# Part to Total Ratios Using Tape Diagrams and Tables

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## Objective

SWBAT write pairs of quantities in equivalent ratios that compare part to part and part to total by creating ratio tables and tape diagrams.

#### Big Idea

Tape diagrams and double number line diagrams can show ratio relationships and be used to reason about solutions to problems.

5 minutes

With their partners, students work on the Think About It problem.  After 3 minutes I ask a pair to identify the relationship between the juices.  I am looking for, "For every 5 cups of grape juice, there are 2 cups of peach juice," as this is the way we've been working on verbalizing ratios.  If a student responds by saying "the ratio is 5 to 2," I will ask them to frame it using our ratio language.

I then ask for students to explain how they determined how many cups of grape juice would be in 28 total cups of this mixture.  I expect that there will be students who were not sure how to figure this out.  There also will be students who were able to use the given table to determine 20 cups of grape juice.  To help students, I suggest that they add a total column to their table, and model where to place it.  I don't fill in any totals.  I give students 45 'thinking seconds' and then again ask how many cups of grape juice would be in 28 total cups of this mixture.

## Intro to New Material

15 minutes

In this lesson, students are using tape diagrams and tables to solve ratio problems.  We've worked with both models in previous lessons, so using these tools is not new for students.

In the Intro to New Material section, I rely on students to help me start the problems.  This section feels more like a guided practice in this lesson, because the strategies are not new for students.

I have students construct a tape diagram for the first problem in this set.  I ask why we can create a tape diagram, and I am looking for students to tell me that the units for the ingredients in the Slimy Gloopy mixture are the same.  Once we've modeled the ratio of glue to starch, I ask students, "what we should divide 85 by?"  In the previous lesson, students were given one term in the second ratio.  In this lesson, students are working with totals.  Once they've named that we need to divide by the total number of parts in our tape diagram, I have students help me finish the problem.

The second problem requires students to use a ratio table.  Together, students and I label each row as 'tshirt,' 'long sleeves,' and 'total.'  I fill in 0 for each quantity in the first column and the ask students to complete their ratio tables.  Students then use the tables to answer the questions.

I use the fourth problem as a chance to have everyone write.  All students respond to the prompt on their papers.  I then have students turn and talk with their partners about whether or not we can add a total column.  I ask students to vote with their thumbs about whether or not we can add a total column.  Finally, I ask 1-2 students share out their thoughts with the class.

## Partner Practice and Discussion

15 minutes

Students work in pairs on the Partner Practice problem set.  As they work, I circulate around the classroom.  I am looking for:

• Are students correctly identifying the terms and putting them in the correct order?
• Are students correctly identifying the proportional relationship?
• Are students correctly filling in all of the equivalent ratios?
• Are students calculating the correct total?

• How did you know what the relationship was?
• How did you know what the rest of the ratios would be?
• What does your ratio represent?
• How do you know the total?
• How do you know if you can find the total?

After 10 minutes of work time, I bring the class back together to discuss problem C.  This problem appears to be more difficult, because the ratio is modeled, rather than stated.  I ask students what the ratio is of cats to dogs in this problem.  I then ask students how they determined the ratio of cats to dogs.

## Independent Practice

18 minutes

Students work on the Independent Practice problem set.  A student sample of the independent work is included. As students are working, I am circulating around the room, looking for and asking the same questions as I did during Independent Practice.  I pay careful attention to whether or not students are using the model asked for in each problem.  I want students to be proficient with both models (and it is important that they carefully read and follow directions).

For Problem 5, because we've been stressing that units must be the same for tape diagrams and for totals, some students will be thrown off.  This problem isn't asking students to find a total, but it does involve miles and minutes.  I watch for puzzled faces on this problem, and guide students through with 'How far does he go in 12 minutes? How far does he go in 24 minutes? How did you figure that out?  Huh.  How can you show that in a model?'

## Closing and Exit Ticket

7 minutes

After 15 minutes of independent work time, I bring the class back together for a discussion.  I ask students if, in Problem 3, we can add a total column to the ratio table.  I have students turn and talk to their partners and share their thoughts.  I then ask for 1-2 students to share out (1 student to answer the question, a 2nd student if we need to add a bit more to the response).

Students work independently on the Exit Ticket to end the lesson.