options: Represent and solve problems involving addition and subtraction.

Work with equal groups of objects to gain foundations for multiplication.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Create equations that describe numbers or relationships

Write expressions in equivalent forms to solve problems.

Work with addition and subtraction equations.

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Understand properties of multiplication and the relationship between multiplication and division.

Use the four operations with whole numbers to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Solve equations and inequalities in one variable.

Rewrite rational expressions.

Solve systems of equations.

Analyze patterns and relationships.

Write and interpret numerical expressions.

Understand solving equations as a process of reasoning and explain the reasoning.

Represent and solve equations and inequalities graphically.

Add and subtract within 20.

Use polynomial identities to solve problems.

Perform arithmetic operations on polynomials.

Generate and analyze patterns.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Represent and solve problems involving multiplication and division.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Gain familiarity with factors and multiples.

Understand and apply properties of operations and the relationship between addition and subtraction.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand the relationship between zeros and factors of polynomials.

Understand the connections between proportional relationships, lines, and linear equations.

Reason about and solve one-variable equations and inequalities.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Interpret the structure of expressions.
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


correct_option_index: 2