### Clarifications

*Clarification 1:* Within this benchmark, transformations are limited to reflections, translations, rotations or dilations of images. *Clarification 2:* Lines of reflection are limited to the x-axis, y-axis or lines parallel to the axes.

*Clarification 3:* Rotations must be about the origin and are limited to 90°, 180°, 270° or 360°.

*Clarification 4:* Dilations must be centered at the origin.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

- NA

### Terms from the K-12 Glossary

- Coordinates
- Coordinate Plane

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students plotted rational-number ordered pairs in all four quadrants as well as identified the $x$- or $y$-axis as a line of reflection when two ordered pairs have an opposite $x$- or $y$-coordinate. In grade 7, students solved mathematical and real-world problems involving scale factors. In grade 8, students apply a single transformation using coordinates and the coordinate plane. In Algebra 1, students will apply a single transformation to functions. In Geometry, students will describe transformations given a preimage and an image and represent the transformation algebraically using coordinates and use them to study congruence and similarity.- Use grid paper to illustrate translations of a line or triangle to demonstrate the relationship between them and a new image. Then, illustrate translations of more complex figures such as polygons.
- Transformations can be noted using the prime notation (′) for the image and its vertices. The preimage and its vertices will not have prime notation.
- For example, the picture below showcases a single transformation.

- For example, the picture below showcases a single transformation.
- Problem types include telling which direction, clockwise or counterclockwise, for rotations.
- Instruction includes looking for patterns to create rules for transformations on the coordinate plane.
- For mastery of this benchmark, single transformations include one vertical translation or one horizontal translation. A vertical and horizontal translation would be considered two transformations.

### Common Misconceptions or Errors

- Students may incorrectly visualize transformation on the coordinate plane. To address this misconception, provide students with manipulatives.
- Students may incorrectly apply rules for transformations. To address this misconception, students should generate examples and non-examples of given transformations.

### Strategies to Support Tiered Instruction

- Teacher supports understanding of transformations on the coordinate place by providing examples using geometric software. Instruction includes the use of manipulatives and graph paper.
- Teacher reminds students when plotting points on a coordinate plane that they can first find the $x$-coordinate on the $x$-axis (horizontal axis) and then find the $y$-coordinate on the $y$-axis (vertical axis).
- Teacher reviews vocabulary discussing the meaning of the terms.
- Translation is a vertical or horizontal slide of the figure. To determine the coordinates of the image of a translated figure you must add or subtract the horizontal distance to the $x$-coordinate of each vertex and add or subtract the vertical distance to the $y$-coordinate of each vertex. (Note that in later courses, students learn that translation can also occur diagonally.)
- Preimage is the figure before any transformations are performed.
- Image is the figure after a transformation is performed.

- Teacher co-creates a graphic organizer to generate examples and non-examples of reflections, translations, rotations, or dilations of images.
- Teacher provides instruction to support understanding of applying the translation to all vertices, not just one vertex.
- Teacher reviews directions of rotations. Clockwise is the direction the hands go on an analog clock . Counterclockwise is the opposite direction of the hands on an analog clock .
- For example, which quadrant would the image be in if you rotated the figure?
- 90 degrees clockwise
- 90 degrees counterclockwise
- 180 degrees clockwise
- 180 degrees counterclockwise

- For example, which quadrant would the image be in if you rotated the figure?
- Teacher reviews which is the $x$-axis and which is the $y$-axis for students that incorrectly reflect across the wrong axis. Teacher co-creates anchor chart explaining different parts of coordinate plane, and how to plot and label points.
- For example, teachers could ask students which quadrant the image would be in if you reflected the figure across the $x$-axis or across the $y$-axis.

- Instruction includes providing students with manipulatives for students that incorrectly visualize transformations on the coordinate plane.

### Instructional Tasks

*Instructional Task 1*

**(MTR.1.1, MTR.2.1)**Use the information you have learned about transformations to complete the task below.

- Part A. Using graph paper, plot the following points to create an image on the coordinate plane. $A$(−3,2), $B$(0,1), $C$(−3, −1) and $D$(−1, −1)
- Part B. Using a different color for each transformation, complete each of the following transformations on the same coordinate plane.

a. A reflection over the $y$-axis

b. A rotation of 180° about the origin - Part C. Will any of the new images include the origin?

### Instructional Items

*Instructional Item 1*

Find the coordinates of the vertices of the image of triangle $I$$J$$K$ after the translation 3 units to the left.

*Instructional Item 2*

Find the coordinates of the vertices of the image of triangle $C$$A$$T$ after a 270° counterclockwise rotation about the origin.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Problem-Solving Tasks

## MFAS Formative Assessments

Students are asked to dilate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Students are asked to reflect two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Students are asked to rotate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Students are asked to translate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

## Student Resources

## Problem-Solving Tasks

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

In this resource, students experiment with the reflection of a triangle in a coordinate plane.

Type: Problem-Solving Task

The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of two-dimensional figures are found by reflecting individual points.

Type: Problem-Solving Task

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections.

Type: Problem-Solving Task

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

In this resource, students experiment with the reflection of a triangle in a coordinate plane.

Type: Problem-Solving Task

The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of two-dimensional figures are found by reflecting individual points.

Type: Problem-Solving Task

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections.

Type: Problem-Solving Task

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Type: Problem-Solving Task