options: Perform arithmetic operations on polynomials.

Generate and analyze patterns.

Represent and solve equations and inequalities graphically.

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

Create equations that describe numbers or relationships

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

Add and subtract within 20.

Use polynomial identities to solve problems.

Work with radicals and integer exponents.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Analyze patterns and relationships.

Write expressions in equivalent forms to solve problems.

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

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Write and interpret numerical expressions.

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

Understand the relationship between zeros and factors of polynomials.

Rewrite rational expressions.

Use the four operations with whole numbers to solve problems.

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

Gain familiarity with factors and multiples.

Interpret the structure of expressions.

Multiply and divide within 100.

Represent and solve problems involving multiplication and division.

Use properties of operations to generate equivalent expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve systems of equations.

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

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

Solve equations and inequalities in one variable.

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

Reason about and solve one-variable equations and inequalities.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.
problem_activity: Optional activity
This activity allows students to practice identifying the base and height of triangles and using them to find areas.
Because there are no directions on which base or height to use, and because not all sides would enable them to calculate area easily, students need to think structurally and choose strategically. All triangles in the problems have either a vertical or a horizontal side. Choosing such a side as the base makes it easier to identify the corresponding height.
In some cases, students may opt to use a combination of area-reasoning strategies rather than finding the base and height of the shaded triangles and applying the formula. For instance, they may enclose a shaded triangle with a rectangle and subtract the areas of extra triangles (with or without using the formula on those extra triangles). Notice students who use such strategies so they could share later.
Launch
Keep students in groups of 2. Explain that they will now practice locating or drawing heights and using them to find area of triangles. Give students 8–10 minutes of quiet think time and time to share their responses with a partner afterwards. Provide access to their geometry toolkits (especially index cards).
If time is limited, consider asking students find the area of two or three triangles instead of all four.
Student Facing
For each triangle, identify and label a base and height. If needed, draw a line segment to show the height.
Then, find the area of the triangle. Show your reasoning. (The side length of each square on the grid is 1 unit.)
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Student Response
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Student Facing
Are you ready for more?
Find the area of this triangle. Show your reasoning.
[IMG]
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students may think that a vertical side of a triangle is the height regardless of the segment used as the base. If this happens, have them use an index card as a straightedge to check if the two segments they are using as base and height are perpendicular.
Some students may not immediately see that choosing a side that is either vertical or horizontal would enable them to find the corresponding height very easily. They may choose a non-vertical or non-horizontal side and not take advantage of the grid. Ask if a different side might make it easier to determine the base-height lengths without having to measure.
Activity Synthesis
Focus the whole-class discussion how students went about identifying bases and heights. Discuss:
“Which side did you choose as the base for triangle B? C? Why?”
“Aside from choosing a vertical or horizontal side as the base, is there another way to find the area of the shaded triangles without using their bases and heights?” (Invite a couple of students who use the enclose-and-subtract method to find the area of B, C, or D to share.)
“Which strategy do you prefer or do you think is more efficient?”
“Can you think of an example where it might be preferable to find the base and height of the triangle of interest?” (Students may point to any of the triangles in the task.)
“Can you think of an example where it might be preferable to enclose the triangle of interest and subtract other areas?” (Students may point to the triangle shown in "Are you ready for more?", where none of the shaded triangle's sides are horizontal or vertical.)
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Provide sentence frames to help students produce statements that describe the strategies they use to identify bases and heights, and to find area. For example, “For triangle ___, I chose side ___ as the base because . . .” or “The next time I need to find the area of a triangle, the strategy I will use is _____, because . . . .”
Design Principle(s): Support sense-making


correct_option_index: 9