# Lesson 2

Ways to Look at Triangles

## Warm-up: Number Talk: Sums and Products (10 minutes)

### Narrative

This Number Talk encourages students to use their understanding of mixed numbers and properties of operations to mentally solve problems. The strategies elicited here will be helpful as students develop their fluency in performing operations on fractions.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$12 + 12 + 75$$
• $$12\frac{1}{2} + 12\frac{1}{2} + 75$$
• $$(2 \times 12\frac{1}{2}) + (4 \times 12\frac{1}{2})$$
• $$7 \times 12\frac{1}{2}$$

### Activity Synthesis

• “How is each expression related to the one before it?”
• “How can the first three expressions help us find the value of the last expression?”
• “Who can restate _____'s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Triangle Hunt (25 minutes)

### Narrative

The purpose of this activity is for students to describe and sort triangles based on the length of their sides and the size of their angles. Students are asked to find all the triangles (from the same set of cards used in the previous lesson) that have specific attributes.

Along the way, students notice attributes that all triangles seem to share (for instance, having three angles and at least two acute angles), attributes that some triangles share (for instance, having two equal angles or two equal sides), and those that no triangles seem to have (for example, parallel sides or multiple obtuse angles). With these repeated observations, students deepen their understanding of the properties of triangles and their sub-groups (MP8).

In the synthesis, students are introduced to right triangles.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed _____ , so I matched . . .” Encourage students to challenge each other when they disagree.
Action and Expression: Internalize Executive Functions. Invite students to discuss the steps they might take to complete the task. For example, students may decide to look at one triangle at a time and decide which attributes it has. Alternatively, they may decide to look at each triangle through the lens of one attribute at a time.
Supports accessibility for: Organization

### Required Materials

Materials to Gather

### Required Preparation

• Each group needs a set of shape cards from the previous lesson. If time permits, separate the triangle cards from each set in advance.
• Gather the Collect and Display chart from the previous lesson for display in the activity synthesis.

### Launch

• Groups of 2
• Give each group a set of cards from the first lesson.
• Provide access to rulers and protractors.
• Make available the chart with vocabulary from the previous lesson for reference during this lesson.
• “Use only the cards with triangles for this activity.”
• “Complete the triangle hunt with a partner. When your group is done, compare responses with another group.”

### Activity

• 5 minutes: group work time
• 3 minutes: discussion with another group
• 2 minutes: individual work time on the last question

### Student Facing

1. From the set of triangle cards, find all the triangles that have each attribute. Record their letter names here.
2. Choose one sentence to complete based on your work.

1. I noticed that some triangles . . .
2. I noticed that all triangles . . .
3. I noticed that no triangles . . .

### Advancing Student Thinking

If students place each triangle in only one category, consider asking:

• “How did you make sure you found all the triangles that have each attribute?
• “How would you check if a triangle has only one of the listed attributes or more than one?”

### Activity Synthesis

• Add new vocabulary and sketches to the chart from the previous lesson.
• “What do you notice about triangles H, L and V? (They all have right angles.)
• “Triangles that have a right angle are right triangles.
• “How did you know whether an angle in a triangle is a right angle?” (I measured it with a protractor or an object with a square corner.)
• “Can you find other right triangles in the set?” (Triangle Y)
• Invite students to share their completed sentences for the last question. Record their responses and invite the class to agree or disagree.
• Some triangles . . .
• All triangles . . .
• No triangles . . .
• If time permits, consider asking: “In our set of triangles, we didn’t see any with more than one right angle or more than one obtuse angle. Do you think it’s possible for a triangle to have:
• Two right angles?” (No, two of the sides won’t meet to make a triangle.)
• Two obtuse angles?” (No, two of the sides won’t meet to make a triangle.)

## Activity 2: The Right Kind of Triangle (10 minutes)

### Narrative

In this activity, students identify right triangles. They also explain why certain given shapes are not right triangles. As they determine whether a triangle is a right triangle, students may use the corner of an index card or a piece of paper, use a protractor, or trace an angle in a triangle to compare with an angle elsewhere that they know to be a right angle. In doing so, students practice choosing tools strategically (MP5).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Provide access to index cards, patty paper, and protractors.

### Activity

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who use tools to check angles and those who estimate them visually. Ask the latter: “How might we check that the angles that look like right angles are $$90^\circ$$?”

### Student Facing

1. Identify all shapes that are right triangles. For each right triangle, mark the right angle with a small square.
2. Explain why the other shapes are not right triangles.

### Advancing Student Thinking

Students may not identify triangles C or D because of the orientation. Consider asking:

• “How can we show that a triangle has a right angle?”
• “Where else do you see triangles that may have perpendicular sides?”
• “What tools could you use to see if each triangle has any right angles?”

### Activity Synthesis

• Invite students to share their responses and how they found the right triangles in the set.
• “Were there any triangles that appeared to be right triangles but are not, or don’t appear to be right triangles but are?” (Triangle H looks like one but isn’t.)

## Lesson Synthesis

### Lesson Synthesis

“Today we analyzed and identified triangles with different attributes.“

“If we classify or group triangles based on side length, what types might we see?” (Triangles with all the same length, all different lengths, or two sides with the same length)

“How can we tell if sides were the same length?” (Use a ruler or patty paper to measure, fold the paper to see if the sides match up perfectly)

“If we classify or group triangles based on angle, what types might we see?” (Triangles with all acute angles, only one obtuse angle, or one right angle)

“Earlier we identified some right triangles. Which of these statements do you think defines a right triangle: ‘a triangle with one right angle’ or ‘a triangle with one pair of perpendicular sides’?” (Both are accurate. Perpendicular sides make a right angle.)

“Take 12 minutes to add any new words from today's lesson to your word walls.”

Tell students that, in the next lesson, they will explore some of the same attributes in quadrilaterals.