length 3
··
4 mi and area 1
··
2 square mi?
stanDaRDs FoR MatheMaticaL PRactice: SMP 1–4, 7, 8 (see page A9 for full text)
Students read a word problem and explore dividing a whole number by a fraction using a model.
steP by steP
• Tell students that this page models dividing whole numbers by fractions using a visual model.
• Have students read the problem at the top of the page.
• Work through Explore It as a class.
• Ask students how they determined whether the number of planters would be greater or less than 4.
Use the diagram on the page to review the words divisor, dividend, and quotient. Throughout the lesson, use models or manipulatives to demonstrate concepts and processes. Allow students to use the models to demonstrate their learning.
eLL support
sMP tip: Students map important quantities in the problem to the diagram as a way of understanding dividing with fractions (SMP 4). Students need many opportunities to explain the connections between different representations. Have students explain how the model helps them solve the problem.
Model dividing by a number less than 1.
• Draw 2 identical circles on the board. Write 2 4 1
··
4 .
• Ask, Is the divisor, 1 ··
4 , greater than or less than 1?
[less]
• How many 1 ··
4 s are in 2 circles? Let a volunteer draw
lines in the circles to show fourths. Ask, Which is
greater: the number of 1 ··
4 parts or the number of
whole circles? [The number of 1 ··
4 parts is more.]
• Write 1 ··
2 4 1
··
4 5 . Ask, Will the quotient be greater
than or less than 1? [greater than 1] Then let
a volunteer draw a model to illustrate.
visual Model • How does using a model help you solve the problem?
Students may answer that drawing the model helps them to see and count the groups.
• Explain, in your own words, dividing fractions with a model.
Responses should discuss drawing the dividend and dividing it into groups of the divisor.
• Why does it make sense that the quotient is greater than the dividend when you divide with a fraction less than 1? If the divisor is a fraction greater than 1, will the quotient be greater or less than the dividend?
Responses should show understanding that taking out groups that are less than one whole will mean that there are more groups than the dividend. Dividing by a fraction greater than 1 will result in fewer groups than the dividend.
in the previous lesson, you learned what dividing by fractions means. in this lesson you will divide with fractions to solve problems. take a look at this problem.
Charlie is growing vegetables in planters. He has 4 bags of soil and uses 2 ··
3 of a bag
of soil to fill each planter. How many planters can he fill?
explore it
use the math you already know to solve the problem.
Think of the number of planters that Charlie can fill as how many 2 ·· 3 s are in 4. Will that
number be greater than or less than 4? Explain your reasoning.
The model below represents the 4 bags of soil. Draw horizontal lines to divide each bag into thirds.
Circle and count groups of 2 ·· 3 in the model. How many did you circle?
Why do you circle groups of 2 ·· 3 to represent this problem?
How many planters can Charlie fill?
Explain how the model helped you solve the problem.
ccLs6.ns.1
the number of planters will be greater than 4. When you divide a given number
by a number less than 1, the answer will be greater than the given number.
6
each planter holds 2 ·· 3 of a bag of soil, so each circled group fills one planter.
you are trying to find how many 2 ·· 3 s are in 4.
Students explore solving a division problem using multiplication.
steP by steP
• Read Find Out More as a class.
• Remind students that a fraction and its reciprocal must have a product of 1.
• Point out that when you multiply fractions you multiply the numerators, then multiply the denominators, and then simplify if possible.
• Discuss Reflect. Guide students to think about dividing as being the same as multiplying by the reciprocal (multiplicative inverse).
use a model to understand using reciprocals in division.
• Help students understand why they can solve any
division problem by multiplying the dividend by
the reciprocal of the divisor.
• Draw this model on the board for 5 4 1 ··
3 :
13
1 1 1 1 1
• Explain that the expression can be read as “how
many groups of 1 ··
3 are in 5?” Show on the model
that 1 contains 3 groups of 1 ··
3 , and there are 5
groups of 1 in 5. The total number of groups of 1 ··
3
in 5 is simply 5 3 3 5 15.
visual Model
Ask students to think of everyday places or situations where people might need to divide by fractions. Encourage them to share their ideas with the class.
Examples: cooking, making crafts, measuring, building structures
When you found the number of 2 ·· 3 s that are in 4, you were dividing. You are solving the
problem 4 4 2 ·· 3 . You can solve this problem by multiplying.
You know that multiplication and division are related. 4 divided by 2 is the same as 1 ·· 2 of 4,
or multiplying 4 by 1 ·· 2 .
4 4 2 5 2
4 3 1 ·· 2 5 2
When dividing with unit fractions, you learned that dividing 4 by 1 ·· 3 is the same as multiplying 4 by 3.
4 4 1 ·· 3 5 12
4 3 3 5 12
Dividing with any fraction works the same way. Dividing 4 by 2 ·· 3 is the same as
multiplying 4 by 3 ·· 2 .
4 4 2 ·· 3 5 6
4 3 3 ·· 2 5 12 ·· 2
5 6
You can solve any division problem using multiplication. To divide by any number, you can multiply by its multiplicative inverse, which is also known as the reciprocal.
Reflect
1 Explain how you can solve this division problem by using multiplication.
6 4 2 ·· 3
Think of 2 as 2 ·· 1 . Dividing by 2 ·· 1 is the
same as multiplying by 1 ·· 2 .
Dividing by 1 ·· 3 is the same as
multiplying by 3 ·· 1 or 3.
Dividing by 2 ·· 3 is the same as
multiplying by 3 ·· 2 .
Dividing by 2 ·· 3 is the same as multiplying by 3 ·· 2 . so, you can solve 6 divided by 2 ·· 3
Students read a word problem and explore how to divide a whole number by a fraction using a bar picture and by modeling the problem using words and an equation.
steP by steP
• Read the problem at the top of the page as a class.
• Read and discuss Picture It. Ask, What does the shaded part of the bar show? [how much of the whole bottle Kelly drank, or two fifths]
• Read and discuss Model It. Walk through each step to be sure students understand how to use the inverse operation.
sMP tip: Students reason abstractly when they analyze a problem and represent it as an equation with a missing factor in order to find a solution (SMP 2). Ask students to explain how their equations or models represent the context of the problem.
Make a model to show how many 2 ·· 3 s are in 4.
Materials: sheet of paper for each student, pencils
• Tell students they will make a model to show how
many 2 ··
3 s are in 4.
• Give each student a sheet of paper. Have them fold it into 4 equal parts and draw lines on the folds to show 4 equal sections. Tell them this will be a model for 4.
• Ask, What is the first thing you could do to the model
to start showing how many 2 ·
3 s are in 4? [Draw lines to
divide each of the 4 whole sections into thirds.]
What might you do next? [Circle each group of 2 of
the thirds and count them.]
• Let students finish their models. Ask them what
the solution is. Suggest they label their model
with the equation solved: 4 4 2 ··
3 5 6.
hands-on activity
• What is another way you could solve the problem regarding Kelly’s water bottle on page 60?
Responses may include dividing 3 cups in half
to find 1 ··
5 of the amount in the bottle, and then
multiplying by 5 to find the total amount in the
bottle.
• How is dividing fractions similar to dividing whole numbers?
Listen for responses that indicate dividing a quantity into groups.
Students revisit the problem on page 60 to learn how to solve it using the bar picture and the equation model.
steP by steP
• Read Connect It as a class. Be sure to point out that the problems refer to the problem on page 60.
• For problem 4, remind students to change 1 1 ··
2 to an
improper fraction before multiplying by 5. Multiply
the denominator times the whole number, and then
add the numerator. The result is the numerator, and
the denominator stays the same.
• In problem 5, review how to change an improper fraction to a mixed number: Divide the numerator by the denominator to get a whole number, the remainder is the new numerator, and the denominator stays the same.
• Have students work through Try It on their own. Then discuss with them how they solved the problems.
tRy it soLutions
7 Solution: 8 servings; Students solve the problem by
using the equation 12 4 3 ··
2 5 ? or a drawing such
as 12 cups with lines dividing each cup into halves
and circled groups of 1 1 ··
2 .
8 Solution: 30 minutes; Students may use the equation
9 4 3 ··
10
5 ? They may use a drawing such as a bar
divided into tenths with 3 ··
10
shaded to model the
distance she went in 9 minutes. This would show
that 1 ··
10
is equal to 3 minutes; 3 3 10 5 30 minutes.
ERROR ALERT: Students who wrote 18 servings forgot to multiply by the reciprocal of the divisor. Remind them that they need to find the reciprocal of the divisor before multiplying.
Students read a word problem and explore how to divide a fraction by a fraction using a double number line and by modeling the problem using words and an equation.
steP by steP
• Read the problem at the top of the page as a class.
• Discuss Picture It:
Ask, What does the top number line represent? [the distance divided into fourths]
Ask, What does the bottom number line represent? [the distance divided into eighths]
• Discuss Model It. Ask, What does the question mark in
the equation represent? [the number of hurdles Eli
jumped over during his 3 ··
4 -mile run]
• Which method for dividing fractions do you prefer? Why? Are there situations when one method may be easier to use than another?
Encourage students to support their opinions and to listen to the opinions of others. Point out that there is no correct answer, and that different students may have different preferences.
• Can you think of another way to describe dividing fractions? Explain.
Encourage students to suggest ideas or knowledge on other ways to divide fractions.
Students revisit the problem on page 62 and solve it using the double number line and the equation model.
steP by steP
• Read and discuss Connect It as a class. Refer to the problem on page 62.
• For problem 13, remind students that they should find the inverse (reciprocal) only of the divisor and not the dividend. Also remind them they should simplify improper fractions to a mixed- or whole-number answer.
• Have students work through Try It on their own. Then discuss their answers and solutions.
tRy it soLutions
15 Solution: 8; Students may draw a double number line or may use the standard algorithm to solve the problem.
2 ··
3 4 1
··
12 5 2
··
3 3 12
··
1 5 24
··
3 5 8
16 Solution: 5; Students may draw a bar picture and shade half of the figure, then draw lines to divide the figure into tenths. 5 parts would be shaded. Or, they may use the standard algorithm.
1 ··
2 4 1
··
10 5 1
··
2 3 10
··
1 5 5
sMP tip: Give students multiple opportunities to solve and model problems. Students use repeated reasoning to understand algorithms and make generalizations about patterns (SMP 8).
ERROR ALERT: Students who wrote 18 pieces of rope may have multiplied the inverse of both the dividend and the divisor. Remind students that they should only multiply by the inverse (reciprocal) of the divisor. Review Find Out More on page 59 of this lesson to help students understand why multiplying by the reciprocal is mathematically valid.
Students explore how to divide a mixed number by a fraction using bar pictures and by modeling the problem using words and an equation.
steP by steP
• Read the problem at the top of the page as a class.
• Discuss Picture It:
Ask, How do the shaded bars in the first picture
represent 1 4 ··
5 pounds of granola? [One whole and 4
more parts are shaded.]
Ask, What does each circle in the second picture
represent? [Each circle represents one 2 ··
5 -pound bag
of granola.]
• In Picture It, some students might be confused by the
circle that contains part of the whole bar and part of
the partial bar. Suggest to students that they think
of the entire 1 bar plus 4 ··
5 bar as one entity.
• Discuss Model It. Remind students that they must write the mixed number in the dividend as an improper fraction before dividing.
• Mari is selling bags of granola to raise money, so she wants to have as many bags as possible to sell. What else might she think about when dividing up the granola?
Listen for responses that show students making connections to personal experiences to make sense of the problem. They might discuss how the size of each bag might make a difference to buyers: Customers might not buy if they think there is not enough granola in a bag. Students might also mention cost to customers or the amount left over after filling the bags.
Students revisit the problem on page 64 and solve it using the bar picture and the equation model.
steP by steP
• Discuss Connect It as a class. Point out that Connect It refers to the problem on the previous page.
• When discussing problem 20, be sure students make the connection between the bar model and the mathematical process for changing a mixed number to an improper fraction. For students having trouble understanding why writing a mixed number as an improper fraction is mathematically valid, ask students to count the total number of shaded squares (9) in the first bar picture and point out that the bars are divided into fifths, so there are 9 fifths altogether.
• Have students work through Try It on their own. Let volunteers share their solutions and answers with the class. Clear up misconceptions and discuss any questions students may have.
Students practice solving problems that require dividing a whole number by a fraction, dividing a fraction by a fraction, and dividing a mixed number by a fraction.
steP by steP
• Ask students to solve the problems individually on pages 66 and 67.
• In the student model at the top of the page, call attention to writing each mixed number as an improper fraction before multiplying.
• When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.
soLutions
Ex Using a model of words and an equation is one way for students to show their solution to the problem.
26 Solution: Lexi will use 5 ··
6 pound of seeds for the
entire garden; Students could solve the problem by
using an equation. 1 ··
2 4 3
··
5 5 1
··
2 3 5
··
3 5 5
··
6
27 Solution: Each person will need to run 6 11 ··
20
miles;
Students could solve the problem by using an
equation.
131 ···
5 4 4 5 131
···
5 3 1
··
4 5 131
···
20 5 6 11
··
20
28 Solution: B; Students must recognize the language as a division problem “how many are in ?”
Explain to students why the other two answer choices are not correct:
Students divide by fractions to solve word problems that might appear on a mathematics test.
steP by steP
• First, tell students that they will divide by fractions to solve word problems. Then have students read the directions and answer the problems independently. Remind students to fill in the correct answer choices on the Answer Form.
• After students have completed the Common Core Practice problems, review and discuss correct answers. Have students record the number of correct answers in the box provided.
soLutions
1 Solution: D; 3 ··
8 4 3
··
2 5 3
··
8 3 2
··
3 5 6
··
24 5 1
··
4
2 Solution: B; 5 3 ··
4 4 1
··
2 5 23
··
4 3 2
··
1 5 46
··
4 5 11 1
··
2 , or
11 books. Students should realize that you can’t
have half of a book.
3 Solution: A; Students should reason that when
a number is divided by a number less than 1, the
quotient will be greater than 1. Students may also
eliminate the other choices or work the problem.
3 ··
4 4 1
··
2 5 3
··
4 3 2
··
1 5 6
··
4 5 1 1
··
2
4 Solution: C; transposed the dividend and divisor.
Materials: index cards or sheets of paper, pencils
Give each student 1–3 index cards or sheets of paper. Tell them to make up a word problem involving division of fractions and mixed numbers for each card. Have students write the word problem on one side of the card and the solution on the other side. Tell them their solution needs to be complete enough so that someone who doesn’t know how to solve the problem can figure it out and why it works. The solution can include such things as a drawing, words explaining the process, or an equation. Have students exchange cards with another student. The other student is to solve the problem and then look at the solution and offer suggestions for changes if the student sees any. Alternatively, let students verbally show and tell how to solve the problems they’ve made up.
challenge activityMake a number line to model division.
Materials: half or whole sheets of paper, pencils
Tell students they will draw a model to solve a
problem. Display this problem: “Mari has 1 1 ··
2 hours
left to prepare for the bake sale. It takes her 1 ··
4 hour to
prepare each item. How many items can she
prepare?” Tell students to draw a number line on
their paper and use tic marks to divide the line into
halves from 0 to 2. They should label each whole and
half number mark (0, 1 ··
2 , 1, 1 1
··
2 , 2). Point out that the
problem asks how many fourths are in one and a half.
Tell students to divide each half on their number line
with a tic mark to create a fourth. Then tell them
to circle each fourth between 0 and 1 1 ··
2 . Ask students
what each circled part represents 3 1 ··
4 hour 4 and how
many circled parts there are [6]. Ask how many
fourths are in one and a half [6]. Ask how many items
Mari can prepare in the time she has left. [6 items]
hands-on activity
• Ask students to evaluate the expression 1 1 ··
2 4 2
··
3 3 2 1
··
4 4 .
• For students who are struggling, use the chart below to guide remediation.
• After providing remediation, check students’ understanding. Ask students to explain their thinking while
evaluating 2 1 ··
3 4 3
··
4 3 3 1
··
9 4 .
• If a student is still having difficulty, use Ready Instruction, Level 6, Lesson 6.
if the error is . . . students may . . . to remediate . . .
1 have failed to find the multiplicative inverse (reciprocal) of the divisor before multiplying.
3 ·· 2 3 2 ·· 3 5 6 ·· 6 5 1
Remind students that they must multiply by the multiplicative
inverse of the divisor. For students not understanding why this
works, write 6 4 1 ·· 4 . Draw 6 circles. Ask how to divide each circle
by 1 fourth. Ask how many fourths altogether. Point out that
6 wholes are each split into 4 parts, so 6 wholes are multiplied by
4 parts to get 24. 6 3 4 is the same as 6 ·· 1 4 1 ·· 4 .
3 ·· 4 have forgotten to include the whole number.
1 ·· 2 4 2 ·· 3 5 1 ·· 2 3 3 ·· 2 5 3 ·· 4
Encourage students to write all mixed numbers as improper fractions as the first step in setting up their computation so they won’t forget.
