Unit 1, Lesson 2
A Triangle’s Sides
Integrated Math 2
2.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-CED.A.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
HSG-CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Instructional Routines
MLR8: Discussion SupportsMaterials
To Copy (from Blackline Masters)
6–12 Blank Math Community Chart
Do Any Three Lengths Make a Triangle Handout
Activity Narrative
Students use their knowledge from an earlier lesson to generate a list of side lengths that do not form triangles. Students will compare this to a list of side lengths that form triangles to find patterns in the examples and generalize (MP8) their understanding of the Triangle Inequality Theorem.
Launch
Give students 1–2 minutes of quiet think time to prepare an answer.
Activity
NoneFind 3 side lengths that would not form a triangle. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to generate a list of side lengths that form triangles alongside a list of side lengths that do not. Ask 1 or 2 students to share their answers to the task and their reasoning. As students share, record and display their responses for all to see. If time allows, generate a longer list by asking 4 or 5 more students to share only their side lengths, without reasoning.
If any students give a set of side lengths that form a triangle, move them to a new list. Leave these lists on display for the next activity or provide students with the blackline master “Do Any Three Lengths Make a Triangle Handout.” If students are using the blackline master, allow them 1–2 minutes to add to it a few of the examples from the class lists.
MLR8 Discussion Supports. Invite students to repeat their reasoning, using mathematical language: “Can you say that again, using the word ‘endpoint’?”
Advances: Speaking, Representing
Advances: Speaking, Representing
Math Community
After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.
Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.
After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.
2.2
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
As students compare their lists of side lengths that will and will not create triangles, students continue to explore the Triangle Inequality Theorem and begin to write clear and precise conjectures in everyday language. In a later lesson, students will write explicit mathematical statements such as “If a triangle has side lengths , , and , then , so it is important to focus on conceptual understanding and communication instead of mathematical symbols.
In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
This is the first time Math Language Routine 1: Stronger and Clearer Each Time is suggested in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. It is not necessary that students finish this draft before moving to the structured partner meetings step. Students then meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions, such as, “What did you mean by ?” and “Can you say that another way?” Finally, students write a second draft of their response, reflecting ideas from partners and improvements on their initial ideas. Students should be encouraged to incorporate any good ideas and words they got from their partners to make their second draft stronger and clearer.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Lesson Synthesis
In this lesson, students used data gathered from their experiments with triangle side lengths to deduce a relationship among triangle side lengths. Here are some questions for discussion to help students clarify their understanding of the Triangle Inequality Theorem:
- “Given two side lengths 2 and 6, what is a third side length that will form a triangle?” (5)
- “Given two side lengths 2 and 6, what is a third side length that will not form a triangle?” (4)
- “Can the third side of a triangle ever equal the sum of the other two sides? Why or why not?” (No, because the three sides would form a line segment.)
Since students may have written their conjectures in many different ways, display this conjecture for all to see and give students brief quiet think time to compare it to their own conjecture:
In a triangle, any one side’s length must be less than the sum of the other two sides’ lengths.
Ask students to raise their hand if they think their conjecture is different from this one. Invite 2–3 students who raise their hands to explain. Discuss to conclude whether those conjectures are the same as the class conjecture but worded differently, and clear up any misconceptions about the class conjecture. If any students thought of the triangle inequality in terms of a difference of sides, such as saying a third side’s length of the triangle must be longer than the difference of the other two, tell these students to keep those ideas ready to discuss in the next lesson.
Student Lesson Summary
To make a conjecture, or educated guess, about an idea in mathematics, we can start by experimenting with different values to find out what is true in a given situation. By trying out different side lengths, we were able to figure out that the length of one side of a triangle must be less than the sum of the lengths of the other two sides. This conjecture turns out to be true in all situations, but we will convince ourselves of this fact in another lesson.
Consider this progression of conjectures about the side lengths of a triangle:
- One side can’t be too long.
- Any one side must be shorter than the other two sides together.
- Any one side’s length must be less than the sum of the other two sides’ lengths.
Which conjecture is the most clear? These statements may look like a progression of ideas used when developing a conjecture. It is helpful to have a partner to share our thoughts with so that we can refine our ideas into stronger and clearer statements.