Lesson 1 | Congruence in Two Dimensions | 10th Grade Mathematics

Objective

Define polygon and identify properties of polygons.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Congruence

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Congruence

G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define a polygon as a closed figure.
Distinguish between convex polygons (all interior angles less than 180°), and concave polygons (one or more interior angles as greater than 180°).
Describe the features of polygons and classify on the basis of the features of a polygon:
1. Number of sides/angles
2. Relative length of sides
3. Parallel or perpendicular sides
4. Presence of right angles
5. Presence of interior angles that are less or greater than 180°
Determine the least amount of description necessary to accurately classify polygons and identify features that discriminate one polygon from another.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson prepares students to access standard G-CO.3, but has a focus on elementary standards for review.
The inclusive definition of trapezoids is used throughout this lesson. Trapezoids are quadrilaterals that contain at least one pair of parallel sides.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Your friend gives you a bunch of shapes and asks you to choose one, but not tell him or her. Choose a figure. What are the features of this polygon that set it apart from the rest of the polygons shown?

Guiding Questions

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Problem 2

Without looking at any resources, define the following terms concisely and accurately.

Polygon
Parallelogram
Triangle

Guiding Questions

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References

Continuous Everywhere But Differentiable Nowhere Attacks and Counterattacks in Geometry

Problem 3

Decide whether each of these statements is always, sometimes, or never true. If it is sometimes true, draw and describe a figure for which the statement is true and another figure for which the statement is not true.

A rhombus is a square
A triangle is a parallelogram
A square is a parallelogram
A square is a rhombus
A parallelogram is a rectangle
A trapezoid is a parallelogram
A trapezoid is a quadrilateral

Guiding Questions

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References

Illustrative Mathematics Always, Sometimes, Never

Always, Sometimes, Never, accessed on March 8, 2017, 11:39 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

A convex polygon with one pair of parallel sides can be generally classified as a trapezoid. What would need to be true about this shape to also be classified as a rhombus?

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Illustrative Mathematics What is a Trapezoid? (Part 2)
Illustrative Mathematics What shape am I?
Illustrative Mathematics Defining Attributes of Rectangles and Parallelograms
Illustrative Mathematics What do these shapes have in Common?
Desmos Polygraph: Polygons
Dan Meyer's Three-Act Math Best Square

Lesson 2

Lesson Map

Topic A: Introduction to Polygons

Define polygon and identify properties of polygons.

G.CO.A.1 G.CO.C.11

Prove interior and exterior angle relationships in triangles.

G.CO.C.10

Describe and apply the sum of interior and exterior angles of polygons.

G.CO.C.11

Topic B: Rigid Motion Congruence of Two-Dimensional Figures

Determine congruence of two dimensional figures by translation.

G.CO.A.2 G.CO.A.4 G.CO.A.5 G.CO.B.7

Reflect two dimensional figures on and off the coordinate plane.

G.CO.A.2 G.CO.A.3 G.CO.A.5 G.CO.B.7

Rotate two dimensional figures on and off the coordinate plane.

G.CO.A.2 G.CO.A.3 G.CO.A.5 G.CO.B.7

Describe a sequence of rigid motions that map a pre-image to an image (specifically triangles, rectangles, parallelograms, and regular polygons).

G.CO.A.5 G.CO.B.6

Describe single rigid motions, or sequences of rigid motions that have the same effect on a figure.

G.CO.A.3 G.CO.A.5 G.CO.B.6

Topic C: Triangle Congruence

Develop the Side Angle Side criteria for congruent triangles through rigid motions.

G.CO.B.7 G.CO.B.8

Prove angle relationships using the Side Angle Side criteria.

G.SRT.B.5

Prove and apply that the points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

G.CO.A.2 G.CO.C.10 G.CO.C.9

Develop Angle, Side, Angle (ASA) and Side, Side, Side (SSS) congruence criteria. Describe how the criteria develop from rigid motions.

G.CO.B.7 G.CO.B.8

Use triangle congruence criteria, rigid motions, and other properties of lines and angles to prove congruence between different triangles.

G.SRT.B.5

Prove triangles congruent using Angle, Angle, Side (AAS), and describe why AAA is not a congruency criteria.

G.CO.B.7 G.CO.B.8 G.CO.C.10

Develop the Hypotenuse- Leg (HL) criteria, and describe the features of a triangle that are necessary to use the HL criteria.

G.CO.B.7 G.CO.C.10

Use criteria for triangle congruence to prove relationships among angles and sides in geometric problems.

G.CO.C.9 G.SRT.B.5

Topic D: Parallelogram Properties from Triangle Congruence

Prove that the opposite sides and opposite angles of a parallelogram are congruent.

G.CO.C.11

Prove theorems about the diagonals of parallelograms.

G.CO.C.11

Congruence in Two Dimensions

Objective

Common Core Standards

Core Standards

Congruence

Congruence

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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