Part to Part Ratios Using Tape Diagrams and Tables

Students work independently on the Think About It problem.  The problem starts by having students practice what they learned in the previous lesson.   It then has students extend their thinking, by asking them to apply the rate to an equivalent ratio.  After 3 minutes of work time, I have students share out how they tried to figure out the number of Wilson racquets in the last part of this problem. Some students will create equivalent fractions, while others will draw models.     

As we conclude this warmup, I frame the lesson by letting students know that we will be using two models in this lesson - tape diagrams and tables. 

In this lesson, students will use a tape diagram or a ratio table to solve for a given value by making equivalent ratios until they find an equivalent ratio pair.  

In all of the problems in this lesson, we will use an additive relationship.  Some students are able to make the jump to a multiplicative relationship, but in this lesson I want students to physically build out the diagrams.  In future lessons, we will just represent the given ratio and then create an equivalent ratio,   but here we stay concrete.

To start the introduction of new material, I model how to create a tape diagram for the Think About It problem. You can see the problem solved step by step. The steps that I follow:

• Write the given units separately.
• Next to the first unit, draw however many equally sized boxes represent the first part of the ratio.
• Next to the second unit, draw however many boxes represent the second part of the ratio (keeping them exactly the same size as the unit above).
• Decide what the known term of the second ratio is.  Decide how many groups of the given ratio you would need to add to get to that term.  Draw them next to the given ratio boxes.
• However many groups you made of the first term, make that many groups of the second term also drawing the boxes.
• Count the boxes for the unknown part of the ratio.  Include units.

Students help me create the tape diagram for the first problem in the Intro to New Material.  For the second problem, I guide students to create a ratio table.

Students work in pairs on the Partner Practice problems.  As they work, I circulate around the room.  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 using the model that the problem asks for?
• Are students drawing tape diagrams correctly (equally sizes boxes, etc.)?

I am asking:

• 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?

I pay close attention to the organization of the tape diagrams.  If students do not line up their boxes, they may lose track of what they need to draw.  I created an example of the potential mistake.  

After 10 minutes of partner work time, students work independently on the final check for understanding problem.  I have one student who feels confident about his/her tape diagram share his/her work on the document camera.  I then have one student who feels confident about his/her ratio table share his/her work on the document camera.

Students work on the Independent Practice problem set.  As they work, I circulate around the room and I look for and ask the same questions I used during the partner practice.

Students need to annotate and take care to organize their work.  A Student Sample is included here. 

After 15 minutes of independent work time, I bring the class back together for a conversation.  We discuss Problem_7 from the Independent Practice problem set.  This problem can be difficult for students because it involves the repeated addition of a fraction.  Strategies that students share may include converting the fraction to a decimal, adding 2 1/2 repeatedly in the margin, or drawing pictures. 

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