Reasoning about Contexts with Tape Diagrams

10 min

Teacher Prep
Setup
Students in groups of 2. Give students 1 minute of quiet think time, followed by partner- and whole-class discussions following first question. 1 minute of quiet work time and whole-class discussion for remaining question.

Narrative

The purpose of this Warm-up is to re-introduce students to tape diagrams as a representation of relationships between quantities. These diagrams will be helpful for reasoning about situations in activities in this lesson. While students may notice and wonder many things about these diagrams, the understanding that the length of a piece of tape is meaningful, and that pieces labeled with the same variable indicate that they have the same length, are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is multiples of identical sections of the tape diagram, the relative lengths of the sections of the diagram, and the total length.

Launch

Arrange students in groups of 2. Display the image for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.

Student Task

What do you notice? What do you wonder?

Two tape diagrams. First diagram has 4 equal parts each labeled, a+b, total c. Second diagram has four equal parts each labeled x, a larger part labeled y, total z.

Sample Response

Answers vary. Sample responses: 

Students may notice:
  • There are two diagrams of rectangles with pieces labeled aa, bb, cc, xx, yy, and zz.
  • The cc and zz appear at the top of the diagrams.
  • Each diagram consists of a large rectangle partitioned into smaller rectangles.
  • In the first diagram, the rectangle contains 4 (a+b)(a+b)’s.
  • In the second diagram, the rectangle contains 4 xx’s and 1 yy. 
Students may wonder:
  • What do the diagrams represent? 
  • What do the pieces of the diagrams represent?
  • Do all of the xx’s represent the same value? All the yy’s? All the zz’s?
  • Are longer pieces “worth” more?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the appropriate parts of the diagrams. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the idea that pieces labeled with the same variable represent the same value does not come up during the conversation, ask students to discuss this idea.

As an extension, ask students to share possible values for the variables in each diagram. Record and display their responses for all to see. If possible, record the values on the displayed diagram.

  • In the first diagram, if a=1a=1 and b=4b=4, and we assume that cc is the total, then c=1+4⋅4=20c=1+4 \boldcdot 4=20.
  • In the second diagram, if x=1x=1 and y=3y=3, and we assume that zz is the total, then z=4⋅1+3=7z=4 \boldcdot 1+3=7.
Standards
Building Toward
  • 7.EE.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
  • 7.EE.B.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

15 min

10 min