Multistep Tape Diagrams, Part 1

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SWBAT solve multi-step ratio problems, using tape diagrams.

Big Idea

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

Think About It

5 minutes

The Think About It problem in this lesson is one that students have difficulty solving.  The goal of this problem is not to have students getting all the way to the answer.  Rather, I want students to read and annotate the problem and persevere.  

Students work in pairs on the Think About It problem.  My expectation is that they annotate the problem and then create a tape diagram to represent the original ratio.  After 3 1/2 minutes of work time, I bring the class together for a discussion about this problem.   I ask students what the problem is asking us to find, what we know, how we can model the given information, and then for any ideas pairs had for finding the original amount of green paint.  

I then use this problem to start off the Intro to New Material section.

Intro to New Material

15 minutes

The Intro to New Material section starts with the Think About It problem.  I model for students how to use a tape diagram to find the original amount of green paint.  Students already have their tape diagrams started, showing the original ratio of yellow to blue paint.  

I ask students what else we know in this problem, beyond the original ratio of yellow to blue paint.  Students identify the fact that more blue paint is added, and that the added blue paint makes the amount of yellow and blue paints equal.

I model for students how to add 4 rectangles to the row of the tape diagram that represents the blue paint.  To help them keep organized, I teach students to leave a small space between the starting blue paint and the added blue paint.  I also have students bracket off the new part of the tape diagram, as another way to help them make sense of the problem.  

Once I've finished the paint problem on my own, we move on to the second problem in the Intro to New Material section.  The scenario in this problem, about Hawaiian pizza, is one that students worked with yesterday.  I guide students through this problem, but have them help me make sense of the problem.  I ask them what we know, how we can represent the information, what do the new rectangles represent, etc.  I do retain most of the control during this problem.  

Guided Practice and Discussion

15 minutes

This lesson can be difficult for students.  Because of this, I facilitate a Guided Practice portion of this lesson, instead of partner practice (which most of my lessons have).

For the first problem in the Guided Practice, I call on students to help work out this problem, step-by-step.  Because I am still building confidence in the process at this point, I call on raised hands.  I am asking:

  • What does the ratio mean in this problem?
  • What is the value of each part?  How do you know?
  • If both terms have the same amount, how would this be represented in a tape diagram?
  • How do you use the value that is added to find each part?
  • What does it look like when we add this to our tape diagram?
  • How do we find the value of each part of the tape diagram?
  • What is the question asking you to find?
  • How do we use the diagram to help us answer this question?


For the second problem, I move to cold-call.  The question I ask students, as I call on them, is 'what next?'  My goal here is to have students own the problem-solving process, from start to finish.

For the third problem, I have pairs work together to solve the problem (if students struggle during the cold-call of problem 2, I will keep the class together for more guided practice).

Finally, students independently complete the Check for Understanding problem.  A sample CFU problem is included here. 


Independent Practice

10 minutes

Students work on the Independent Practice problem set.  As they work, I am looking for:

  • Are students correctly creating tape diagram from a given ratio with equally sized rectangles and labeling?
  • Are students correctly drawing the added quantity to the tape diagram with equal parts for both terms?
  • Are students including a bracket and label for the new portion?
  • Are students correctly solving for each part or total using division or multiplication?
  • Are students showing clear, logical work?


I pay special attention to whether students are answering the specific question(s) being asked in each problem.  Some students may try to always identify the original amount.

Closing and Exit Ticket

7 minutes

After 10 minutes of independent work time, I bring the class back together for a debrief.  I ask for 2 students to show off a problem solution, allowing them to pick from problems 1 - 4.  Each students shows his/her work on the document camera, and explains the solution to the class. 

Students then work independently on the Exit Ticket to end the lesson.  The sample exit ticket shows a strong student response.