Lesson 10
Dilations on a Square Grid
10.1: Dilations on a Grid (5 minutes)
Warmup
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
Provide access to geometry toolkits.
Supports accessibility for: Socialemotional skills; Conceptual processing
Student Facing
 Find the dilation of triangle \(QRS\) with center \(T\) and scale factor 2.
 Find the dilation of triangle \(QRS\) with center \(T\) and scale factor \(\frac{1}{2}\).
Student Response
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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.
Supports accessibility for: Conceptual processing; Organization
Design Principle(s): Support sensemaking; Maximize metaawareness
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 Response
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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.
Student Response
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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.
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.
Supports accessibility for: Memory; Organization
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:

Silently read your card and think about what information you need to be able to answer the question.

Ask your partner for the specific information that you need.

Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.

Share the problem card 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.

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.

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

Read the problem card and solve the problem independently.

Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Student Response
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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 \(D\).
 What would the image of triangle \(ABC\) look like under a dilation with scale factor 0?

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.

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.
Student Response
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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).
10.4: Cooldown  Identifying a Dilation (5 minutes)
CoolDown
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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 \((\text1, \text 2)\), \((3,1)\), 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:
 The coordinates of \(A\), \(B\), \(C\), and \(D\)
 The coordinates of the center of dilation, \(P\)
 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.