### Clarifications

*Clarification 1*: Instruction includes showing that the corresponding sides are proportional, and the corresponding angles are congruent.

**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 identify the scale factor of a dilation given the preimage and the image and learn that when the transformation is a dilation, this transformation does not preserve congruence. The preimage is mapped onto a scaled copy of itself. In Geometry, students determine whether two figures are similar and justify their answers using a dilation (non-rigid motion) or a sequence that includes at least one dilation. This leads to the definition of similarity in terms of dilations and rigid transformations.*(MTR.5.1)*

- When identifying the dilation, specify the center and scale factor of the dilation.
- Instruction includes describing the transformation using words and using coordinates.
- It is important to identify corresponding parts between the preimage and the image leading to the similarity statement and the congruence of the corresponding angles and the proportionality of the corresponding sides.
- Instruction includes using examples to compare transformations. Include situations where the preimage and the image are not similar to show how dilations will fail mapping one figure onto the other.
*(MTR.4.1)* - When proving two triangles are similar, it is important to discuss with the students the effects of choosing which one of the triangles is the pre-image. This affects the scale factor of the dilation.
- For example, when proving that Δ
*ABC*and Δ*PQR*are similar, the scale factor of the dilation that maps Δ*ABC*onto Δ*PQR*is $k$, while the scale factor of the dilation that maps Δ*PQR*onto Δ*ABC*is $\frac{\text{1}}{\text{k}}$.

- For example, when proving that Δ

### Common Misconceptions or Errors

- When determining the scale factor of a dilation, students may misidentify the preimage and image, leading to an incorrect scale factor.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1)*

- A dilation with scale factor 3 was used to map polygon
*ABCD*onto polygon*A'B'C'D'*

- Part A. Fill in the blanks with either
*congruent or proportional.*

If the figures are similar, the corresponding sides are _______ and corresponding angles are________. - Part B. Identify the sequence of rigid and non-rigid transformations that maps polygon
*ABCD*onto polygon*A'B'C'D'*. - Part B. Use the definition of similarity to prove that polygon
*ABCD*is similar to polygon*A'B'C'D'*. You may need to decompose the polygon into triangles and rectangles.

### Instructional Items

*Instructional Item 1*

- In triangles
*ABD*and*JKL*, $m$∠*A*= $m$∠*J*, $m$∠*C*= $m$∠*L*, and*AC*=*2JL*

- Part A. Describe a sequence of transformations that maps Δ
ABConto ΔJKL.- Part B. Based on the transformations chosen, determine whether Δ
ABCis congruent or similar to ΔJKL.

**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

## Perspectives Video: Professional/Enthusiast

## Problem-Solving Task

## Text Resource

## MFAS Formative Assessments

Students are asked to justify statements of a proof of the AA Similarity Theorem.

Students will indicate a complete proof of the AA Theorem for triangle similarity.

Students are asked to use the definition of similarity in terms of similarity transformations to determine whether or not two quadrilaterals are similar.

Students are asked to use the definition of similarity in terms of similarity transformations to determine whether or not two triangles are similar.

## Student Resources

## Perspectives Video: Professional/Enthusiast

<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Task

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

## Parent Resources

## Perspectives Video: Professional/Enthusiast

<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Task

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task