Lesson 12

Practice With Proportional Relationships

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12.1: Vegetable Garden (5 minutes)

CCSS Standards


Addressing

Routines and Materials


Required Materials

Warm-up

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.

Student Facing

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

Scale: 1 unit = 2.8 ft

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.

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

Student Response

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Activity Synthesis

Invite students to share:

  • equations written in the form of equivalent ratios
  • equations written in the form \(y=kx\)
  • equivalent ratios in which each ratio compares side lengths within each triangle
  • equivalent ratios in which each ratio compares pairs of corresponding sides

12.2: Card Sort: Corresponding Parts (15 minutes)

CCSS Standards


Addressing

Routines and Materials


Instructional Routines

Required Materials

Optional 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

Arrange students in groups of 2. Distribute pre-cut slips.

Conversing: MLR8 Discussion Supports. Use this routine to help students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin with this sentence frame: “Card _____ matches with Card _____ because....” Invite the listener, Partner B, to press for additional details referring to specific features of the triangles, such as the corresponding parts and scale factor between the triangles. Students should switch roles for each figure. This will help students justify how the features of triangles can be used to identify similar triangles.
Design Principle(s): Support sense-making; Cultivate conversation
Representation: Internalize Comprehension. Differentiate the degree of difficulty or complexity by beginning with an example with more accessible values. Display three triangles and invite students to practice explaining why one pair is similar and why another pair is not similar.
Supports accessibility for: Conceptual processing

Student Facing

Your teacher will give you a set of cards. Group them into pairs of similar figures. For each pair, determine:

  1. a similarity statement
  2. the scale factor between the similar figures
  3. the missing lengths

Student Response

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Anticipated Misconceptions

Encourage students to physically rotate the cards to line up corresponding parts.

Activity Synthesis

There are many nuances to hit in discussion. One important idea to point out is that to calculate the scale factor, students had to calculate a ratio of side lengths. Because the scale factor is always the same, and there are many possible ratios to construct, it means all of those ratios are equivalent. It’s likely that students will assume the scale factor goes from small to large. Point out that we could also dilate by the reciprocal to scale from large to small, in which case, it’s also true that the reciprocals of the ratios are also all equivalent.

12.3: Quilting Questions (15 minutes)

CCSS Standards


Addressing

Routines and Materials


Instructional Routines

Optional activity

In both subsequent lessons and units, students will work extensively with similar right triangles. This activity gives students opportunities to practice the Pythagorean Theorem, and apply similar triangles in a contextual problem.

Launch

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.
Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to increase awareness of the language used to talk about similar right triangles. Before revealing the questions in this activity, display the image of the quilt design made of right isosceles triangles. Ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the features of the right triangles. For example, “Are all the right triangles similar?”, “What is the scale factor between the small and large right triangle?”, and “What are the side lengths of the right triangles?”
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

  1. 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.
    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.
  2. Are the triangles similar? If so, what are the scale factors?
  3. This quilt is meant for a baby (1 unit = 6 inches). To make a quilt for a queen-size bed, it needs to be 90 inches wide. What dimensions should the center squares of the big quilt have to reach that width?

Student Response

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Student Facing

Are you ready for more?

  1. Here is a quilt design made of right triangles. Find as many different size triangles as you can.
    A shape made up of many right triangles. Diamond-shape C F Q L is divided into 16 equal-sized right triangles. Right triangles C L D and C F E extend out from the diamond-shape like wings.
  2. Write similarity statements for 2 pairs of triangles.
  3. The smallest triangles have legs 2 units and 3 units long. Write some equivalent ratios or equations that will help you determine the dimensions of triangle \(LCQ\). Use your equivalent ratios or equations to find the dimensions of triangle \(LCQ\).

Student Response

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Anticipated Misconceptions

Students who struggle with the number of similar triangles in the image can trace each size triangle separately onto tracing paper and label the corresponding sides using colored pencils.

Activity Synthesis

A key point to emphasize in this discussion is that ratios in which each ratio compares side lengths within each figure are also equivalent across similar figures. That is, for a triangle with side lengths \(a, b, c\), and a similar triangle with corresponding side lengths \(x,y,z\), we have equivalent ratios such as \(\frac{a}{b} = \frac{x}{y}\).

Invite students to share strategies for scaling the figure up from a baby blanket to a quilt for a queen-size bed.

Lesson Synthesis

Lesson Synthesis

Ask students to write several equations about this pair of triangles. Display the image for all to see:

\(\angle Y \cong \angle W, \overline{JD} \perp \overline{DY},\overline{BL} \perp \overline{BW} \)

Similar triangles D J Y and B L W.

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

  • equations written in the form \(y=kx\) (\(WL=\frac12 \boldcdot 2\))
  • equivalent ratios in which each ratio compares side lengths within each triangle (\(\frac21=\frac{WL}{0.5}\))
  • equivalent ratios in which each ratio compares pairs of corresponding sides (\(\frac{1}{0.5}=\frac{2}{WL}\))

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

12.4: Cool-down - Ratios Galore (5 minutes)

CCSS Standards


Addressing

Cool-Down

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Student Lesson Summary

Student Facing

When 2 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 \(ABC\) is similar to triangle \(DEF\), we know that pairs of corresponding side lengths are in the same proportion. \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\)

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.

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.

\(\frac{AB}{BC}=\frac{DE}{EF}\)

\(\frac{AB}{AC}=\frac{DE}{DF}\)

\(\frac{AC}{BC}=\frac{DF}{EF}\)

We can use these equivalent ratios to find unknown side lengths. Which equivalent ratios would work to find \(DE\) and \(DF\)?

We can use \(\frac{AB}{BC}=\frac{DE}{EF}\) to find \(DE\). Then \(\frac34=\frac{DE}{6}\), which gives \(DE=4.5\).

Or we can use \(\frac{AB}{DE}=\frac{BC}{EF}\) to find \(DE\). In this case, we get \(\frac{3}{DE}=\frac46\), which also gives \(DE=4.5\).