## Notes on Dilations - Section 2: Notes on Dilations

# Experimenting with Dilations

Lesson 1 of 8

## Objective: SWBAT apply properties of dilations

## Big Idea: Time for pupils to dilate...and we're not talking optometry. In this lesson, students get familiar with the major characteristics of dilations.

#### Performing Dilations

*25 min*

While I don't quite intend for this to be a Discovery Lesson, I do want my students to have an independent experience with dilation before I unveil all formal definitions and such. I want them to do some inductive reasoning based on firsthand experience. I want to give my students a point of reference before I introduce formal definitions.

To begin the lesson section, I hand out the Experimenting With Dilations resource. For this activity my students will also need a calculator and a ruler with centimeter markings. I ask my students to take time to read and understand the directions at the top of the first page of the resource. I expect my students will struggle to interpret the instruction. I intentionally wrote them using mathematical notation and vocabulary, so my students can work on decoding mathematical text.

After students have read the instructions at their seats, I will have them move their desks together with their A-B (elbow) partner. Then, I will instruct the A partner to leave their resource with directions on the first page showing. I will direct the B partner to open to the second page so that the diagram is showing. Then, I'll say, "Look at the directions and diagram side-by-side and work together to make sense of the "Let" statement and the two conditions they must satisfy. When you have agreed on the meaning of the directions, proceed to follow them." My only other directions are to measure lengths in centimeters and to be precise to the nearest tenth of a centimeter.

Walking around, I am looking for pairs that have not yet understood the directions. I stop to consult with these students. First, I typically ask "What does it mean to let P be any vertex?" Then I ask the students to select a vertex to use as an example. Then I say something like "So if A is our P, and P' has to be collinear with I and P, where does A' have to be?" When we've established that A' has to be on ray IA, then we talk about the statement IP' = 1.8 IP. This usually gets my students unstuck.

Once the whole class is working to perform the dilation, I will give a deadline (e.g., "In the next five minutes, let's all have our drawings finished and the questions on the first page answered completely.") I will also advise students to use color to make their images "pop" (i.e., stand apart from their pre-images).

When the deadline passes, I will bring the students' attention to the front of the class as we prepare for whole-class instruction. I will give the students an extra 1-minute to review the answers to the questions with their partners. I will let them know that I will be calling random non-volunteers to read what they have written on their papers.

After about a minute, I will call random students to read their answers. I listen, re-voice answers and clarify/elaborate as needed. For example, I ensure that students know the geometric term for the figures is **similar**. When we get to the item that asks students to conjecture about the relationship between the segment lengths and angle measures, my students tend to say "The angles are the same and the sides are longer." When this happens, I pause for a **pair-share** using the prompt:

**So we seem to agree that the sides are longer: How can we make a more precise conjecture about the relationship between the side lengths?**

After this prompt, students by and large predict that the segment lengths are 1.8 times longer. Finally I model for students what/how they should have measured to test their conjectures.

#### Resources

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#### Notes on Dilations

*5 min*

Now that my students have had an experience performing a** pencil-and-paper dilation**, I want to give them more precise language to describe the experience. I put my Notes on Dilations under the document camera so that students can see it and follow along with my presentation making notes. (I also post it online so that students have access to it later.)

As I review my notes with the class, my main message to them is that this is not new knowledge. I emphasize that they have just experienced everything that is in these notes and that the purpose of the notes is just to create a record of what they have just experienced.

As I finish talking, I instruct students to finish taking the notes and then, proceed to work through the rest of the Experimenting With Dilations packet. I explain that they have learned everything they will need to know in order to complete the tasks; they will just have to apply what they have learned in order to complete each challenge. **I also exhort them to document their work in an organized manner using precise terminology and notation. **I remind them that their work should speak for itself when someone picks it up to read it.

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#### Dilation Challenges

*25 min*

Pages 3-5 of Experimenting With Dilations present my students with three dilation challenges, each more complex than the last. I expect my students to complete them at their own pace and I have particular ideas for interacting with Pairs of students as they work on each challenge.

**Challenge 1: Finish the Dilation**

Students have just performed a dilation given a center and scale factor. In this challenge, they are given a center and the image of one vertex. From that they must finish the dilation.

For students who are stuck, my line of questioning is, 1) "What two things must we know in order to complete a dilation? [center and scale factor] and 2) "What are we missing here?" [scale factor]. Some students also are not clear on where B' should be since B is the center of dilation. A question about the distance from B to the center of dilation [zero] usually gets students thinking correctly.

With regard to precision, students should write something along the lines of Scale Factor = BD'/BD. For those that don't communicate this, I prompt them to be more precise.

**Challenge 2: Describe the Dilation Precisely**

In this challenge, students are given a preimage and its image under dilation. They must decide what minimum information is necessary to specify the dilation [center and scale factor] and then how to go about determining this information.

When my students are stuck, I usually emphasize the idea of "minimum information" and clarify that we are attempting to provide the information someone would need if they could not see the image.

With regard to precision, I want to make sure that my students properly explain how they determined the center and scale factor. For the center of dilation, they should touch upon how they knew that O, and not G, for example, was the center. For their explanation of the scale factor, I look to see if they have been as concise as possible, using notation, for example, instead of words.

**Challenge 3: Satisfy the Constraints**

In this challenge, students must choose from all possible dilations with scale factor 2/5 in order to find one that will satisfy two constraints. Students will need to know that (from challenge 1) that a dilation leaves a point that is already at the center of dilation unchanged and they will need to predict the effects of choosing different centers on the location of the image. Since this is the last challenge, I allow students to struggle with it until they get it. Some students satisfy only one of the constraints. Some satisfy neither of them. This is ok as they will have the opportunity to learn from others later in the lesson.

#### Resources

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#### Presentations

*15 min*

As I have been walking around the classroom looking at student work, I have pre-selected students to present their work on the challenges. Each student will come up to the front of the class and put their paper under the document camera. Then he/she will explain how they completed the challenge.

For **Challenge 1**, I plan to choose a student who's work is visually appealing and who has documented their process in an organized manner with attention to precise terminology and notation. They explain how they got the scale factor and how they used it.

For **Challenge 2**, I will select a student who has done a good job of explaining not only how they determined that O was the center, but also how they ruled out other points (thoroughness). Since this item requires a written narrative, I'm also looking for one that is legible, concise, and precise.

For **Challenge 3**, I will select two students. The first is a student who has satisfied only one of the constraints. The second will have satisfied both constraints and will explain how they knew where to put the center in order to satisfy both constraints.

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: Experimenting with Dilations
- LESSON 2: Verifying Properties of Dilations
- LESSON 3: Dilation Tasks
- LESSON 4: Triangle Similarity Criteria
- LESSON 5: Proving Theorems involving Similar Triangles
- LESSON 6: Altitude to the Hypotenuse
- LESSON 7: Proving Pythagorean Theorem Using Similar Triangles
- LESSON 8: The Golden Ratio