### Clarifications

*Clarification 1*: Transformations include translations, dilations, rotations and reflections described using words or using coordinates.

*Clarification 2*: Instruction includes two or more transformations.

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

**Grade:**912

**Strand:**Geometric Reasoning

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

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Coordinate Plane
- Dilation
- Origin
- Reflection
- Rigid Transformation
- Rotation
- Scale Factor
- Translation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students learned about the effects of translations, rotations, reflections, and dilations on geometric figures. In Algebra 1, students extended this knowledge of transformations to transforming functions using tables, graphs and equations. In Geometry, students are given the preimage and a sequence of two or more transformations (a composition) and draw the image on a coordinate plane. In later courses, types of transformations will be expanded to include stretches that can transform functions and conic sections and conversions between rectangular and polar coordinates.- The purpose of this benchmark is that students produce an image on the coordinate plane as the result of combining transformations (translations, dilations, rotations and reflections).
- Instruction includes transformations described using words and using coordinates.
- Instruction includes understanding that a single transformation can be done as a sequence of transformation. Likewise, some sequences can be described as a single transformation.
- For example, if the transformation sequence is to translate 1 unit to the left, then rotate 90-degrees clockwise about the origin, then translate 1 unit to the right; the result is the same as rotating the figure 90-degrees clockwise about the point (1,0). This can be described in coordinates as ($x$, $y$) → ($x$ − 1, $y$), then ($x$ − 1, $y$) → ($y$, −($x$ − 1)), then ($y$, −($x$ − 1)) → ($y$ + 1, −($x$ − 1)), and the result of rotating the figure 90-degrees clockwise about the point (1,0) can be described in coordinates as ($x$, $y$) → ($y$ + 1, −$x$ + 1).

- Instruction includes discussing the resulting images using the same preimage and a set of transformations in different orders and understanding that transformations may or not be commutative based on the sequence and type of transformations.
- For example, if the transformation is the result of a vertical and a horizontal translation, the sequence is commutative.
- For example, if the transformation is the result of a 180° rotation and a reflection over the $x$-axis, the sequence is commutative.
- For example, if the transformation is the result of a 90° rotation and a reflection over the $x$-axis, the sequence is not commutative.
- Instruction includes the understanding that if a dilation is part of a sequence, it can be applied at any time within the sequence of transformations without changing the final result.

### Common Misconceptions or Errors

- Students may think order of transformations doesn’t matter.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1, MTR.5.1)*

- Part A. On the coordinate plane, draw the resulting figure after transforming quadrilateral
*ABCD*through the following sequence below.- Reflect quadrilateral
*ABCD*over the line $y$ = $x$ . - Translate horizontally and vertically the resulting figure using ($x$, $y$) → ($x$ + 3, $y$ − 2).

- Reflect quadrilateral

- Part B. Would the resulting figure be the same if the transformations were reversed? How did you come to your conclusion?

### Instructional Items

*Instructional Item 1*

- Perform the following sequence of transformations on the polygon
*ABCDEF*on the coordinate plane.- Rotate 180° counterclockwise about the origin.
- Then, translate horizontally 2 units to the left and vertically 3 units down.

*Instructional Item 2*

- Draw the resulting figure after quadrilateral
*ABCD*is transformed using ($x$, $y$) → (−$x$, $y$ − 3).

**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

## Perspectives Video: Professional/Enthusiast

## MFAS Formative Assessments

Students are asked to dilate a line segment and describe the relationship between the original segment and its image.

Students are asked to graph the image of two points on a line after a dilation using a center on the line and to generalize about dilations of lines when the line contains the center.

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Students are asked to describe what happens to a triangle after repeated reflections and rotations.