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Unit 1, Lesson 4

Big Angles, Long Sides. Small Angles, Short Sides.

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Integrated Math 2

4.1

Warm-up

10 mins

Which Three Go Together: Tri-"angles"

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Which Three Go Together?

Materials

None

Activity Narrative

This is the first Which Three Go Together? routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together? Why do they go together?”

Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepens their awareness of connections across representations.

This Warm-up prompts students to compare four triangles. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Launch

Arrange students in groups of 2–4. Display the triangles for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three triangles that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.

Activity

None

Student Task Statement

Which three go together? Why do they go together?

Student Response

to access Student Response.

Activity Synthesis

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and make sure the reasons given are correct.

During the discussion, prompt students to explain the meaning of any vocabulary they use, such as “right,” “equilateral,” or “scalene,” and to clarify their reasoning as needed. Consider asking:

“How do you know ?”
“What do you mean by ?”
“Can you say that in another way?”

4.2

Activity

15 mins

Let’s Include Angles

Instructional Routines

MLR8: Discussion Supports

Materials

4.3

Activity

15 mins

A Few More Triangles

Instructional Routines

None

Materials

To Gather

Lesson Synthesis

Display these triangles for all to see. Ask students the following questions:

“In triangle , which side is the longest? () The shortest? () How do you know?” (These sides are opposite the largest and smallest angles, respectively.)
“In triangle , list the angles in order from smallest to largest. Explain your reasoning.” (Angle , angle , angle . The shortest side is , which is across from angle . The longest side is , which is across from angle .)

State the following conjecture and ask students if they agree:

“In a triangle, larger angles are opposite longer sides and longer sides are opposite larger angles.”

Invite a few students to share their thinking. Explain to students that they have justified this conjecture only experimentally, which is an informal way to use examples to verify ideas. In future lessons they will prove theorems formally.

to access Cool-down.

Student Lesson Summary

In a triangle, larger angles are always opposite, or across from, longer sides. In triangle , the largest angle is and it is opposite the longest side . Similarly, the smallest angle, , is opposite the shortest side, . The converse is also true—longer sides are opposite larger angles. Since angle is the largest angle, we know side is the longest.

In triangle , which side is the shortest? Using the Triangle Angle Sum Theorem, we find angle is 32 degrees, so side must be the shortest side in the triangle.

In right triangles, we have additional methods to help us figure out information about the angles and sides. You may have learned in an earlier grade that the hypotenuse of a right triangle is always the longest side. Now we can say that the hypotenuse is the longest side in a right triangle because it is opposite the right angle, which is always the biggest angle in a right triangle.

In triangle , which side is shorter: or ?

Method 1

Using Pythagorean Theorem, we can solve for .

This method is helpful if you need to know the length of and you have a right triangle.

Method 2

Using the Triangle Angle Sum Theorem, we know the measure of angle is 56 degrees.

Since angle is the smallest angle, the shortest side must be .

This method doesn’t give us the length of , but it does answer the question of which side in the triangle is shortest.

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Lesson 4

Big Angles, Long Sides. Small Angles, Short Sides.

Warm-up

Which Three Go Together: Tri-"angles"

Standards Alignment

HSG-CO.C.10

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Activity Synthesis

Activity

Let’s Include Angles

Instructional Routines

Materials

Activity

A Few More Triangles

Instructional Routines

Materials

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Building on Student Thinking

Are You Ready for More?

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

7.G.A.2

HSG-CO.C.10

Standards Alignment

8.G.B

HSG-CO.C.10

Student Response

Building on Student Thinking

Are You Ready for More?