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Unit 1, Lesson 4

Creating Cross-Sections by Dilating

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Integrated Math 3

4.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

To Gather

Math Community Chart Sticky notes

Activity Narrative

This is the first Notice and Wonder activity in the course. Students dilate a triangle from a center of dilation and are then asked: “What do you notice? What do you wonder?”

Students are given time to write down what they notice and wonder about their completed drawings and then time to share their thoughts. Their responses are recorded for all to see. Often, the goal is to elicit observations and curiosities about a mathematical idea that students are about to explore. Pondering the two open questions allows students to build interest about and gain entry into an upcoming task.

The purpose of the activity is to elicit the idea that the triangle and its dilation can be viewed as representations of the cross-sections of a three-dimensional pyramid. This will help prepare students for the next activity, in which they create a representation of a pyramid by suspending several dilations of a rectangle. While students may notice and wonder many things about the figures they draw, the three-dimensional visualization is the most important discussion point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Launch

Arrange students in groups of 2. Tell students to draw the dilated figure. Then ask them to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.

Activity

None

Dilate triangle using center and a scale factor of 2.

<p>Triangle BCD dilated using center P and a scale factor of 2.</p>

Look at your drawing. What do you notice? What do you wonder?

Activity Synthesis

Choose a student’s drawing, and display it for all to see. Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the displayed drawing. Next, ask students, “Is there anything on the list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the idea that the drawing resembles a pyramid does not come up during the conversation, ask students to discuss this idea. Remind students that pyramids, like prisms, are named for their bases, so this pyramid should be called a “triangular” pyramid.

Math Community

After the Warm-up, display the class Math Community Chart for all to see and explain that the listed “Doing Math” actions come from the sticky notes students wrote in the first exercise. Give students 1 minute to review the chart. Then invite students to identify something on the chart they agree with and hope for the class or something they feel is missing from the chart and would like to add. Record any additions on the chart. Tell students that the chart will continue to grow and that they can suggest other additions that they think of throughout today’s lesson during the Cool-down.

4.2

Activity

Instructional Routines

Extend It MLR8: Discussion Supports

Materials

To Gather

Blank paper Rulers Scientific calculators Scissors String

Activity Narrative

In this activity, students dilate rectangles and suspend them to resemble cross-sections of a pyramid. This work connects to later lessons in which students use areas of cross-sections in the development of the pyramid volume formula.

This task reinforces the idea that scaling by a factor of adjusts all distances in a figure by a factor of , and foreshadows the upcoming concept that if a figure is scaled by a factor of , then its area changes by a factor of .

Monitor for groups who choose different lengths of string, therefore creating pyramids of different heights.

Lesson Synthesis

The goal of the discussion is to extend students’ understanding of cross-sections and dilation. Display the pyramids from 2 groups with different heights. Here are some questions for discussion:

“Where was the scale factor rectangle placed in each pyramid?” (It was exactly halfway up the pyramid in each case.)
“What is the difference between using the center of the rectangle as the center of dilation, and using the top vertex of the pyramid as the center?” (When we use the center of the rectangle, the dilation stays in the same plane. That’s what we did, but then we assembled those into a pyramid. The top vertex of the pyramid can be viewed as the center of dilation in 3 dimensions.)
“What would happen if we dilated the rectangle by a scale factor of or , using the top vertex of the pyramid as the center of dilation?” (The rectangle would be a cross-section near the base. The rectangle would be a cross-section near the top vertex of the pyramid.)
“What is the range of scale factors that create cross-sections of the pyramid?” (The scale factors between 0 and 1 create cross-sections. Larger scale factors extend outside of the pyramid.)

to access Cool-down.

Student Lesson Summary

Imagine a triangle lying flat on the desk, and a point directly above the triangle. If we dilate the triangle using center and scale factor or 0.5, together the triangles resemble cross-sections of a pyramid.

<p><strong>Image of pyramid. Apex, point p. Halfway from apex to base, small green triangular cross section. Base, larger blue triangle.</strong></p>

We can add in more cross-sections. This image includes two more cross-sections, one with scale factor and one with scale factor . The triangle with scale factor is the base of the pyramid, and if we dilate with scale factor , we get a single point at the very top of the pyramid.

<p>A triangle laying flat with point P above it.</p>

Each triangle’s side lengths are a factor of times the corresponding side length in the base. For example, for the cross-section with , each side length is half the length of the base’s side lengths.

Launch

Arrange students in groups of 4.

Draw a triangle and a square on the board. For each shape, hold a ball or another small object representing point off the surface of the board, projected directly out from the center of the shape. Invite students to imagine the three-dimensional objects created by connecting to each vertex of the shape: a triangular pyramid and a square pyramid.

Display this image for all to see. Tell students that it is a top-down view of the three-dimensional shapes they just imagined.

<p>Birds eye view of square and triangular pyramids </p>

Ask students how they could dilate each two-dimensional figure by a scale factor of using as the center of dilation. With students’ guidance, complete the dilations. Be sure students see that the dilation is created by measuring the distance from to each vertex of the “base” and multiplying that distance by to find the halfway point. Ask students what the dilations could represent: cross-sections of each solid.

<p>Birds eye view of square and triangular pyramids with cross sections </p>

Ask students if they are familiar with mobiles. (Mobiles can be decorations for babies. A mobile is a kind of sculpture in which materials are suspended in the air.) Share that the first mobiles were wind chimes, which have been found around the world, dating back to 3000 BCE in Southeast Asia. Tell students they’ll be using the concepts just discussed to make a pyramid mobile.

MLR8 Discussion Supports. Invite students to repeat their strategy using mathematical language: "Can you say that again, using the phrase 'scale factor'?"
Advances: Speaking, Representing

Action and Expression: Provide Access for Physical Action. Provide access to pre-cut materials to reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.
Supports accessibility for: Fine Motor Skills, Visual-Spatial Processing

Activity

None

Your teacher will give you sheets of paper. Each student in the group should take one sheet of paper and complete these steps:

Locate and mark the center of your sheet of paper by drawing diagonals (or by another method).
Each student should choose one scale factor from the table. Make sure that each scale factor is chosen. On your paper, draw a dilation of the entire sheet of paper, using the center you marked as the center of dilation.
Measure the length and width of your dilated rectangle and calculate its area. Record the data in the table.
Cut out your dilated rectangle, and make a small hole in the center.

scale factor,	length of scaled rectangle	width of scaled rectangle	area of scaled rectangle

Now the group as a whole should complete the remaining steps:

Cut 1 long piece of string (more than 30 centimeters) and 4 shorter pieces of string. Make 4 marks, an equal distance apart, on the long piece of string.
Thread the long piece of string through the hole in the largest rectangle. Tie a shorter piece of string beneath it where you made the first mark on the string. This will hold up the rectangle so that it doesn't slide down the long string.
Thread the remaining pieces of paper onto the string from largest to smallest, tying a short piece of string beneath each one at the marks you made.
Hold up the end of the string to make your cross-sections resemble a pyramid. As a group, you may have to steady the cross-sections for the pyramid to clearly appear.

Student Response

to access Student Response.

Building on Student Thinking

If students struggle to create the dilations, remind them of the demonstration from the activity Launch. Suggest that they draw lines connecting the center of the paper to each vertex, and measure the lines. Then invite them to think about how the scale factor they’re using will apply to these distances.

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Audience

Assessment Access

Lesson 4

Creating Cross-Sections by Dilating

Warm-up

Standards Alignment

Building On

HSG-SRT.A

Addressing

HSG-GMD.B.4

Building Toward

HSG-GMD.A.1

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

HSG-SRT.A

Addressing

HSG-GMD.B.4

Building Toward

HSG-GMD.A.1