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

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

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.

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

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

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.

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.

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

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.

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.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
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.
[IMG]
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: 2

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