options: Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
problem_activity: Warm-up
The purpose of this warm-up is for students to revisit ideas they learned in the previous lesson about balanced hangers:
You can add or subtract the same thing on each side and the hanger stays in balance.
You can divide each side by the same number and the hanger stays in balance.
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
Figures A, B, C, and D show the result of simplifying the hanger in Figure A by removing equal weights from each side.
[IMG]
Here are some equations. Each equation represents one of the hanger diagrams.
\(2(x + 3y) = 4x + 2y\)
\(2y = x\)
\(2(x + 3y) + 2z = 2z + 4x + 2y\)
\(x + 3y = 2x + y\)
Write the equation that goes with each figure:
A:
B:
C:
D:
Each variable (
\(x\)
,
\(y\)
, and
\(z\)
) represents the weight of one shape. Which goes with which?
Explain what was done to each equation to create the next equation. If you get stuck, think about how the hangers changed.
Student Response
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Anticipated Misconceptions
Some students may think a variable stands for more than one object. Tell these students that a variable only stands for one object, as it also only represents one number.
Activity Synthesis
Ask students to explain how they decided on the matching equation. As students discuss the final questions, highlight responses that emphasize the same objects being removed from each side creates the next figure in line. The exception to this is the move from Hanger B to Hanger C, where the number of objects on each side is halved. Ask students to explain why this is an okay move, even though different objects are being removed from each side (1 square and 3 triangles on the left, 2 squares and 1 triangle on the right). Refer to MLR 2 (Collect and Display).


correct_option_index: 9