Reasoning about Solving Equations (Part 1)

5 min

Narrative

The purpose of this Warm-up is to elicit the idea that different weights on each side of a hanger cause the hanger to be balanced or unbalanced, which will be useful when students use  hanger diagrams to develop general strategies for solving equations in a later activity. While students may notice and wonder many things about this image, the possible weights on each hanger and the fact that one hanger is balanced and the other is not are the important discussion points.

Launch

Tell students to close their books or devices (or to keep them closed). Display this image for all to see.

Socks with items inside hanging from hangers. The left hanger is balanced with 2 red socks that have balls inside. The right hanger is unbalanced. A navy sock has a rectangular object inside and the yellow sock has a triangle inside.

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. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image.

If needed, explain that the photo shows two clothes hangers with a sock hung from each end of each hanger. The socks have different objects inside them that have different weights. If the contrast of “balanced” and “unbalanced” hangers does not come up during the conversation, ask students to discuss this idea. If possible, use a real clothes hanger to demonstrate.

Give students 3 minutes of quiet work time followed by a whole-class discussion.

Student Task

In the two diagrams, all the triangles weigh the same and all the squares weigh the same.

For each diagram, come up with . . .

  1. One thing that must be true 
  2. One thing that could be true
  3. One thing that cannot possibly be true 

Two hanger diagrams.
Two hanger diagrams. First hanger, unbalanced with left side lower, left side, green triangle, right side, blue square. Second hanger, balanced, left side, green triangle, right side, three blue squares, one red circle.

 

Sample Response

Sample responses:

  1. The triangle is heavier than the square; 1 triangle weighs the same as 3 squares and a circle.
  2. The triangle weighs 32 ounces, the square weighs 10 ounces, and the circle weighs 2 ounces.
  3. The triangle and the square weigh the same.

Synthesis

The purpose of this discussion is to understand how the hanger diagrams work. Some possible questions for discussion:

  • “What are some things that must be true, could be true, and cannot possibly be true about the diagrams?”
  • “What does it mean when the diagram is balanced?” (The weight on either side is equal.)
  • “What does it mean when the diagram is unbalanced?” (The weight on the lower side is heavier than the weight on the higher side.)
Standards
Building Toward
  • 7.EE.4.a·Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <em>For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?</em>
  • 7.EE.B.4.a·Solve word problems leading to equations of the form <span class="math">\(px + q = r\)</span> and <span class="math">\(p(x + q) = r\)</span>, where <span class="math">\(p\)</span>, <span class="math">\(q\)</span>, and <span class="math">\(r\)</span> are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <span>For example, the perimeter of a rectangle is <span class="math">\(54\)</span> cm. Its length is <span class="math">\(6\)</span> cm. What is its width?</span>

15 min

15 min