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Unit 2, Lesson 14

Practice with Proportional Relationships (Optional)

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Integrated Math 2

14.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Four-function calculators

Activity Narrative

In previous lessons in this unit, students studied scale drawings. This activity provides additional practice finding an unknown side in a scaled copy. Students will do more work with finding unknown values in proportional relationships throughout this lesson.

Launch

Invite students to share any experiences they have with gardening. Then ask students if they can think of a situation in which a garden plot might need to be in the shape of a triangle rather than a rectangle.

Tell students that in this situation, that a school has a square courtyard in the middle of the school and there is a diamond-shaped sitting area in the middle, leaving 4 triangle areas in the corners for different groups to plant a garden.

Activity

None

These are the plans for a vegetable garden that a school is designing.

Scale: 1 unit = 2.8 ft

<p>Triangle A B C. Length of A B is 4, A C is 3, and B C is 5. At the top of the triangle are 3 potatoes, 6 carrots underneath, then 3 pumpkins at the bottom.</p>

Write at least 3 equivalent ratios or equations using lengths from both the diagram and the full-size garden.

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Invite students to share:

Equations written in the form of equivalent ratios.
Equations written in the form .
Equivalent ratios in which each ratio compares side lengths within each triangle.
Equivalent ratios in which each ratio compares pairs of corresponding sides.

14.2

Activity

Optional

Instructional Routines

Card Sort MLR8: Discussion Supports

Materials

To Copy (from Blackline Masters)

Corresponding Parts Cards

Activity Narrative

Students sort different images during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

As students work, encourage them to justify their decisions about which triangles are similar using more precise language (MP6).

Launch

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2, and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:

Give students 1 minute to place all the cards face up and to start thinking about possible ways to sort the cards into categories.
Pause the class and select 1–3 students to share the categories they identified.
Discuss as many different categories as time allows.

Attend to the language that students use to describe their categories and images, giving them opportunities to describe their categories more precisely. Highlight the use of terms like “similar,” “scaled copy,” and “corresponding parts.” After a brief discussion, invite students to open their books or devices and continue with the activity.

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.
Advances: Conversing, Representing

Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for finding corresponding parts of the triangle to write similarity statements. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Attention, Social-Emotional Functioning

14.3

Activity

Optional

Instructional Routines

MLR6: Three Reads

Materials

None

Activity Narrative

This lesson gives students an opportunity to practice using the Pythagorean Theorem along with similar triangles in a real-world situation. In later lessons students will prove the Pythagorean Theorem using some of the ideas from this activity.

This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Lesson Synthesis

Display the image, and ask students to write several equations about this pair of triangles:

<p>Similar triangles D J Y and B L W.</p>

Listen for and invite students to share (or add if no students mention):

Equations written in the form ().
Equivalent ratios in which each ratio compares side lengths within each triangle ().
Equivalent ratios in which each ratio compares pairs of corresponding sides ().

After students share multiple equations involving , discuss that the equations should all have the same solution. Encourage students to make sure that there is at least one way of finding unknown values in equivalent ratios that they feel comfortable with.

to access Cool-down.

Student Lesson Summary

When two figures are similar, there are lots of equivalent ratios between the triangles and within the triangles. We can use those relationships to find missing lengths. For example, if we know that triangle is similar to triangle , we know that pairs of corresponding side lengths are in the same proportion.

<p>Two similar triangles A B C and D E F. Length of A B is 3, B C is 4, A C is 5, and E F is 6.</p> — $\triangle ABC \sim \triangle DEF$

We also know that pairs of side lengths in one triangle are in the same proportion as pairs of side lengths in the other triangle.

We can use these equivalent ratios to find unknown side lengths. Which equivalent ratios would work to find ?

We can use to find . Then , which gives .

Or we can use to find . In this case, we get , which also gives .

Launch

Invite students to relate to the context by asking if any students have a quilt that is special to their family. Quilts are an ancient practical art that have been used for a variety of things, including telling a family history and raising money to support various causes.

Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and image without revealing the questions.

In the first read, students read the problem with the goal of comprehending the situation.
For the first read, read the problem aloud while everyone else reads along, and then ask, “What is this situation about? What is going on here?” Allow 1 minute to discuss with a partner and then share with the whole class. A typical response may be, “Someone is making a quilt from squares and triangles.” Listen for and clarify any questions about the context.
In the second read, students analyze the mathematical structure of the story by naming quantities.
Invite students to read the problem aloud with their partner, or select a student to read to the class, then prompt students by asking, “What can be counted or measured in this situation?” Give students 30 seconds of quiet think time, followed by another 30 seconds to share with their partner. A typical response may be: “I know the size of the smallest squares in the center” or "I know the legs of an isosceles right triangle are the same length."
In the third read, students brainstorm possible starting points for answering the questions.
Invite students to read the problem aloud with their partner, or select a different student to read to the class. After the third read, reveal the first instruction on finding the dimensions of the triangles and ask, “What are some ways we might get started on this?” Instruct students to think of ways to approach the questions without actually solving. Give students 1 minute of quiet think time followed by another minute to discuss with their partner. Invite students to name some possible strategies referencing quantities from the second read. Provide these sentence frames as partners discuss:
- “To find the side lengths of the central squares, I would _____.”
- “Once I know the lengths of the sides of the central squares, I could find _____.”
- “When I know the length of the hypotenuse of an isosceles right triangle, I can find the length of the legs by _____.”
As partners are discussing their solution strategies, select 1–2 students to share their ideas with the whole class. As students are presenting their strategies to the whole class, create a display that summarizes starting points for the question. (Stop students as needed before they share complete solutions or answers.)

Give students time to complete the rest of the activity, and follow with a whole-class discussion.

Activity

None

Here is a quilt design made of right isosceles triangles. The smallest squares in the center have an area of 1 square unit.

Find the dimensions of the triangles.
Are the triangles similar? If so, what are the scale factors?
The instructions that came with this pattern are for a baby’s quilt. The instructions say “1 unit = 6 inches.” To make this quilt for a queen-sized bed, the length of one side of the quilt needs to be 90 inches. What dimensions should the center squares of the big quilt have to reach that width?

<p>A small square with isosceles right triangles on its sides, forming a larger square. Isosceles right triangles are on those sides, forming another square. There is one more iteration of the pattern.</p>

Student Response

to access Student Response.

Building on Student Thinking

If students struggled to use the Pythagorean Theorem in a previous lesson:

Begin by asking students what they notice and wonder about the image from this activity.
Draw students’ attention to the right triangles, and ask them what must be true about the side lengths.
Provide some possible lengths for sides, and have students practice finding an unknown side length in an isosceles right triangle to allow students to practice working with the square root of 2.

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Lesson 14

Practice with Proportional Relationships (Optional)

Warm-up

Standards Alignment

Building On

Addressing

HSG-SRT.B.5

Building Toward

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Copy (from Blackline Masters)

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSG-SRT.B.5

Building Toward

Standards Alignment

Building On

Addressing

HSG-SRT.B.5

HSN-Q.A.1

Building Toward