Unit 3, Lesson 6
Reasoning about Solving Equations (Part 1)
Accelerated 7
6.1
Warm-up
Hanger Diagrams
Instructional Routines
Notice and WonderMaterials
NoneActivity 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.
Two clothes hangers hanging from a rod. One clothes hanger has a pink sock on both ends and is level. The other clothes hanger has a blue sock on both ends and is not level.
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.
Activity
NoneIn the two diagrams, all the triangles weigh the same and all the squares weigh the same.
For each diagram, come up with . . .
- One thing that must be true
- One thing that could be true
- One thing that cannot possibly be true
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.
Activity 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.)
Student Lesson Summary
In this lesson, we worked with two ways to show that two amounts are equal: a balanced hanger and an equation. We can use think about the weights on a balanced hanger to understand steps we can use to find an unknown amount in a matching equation.
This hanger diagram shows a total weight of 7 units on one side that is balanced with 3 equal, unknown weights and a 1-unit weight on the other. An equation that represents the relationship is
We can remove a weight of 1 unit from each side and the hanger will stay balanced. This is the same as subtracting 1 from each side of the equation.
Balanced hanger, left side, 6 blue squares and one red squared being removed. Right side, 3 green squares, and one red square being removed. To the side, an equation 7 minus 1 = 3 x + 1 minus 1, with each minus 1 written in red.
An equation for the new balanced hanger is
We can make 3 equal groups on each side and the hanger will stay balanced. This is the same as dividing each side of the equation by 3 (or multiplying each side by
Balanced hanger, left side 6 blue squares, right side, three green circles. A dotted line is drawn around each of 3 groups, each groups consists of two blue square s from the left side and one green circle from the right side. To the side, an equation says 1/3 * 6 = 1/3 * 3 x.
The two sides of the hanger balance with two 1-unit weights on one side and 1 weight of unknown size on the other side. So, the unknown weight is 2 units.
Here is a concise way to write the steps above:
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