Unit 4 Dividing Fractions — Unit Plan

TitleTakeawaysVisual / Anchor ChartAssessment
Lesson 1
Size of Divisor and Size of Quotient
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Result of Division (1 problem)

Without computing, decide whether the value of each expression is much smaller than 1,
close to 1, or much larger than 1.

  1. 1,000,001÷991,000,001 \div 99
  2. 3.7÷4.23.7 \div 4.2
  3. 1÷8351 \div 835
  4. 100÷1100100 \div \frac{1}{100}
  5. 0.006÷6,0000.006 \div 6,000
  6. 50÷501450 \div 50\frac14
Show Solution
  1. much larger than 1
  2. close to 1
  3. much smaller than 1
  4. much larger than 1
  5. much smaller than 1
  6. close to 1
Lesson 2
Meanings of Division
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Groups on a Field Trip (1 problem)

During a field trip, 60 students are put into equal-size groups.

  1. Describe two ways to interpret 60÷560 \div 5 in this situation.

  2. Find the value of the expression. Explain what it could mean in this situation.
  3. Write a multiplication equation that can describe the same situation.
Show Solution
  1. 60÷560 \div 5 could represent:
    • "How many students are in each group if there are 5 groups?"
    • "How many groups can be formed if there are 5 students per group?")
  2. 12. It could mean that there are 12 students in each of the 5 groups, or that there are 12 groups with 5 students in each group.
  3. Any of the following equations are acceptable:
    • 5⋅12=605 \boldcdot 12 = 60
    • 12⋅5=6012 \boldcdot 5 = 60
    • 5⋅?=605 \boldcdot {?} = 60

    • ?⋅5=60? \boldcdot 5 = 60

Lesson 3
Interpreting Division Situations
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Rice in Bags (1 problem)

Andre poured 27 ounces of rice into 6 bags. If all bags have the same amount of rice, how many ounces are in each bag?

  1. Write an equation to represent the situation. Use a "?" to represent the unknown quantity.
  2. Find the unknown quantity. Show your reasoning.
Show Solution
  1. Sample responses:
    • 6⋅?=276 \boldcdot ? = 27
    • ?⋅6=27? \boldcdot 6 = 27
    • 27÷6=?27 \div 6 = ?
  2. 4124\frac{1}{2} ounces (or equivalent). Sample reasoning:
    •  
    • If Andre put 4 ounces in each bag, that’s 24 ounces in 6 bags. Splitting the remaining 3 ounces into 6 bags means putting 12\frac{1}{2} ounce more in each bag.

    • If there were 3 bags, each bag would have 27÷327 \div 3 or 9 ounces. Splitting each 9 ounces into 2 bags gives 6 bags with 4.5 ounces in each.

Section A Check
Section A Checkpoint
Problem 1

Han arranged 28 photos in a photo album. He put the same number of photos on each page.

  1. What can the expression 28÷728\div 7 mean in this situation? Describe two ways to interpret it.

  2. Write a multiplication equation that can describe the same situation.
Show Solution
  1. 28÷728 \div 7 could represent:
    • How many photos did Han put on each page if he placed 28 photos on 7 pages?
    • How many pages did Han use if he placed 7 photos on each page and 28 photos in total?
  2. Any of the following equations are acceptable:
    • 7⋅?=287 \boldcdot {?} = 28
    • 7⋅4=287 \boldcdot 4 = 28
    • ?⋅7=28{?} \boldcdot 7 = 28
    • 4⋅7=284 \boldcdot 7 = 28
Problem 2
Select all representations that describe the same relationship as 6⋅?=546 \boldcdot {?} = 54 does.
Show Solution
A, C, E, F
Lesson 4
How Many Groups? (Part 1)
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Halves, Thirds, and Sixths (1 problem)
  1. The hexagon represents 1 whole.

    Four pattern blocks: One large yellow hexagon, one blue rhombus, one red trapezoid, and one green triangle.

    Draw a pattern-block diagram that represents the equation 4⋅13=1134 \boldcdot \frac13= 1\frac 13.

  2. Answer the following questions. If you get stuck, consider using pattern blocks.

    1. How many 12\frac12s are in 3123\frac12?
    2. How many 13\frac13s are in 2232\frac23?
    3. How many 16\frac16s are in 23\frac23?
Show Solution
  1.  
    <p>Pattern blocks.</p>
    1. There are seven 12\frac12s in 3123\frac12.
    2. There are eight 13\frac13s in 2232\frac23.
    3. There are four 16\frac16s in 23\frac23.
Lesson 5
How Many Groups? (Part 2)
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Two-fifths in 4 (1 problem)

How many 25\frac{2}{5}s are in 4?

  1. Answer the question and show your reasoning.

  2. Select all equations that represent the situation.

    1. 4⋅25=?4 \boldcdot \frac25={?}
    2. ?⋅25=4{?} \boldcdot \frac25=4
    3. 25÷4=?\frac25 \div 4={?}
    4. 4÷25=?4 \div \frac25 ={?}
    5. ?÷25=4{?} \div \frac25 = 4
Show Solution
  1. 10. Sample reasoning:
    • Ten groups of 25\frac{2}{5} make 205\frac{20}{5}, which is 4.
    • There are 20 fifths in 4, so that means 10 groups of two-fifths.
  2. B and D
Lesson 6
Using Diagrams to Find the Number of Groups
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How Many in 2? (1 problem)

How many 34\frac34s are in 2?

  1. Write a multiplication equation and a division equation that can be used to answer the question.

  2. Draw a tape diagram, and answer the question. Use the grid to help you draw, if needed.

    A blank grid with a height of 7 units and length of 16 units.

Show Solution
  1. ?⋅34=2? \boldcdot \frac34 = 2. 2÷34=?2 \div \frac34 = ?
  2. There are two and two-thirds 34\frac 34s in 2.

<p>Fraction bar diagram. </p>

Lesson 7
What Fraction of a Group?
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A Partially Filled Fish Tank (1 problem)

There are 6 gallons of water in a 20-gallon fish tank. What fraction of the tank is filled?

  1. Write a multiplication equation and a division equation to represent the situation.

  2. Answer the question. You can draw a tape diagram if you find it helpful.

Show Solution

  1. ?⋅20=6? \boldcdot 20 =6 and 6÷20=?6 \div 20 = ?

  2. 620\frac{6}{20} or 310\frac{3}{10} of the tank

Lesson 8
How Much in Each Group? (Part 1)
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Ice Cubes and Bus Seats (1 problem)

Answer each question, and show your reasoning.

  1. Kiran filled 1121\frac{1}{2} ice trays with water and made 24 ice cubes. How many ice cubes are in 1 ice tray?
  2. There are 24 people on a bus. They fill 25\frac{2}{5} of the seats on the bus. How many seats are on the bus?
Show Solution

  1. 16. Sample reasoning:

  2. 60. Sample reasoning: If 25\frac{2}{5} of the number of seats is 24, then 15\frac{1}{5} of it is 12, and all of it is 5⋅125 \boldcdot 12, which is 60.
Lesson 9
How Much in Each Group? (Part 2)
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Refilling a Soap Dispenser (1 problem)

Noah is filling soap dispensers with liquid soap. He fills 3123\frac{1}{2} dispensers with 78\frac{7}{8} gallon of liquid soap.

How much liquid soap fills 1 dispenser? Show your reasoning.

Show Solution

28\frac{2}{8} or 14\frac{1}{4} gallon. Sample reasoning:

  • If 78\frac{7}{8} gallon fills 3123\frac{1}{2} dispensers, then 18\frac{1}{8} gallon fills 12\frac{1}{2} dispenser, and 28\frac{2}{8} gallon fills 1 dispenser.
  •  

Section B Check
Section B Checkpoint
Problem 1

An artist is making a paste for a sculpture. She uses 85\frac{8}{5} kilograms of flour to make 23\frac{2}{3} of a batch. How much flour is needed to make a full batch?

  1. Draw a diagram and label it to represent the situation.
  2. Find the answer and show your reasoning.
Show Solution
  1. Sample response:

  2. 125\frac{12}{5} or 2252\frac{2}{5} kilograms. Sample reasoning: If there are 85\frac{8}{5} kilograms in 23\frac{2}{3} of a batch, then there is 45\frac{4}{5} kilogram in 13\frac{1}{3} of a batch and 125\frac{12}{5} kilograms in 1 whole batch.
Problem 2

For each experiment, a scientist needs 310\frac{3}{10} liter of a liquid. If the scientist has 4124 \frac{1}{2} liters of the liquid, how many experiments can be done?

  1. Write a multiplication equation and a division equation to represent the question. Use a “?” for the unknown value.
  2. Explain or show that the answer is 15 experiments.
Show Solution
  1. ?⋅310=412{?} \boldcdot \frac{3}{10} = 4\frac{1}{2} and 412÷310=?4\frac{1}{2} \div \frac{3}{10} = {?}
  2. Sample response: 
    • 15⋅310=451015 \boldcdot \frac{3}{10} = \frac{45}{10}, which is 4124\frac{1}{2}.
    • Ten experiments can be done with 3 liters (10⋅310=310 \boldcdot \frac{3}{10} = 3) and 5 more can be done with 1121\frac{1}{2} liters.
    • There are 45 tenths in 4124\frac{1}{2} and there are 15 groups of 3 tenths in 45 tenths.
Lesson 10
Dividing by Unit and Non-Unit Fractions
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Dividing by $\frac13$ and $\frac35$ (1 problem)

  1. Explain or show how you could find 5÷135 \div \frac 13. You can use this diagram if it is helpful.

    A tape diagram of five equal parts. From the beginning of the diagram to the end of the diagram a brace is drawn and labeled 5.

  2. Find 12÷3512 \div \frac 35. Try not to use a diagram, if possible. Show your reasoning.
Show Solution
  1. Sample reasoning: 5÷135 \div \frac 13 can mean “How many 13\frac 13s (thirds) are in 5?” There are 3 thirds in 1, so in 5, there are 5 times as many thirds. Five times as many is 5⋅35 \boldcdot 3, so there are 15 thirds in 5.

    <p>Fraction bar diagram.</p>

  2. 20. Sample reasoning: 12÷35=12⋅5⋅13=2012 \div \frac 35 = 12 \boldcdot 5 \boldcdot \frac 13 = 20
Lesson 11
Using an Algorithm to Divide Fractions
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Finding Quotients of Fractions (1 problem)

Calculate each quotient. Show your reasoning.

  1. 2425÷45\frac{24}{25} \div \frac{4}{5}

  2. 4÷274 \div \frac{2}{7}
Show Solution
  1. 65\frac{6}{5} (or equivalent). Sample reasoning:
    • 45\frac{4}{5} is 2025\frac{20}{25}. There is 1 full group of 2025\frac{20}{25} in 2425\frac{24}{25}. The leftover 425\frac{4}{25} is 15\frac{1}{5} of a group. There is a total of 1151\frac{1}{5} groups.
    • 2425⋅54=120100=65\frac{24}{25} \boldcdot \frac{5}{4} = \frac{120}{100} = \frac{6}{5}
  2. 14 (or equivalent). Sample reasoning:
    • 4⋅72=282=144 \boldcdot \frac{7}{2} = \frac{28}{2} = 14
    • There are 28 one-sevenths in 4 so there are half as many two-sevenths. Half of 28 is 14.
Section C Check
Section C Checkpoint
Problem 1
Select all the expressions that can give the value of 6÷326 \div \frac{3}{2}.
Show Solution
B, C, D, F
Problem 2

Calculate each quotient. Show your reasoning.

  1. 98÷32\frac{9}{8} \div \frac{3}{2}

  2. 2110÷152 \frac{1}{10} \div \frac{1}{5}
Show Solution
  1. 34\frac{3}{4} (or equivalent). Sample reasoning:
    • 98÷32=98⋅23=1824=34\frac{9}{8} \div \frac{3}{2} = \frac{9}{8} \boldcdot \frac{2}{3} = \frac{18}{24} = \frac{3}{4}
    • The quotient is the unknown factor in the multiplication equation 32⋅?=98\frac{3}{2} \boldcdot {?} = \frac{9}{8}. That number is 34\frac{3}{4}.
  2. 212\frac{21}{2} or 101210\frac{1}{2} (or equivalent). Sample reasoning:
    • 2110÷15=2110⋅5=10510=10510\frac{21}{10} \div \frac{1}{5} = \frac{21}{10} \boldcdot 5 = \frac{105}{10} = 10\frac{5}{10}
    • There are 10 groups of 15\frac{1}{5} in 2 wholes, and 110\frac{1}{10} is 12\frac{1}{2} a group of 15\frac{1}{5}, so there are 101210\frac{1}{2} groups of 15\frac{1}{5} in 21102\frac{1}{10}.
Lesson 12
Fractional Lengths
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Building A Fence (1 problem)

A builder was building a fence. In the morning, he worked for 25\frac25 of an hour. In the afternoon, he worked for 910\frac{9}{10} of an hour. How many times as long as in the morning did he work in the afternoon?

Write a division equation to represent this situation, then answer the question. Show your reasoning. If you get stuck, consider drawing a diagram.

Show Solution

Division equation: 910÷25=?\frac {9}{10} \div \frac 25 = {?} (or 910÷?=25\frac {9}{10} \div {?} = \frac 25). In the afternoon, he worked 2142\frac14 times as long as he did in the morning. Sample reasoning: 910÷25= 910⋅52=4520=94\frac {9}{10} \div \frac 25 = \frac {9}{10} \boldcdot \frac 52 = \frac {45}{20} = \frac94.

Lesson 13
Rectangles with Fractional Side Lengths
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Two Frames (1 problem)

Two rectangular picture frames have the same area of 45 square inches but have different side lengths. Frame A has a length of 6346 \frac34 inches, and Frame B has a length of 7127\frac12 inches.

  1. Without calculating, predict which frame has the shorter width. Explain your reasoning.
  2. Find the width that you predicted to be shorter. Show your reasoning.
Show Solution
  1. Frame B has a longer length, so its width is shorter if the two pairs of side lengths produce the same product of 45.
  2. 6 inches. Sample reasoning: 45÷712 =45÷152=45⋅215=645\div 7\frac12 = 45 \div \frac {15}{2} = 45 \boldcdot \frac {2}{15} = 6 
Lesson 14
Fractional Lengths in Triangles and Prisms
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Triangles and Cubes (1 problem)
    1. How many cubes with edge lengths of 13\frac 13 inch are needed to build a cube with an edge length of 1 inch?
    2. What is the volume, in cubic inches, of one cube with an edge length of 13\frac 13 inch?
  2. A triangle has a base of 3253\frac{2}{5} (or 175\frac{17}{5}) inches and an area of 51105\frac{1}{10} (or 5110\frac{51}{10}) square inches. Find the height of the triangle. Show your reasoning.
Show Solution
    1. 27 cubes
    2. 127\frac{1}{27} in3
  2. 3 inches. Sample reasoning: 12⋅175⋅h=5110\frac{1}{2} \boldcdot \frac{17}{5} \boldcdot h = \frac{51}{10}, so 1710⋅h=5110\frac{17}{10} \boldcdot h = \frac{51}{10}. There are 3 groups of 1710\frac{17}{10} in 5110\frac{51}{10}.
Lesson 15
Volume of Prisms
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Storage Box (1 problem)

A storage box has a base that measures 3 inches by 4 inches and a height of 1121\frac{1}{2} inches. The box can be packed with 144 cubes with an edge length of 12\frac{1}{2} inch.

  1. Find the volume of the box in cubic inches. Show your reasoning.

  2. Describe a different way to find the volume of the box. (It is not necessary to do the calculation.)
Show Solution
  1. 18 cubic inches. Sample reasoning: 3⋅4⋅112=183 \boldcdot 4 \boldcdot 1\frac{1}{2} = 18
  2. Sample response: Find the volume of a 12\frac{1}{2}-inch cube and multiply it by 144. The volume of 1 cube is 13\frac{1}{3} cubic inch, so the volume of the prism is 144⋅18144 \boldcdot \frac{1}{8}, which is 1448\frac{144}{8} (or 18) cubic inches.
Section D Check
Section D Checkpoint
Problem 1
A rectangular piece of paper has an area of 5585\frac{5}{8} square feet and a side length of 1141\frac{1}{4} feet. What is its width in feet?

Show Solution
4124\frac{1}{2} feet.
Problem 2

A rectangular prism that measures 2122\frac{1}{2} inches in length, 2 inches in width, and 3 inches in height is packed with 12\frac{1}{2}-inch cubes.

Select all the strategies for finding the volume of the prism in cubic inches.

Show Solution
B, D, E
Problem 3

A pool in the shape of a rectangular prism holds 11 cubic meters of water. The area of the base of the pool is 8458\frac{4}{5} square meters.

What is the height of the water in meters? Show your reasoning.

Show Solution
1141\frac{1}{4} meters. Sample reasoning: 8458\frac{4}{5} is 445\frac{44}{5}. Dividing the volume by the area of the base gives the height: 11÷445=11⋅544=5544=5411 \div \frac{44}{5} = 11 \boldcdot \frac{5}{44} = \frac{55}{44} = \frac{5}{4}
Lesson 16
Solving Problems Involving Fractions
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A Box of Pencils (1 problem)

A box of pencils is 5145\frac14 inches wide. Seven pencils, laid side by side, take up 2582 \frac 58 inches of the width.

  1. How many inches of the width of the box is not taken up by the pencils? Explain or show your reasoning.

  2. All 7 pencils have the same width. How wide is each pencil? Explain or show your reasoning.
Show Solution
  1. 2582 \frac 58 inches, because 514− 258=2585\frac14 - 2 \frac 58 = 2 \frac 58
  2. 38\frac 38 inch, because 258÷7= 218 ⋅17=382 \frac 58 \div 7 = \frac {21}{8} \boldcdot \frac17 = \frac38
Lesson 17
Fitting Boxes into Boxes
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No cool-down
Unit 4 Assessment
End-of-Unit Assessment