# Cluster 1: Understand congruence and similarity using physical models, transparencies, or geometry software. (Major Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.8.G.1
Title: Understand congruence and similarity using physical models, transparencies, or geometry software. (Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Geometry

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.8.G.1.AP.1a
Perform rotations, reflections, and translations using pattern blocks.
MAFS.8.G.1.AP.1b
Draw rotations, reflections, and translations of polygons.
MAFS.8.G.1.AP.2a
Demonstrate that two-dimensional polygons that are rotated, reflected, or translated are still congruent using area, perimeter, and length of sides on a coordinate plane.
MAFS.8.G.1.AP.3a
Dilate common polygons using graph paper and identifying the coordinates of the vertices.
MAFS.8.G.1.AP.4a
Recognize congruent and similar figures.
MAFS.8.G.1.AP.4b
Identify two-dimensional figures as similar or congruent given coordinate plane representations.
MAFS.8.G.1.AP.4c
Compare area and volume of similar figures.
MAFS.8.G.1.AP.5a
Use angle relationships to find the value of a missing angle.
MAFS.8.G.1.AP.3b
Given two figures on a coordinate plane, identify if the image is dilated, translated, rotated, or reflected.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Educational Game

Transformation Complete:

Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.

Type: Educational Game

## Educational Software / Tools

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

Glossary:

This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

Type: Educational Software / Tool

## Formative Assessments

What Is the Triangle Relationship?:

Students are asked to write an informal justification of the AA Similarity Theorem.

Type: Formative Assessment

Same Side Interior Angles:

Students are asked to describe and justify the relationship between same side interior angles.

Type: Formative Assessment

Justifying the Triangle Sum Theorem:

Students are asked to provide an informal justification of the Triangle Sum Theorem.

Type: Formative Assessment

Justifying the Exterior Angle of a Triangle Theorem:

Students are asked to apply the Exterior Angle of a Triangle Theorem and provide an informal justification.

Type: Formative Assessment

Justifying Angle Relationships:

Students are asked to describe and justify the relationship between corresponding angles and alternate interior angles.

Type: Formative Assessment

Similarity - 2:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Type: Formative Assessment

Similarity - 1:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Type: Formative Assessment

Similarity - 3:

Students are asked to describe a sequence of transformations that demonstrates two polygons are similar.

Type: Formative Assessment

Proving Similarity:

Students are asked to explain similarity in terms of transformations.

Type: Formative Assessment

Angle Transformations:

Students are given the opportunity to experimentally verify the properties of angle transformations (translations, reflections, and rotations).

Type: Formative Assessment

Segment Transformations:

Students are given the opportunity to experimentally verify the properties of segment transformations (translations, reflections, and rotations).

Type: Formative Assessment

Proving Congruence:

Students are asked to explain congruence in terms of rigid motions.

Type: Formative Assessment

Multistep Congruence:

Students are asked to describe a sequence of rigid motions to demonstrate the congruence of two polygons.

Type: Formative Assessment

Parallel Line Transformations:

Students are given the opportunity to experimentally verify the properties of parallel line transformations (translations, reflections, and rotations).

Type: Formative Assessment

Rigid Motion - 3:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Type: Formative Assessment

Rigid Motion - 2:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Type: Formative Assessment

Rigid Motion - 1:

Students are asked to describe a rigid motion to demonstrate that two polygons are congruent.

Type: Formative Assessment

Translation Coordinates:

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

Type: Formative Assessment

Rotation Coordinates:

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

Type: Formative Assessment

Reflection Coordinates:

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

Type: Formative Assessment

Dilation Coordinates:

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

Type: Formative Assessment

## Image/Photograph

Angles (Clipart ETC):

This large collection of clipart contains images of angles that can be freely used in lesson plans, worksheets, and presentations.

Type: Image/Photograph

## Lesson Plans

Identifying Similar Triangles:

This 105-minute lesson series helps teachers assess how students reason about geometry, including how they use facts about the angle sum and exterior angles of triangles to calculate missing angles, apply angle theorems to parallel lines cut by a transversal, and interpret geometrical diagrams using mathematical properties to identify similarity of triangles. In order to complete this lesson, students will need whiteboards, pens, wipes, copies of the assessment tasks, pencils, markers, scissors, glue sticks, and poster paper.

Type: Lesson Plan

Coding Geometry Challenge #10 & 11:

This set of geometry challenges focuses on scaled drawings and area as students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor.

Type: Lesson Plan

Coding Geometry Challenge #23 & 24:

This set of geometry challenges focuses on using transformations to show similarity and congruence of polygons and circles. Students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor.

Type: Lesson Plan

"Triangle Congruence Show" Starring Rigid Transformations:

Students will be introduced to the definition of congruence in terms of rigid motion and use it to determine if two triangles are congruent.

Type: Lesson Plan

Triangles on a Lattice:

In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible.

Type: Lesson Plan

Rotations and Reflections of an Equilateral Triangle:

Students will apply simple transformations (rotation and reflection) to an equilateral triangle, then determine the result of the action of two successive transformations, eventually determining whether the action satisfies the commutative and associate properties.

Type: Lesson Plan

Transformations... Geometry in Motion:

Transformations... Geometry in Motion is designed for students to practice their knowledge of transformations. Students will represent transformations in the plane, compare transformations, and determine which have isometry. Students should have a basic understanding of the rules for each transformation as they will apply these rules in this activity. There is a teacher-led portion in this lesson followed by partner-activity. Students will be asked to explain and justify reasoning, as well.

Type: Lesson Plan

A Transformation's Adventure with Patty Paper: Exploring Translations, Reflections and Rotations.:

Students are introduced to isometric transformations using patty paper. Translations, reflections, and rotations will be explained and practiced, emphasizing the properties preserved during those transformations and, without sacrificing precision, allowing students to differentiate between these isometries. The lesson can also be taught using GeoGebra free software.

Type: Lesson Plan

Slide to the Left... Slide to the Right!:

Students will identify, review, and analyze transformations. They will demonstrate their understanding of transformations in the coordinate plane by creating original graphs of polygons and the images that result from specific transformations.

Type: Lesson Plan

Identifying Similar Triangles:

This 105-minute lesson series helps teachers assess how students reason about geometry, including how they use facts about the angle sum and exterior angles of triangles to calculate missing angles, apply angle theorems to parallel lines cut by a transversal, and interpret geometrical diagrams using mathematical properties to identify similarity of triangles. In order to complete this lesson, students will need whiteboards, pens, wipes, copies of the assessment tasks, pencils, markers, scissors, glue sticks, and poster paper.

Type: Lesson Plan

Scientific calculations from a distant planet:

Students will act as mathematicians and scientists as they use models, observations and space science concepts to perform calculations and draw inferences regarding a fictional solar system with three planets in circular orbits around a sun. Among the calculations are estimates of the size of the home planet (using a method more than 2000 years old) and the relative distances of the planets from their sun.

Type: Lesson Plan

Shape It Up:

Students will derive the formula for the sum of the interior angles of a polygon by drawing diagonals and applying the Triangle Sum Theorem. The measure of each interior angle of a regular polygon is also determined.

Type: Lesson Plan

Dilly Dallying with Dilations:

Students will understand the concept of dilation by constructing similar polygons on a coordinate grid using coordinate notation of dilation . students use similar figures to determine the scale factor. students use proportions to determine side lengths of similar figures.

Type: Lesson Plan

The Ins and Outs of Polygons:

In this lesson, students will explore how to find the sum of the measures of the angles of a triangle, then use this knowledge to find the sum of the measures of angles of other polygons. They will also be able to find the sum of the exterior angles of triangles and other polygons. Using both concepts, students will be able to find missing measurements.

Type: Lesson Plan

Triangles: Finding Interior Angle Measures:

The lesson begins with a hands-on activity and then an experiment with a GeoGebra-based computer model to discover the Triangle Angle Sum Theorem. The students write and solve equations to find missing angle measures in a variety of examples.

Type: Lesson Plan

Polygon Transformers:

This guided discovery lesson introduces students to the concept that congruent polygons can be formed using a series of transformations (translations, rotations, reflections). As a culminating activity, students will create a robot out of transformed figures.

Type: Lesson Plan

Special Angle Pairs Discovery Activity:

This lesson uses a discovery approach to identify the special angles formed when a set of parallel lines is cut by a transversal. During this lesson students identify the angle pair and the relationship between the angles. Students use this relationship and special angle pairs to make conjectures about which angle pairs are considered special angles.

Type: Lesson Plan

How Many Degrees?:

This lesson facilitates the discovery of a formula for the sum of the interior angles of a regular polygon. Students will draw all the diagonals from one vertex of various polygons to find how many triangles are formed. They will use this and their prior knowledge of triangles to figure out the sum of the interior angles. This will lead to the development of a formula for finding the sum of interior angles and the measure of one interior angle.

Type: Lesson Plan

Help me Find my Relationship!:

In this lesson, students will investigate the relationship between angles when parallel lines are cut by a transversal. Students will identify angles, and find angle measures, and they will use the free application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of angle relationships.

Type: Lesson Plan

Exploring Rotations with GeoGebra:

This lesson will help students understand the concept of a geometric rotation. The teacher/students will use a GeoGebra applet to derive the rules for rotating a point on the coordinate plane about the origin for a 90 degree, 180 degree, and a 270 degree counterclockwise rotation.

Type: Lesson Plan

An Investigation of Angle Relationships Formed by Parallel Lines and a Transversal Using GeoGebra:

In this lesson, students will discover angle relationships formed when two parallel lines are cut by a transversal (corresponding, alternate interior, alternate exterior, same-side interior, same-side exterior). They will establish definitions and identify whether these angle pairs are supplementary or congruent.

Type: Lesson Plan

Rotations and Reflections of an Equilateral Triangle:

Students will apply simple transformations (rotation and reflection) to an equilateral triangle, then determine the result of the action of two successive transformations, eventually determining whether the action satisfies the commutative and associate properties.

Type: Lesson Plan

Triangles on a Lattice:

In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible.

Type: Lesson Plan

## Original Student Tutorial

Home Transformations:

Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

All Circles Are Similar- Especially Circular Pizza!:

What better way to demonstrate that all circles are similar then to use pizzas! Gaines Street Pies explains how all pizza pies are similar through transformations.

Type: Perspectives Video: Professional/Enthusiast

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Congruent Triangles:

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.''

Reflecting Reflections:

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

Point Reflection:

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.

Reflecting a Rectangle Over a Diagonal:

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.

Find the Angle:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Find the Missing Angle:

This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Triangle congruence with coordinates:

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.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Rotating polygons 180 degrees about their center:

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Finding Missing Angle Measures:

In this video, we find missing angle measures from a variety of examples.

Type: Tutorial

Proving congruent angles:

In this tutorial, students are asked to prove two angles congruent when given limited information. Students need to have a foundation of parallel lines, transversals and triangles before viewing this video.

Type: Tutorial

Introduction to Transformations:

This video introduces the concept of rigid transformation and congruent figures.

Type: Tutorial

Scaling Down a Triangle by Half:

This video demonstrates the effect of a dilation on the coordinates of a triangle.

Type: Tutorial

Testing Similarity Through Transformations:

This video shows testing for similarity through transformations.

Type: Tutorial

Sum of measures of triangles proof:

This video gives the proof of sum of measures of angles in a triangle. This video is beneficial for both Algebra and Geometry students.

Type: Tutorial

## Virtual Manipulative

Rotation of a Point:

This virtual manipulative is an interactive visual presentation of the rotation of a point around the origin of the coordinate system. The original point can be dragged to different positions and the angle of rotation can be changed with a 90° increment.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorial

Home Transformations:

Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

Type: Original Student Tutorial

## Educational Game

Transformation Complete:

Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.

Type: Educational Game

## Educational Software / Tools

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

Glossary:

This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

Type: Educational Software / Tool

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Congruent Triangles:

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.''

Reflecting Reflections:

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

Point Reflection:

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.

Reflecting a Rectangle Over a Diagonal:

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.

Find the Angle:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Find the Missing Angle:

This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Triangle congruence with coordinates:

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.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorials

Rotating polygons 180 degrees about their center:

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Finding Missing Angle Measures:

In this video, we find missing angle measures from a variety of examples.

Type: Tutorial

Proving congruent angles:

In this tutorial, students are asked to prove two angles congruent when given limited information. Students need to have a foundation of parallel lines, transversals and triangles before viewing this video.

Type: Tutorial

Introduction to Transformations:

This video introduces the concept of rigid transformation and congruent figures.

Type: Tutorial

Scaling Down a Triangle by Half:

This video demonstrates the effect of a dilation on the coordinates of a triangle.

Type: Tutorial

Testing Similarity Through Transformations:

This video shows testing for similarity through transformations.

Type: Tutorial

Sum of measures of triangles proof:

This video gives the proof of sum of measures of angles in a triangle. This video is beneficial for both Algebra and Geometry students.

Type: Tutorial

## Virtual Manipulative

Rotation of a Point:

This virtual manipulative is an interactive visual presentation of the rotation of a point around the origin of the coordinate system. The original point can be dragged to different positions and the angle of rotation can be changed with a 90° increment.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

## Educational Software / Tool

Glossary:

This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

Type: Educational Software / Tool

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Congruent Triangles:

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.''

Reflecting Reflections:

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

Point Reflection:

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.

Reflecting a Rectangle Over a Diagonal:

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.

Find the Angle:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Find the Missing Angle:

This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.