# Lesson 10

Dilations on a Square Grid

## 10.1: Dilations on a Grid (5 minutes)

### Warm-up

In a previous lesson, students performed dilations on a circular grid and with no grid. In this activity, they perform dilations on a square grid. A square grid is particularly helpful if the center of dilation and the points being dilated are grid points. When the extra structure of coordinates is added, as in the next activity, the grid provides an extremely convenient tool for naming points and describing the effects of dilations using coordinates. As in previous lessons, students will again see that scale factors greater than 1 produce larger copies while scale factors less than 1 produce smaller copies.

Monitor for how students find the dilated points and the language they use to describe the process. In particular:

• using a ruler or index card to measure distances along the rays emanating from the center of dilation
• taking advantage of the grid and counting how many squares to the left or right, up or down

### Launch

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, reference examples from the previous lessons on methods for dilating points on a circular grid and on no grid to provide an entry point into this activity.
Supports accessibility for: Social-emotional skills; Conceptual processing

### Student Facing

1. Find the dilation of triangle $$QRS$$ with center $$T$$ and scale factor 2.
2. Find the dilation of triangle $$QRS$$ with center $$T$$ and scale factor $$\frac{1}{2}$$.

### Activity Synthesis

Select students to show how they found the dilations. First, select any students who used the same methods as when there was no grid, followed by students who noticed they could use the structure of the grid. Draw connections between these two methods—show that when you measure with a ruler or by making markings on an index card, the dilated point ends up in the same place as by reasoning about the grid.

Expect students to use expressions like “moving over two and up one.” These measurements can be multiplied by the scale factor in order to find the location of the dilated point.

Tell students that moving forward they will do work on the grid with the added structure of coordinates. The method of performing dilations is the same. The only change is that the coordinates give a concise way to name points.

## 10.2: Card Sort: Matching Dilations on a Coordinate Grid (15 minutes)

### Activity

In the previous task, students worked on a square grid without coordinates. This activity adds the structure of coordinates and this extra structure plays a key role, allowing students to name points. Students match figures with their dilated images, using coordinates to describe the center of dilation and the vertices. The same strategies that were used previously in dilating images,on a circular grid and with no grid, will be useful here.

Monitor for students who identify that the dilation of a circle is a circle and similarly for triangles and quadrilaterals. This will help them eliminate certain possibilities for each match. Because there is one card that does not match, students should verify the other matches by performing the dilations. Once the card without a match has been identified, reasoning based on eliminating possibilities (without performing the dilations) is correct. Monitor for students who systematically perform the dilations to help identify a match versus those who reason by structure and elimination of possibilities. Invite both to share during the discussion.

### Launch

Students practice matching an original figure and dilation description to information about the dilated images using the coordinate plane. Distribute one set (numbers 1 through 6 and letters A through F) of cards to each student.

There is one extra option that does not have a match. Students should draw the dilated image for that option themselves.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Conversing, Reading: MLR2 Collect and Display. As students work in pairs on the task, circulate and listen to pairs as they decide whether two cards match. Write down the words and phrases students use to justify why an original figure card matches with a dilated figure card. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “Card 1 matches with Card C because they are both trapezoids” can be clarified by asking students to explain why Card 1 does not match with Card A even though both are trapezoids. Listen for students who state that the scale factor and center of dilation must also be considered when matching the cards. Write down the language students use to describe how the scale factor and center of dilation affect the dilated figure. This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness

### Student Facing

Your teacher will give you some cards. Each of Cards 1 through 6 shows a figure in the coordinate plane and describes a dilation.

Each of Cards A through E describes the image of the dilation for one of the numbered cards.

Match number cards with letter cards. One of the number cards will not have a match. For this card, you’ll need to draw an image.

### Student Facing

#### Are you ready for more?

The image of a circle under dilation is a circle when the center of the dilation is the center of the circle. What happens if the center of dilation is a point on the circle? Using center of dilation $$(0,0)$$ and scale factor 1.5, dilate the circle shown on the diagram. This diagram shows some points to try dilating.

### Anticipated Misconceptions

If students are having trouble finding accurate matches, suggest that they identify the center of dilation and consider if the dilation will result in a smaller or larger sized image.

### Activity Synthesis

Share the correct answers and invite selected students to share the strategies they used to solve the problems. This is a matching problem, so students may not have dilated the entire image to find the correct answer among the choices. Important points to bring out include:

• A dilation maps a circle to a circle, a quadrilateral to a quadrilateral, and a triangle to a triangle.
• If the center of dilation for a polygon is one of the vertices, then that vertex is on the dilated polygon.
• If the scale factor is less than 1 then the dilated image is smaller than the original figure.
• If the scale factor is larger than 1 then the dilated image is larger than the original figure.

## 10.3: Info Gap: Dilations (20 minutes)

### Activity

This info gap activity gives students an opportunity to determine and request the information needed for a dilation, and to realize that using coordinates greatly simplifies talking about specific points. In order to perform a dilation, students will need to know the center of dilation (which can be communicated using the coordinate grid), the coordinates of the polygon that they are dilating (also communicated using the coordinate grid), and the scale factor. With this information, they can find the dilation as in previous activities.

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).

Here is the text of one of the cards for reference and planning:

Listen 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. In addition to the location of points, listen for how students use “center of dilation” and “scale factor” in order to communicate this essential information.

Teacher Notes for IM 6–8 Math Accelerated
Adjust this activity to 15 minutes.

### Launch

Tell students they will continue to practice describing and drawing dilations using coordinates. Explain the Info Gap and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. Provide access to geometry toolkits. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for the second problem and instruct them to switch roles.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to dilate a polygon on a coordinate grid. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?”
Design Principle(s): Cultivate Conversation

### Student Facing

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:

3. Explain how you are using the information to solve the problem.

Continue to ask questions until you have enough information to solve the problem.

4. Share the problem card and solve the problem independently.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

4. Read the problem card and solve the problem independently.

5. Share the data card and discuss your reasoning.

### Student Facing

#### Are you ready for more?

Triangle $$EFG$$ was created by dilating triangle $$ABC$$ using a scale factor of 2 and center $$D$$. Triangle $$HIJ$$ was created by dilating triangle $$ABC$$ using a scale factor of $$\frac12$$ and center $$D$$.

1. What would the image of triangle $$ABC$$ look like under a dilation with scale factor 0?
2. What would the image of the triangle look like under dilation with a scale factor of -1? If possible, draw it and label the vertices $$A’$$, $$B’$$, and $$C’$$. If it’s not possible, explain why not.

3. If possible, describe what happens to a shape if it is dilated with a negative scale factor. If dilating with a negative scale factor is not possible, explain why not.

### Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Some guiding questions:

• “Other than the answer, what information would have been nice to have?”
• “How did using coordinates help in talking about the problem?”
• “If this same problem had a figure on a grid without coordinates, how would you talk about the points?”
• “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?”

Highlight 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

### Lesson Synthesis

The purpose of this discussion is for students to reflect on the dilations they have done so far and what advantages a coordinate grid offers. Ask students, “Why are coordinates useful?” After a brief quiet think time, select 2–3 students to share their ideas. Some points to highlight are:

• Coordinates are an exceptionally powerful tool for communicating precisely the location of points 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 $$(\text{-}3,2)$$.
• The location of a polygon is determined by the location and order of its vertices. On a coordinate plane, these can be communicated by giving their coordinates.
• When we perform a dilation, we need to know the center of dilation (a point) and the scale factor (a number). On the coordinate plane, all of the information we need to dilate a polygon can be communicated unambiguously using coordinates and the scale factor.

Coordinates in the plane are much like an address: they tell you where to go unambiguously. Because the plane is laid out in a grid, these “addresses” are particularly simple, consisting of two signed numbers. Tell students that we will use coordinates much more in this unit and beyond, not only to describe individual points but also to describe relationships (like proportional relationships and other new types of relationships).

## Student Lesson Summary

### Student Facing

Square grids can be useful for showing dilations. The grid is helpful especially when the center of dilation and the point(s) being dilated lie at grid points. Rather than using a ruler to measure the distance between the points, we can count grid units.

For example, suppose we want to dilate point $$Q$$ with center of dilation $$P$$ and scale factor $$\frac{3}{2}$$. Since $$Q$$ is 4 grid squares to the left and 2 grid squares down from $$P$$, the dilation will be 6 grid squares to the left and 3 grid squares down from $$P$$ (can you see why?). The dilated image is marked as $$Q’$$ in the picture.

Sometimes the square grid comes with coordinates. The coordinate grid gives us a convenient way to name points, and sometimes the coordinates of the image can be found with just arithmetic.

For example, to make a dilation with center $$(0,0)$$ and scale factor 2 of the triangle with coordinates $$(\text-1, \text- 2)$$, $$(3,1)$$, and $$(2, \text- 1)$$, we can just double the coordinates to get $$(\text- 2, \text- 4)$$, $$(6,2)$$, and $$(4, \text- 2)$$.

In general, an important use of coordinates is to communicate geometric information precisely. Let’s consider a quadrilateral $$ABCD$$ in the coordinate plane. Performing a dilation of $$ABCD$$ requires three vital pieces of information:

1. The coordinates of $$A$$, $$B$$, $$C$$, and $$D$$
2. The coordinates of the center of dilation, $$P$$
3. The scale factor of the dilation

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