options: 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.

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
problem_activity: Activity
This activity continues the work of using a balanced hanger to develop strategies for solving equations. Students are presented with balanced hangers and are asked to write equations that represent them. They are then asked to explain how to use the diagrams, and then the equations, to reason about a solution. Students notice the structure of equations and diagrams and find correspondences between them and between solution strategies.
Launch
Draw students’ attention to the diagrams in the task statement. Ensure they notice that the hangers are balanced and that each object is labeled with its weight. Some weights are labeled with numbers but some are unknown, so they are labeled with a variable.
Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.
Student Facing
Here are some balanced hangers where each piece is labeled with its weight. For each diagram:
Write an equation.
Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.
Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.
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Student Response
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Activity Synthesis
Invite students to demonstrate, side by side, how they reasoned with both the diagram and the equation. For example, diagram A can be shown next to the equation
\(7=3x+1\)
. Cross out a piece representing 1 from each side, and write
\(7-1=3x+1-1\)
, followed by
\(6=3x\)
. Encircle 3 equal groups on each side, and write
\(6 \div 3 = 3x \div 3\)
, followed by
\(2=x\)
. Repeat for as many diagrams as time allows. If diagrams A and B did not present much of a challenge for students, spend most of the time on diagrams C and D.
We want students to walk away with two things:
An instant recognition of the structure of equations of the form
\(px+q=r\)
where
\(p\)
,
\(q\)
, and
\(r\)
are specific, given numbers.
A visual representation in their minds that can be used to support understanding of why for equations of this type, you can subtract
\(q\)
from each side and then divide each side by
\(p\)
to find the solution.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following terms and maintain the display for reference throughout the unit: hanger diagram. For example, display an example of a balanced hanger. With class participation, create step-by-step instructions on how to write and solve an equation based on the hanger.
Supports accessibility for: Memory; Language
Representing, Speaking: MLR8 Discussion Supports.
To invite participation in the whole-class discussion, use sentence frames to support students’ explanations for how they found the weights by reasoning about the hangers and equations. For example, provide the frame “First, I ___ because . . .”, “Then I ___ because . . . ." Be sure to verbalize and amplify mathematical language in the students’ explanations (e.g., “subtracting the constant”, and “dividing by the coefficient”). This will help students explain their reasoning with the diagram and the equation.
Design Principle(s): Optimize output (for explanation); Cultivate conversation


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