Tape Diagrams - Part to Part and Part to Total Ratios

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SWBAT represent an equivalent ratio when given a part to determine the other part or total by creating a tape diagram.

Big Idea

There are multiple ways to represent proportional relationships and reason about solutions to problems.

Think About It

8 minutes

Students work in partners on the Think About It problem.  Students can use any method to solve this problem.  The numbers are small enough that an extended tape diagram, from the previous lesson, is easy to construct.

After a few minutes of work time, I have a student who used a tape diagram show the work on the document camera.  I then also have a student who used a ratio table show the work on the document camera.  I then let students know that in this lesson, we will focus exclusively on the tape diagram with our part-to-part ratio work.

Intro to New Material

15 minutes

To start the Intro to New Material section of this lesson, I ask students to start the tape diagram for Problem #1.  Students represent the ratio of 7:3.  I tell students that today, we are going to work more efficiently.  Instead of modeling each student in our tape diagram for this problem, we are going to use the boxes to represent a multiple of each type of student.  

In this problem, there are 42 students who take the bus and 7 parts in the tape diagram for this part of the ratio: 42 ÷ 7 = 6. We write the number 6 in each box of the tape diagram, including the boxes for the students who walk, because each box of the tape diagram represents the same amount. I hope students will make the following connection: if we were going to represent all of the students in the tape diagram, we would have needed to draw six groups of 7 boxes.  

We then use the labeled boxes to answer the question (in this case, a question about the unknown term of the second ratio).

I guide students through Problem #2, asking them to provide me with the steps I need to take to create the tape diagram. 


Partner Practice and Discussion

20 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 creating tape diagram from a given ratio with equally sized rectangles and labeling?
  • Are students correctly solving for each part or total using division or multiplication?
  • Are students answering the question asked (finding a part vs. finding a total)?
  • Are students showing clear, logical work

I'm asking students:

  • What does the ratio mean in this problem?
  • How will you represent this ratio in your model?
  • What is the value of each part?  How do you know?
  • How can you determine each part given a total for one term?
  • How can you determine a total given a total amount for one term?


After 15 minutes of partner work time, I bring the class back together to discuss problem 3.  In this problem, the known part of the second ratio is the 2nd term in the tape diagram.  A common error is for students to assume that they are always given number of the first term.  I use this opportunity to reinforce annotation and close reading.

We also discus problem 5.  I have students vote with a thumbs up or down about whether or not they think the situation in the problem can be represented with a tape diagram.  

Independent Practice

15 minutes

Students work on the Independent Practice problem set.  As they work, I circulate around the classroom.  I am looking for and asking the same questions I used during the partner practice. 

I pay close attention to the organization of the tape diagrams.  I check to be sure that students are creating boxes of equal size and that they are labeling all parts.

I want the work that students produce to look like the student work sample.


Closing and Exit Ticket

7 minutes

After independent work time, I bring the class back together for a conversation about Problem 8. For some students, dividing 135 by 15 will be difficult.  I highlight an exemplar student response (which I identify as I am circulating during work time).  I then ask students strategies they used to fond the quotient of 135 and 15.  Some students will jump right into long division.  Others will use repeated addition of 15 or a combination of adding 30 and 15.  Other students will guess and check multiplication facts, knowing that 15*10 is 150, so the factor they are looking for must be smaller than 10.  Not all of my students are fluent with their division facts, and I want to be sure that they have multiple strategies for attacking a problem that involves multi-digit divisors. 

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