options: Interpret the structure of expressions.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Use the four operations with whole numbers to solve problems.

Reason about and solve one-variable equations and inequalities.

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.

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

Use properties of operations to generate equivalent expressions.

Solve systems of equations.

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

Work with addition and subtraction equations.

Understand the relationship between zeros and factors of polynomials.

Write and interpret numerical expressions.

Represent and solve problems involving multiplication and division.

Generate and analyze patterns.

Represent and solve equations and inequalities graphically.

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

Perform arithmetic operations on polynomials.

Represent and solve problems involving addition and subtraction.

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

Write expressions in equivalent forms to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Add and subtract within 20.

Create equations that describe numbers or relationships

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

Work with radicals and integer exponents.

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

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

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

Understand solving equations as a process of reasoning and explain the reasoning.
problem_activity: Activity
In this activity, students apply the expression they previously generated to find the areas of various triangles. Each diagram is labeled with two or three measurements. Before calculating, students think about which lengths can be used to find the area of each triangle.
As students work, notice students who choose different bases for Triangles B and D. Invite them to contribute to the discussion about finding the areas of right triangles later.
Launch
Explain to students that they will now practice using their expression to find the area of triangles without a grid. For each triangle, ask students to be prepared to explain which measurement they choose for the base and which one for the corresponding height and why.
Keep students in groups of 2–4. Give students 5 minutes of quiet think time, followed by 1–2 minutes for discussing their responses in their group.
Representing: MLR2 Collect and Display.
While pairs are working, circulate and listen to student talk about identifying the bases and corresponding heights for each of triangles. Write down common or important phrases you hear students say about each triangle, specifically focusing on how students make sense of the base and height of each triangle. Record the words students use to refer to each triangle and display them for all to see during the whole-class discussion.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
For each triangle, circle a base measurement that you can use to find the area of the triangle. Then, find the area of any
three
triangles. Show your reasoning.
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Student Response
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Anticipated Misconceptions
The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide the measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the warm-up activity.
Activity Synthesis
The aim of this whole-class discussion is to deepen students’ awareness of the base and height of triangles. Discuss questions such as:
For Triangle A, can we say that the 6-cm segment is the base and the 5-cm segment is the height? Why or why not? (No, the base of a triangle is one of its sides.)
What about for Triangle C:  Can the 3-cm segment serve as the base? Why or why not? (No, that segment is not a side of the triangle.)
Can the 3.5-cm side in Triangle C serve as the base? Why or why not? (Yes, it is a side of the triangle, but because we don't have the height that corresponds to it, it is not helpful for finding the area here.)
More than two measurements are given for Triangles C, D, and E. Which ones are helpful for finding area? (We need a base and a corresponding height, which means the length of one side of the triangle and the length of a perpendicular segment between that side and the opposite vertex.)
When it comes to finding area, how are right triangles—like B and D—unique? (Either of the two sides that form the right angle could be the base or the height. In non-right triangles—like A, C, and E—the height segment is not a side of the triangle; a different line segment has to be drawn.)


correct_option_index: 5