**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
- Origin
- Reflection
- Rigid Transformation
- Rotation
- Translation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students first learned to associate congruence and similarity with reflections, rotations, translations and dilations. In Geometry, students learn that these transformations provide an equivalent alternative to using congruence and similarity criteria for triangles.- Instruction includes the connection to the Side-Angle-Side, Angle-Side-Angle (and Angle-Angle-Side), Hypotenuse-Leg and Side-Side-Side congruence criteria. It may be helpful for students to work through one transformation at a time to determine congruence. One approach would be to begin with a translation to map one of the vertices of one triangle to the corresponding vertex of the other. Then, if necessary, a rotation about the first vertex can be used to map a second vertex of one triangle to the corresponding vertex of the other. Finally, it may be necessary to use a reflection to match the corresponding third vertices.
- For example, triangles
*ABC*and*PQR*are given below, with*AB*≅*PQ*,*BC*≅*QR*and*AC*≅*PR*.One method to prove the triangles are congruent is as follows: The first step is to discuss the transformations that would map*AB*onto*PQ*. Students should be able to identify a convenient sequence of a translation and a rotation, both rigid motions, that will accomplish it. Therefore*PQ*≅*A′B′*since rigid motions preserve distances.Applying the same sequence of transformations to point*C*only, results in*AB*≅*A′B′*,*BC*≅*B′C′*and*AC*≅*A′C′*.In order to prove that the triangles are congruent, there may be two situations.

Situation A: Point*C*has already been mapped to point*R*from the previous sequence. In this case, use the transitive property of congruence, if*BC*≅*QR*and*BC*≅*B′C′*, then*QR*≅*B′C′*, and if*AC*≅*PR*and*AC*≅*A′C′*, then*PR*≅*A′C′*proving triangles*ABC*and*PQR*are congruent using transformations.

Situation B: If Point*C*has not already been mapped to point*R*from the previous sequence, like shown below, one must use a reflection.To prove that*C*′ can be mapped onto*R*using a reflection, use the converse of the Perpendicular Bisector Theorem.*A′B′*is the perpendicular bisector of*C′R′*since*PR*≅*A′C′*(and*QR*≅*B′C′*). Therefore,*C′D′*≅*DR*and*C*′ can be mapped onto*R*using a reflection over*A′B′*proving triangles*ABC*and*PQR*are congruent using transformations.

- For example, triangles

### Common Misconceptions or Errors

- Students may have difficulty understanding the value of having two different, equivalent approaches to proving congruence and similarity. Discuss how approaching a situation in different ways deepens understanding.
*(MTR.2.1)*

### Instructional Tasks

*Instructional Task 1 (*

*MTR.2.1*, MTR.4.1)- Triangles
*ABC*and*DEF*are shown below and*AB*≅*EF*, ∠*B*≅ ∠*E*and ∠*A*≅ ∠*F*.- Part A. Describe a sequence of rigid transformations that maps triangle
*ABC*onto triangle*DEF*. - Part B. Compare your sequence with a partner. What do you notice? What information about these triangles makes it possible to determine the rigid transformations in Part A?

- Part A. Describe a sequence of rigid transformations that maps triangle

Instructional Task 2 (

Instructional Task 2 (

*MTR.4.1*)- When applying the transformation ($x$, $y$) → ($x$ + 4, $y$ − 6), segment
*AB*maps onto segment*LN*and segment*AC*maps onto segment*LM*.- Part A. How can Δ
*ACB*≅ Δ*LMN*be proved using one of the congruence criteria? - Part B. How can Δ
*ACB*≅ Δ*LMN*be proved using rigid transformations?

- Part A. How can Δ

### Instructional Items

*Instructional Item 1*

- Use the image below to complete the sentence.If
*NP*≅ , ∠*P*≅ and*MP*≅ , then there is a sequence of rigid transformations that maps Δ*KJL*onto .

**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 given two congruent triangles and asked to determine the corresponding side lengths and angle measures and to use the definition of congruence in terms of rigid motion to justify their reasoning.

Students are asked to prove two triangles congruent given that all pairs of corresponding sides and angles are congruent.

Students are given two triangles in which all pairs of corresponding parts are congruent and are asked to use the definition of congruence in terms of rigid motion to show the triangles are congruent.

Students are given two triangles in which all pairs of corresponding parts are congruent and are asked to use the definition of congruence in terms of rigid motion to show the triangles are congruent.

Students are asked to use the definition of congruence in terms of rigid motion to show that two triangles are congruent in the coordinate plane.

## Student Resources

## Problem-Solving Tasks

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Type: Problem-Solving Task

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Type: Problem-Solving Task

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Type: Problem-Solving Task

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Type: Problem-Solving Task

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Type: Problem-Solving Task