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[# Problem Set Sample Solutions]

1. Using the floor tiles design shown below, create  different ratios related to the image.  Describe the ratio relationship, and write the ratio in the form  or the form  to .
   

For every  tiles, there are white tiles.  

The ratio of the number of black tiles to the number of white tiles is to .

(Answers will vary.)

2. Billy wanted to write a ratio of the number of apples to the number of peppers in his refrigerator.  He wrote .  Did Billy write the ratio correctly?  Explain your answer.
   

Billy is incorrect.  There are  apples and  pepper in the picture.  The ratio of the number of apples to the number of peppers is .

[materials]

Problem Set

1. Using the floor tiles design shown below, create  different ratios related to the image.  Describe the ratio relationship, and write the ratio in the form  or the form  to .

2. Billy wanted to write a ratio of the number of apples to the number of peppers in his refrigerator.  He wrote .  Did Billy write the ratio correctly?  Explain your answer.

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[# Example 1  (15 minutes)]

Read the example aloud:

The coed soccer team has four times as many boys on it as it has girls.  We say the ratio of the number of boys to the number of girls on the team is .  We read this as four to one.

• Let’s create a table to show how many boys and how many girls could be on the team.

Create a table like the one shown below to show possibilities of the number of boys and girls on the soccer team.  Have students copy the table into their student materials.

• So, we would have four boys and one girl on the team for a total of five players.  Is this big enough for a team?  

[indent]◦ Adult teams require  players, but youth teams may have fewer.  There is no right or wrong answer; just encourage reflection on the question, thereby having students connect their math work back to the context. 

• What are some other ratios that show four times as many boys as girls, or a ratio of boys to girls of  to ?  

[indent]◦ Have students add each ratio to their table.

• From the table, we can see that there are four boys for every one girl on the team.

Read the example aloud:

Suppose the ratio of the number of boys to the number of girls on the team is .

Create a table like the one shown below to show possibilities of the number of boys and girls on the soccer team.  Have students copy the table into their student materials.

• What are some other team compositions where there are three boys for every two girls on the team?

• I can’t say there are  times as many boys as girls.  What would my multiplicative value have to be?  There are _____________ as many boys as girls.  

Encourage students to articulate their thoughts, guiding them to say there are  as many boys as girls.  

• Can you visualize  as many boys as girls?
• Can we make a tape diagram (or bar model) that shows that there are  as many boys as girls?  

• Which description makes the relationship easier to visualize:  saying the ratio is  to  or saying there are  halves as many boys as girls?  

[indent]◦ There is no right or wrong answer.  Have students explain why they picked their choices.

[materials]

Example 1

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[# Exercise 2  (15 minutes)]

[task #]

Shanni and Mel are using ribbon to decorate a project in their art class.  The ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon is .

Draw a tape diagram to represent this ratio.

Ask students to read the problem and then describe in detail what the problem is about without looking back at the description, if possible.  This strategy encourages students to really internalize the information given as opposed to jumping right into the problem without knowing the pertinent information.

• Let’s represent this ratio in a table.

• We can use a tape diagram to represent the ratio of the lengths of ribbon.  Let’s create one together.

Walk through the construction of the tape diagram with students as they record.

• How many units should we draw for Shanni’s portion of the ratio?  

[indent]◦ Seven

• How many units should we draw for Mel’s portion of the ratio?  

[indent]◦ Three^[a][b]

• What does each unit on the tape diagram represent?  

[indent]◦ Allow students to discuss; they should conclude that they do not really know yet, but each unit represents some unit that is a length.

• What if each unit on the tape diagrams represents  inch?  What are the lengths of the ribbons?  

[indent]◦ Shanni’s ribbon is  inches; Mel’s ribbon is  inches.

• What is the ratio of the lengths of the ribbons?  

[indent]◦  (Make sure that students feel comfortable expressing the ratio itself as simply the pair of numbers  without having to add units.)

• What if each unit on the tape diagrams represents  meters?  What are the lengths of the ribbons?  

[indent]◦ Shanni’s ribbon is  meters; Mel’s ribbon is meters.

• How did you find that?  

Allow students to verbalize and record using a tape diagram.

• What is the ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon now?  (Students may disagree; some may say it is , and others may say it is still .)  

Allow them to debate and justify their answers.  If there is no debate, initiate one:  A friend of mine told me the ratio would be (provide the one that no one said, either  or ).  Is she right?

• What if each unit represents  inches?  What are the lengths of the ribbons?  (Record.  Shanni’s ribbon is  inches; Mel’s ribbon is  inches.)  Why?  

[indent]◦  times  equals ;  times  equals .

• If each of the units represents  inches, what is the ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon?  

Allow for discussion as needed.

• We just explored three different possibilities for the length of the ribbon; did the number of units in our tape diagrams ever change?  

[indent]◦ No.

• What did these three ratios, , , , all have in common?  

Write the ratios on the board.  Allow students to verbalize their thoughts without interjecting a definition.  Encourage all to participate by asking questions of the class with respect to what each student says, such as, “Does that sound right to you?”

• Mathematicians call these ratios equivalent.  What ratios can we say are equivalent to ?
  

Shanni and Mel are using ribbon to decorate a project in their art class.  The ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon is .

Draw a tape diagram to represent this ratio.

[materials]

Exercise 2

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[# Exercise 4  (7 minutes)]

Allow students to work the exercise independently and then compare their answers with a neighbor’s answer.

Josie took a long multiple-choice, end-of-year vocabulary test.  The ratio of the number of problems Josie got incorrect to the number of problems she got correct is .

a. If Josie missed  questions, how many did she get correct?  Draw a tape diagram to demonstrate how you found the answer.

b. If Josie missed  questions, how many did she get correct?  Draw a tape diagram to demonstrate how you found the answer.

c. What ratios can we say are equivalent to ?

 and

d. Come up with another possible ratio of the number Josie got incorrect to the number she got correct.

e. How did you find the numbers?

Multiplied  and .

f. Describe how to create equivalent ratios.

Multiply both numbers of the ratio by the same number (any number you choose).

[materials]

Group Activity

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[# Closing  (5 minutes)]

Provide students with this description:

A ratio is an ordered pair of nonnegative numbers, which are not both zero.  The ratio is denoted  or  to  to indicate the order of the numbers.  In this specific case, the number  is first, and the number  is second.

• What is a ratio?  Can you verbally describe a ratio in your own words using this description?

[indent]◦ Answers will vary but should include the description that a ratio is an ordered pair of numbers, which are both not zero.

• How do we write ratios?  

[indent]◦  colon  () or  to .

• What are two quantities you would love to have in a ratio of  but hate to have in a ratio of ?

[indent]◦ Answers will vary.  For example, I would love to have a ratio of the number of hours of playtime to the number of hours of chores be  but I would hate to have a ratio of the number of hours of television time to the number of hours of studying be .  

Lesson Summary

A ratio is an ordered pair of numbers, which are not both zero.  

A ratio is denoted  to indicate the order of the numbers—the number A is first and the number B is second.

The order of the numbers is important to the meaning of the ratio.  Switching the numbers changes the relationship.  The description of the ratio relationship tells us the correct order for the numbers in the ratio.

Ask students to share their answers to Part (f); then, summarize by presenting the definition of equivalent ratios provided in the Lesson Summary below.

Note that if students do not have a sufficient grasp of algebra, they should not use the algebraic definition.  It is acceptable to use only the second definition.

Two ratios A: B and C:D are equivalent ratios if there is a nonzero number c such that C = cA and D = cB.  For example, two ratios are equivalent if they both have values that are equal.

Ratios are equivalent if there is a nonzero number that can be multiplied by both quantities in one ratio to equal the corresponding quantities in the second ratio.

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[# Exit Ticket  (5 minutes)]

[# Exit Ticket ]

Pam and her brother both open savings accounts. Each begin with a balance of zero dollars. For every two dollars that Pam saves in her account, her brother saves five dollars in his account.

1. Determine a ratio to describe the money in Pam’s account to the money in her brother’s account.
   
   𝟐 : 𝟓
   
2. If Pam has 40 dollars in her account, how much money does her brother have in his account? Use a tape diagram to support your answer.
   
   
3. Record the equivalent ratio.
   
   𝟒𝟎 : 𝟏𝟎𝟎
   
4. Create another possible ratio that describes the relationship between the amount of money in Pam’s account and the amount of money in her brother’s account.
   
   Answers will vary. 𝟒 : 𝟏𝟎, 𝟖 : 𝟐𝟎, etc.

[materials]

Exit Ticket

[# Problem Set Sample Solutions ]

1. Write two ratios that are equivalent to 𝟏:𝟏.
   Answers will vary. 𝟐:𝟐, 𝟓𝟎:𝟓𝟎, etc.
   
2. Write two ratios that are equivalent to 𝟑:𝟏𝟏.
   Answers will vary. 𝟔:𝟐𝟐, 𝟗:𝟑𝟑, etc.
   

a. The ratio of the width of the rectangle to the height of the rectangle is 𝟗 to 𝟒 .

b. If each square in the grid has a side length of 𝟖 𝐦𝐦, what is the width and height of the rectangle?
𝟕𝟐 𝐦𝐦 wide and 𝟑𝟐 𝐦𝐦 high

4. For a project in their health class, Jasmine and Brenda recorded the amount of milk they drank every day. Jasmine drank 𝟐 pints of milk each day, and Brenda drank 𝟑 pints of milk each day.

a. Write a ratio of the number of pints of milk Jasmine drank to the number of pints of milk Brenda drank each day.
𝟐 : 𝟑

b. Represent this scenario with tape diagrams. 

c. If one pint of milk is equivalent to 𝟐 cups of milk, how many cups of milk did Jasmine and Brenda each drink? How do you know?
Jasmine drank 𝟒 cups of milk, and Brenda drank 𝟔 cups of milk. Since each pint represents 𝟐 cups, I multiplied Jasmine’s 𝟐 pints by 𝟐 and multiplied Brenda’s 𝟑 pints by 𝟐.

d. Write a ratio of the number of cups of milk Jasmine drank to the number of cups of milk Brenda drank.
𝟒 : 𝟔

e. Are the two ratios you determined equivalent? Explain why or why not.
𝟐 : 𝟑 and 𝟒 : 𝟔 are equivalent because they represent the same value. The diagrams never changed, only the value of each unit in the diagram.

[materials]

Homework

1. Write two ratios that are equivalent to .

2. Write two ratios that are equivalent to .

[indent] a. The ratio of the width of the rectangle to the height of the rectangle is __________to_________.

[indent]b. If each square in the grid has a side length of , what is the width and height of the rectangle?  

4. For a project in their health class, Jasmine and Brenda recorded the amount of milk they drank every day.  Jasmine drank  pints of milk each day, and Brenda drank  pints of milk each day.

[indent]a. Write a ratio of the number of pints of milk Jasmine drank to the number of pints of milk Brenda drank each day.

[indent]b. Represent this scenario with tape diagrams.

[indent]c. If one pint of milk is equivalent to  cups of milk, how many cups of milk did Jasmine and Brenda each drink?  How do you know?

[indent]d. Write a ratio of the number of cups of milk Jasmine drank to the number of cups of milk Brenda drank.

[indent]e. Are the two ratios you determined equivalent?  Explain why or why not.

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[# Concept Development  (15 minutes)]

Materials:        (S) Personal white board

T:        Here is unit form.  (Write 1 half.)  Here it is written in fraction form.  (Write .)  What does the 2 mean?

S:        2 is the number of equal parts that the whole is cut into.

T:        What does the 1 mean?

S:        We are talking about 1 of the equal parts.

Shape 1:  

T:        (Project or draw a circle partitioned into thirds.)  This is 1 whole.

T:        What unit is it partitioned into?

S:        Thirds.

T:        What is the unit fraction?

S:        1 third.

T:        (Shade 1 third.)  I’m going to make a copy of my shaded unit fraction.  (Shade one more unit.)  
How many units are shaded now?

S:        2 thirds.

T:        Let’s count them.

S:        1 third, 2 thirds.

T:        We can write the number 2 thirds in unit form (write 2 thirds under the circle) or fraction form  (write  under the circle).  What happened to our unit fraction when we made a copy?  Turn and share.

S:        We started with one unit shaded and then shaded in another unit to make a copy.  2 copies make 2 thirds.  → True.  That’s why we changed 1 third to 2 thirds.  Now we’re talking about 2 copies.

T:        Yes! Just like 2 copies of one make 2, we can make 2 copies of 1 third to make 2 thirds.  

Continue with the following suggested shapes.  Students identify the unit fraction and then make copies to build the new fraction.

Shape 2:

Shape 3:

Students transition into guided practice using personal white boards.

Give the following directions:  

• Draw a unit fraction (select examples).
• Make copies of the unit fraction to build a new fraction.
• Count the unit fractions.
• Identify the new fraction both in unit form and fraction form.

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[# Problem Set  (7 minutes) ]

Students should do their personal best to complete the Problem Set within the allotted time.  For some classes, it may be appropriate to modify the assignment by specifying which problems they work on first.  Some problems do not specify a method for solving.  Students should solve these problems using the RDW approach used for Application Problems.

[materials]

Problem Set

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[# Student Debrief  (5 minutes)]

Lesson Objective:  Build non-unit fractions less than one whole from unit fractions.

The Student Debrief is intended to invite reflection and active processing of the total lesson experience.

Invite students to review their solutions for the Problem Set.  They should check work by comparing answers with a partner before going over answers as a class.  Look for misconceptions or misunderstandings that can be addressed in the Debrief.  Guide students in a conversation to debrief the Problem Set and process the lesson.

• Through discussion, guide students to articulate the idea that to show non-unit fractions, they create copies of unit fractions.  This resembles counting 3 ones to make 3, or counting by eights to make copies of 8.

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[# Exit Ticket  (3 minutes)]

After the Student Debrief, instruct students to complete the Exit Ticket.  A review of their work will help with assessing students’ understanding of the concepts that were presented in today’s lesson and planning more effectively for future lessons.  The questions may be read aloud to the students.

[materials]

Exit Ticket

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Apples to Apples

Task

Alice and Claire go apple picking. When they are done, Claire has 3 times as many apples in her basket as Alice has in hers. All of the apples are whole.

[indent]a. What are three different possibilities for numbers of apples that could be in the baskets?

[indent]b. What is the ratio of Alice's apples to Claire's apples?

Alice and Claire's mom measures each of their heights in inches, rounded to the nearest whole inch. She remarks, "Wow! Alice's height is exactly three fourths of Claire's height!"

[indent]c. What are three different reasonable possibilities for their heights?

[indent]d. What is the ratio of Claire's height to Alice's height?

IM Commentary

The purpose of this task is to connect students' understanding of multiplicative relationships to their understanding of equivalent ratios.

Creating tables of equivalent ratios isn't required for this task, but it is a convenient way to organize them. Students should eventually come to view ratio tables as collections of equivalent ratios, so this might be a good opportunity to suggest such a tool.

In Parts (b) and (d), we ask "What is the ratio?" The meaning of this question can be misinterpreted by teachers, because experienced doers-of-math usually default to lowest terms, so it would be easy to mistakenly conclude that 3:1 is the only acceptable answer for (b) and 4:3 is the only acceptable answer for (d). However, any ratio equivalent to those is acceptable.

In Part (c), reasonable height values aren't determined by the task statement, but they are suggested by the context, and students should explain the thinking behind the reasonableness of their values. Paying attention to the meaning of numbers in a context and evaluating them for reasonableness is important to the standards for mathematical practice, especially MP2, MP4, and MP8.

In this task, the units for each quantity are the same and students are essentially given the values of the unit rates and asked to determine possible ratios. The task is meant to come before students have studied unit rates in order to provide a precursor to the idea that unit rates characterize sets of equivalent ratios, but without having to deal with the complexities of the units. If students have already studied unit rates, they might notice that in this case we could say, "There are 3 of Claire's apples for every 1 of Alice's apples" (so there are "apples per apple"), and “There are 3/4 as many inches in Alice's height to every 1 inch in Claire's height” (so there are "inches per inch").

Solution

[indent] a. Here are some possibilities for the numbers of apples in each basket. (For clarity, these are organized in a table, but they don't necessarily have to be. Any response in a ratio of 3:1 is acceptable.)

[indent] b. The ratio of Alice's apples to Claire's apples is 1:3. Any equivalent statement is also acceptable, for instance, "There are 4 apples in Alice's basket for every 12 apples in Claire's."

[indent] c. Here are some possibilities for their respective heights. (For clarity, these are organized in a table, but they don't necessarily have to be.) Example reasoning about reasonable heights: the average height (usually called "length") of a newborn is 20 inches, but since Alice and Claire were presumably walking around and picking apples, we'll take 33 inches, the height of a short two-year-old, to be the least possible height. (These heights are easily googleable.) Since they are being measured by their mom, they are likely young children, so we'll say that 5 feet or 60 inches is the greatest possible height. (Students may set different bounds on the heights based on different reasoning.)

[indent] d. The ratio of Claire's height to Alice's height is 48:36 (or equivalent).

[materials]

Apples to Apples

Alice and Claire go apple picking. When they are done, Claire has 3 times as many apples in her basket as Alice has in hers. All of the apples are whole.

[indent] a. What are three different possibilities for numbers of apples that could be in the baskets?

[indent] b. What is the ratio of Alice's apples to Claire's apples?

Alice and Claire's mom measures each of their heights in inches, rounded to the nearest whole inch. She remarks, "Wow! Alice's height is exactly three fourths of Claire's height!"

[indent] c. What are three different reasonable possibilities for their heights?

[indent] d. What is the ratio of Claire's height to Alice's height?