Unit 4 Dividing Fractions — Unit Plan

Title	Assessment
Lesson 1 Size of Divisor and Size of Quotient	Result of Division Without computing, decide whether the value of each expression is much smaller than 1, close to 1, or much larger than 1. $1,000,001 \div 99$ $3.7 \div 4.2$ $1 \div 835$ $100 \div \frac{1}{100}$ $0.006 \div 6,000$ $50 \div 50\frac14$ Show Solution much larger than 1 close to 1 much smaller than 1 much larger than 1 much smaller than 1 close to 1
Lesson 2 Meanings of Division	Groups on a Field Trip During a field trip, 60 students are put into equal-size groups. Describe two ways to interpret $60 \div 5$ in this situation. Find the value of the expression. Explain what it could mean in this situation. Write a multiplication equation that can describe the same situation. Show Solution $60 \div 5$ 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?") 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. Any of the following equations are acceptable: $5 \boldcdot 12 = 60$ $12 \boldcdot 5 = 60$ $5 \boldcdot {?} = 60$ $? \boldcdot 5 = 60$
Lesson 3 Interpreting Division Situations	Rice in Bags 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? Write an equation to represent the situation. Use a "?" to represent the unknown quantity. Find the unknown quantity. Show your reasoning. Show Solution Sample responses: $6 \boldcdot ? = 27$ $? \boldcdot 6 = 27$ $27 \div 6 = ?$ $4\frac{1}{2}$ 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 $\frac{1}{2}$ ounce more in each bag. If there were 3 bags, each bag would have $27 \div 3$ 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. What can the expression $28\div 7$ mean in this situation? Describe two ways to interpret it. Write a multiplication equation that can describe the same situation. Show Solution $28 \div 7$ 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? Any of the following equations are acceptable: $7 \boldcdot {?} = 28$ $7 \boldcdot 4 = 28$ ${?} \boldcdot 7 = 28$ $4 \boldcdot 7 = 28$ Problem 2 Select all representations that describe the same relationship as $6 \boldcdot {?} = 54$ does. A.A farmer placed 54 eggs into cartons. She placed 6 eggs in each carton. B. ${?}\div 54 = 6$ C. D.Kiran has 6 bags of marbles with 54 marbles in each. E. $54 \div {?} = 6$ F. $54 \div 6 = {?}$ Show Solution A, C, E, F

Lesson 4 How Many Groups? (Part 1)	Halves, Thirds, and Sixths The hexagon represents 1 whole. Draw a pattern-block diagram that represents the equation $4 \boldcdot \frac13= 1\frac 13$ . Answer the following questions. If you get stuck, consider using pattern blocks. How many $\frac12$ s are in $3\frac12$ ? How many $\frac13$ s are in $2\frac23$ ? How many $\frac16$ s are in $\frac23$ ? Show Solution There are seven $\frac12$ s in $3\frac12$ . There are eight $\frac13$ s in $2\frac23$ . There are four $\frac16$ s in $\frac23$ .
Lesson 5 How Many Groups? (Part 2)	Two-fifths in 4 How many $\frac{2}{5}$ s are in 4? Answer the question and show your reasoning. Select all equations that represent the situation. $4 \boldcdot \frac25={?}$ ${?} \boldcdot \frac25=4$ $\frac25 \div 4={?}$ $4 \div \frac25 ={?}$ ${?} \div \frac25 = 4$ Show Solution 10. Sample reasoning: Ten groups of $\frac{2}{5}$ make $\frac{20}{5}$ , which is 4. There are 20 fifths in 4, so that means 10 groups of two-fifths. B and D
Lesson 6 Using Diagrams to Find the Number of Groups	How Many in 2? How many $\frac34$ s are in 2? Write a multiplication equation and a division equation that can be used to answer the question. Draw a tape diagram, and answer the question. Use the grid to help you draw, if needed. Show Solution $? \boldcdot \frac34 = 2$ . $2 \div \frac34 = ?$ There are two and two-thirds $\frac 34$ s in 2.
Lesson 7 What Fraction of a Group?	A Partially Filled Fish Tank There are 6 gallons of water in a 20-gallon fish tank. What fraction of the tank is filled? Write a multiplication equation and a division equation to represent the situation. Answer the question. You can draw a tape diagram if you find it helpful. Show Solution $? \boldcdot 20 =6$ and $6 \div 20 = ?$ $\frac{6}{20}$ or $\frac{3}{10}$ of the tank
Lesson 8 How Much in Each Group? (Part 1)	Ice Cubes and Bus Seats Answer each question, and show your reasoning. Kiran filled $1\frac{1}{2}$ ice trays with water and made 24 ice cubes. How many ice cubes are in 1 ice tray? There are 24 people on a bus. They fill $\frac{2}{5}$ of the seats on the bus. How many seats are on the bus? Show Solution 16. Sample reasoning: 60. Sample reasoning: If $\frac{2}{5}$ of the number of seats is 24, then $\frac{1}{5}$ of it is 12, and all of it is $5 \boldcdot 12$ , which is 60.
Section B Check Section B Checkpoint	Problem 1 An artist is making a paste for a sculpture. She uses $\frac{8}{5}$ kilograms of flour to make $\frac{2}{3}$ of a batch. How much flour is needed to make a full batch? Draw a diagram and label it to represent the situation. Find the answer and show your reasoning. Show Solution Sample response: $\frac{12}{5}$ or $2\frac{2}{5}$ kilograms. Sample reasoning: If there are $\frac{8}{5}$ kilograms in $\frac{2}{3}$ of a batch, then there is $\frac{4}{5}$ kilogram in $\frac{1}{3}$ of a batch and $\frac{12}{5}$ kilograms in 1 whole batch. Problem 2 For each experiment, a scientist needs $\frac{3}{10}$ liter of a liquid. If the scientist has $4 \frac{1}{2}$ liters of the liquid, how many experiments can be done? Write a multiplication equation and a division equation to represent the question. Use a “?” for the unknown value. Explain or show that the answer is 15 experiments. Show Solution ${?} \boldcdot \frac{3}{10} = 4\frac{1}{2}$ and $4\frac{1}{2} \div \frac{3}{10} = {?}$ Sample response: $15 \boldcdot \frac{3}{10} = \frac{45}{10}$ , which is $4\frac{1}{2}$ . Ten experiments can be done with 3 liters ( $10 \boldcdot \frac{3}{10} = 3$ ) and 5 more can be done with $1\frac{1}{2}$ liters. There are 45 tenths in $4\frac{1}{2}$ and there are 15 groups of 3 tenths in 45 tenths.

Lesson 10 Dividing by Unit and Non-Unit Fractions	Dividing by $\frac13$ and $\frac35$ Explain or show how you could find $5 \div \frac 13$ . You can use this diagram if it is helpful. Find $12 \div \frac 35$ . Try not to use a diagram, if possible. Show your reasoning. Show Solution Sample reasoning: $5 \div \frac 13$ can mean “How many $\frac 13$ s (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 $5 \boldcdot 3$ , so there are 15 thirds in 5. 20. Sample reasoning: $12 \div \frac 35 = 12 \boldcdot 5 \boldcdot \frac 13 = 20$
Lesson 11 Using an Algorithm to Divide Fractions	Finding Quotients of Fractions Calculate each quotient. Show your reasoning. $\frac{24}{25} \div \frac{4}{5}$ $4 \div \frac{2}{7}$ Show Solution $\frac{6}{5}$ (or equivalent). Sample reasoning: $\frac{4}{5}$ is $\frac{20}{25}$ . There is 1 full group of $\frac{20}{25}$ in $\frac{24}{25}$ . The leftover $\frac{4}{25}$ is $\frac{1}{5}$ of a group. There is a total of $1\frac{1}{5}$ groups. $\frac{24}{25} \boldcdot \frac{5}{4} = \frac{120}{100} = \frac{6}{5}$ 14 (or equivalent). Sample reasoning: $4 \boldcdot \frac{7}{2} = \frac{28}{2} = 14$ 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 $6 \div \frac{3}{2}$ . A. $6 \boldcdot \frac{3}{2}$ B. $(6 \boldcdot 2) \div 3$ C. $(6 \div \frac{1}{2}) \div 3$ D. $6 \boldcdot 2 \boldcdot \frac{1}{3}$ E. $(6 \div 2) \div 3$ F. $6 \boldcdot \frac{2}{3}$ Show Solution B, C, D, F Problem 2 Calculate each quotient. Show your reasoning. $\frac{9}{8} \div \frac{3}{2}$ $2 \frac{1}{10} \div \frac{1}{5}$ Show Solution $\frac{3}{4}$ (or equivalent). Sample reasoning: $\frac{9}{8} \div \frac{3}{2} = \frac{9}{8} \boldcdot \frac{2}{3} = \frac{18}{24} = \frac{3}{4}$ The quotient is the unknown factor in the multiplication equation $\frac{3}{2} \boldcdot {?} = \frac{9}{8}$ . That number is $\frac{3}{4}$ . $\frac{21}{2}$ or $10\frac{1}{2}$ (or equivalent). Sample reasoning: $\frac{21}{10} \div \frac{1}{5} = \frac{21}{10} \boldcdot 5 = \frac{105}{10} = 10\frac{5}{10}$ There are 10 groups of $\frac{1}{5}$ in 2 wholes, and $\frac{1}{10}$ is $\frac{1}{2}$ a group of $\frac{1}{5}$ , so there are $10\frac{1}{2}$ groups of $\frac{1}{5}$ in $2\frac{1}{10}$ .

Lesson 12 Fractional Lengths	Building A Fence A builder was building a fence. In the morning, he worked for $\frac25$ of an hour. In the afternoon, he worked for $\frac{9}{10}$ 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: $\frac {9}{10} \div \frac 25 = {?}$ (or $\frac {9}{10} \div {?} = \frac 25$ ). In the afternoon, he worked $2\frac14$ times as long as he did in the morning. Sample reasoning: $\frac {9}{10} \div \frac 25 = \frac {9}{10} \boldcdot \frac 52 = \frac {45}{20} = \frac94$ .
Lesson 13 Rectangles with Fractional Side Lengths	Two Frames Two rectangular picture frames have the same area of 45 square inches but have different side lengths. Frame A has a length of $6 \frac34$ inches, and Frame B has a length of $7\frac12$ inches. Without calculating, predict which frame has the shorter width. Explain your reasoning. Find the width that you predicted to be shorter. Show your reasoning. Show Solution Frame B has a longer length, so its width is shorter if the two pairs of side lengths produce the same product of 45. 6 inches. Sample reasoning: $45\div 7\frac12 = 45 \div \frac {15}{2} = 45 \boldcdot \frac {2}{15} = 6$
Lesson 14 Fractional Lengths in Triangles and Prisms	Triangles and Cubes How many cubes with edge lengths of $\frac 13$ inch are needed to build a cube with an edge length of 1 inch? What is the volume, in cubic inches, of one cube with an edge length of $\frac 13$ inch? A triangle has a base of $3\frac{2}{5}$ (or $\frac{17}{5}$ ) inches and an area of $5\frac{1}{10}$ (or $\frac{51}{10}$ ) square inches. Find the height of the triangle. Show your reasoning. Show Solution 27 cubes $\frac{1}{27}$ in³ 3 inches. Sample reasoning: $\frac{1}{2} \boldcdot \frac{17}{5} \boldcdot h = \frac{51}{10}$ , so $\frac{17}{10} \boldcdot h = \frac{51}{10}$ . There are 3 groups of $\frac{17}{10}$ in $\frac{51}{10}$ .
Lesson 15 Volume of Prisms	Storage Box A storage box has a base that measures 3 inches by 4 inches and a height of $1\frac{1}{2}$ inches. The box can be packed with 144 cubes with an edge length of $\frac{1}{2}$ inch. Find the volume of the box in cubic inches. Show your reasoning. Describe a different way to find the volume of the box. (It is not necessary to do the calculation.) Show Solution 18 cubic inches. Sample reasoning: $3 \boldcdot 4 \boldcdot 1\frac{1}{2} = 18$ Sample response: Find the volume of a $\frac{1}{2}$ -inch cube and multiply it by 144. The volume of 1 cube is $\frac{1}{3}$ cubic inch, so the volume of the prism is $144 \boldcdot \frac{1}{8}$ , which is $\frac{144}{8}$ (or 18) cubic inches.
Section D Check Section D Checkpoint	Problem 1 A rectangular piece of paper has an area of $5\frac{5}{8}$ square feet and a side length of $1\frac{1}{4}$ feet. What is its width in feet? Show Solution $4\frac{1}{2}$ feet. Problem 2 A rectangular prism that measures $2\frac{1}{2}$ inches in length, 2 inches in width, and 3 inches in height is packed with $\frac{1}{2}$ -inch cubes. Select all the strategies for finding the volume of the prism in cubic inches. A.Multiply 5 by 4, and then multiply by 6. B.Multiply 5 by 3. C.Find the number of $\frac{1}{2}$ -inch cubes that can be packed in the prism. D.Multiply $2\frac{1}{2}$ by 2, and then multiply by 3. E.Multiply the number of $\frac{1}{2}$ -inch cubes in the prism by $\frac{1}{8}$ . 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 $8\frac{4}{5}$ square meters. What is the height of the water in meters? Show your reasoning. Show Solution $1\frac{1}{4}$ meters. Sample reasoning: $8\frac{4}{5}$ is $\frac{44}{5}$ . Dividing the volume by the area of the base gives the height: $11 \div \frac{44}{5} = 11 \boldcdot \frac{5}{44} = \frac{55}{44} = \frac{5}{4}$

Lesson 16 Solving Problems Involving Fractions	A Box of Pencils A box of pencils is $5\frac14$ inches wide. Seven pencils, laid side by side, take up $2 \frac 58$ inches of the width. How many inches of the width of the box is not taken up by the pencils? Explain or show your reasoning. All 7 pencils have the same width. How wide is each pencil? Explain or show your reasoning. Show Solution $2 \frac 58$ inches, because $5\frac14 - 2 \frac 58 = 2 \frac 58$ $\frac 38$ inch, because $2 \frac 58 \div 7 = \frac {21}{8} \boldcdot \frac17 = \frac38$
Lesson 17 Fitting Boxes into Boxes	No cool-down
Unit 4 Assessment End-of-Unit Assessment	Problem 1 Mai biked $6 \frac34$ miles today, and Noah biked $4 \frac12$ miles. How many times the length of Noah’s bike ride was Mai’s bike ride? A. $\frac23$ times as far B. $\frac32$ times as far C. $\frac94$ times as far D. $\frac{243}{8}$ times as far Show Solution $\frac32$ times as far Problem 2 Priya opened a new bag of soil and used $2\frac{1}{2}$ pounds of soil, which is $\frac{5}{8}$ of the bag. How many pounds of soil were in the full bag? Write a multiplication equation or a division equation to represent the question, and then find the answer. Show Solution Sample response: $\frac{5}{8} \boldcdot {?} = 2\frac{1}{2}$ , or $2\frac{1}{2} \div \frac{5}{8} = {?}$ There were 4 pounds of soil in the bag, because $2\frac{1}{2} \boldcdot \frac{8}{5} = \frac{40}{10} = 4$ . Problem 3 Select all statements that show correct reasoning for finding $15 \div \frac29$ . A. Multiply 15 by 2, then divide by 9. B. Multiply 15 by 9, then divide by 2. C. Multiply 15 by $\frac19$ , then multiply by 2. D. Multiply 15 by 9, then multiply by $\frac12$ . E.Multiply $\frac{2}{9}$ by 1, then multiply by $\frac{1}{15}$ . Show Solution B, D Problem 4 Divide. $\frac34 \div \frac15$ $\frac92 \div \frac34$ $\frac{4}{9} \div \frac{8}{15}$ $5 \frac23 \div \frac32$ Show Solution $\frac{15}{4}$ or $3 \frac34$ (or equivalent) 6 (or equivalent) $\frac{60}{72}$ or $\frac{5}{6}$ (or equivalent) $\frac{34}{9}$ or $3 \frac79$ (or equivalent) Problem 5 Andre draws this tape diagram for $3 \div \frac{2}{3}$ : Andre says that $3 \div \frac{2}{3} = 4 \frac{1}{3}$ because there are 4 groups of $\frac{2}{3}$ , with 1 group of $\frac{1}{3}$ left over. Do you agree with Andre? Explain or show your reasoning. Show Solution Sample response: No, I disagree. There are 4 groups of $\frac{2}{3}$ in $2 \frac{2}{3}$ . Then there is $\frac{1}{3}$ left and this makes $\frac{1}{2}$ of another group. So, there are $4 \frac{1}{2}$ groups of $\frac{2}{3}$ in 3. Minimal Tier 1 response: Work is complete and correct. Sample: No, because $3 \div \frac 2 3 = 4 \frac 1 2$ , not $4 \frac 1 3$ . Tier 2 response: Work shows general conceptual understanding and mastery, with some errors. Sample errors: Disagreement with Andre but a minor error in logic or calculation leads to a different result other than $4 \frac 1 2$ ; agreement with Andre with a reasonable but flawed argument. Tier 3 response: Significant errors in work demonstrate lack of conceptual understanding or mastery. Sample errors: Disagreement with Andre based on a major flaw in logic or calculation; agreement with Andre with a badly flawed argument; agreement or disagreement without any stated justification. Problem 6 A box has a width of $2\frac{2}{3}$ inches, a length of $3\frac{1}{3}$ inches, and a height of $2\frac{1}{3}$ inches. How many $\frac{1}{3}$ -inch cubes does it take to fill the box? Show Solution 560 (8 cubes fit along the width of the box, 10 cubes fit along the length, and 7 cubes fit vertically.) Problem 7 Lin has two small baking pans, each shaped like a rectangular prism. For each question, explain or show your reasoning. The base of Lin’s first pan has an area of $11\frac{1}{4}$ square inches. The length of the pan is $4\frac{1}{2}$ inches. What is the width of the pan? Lin’s second pan has a length of $\frac 8 3$ inches, a width of $\frac{15} 4$ inches, and a height of $\frac 3 2$ inches. What is the volume of the second pan? Show Solution $2 \frac 1 2$ inches (or equivalent). The width is the solution to $4 \frac 1 2 \boldcdot w = 11 \frac 1 4$ . By writing each mixed number as a fraction, the problem is made simpler: $\frac 9 2 w = \frac {45} 4$ . Then $w = \frac 5 2$ . 15 cubic inches (or equivalent). The volume is the product of the pan’s length, width, and height: $\frac 3 2 \boldcdot \frac 8 3 \boldcdot \frac{15} 4 = 15$ . Minimal Tier 1 response: Work is complete and correct, with complete explanation or justification. Sample: $\frac 5 2$ inches, because $11 \frac 1 4 \div 4 \frac 1 2 = \frac 5 2$ . 15 cubic inches, because $\frac 3 2 \boldcdot \frac 8 3 \boldcdot \frac {15} 4 = 15$ . Tier 2 response: Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification. Sample errors: Correct answers without justification; one or two errors in calculation, such as incorrect rewriting of mixed numbers, but correct equations or representations used. Tier 3 response: Work shows a developing but incomplete conceptual understanding, with significant errors. Sample errors: One incorrect answer with invalid work or no work shown; any incorrect choice of multiplication or division; invalid method used to multiply or divide fractions or mixed numbers; more than two errors in calculation. Tier 4 response: Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery. Sample errors: Two incorrect answers with invalid work or no work shown; consistently incorrect choices of multiplication or division; repeated use of invalid methods to multiply or divide.