Unit 1, Lesson 3
Grid Moves
Grade 8
3.1
Warm-up
10 mins
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The purpose of this Warm-up is to familiarize students with an isometric grid, which will be useful when students transform figures on an isometric grid in a later activity. While students may notice and wonder many things, characteristics such as the measures of the angles in the grid and the diagonal parallel lines are the important discussion points. Students are not expected to know that each angle in an equilateral triangle is 60 degrees, but after previous experience with supplementary angles, circles, and rotations, they may be able to explain why each smaller angle is 60 degrees. Many things they notice may be in comparison to the square grid paper which is likely more familiar.
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.
Arrange students in groups of 2. Display the isometric grid for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image, and show where each of the features students notice is located on the actual grid itself, such as triangles, angles, and line segments. 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. If angle measures do not come up during the conversation, ask students to think about how they could figure out the measure of each angle. Some may measure with a protractor, and some may argue that since 6 angles share a vertex where each angle is identical, each angle measures
Math Community
After the Warm-up, display the class Math Community Chart for all to see and explain that the listed “doing math” actions come from the sticky notes students wrote in the first exercise. Give students 1 minute to review the chart. Then invite students to identify something on the chart they agree with and hope for the class or something they feel is missing from the chart and would like to add. Record any additions on the chart. Tell students that the chart will continue to grow and that they can suggest other additions that they think of throughout today’s lesson during the Cool-down.
Display this image for all to see:
Then read or display this statement:
“Mai says that triangle
Priya says that triangle
Do you agree with either of them? Why or why not?”
Give students 1 minute of quiet think time, then 2 minutes to discuss with a partner, followed by a whole-class discussion.
When a figure is on a grid, we can use the grid to describe a move. For example, here is a figure and an image of the figure after a move.
Two identical quadrilaterals on a grid labeled B D C A and B prime D prime C prime A prime. In quadrilateral B D C A, point B is 3 units right and 3 units down from the edge of the grid. Point D is 1 unit left and 1 unit up from point B. Point C is 2 units up from point B. Point A is 2 units right from point B. In quadritaleral B prime D prime C prime A prime, point B prime is 3 units down and 4 units right from point B. Point D prime is 3 units down and 4 units right from point D. Point C prime is 3 units down and 4 units right from point C. Point A prime is 3 units down and 4 units right from point A.
Quadrilateral
This type of grid is called an isometric grid. The isometric grid is made up of equilateral triangles. The angles in the triangles all measure
Here is quadrilateral
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Point out
Give students 6–8 minutes of quiet work time for the first set of transformations. Invite students to share a strategy with the whole class for each transformation. Give students an additional 5–6 minutes of quiet work time for the last set of transformations.
Your teacher will give you tracing paper to carry out the moves specified. Use
Four triangle A B C figures on a grid. Let (0 comma 0) be the bottom left corner of each figure. Figure 1 has point A prime at (7 comma 5) and the coordinates of triangle A B C are: (1 comma 3), B(3 comma 2) and C(5 comma 4). Figure 2 has point C prime at (7 comma 5) and the coordinates of triangle A B C are: (1 comma 3), B(3 comma 2) and C(5 comma 4). Figure 3 has point O at (6 comma 6) and the coordinates of triangle A B C are: (1 comma 4), B(3 comma 3) and C(5 comma 5). Figure 4 has line l at x = 6 and the coordinates of triangle A B C are: (1 comma 5), B(3 comma 4) and C(5 comma 6).
In Figure 1, translate triangle
In Figure 2, translate triangle
In Figure 3, rotate triangle
In Figure 4, reflect triangle
Four identical quadrilateral A B C D figures on a triangular grid. Figures are labeled Figure 5, Figure 6, Figure 7 and Figure 8. Each figure is in the upper left half of the figure box. Figure 7 has dashed line l parallel to the D C side across the figure box.
In Figure 5, rotate quadrilateral
In Figure 6, rotate quadrilateral
In Figure 7, reflect quadrilateral
In Figure 8, translate quadrilateral
Students may struggle to understand the descriptions of the transformations to carry out. For these students, explain the transformations using the words they used in earlier activities, such as “slide,” “turn,” and “mirror image,” to help them get started. Students may also struggle with reflections that are not over horizontal or vertical lines.
Some students may need to see an actual mirror to understand what reflections do and the role of the reflection line. If rectangular plastic mirrors are accessible, students can check their work by placing the mirror along the proposed mirror line.
Working with the isometric grid may be challenging, especially rotations and reflections across lines that are not horizontal or vertical. For the rotations, students can be asked what they know about the angle measures in an equilateral triangle. For reflections, the approach of using a mirror can work or students can look at individual triangles in the grid, especially those with a side on the line of reflection, and see what happens to them. After checking several triangles, they develop a sense of how these reflections behave.