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Provide access to geometry toolkits. Clarify that “estimate” doesn’t mean “guess” and encourage students to use any tools available to make a precise estimate (MP5). Give students 2 minutes of quiet work time followed by a whole-class discussion.
Point
Estimate
The goal of this discussion is to review different ways to determine scale factor. Discuss with students:
“How did you measure distances?” (I used a ruler. I used an index card and marked off the distances.)
“Is the scale factor greater than 1?” (Yes.) “How do you know?” (The point
“Is the scale factor greater than 2?” (Yes.) “How do you know?” (The distance from
“Is the scale factor greater than 3?” (No.) “How do you know?” (The distance from
In this activity, students perform dilations on a square grid, which is particularly helpful when the center of dilation and the points being dilated are grid points. Students will again see that scale factors greater than 1 produce larger copies while scale factors less than 1 produce smaller copies.
Monitor for students who use these strategies to find the dilated points:
Use a ruler or index card to measure distances along the rays emanating from the center of dilation
Count how many squares to the left or right and up or down each point is from the center of dilation
Arrange students in groups of 2. Provide access to geometry toolkits.
Select students who used each strategy described in the Activity Narrative to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven't shared recently.
Point T, triangle Q R S and three projection rays on a square grid. Let the lower left corner be (0 comma 0). Then triangle Q R S is Q(5 comma 7), R(6 comma 5), S(4 comma 4) and point T is T(2 comma 6). The dashed projection rays are T Q, T R and T S.
The goal of the discussion is to make connections between the 2 strategies described in the Activity Narrative for finding a dilation. Measuring with a ruler or index card and reasoning about the grid and counting spaces will both result in the dilated point ending up in the same location. Display 2–3 approaches or representations from previously selected students for all to see. If time allows, invite students to briefly describe their approach orrepresentation. Use Compare and Connect to help students compare, contrast, and connect the different approaches and representations. Here are some questions for discussion:
“What do the 2 strategies have in common? How are they different?” (Both strategies are measuring distance, but one method measures distance in a straight line while the other measures distance by counting horizontal and vertical squares.)
“How does the scale factor show up in each method?” (When using a ruler or index card, the scale factor shows up as how many times farther the new point will be. When counting grid squares, the scale factor tells us how many more squares we need to count from the center of dilation.)
The purpose of this discussion is to review how to perform a dilation on a square grid. Show the figure for all to see and explain how grid lines can be used as a way to measure distance and direction between points. Ask students to think about how they would dilate
Show the following figure for all to see and ask students how this image is different from the previous figure. (This one has
Square grids can be useful for showing dilations, 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, the dilation of point
Sometimes the square grid comes with coordinates, giving us a convenient way to name points. Sometimes the coordinates of the image can be found just using arithmetic, without having to measure.
Two triangles on a coordinate plane, origin O. Horizontal axis scale negative 7 to 7 by 1’s. Vertical axis scale negative 5 to 5 by 1’s. The coordinates of the triangle are (negative 1 comma negative 2), (3 comma 1), (2 comma negative 1 ). The coordinates of the image are(negative 2 comma negative 4), (6 comma 2), (4 comma negative 2).
Help us improve by sharing suggestions or reporting issues.
Some students may have trouble finding accurate matches. Prompt them to identify the center of dilation and then consider if the dilation will result in a smaller or larger sized image.