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6-8 Grade 8 Unit 2 Lesson 5

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IM K-5 Math
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Lesson 5

5.1 Warm-up 5.2 Activity Lesson Synthesis Student Lesson Summary

Unit 2, Lesson 5

More Dilations

Grade 8

5.1

Warm-up

Instructional Routines

Notice and Wonder

Materials

To Gather

Math Community Chart

Activity Narrative

In this Warm-up, students think about how the size of the scale factor impacts the dilation of a figure. While students may notice and wonder many things about the image, the fact that it shows several dilations of one triangle, all with the same center but different scale factors is the 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.

In the digital version of the activity, students use an applet to adjust the scale factor of a dilation. The applet allows students to instantly see how increasing or decreasing a scale factor impacts the dilation of a triangle.

Launch

Arrange students in groups of 2. Display the image for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.

Activity

None

All of the triangles are dilations of Triangle D. What do you notice? What do you wonder?

A triangle D, six images after dilation, point P and three dashed projection rays.

Student Response

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Activity Synthesis

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 image. Next, ask students, “Is there anything on this 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.

Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, plan to discuss the remaining suggestions over the next few lessons.

Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves and math learners.

5.2

Activity

Standards Alignment

Building On

Addressing

8.G.A.3

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

View Full Standard

Building Toward

Instructional Routines

MLR4: Information Gap Cards

Materials

To Copy (from Blackline Masters)

Dilations Cards

To Gather

Geometry toolkits

Activity Narrative

This activity gives students an opportunity to determine and request the information needed to perform a dilation. Students will need to know the center of dilation, the coordinates of the polygon that they are dilating, and the scale factor.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for the information they need to solve the problem. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Monitor for how students request (and supply) information about the center of dilation and the location of the polygon that is being dilated. The coordinate grid helps name and communicate the location of points, which is essential in this activity.

This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.

Launch

Provide access to geometry toolkits. Tell students they will perform dilations on polygons. Display the Info Gap graphic that illustrates a framework for the routine for all to see.

Remind students of the structure of the Info Gap routine, and consider demonstrating the protocol if students are unfamiliar with it. There is an extra set of cards available for demonstration purposes.

Arrange students in groups of 2 or 4. If students are new to the Info Gap routine, allowing them to work in groups of 2 for each role supports communication and understanding. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem and instruct them to switch roles.

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Info Gap graphic visible throughout the activity or provide students with a physical copy.
Supports accessibility for: Memory, Organization

Activity

None

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

Silently read your card, and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”

Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.

If your teacher gives you the data card:

Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Only give information that is on your card. Do not figure out anything for your partner!

These steps may be repeated.
Once your partner says they have enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.

Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

“Other than the answer, what information did you ask for in order to solve the problem?” (the location of the vertices of the original image, the center of dilation, the scale factor)
“How did using coordinates help in talking about the problem?” (Coordinates made it easy to name and locate points.)
“If this same problem had a figure on a grid without coordinates, how would you talk about the points?” (We could describe points in relation to other points. For example a point might be 2 units to the right and 1 to the left of a known point.)
“What if there had been no grid at all? Would you still have been able to request or provide the needed information to perform the transformation?” (Maybe, but it would have been harder and not as precise.)

Highlight for students that coordinates allow us to unambiguously provide the location of a polygon’s vertices. In addition, reinforce the idea that in order to perform a dilation, we need to know the scale factor and the center of dilation. The coordinate grid again provides an efficient means to communicate the center of dilation.

Lesson Synthesis

The goal of this discussion is for students to see how coordinates are useful, especially for communicating location of points in the plane. Here are some questions for discussion:

“Why are coordinates useful? What are they good for?” (Coordinates are an exceptionally powerful tool for communicating the location of points in the plane.)
“How do coordinates communicate the specific location of a point in the plane?” (There is only one point 3 units to the left of the origin and 2 units up from the origin, the point .)
“When performing a dilation, when can we use coordinates?” (The location of a polygon can be communicated by giving the coordinates of the vertices. In a dilation, we also need to know the center of dilation [another point] and the scale factor.)

Student Lesson Summary

One important use of coordinates is to communicate geometric information precisely. Like an address in a city, they tell you exactly where to go. Because the plane is laid out in a grid, these “addresses” are simple, consisting of 2 signed numbers.

Consider a quadrilateral in the coordinate plane. Performing a dilation of requires 3 vital pieces of information:

The coordinates of , , , and
The coordinates of the center of dilation
The scale factor

With this information, we can dilate each of the vertices , , , and and then draw the corresponding segments to find the dilation of . Without coordinates, describing the location of the new points would likely require sharing a picture of the polygon and the center of dilation.

Standards Alignment

Building On

Addressing

8.G.A

Understand congruence and similarity using physical models, transparencies, or geometry software.

View Full Standard

Building Toward

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Building on Student Thinking

Student Response

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Building on Student Thinking

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