- Define the terms transformation, translation, reflection, and rotation
- Indicate that two shapes are congruent if their side lengths and angle measurements are equal
- Determine whether or not two shapes are congruent, both on coordinate grids and not on coordinate grids
- Turn Up the Music Student Worksheet printable
- Answer Key: Designing With Geometry printable
- Standards Chart: Geometry printable
- Graph paper
- Whiteboard or large graph paper and markers
- Make a class set of the Turn Up the Music Student Worksheet printable.
- Print a copy of the Answer Key: Designing With Geometry printable for your use.
Step 1: Draw a non-rectangular parallelogram on the board. Indicate that the angles are 60º and 120º and the lengths of the sides are 4 units and 6 units, and label it Parallelogram A. Draw two more parallelograms. Parallelogram B has the same side lengths as Parallelogram A but has angles that measure 30º and 150º. Parallelogram C has the same side lengths and angle measurements as Parallelogram A.
Step 2: Introduce the term congruent and the symbol for congruence. Indicate that two shapes are congruent if they have the same side lengths and angle measurements. Ask if either parallelogram B or C is congruent to parallelogram A. Show that parallelogram C is congruent to A because it has the same side lengths and angle measurements; parallelogram B is not congruent to A because, although the side lengths are equal, the angle measurements are not.
Step 3: Note that when shapes undergo transformations, the original shape (the pre-image) and the shape after the transformation (the image) are congruent. Demonstrate how this works with a translation. Explain that a translation is a transformation that occurs when a polygon slides in one direction. Demonstrate this with a 30º, 60º, 90º triangle labeled ABC. Draw a congruent triangle slid to the right, labeled A'B'C'. Explain how the image is named "A prime, B prime, C prime" and explain the prime symbol. Mention that a translation is only a slide without any other transformation taking place. Show how the two triangles are congruent because the side lengths and angle measurements are the same for the pre-image and the image.
Step 4: Demonstrate how a reflection works. Explain that a reflection occurs when a polygon is flipped across a line. Draw the following parallelogram on a grid on the board: A (-1, 2), B (-2, -1), C (-4, -1), and D (-3, 2). Then draw a congruent parallelogram flipped across the y-axis: A' (1, 2), B' (2, -1), C' (4, -1), and D' (3, 2). Demonstrate that the parallelograms are congruent by measuring the corresponding sides and angles and showing that they are equal. Also, show how the points flipped across the y-axis, e.g., that A is one unit to the left of the y-axis while A' is one unit to the right. Point out that reflections can also occur when shapes are flipped across the x-axis or another line.
Step 5: Finally, demonstrate how a rotation creates an image that is congruent to the pre-image. Draw the following parallelogram: A (-3, 1), B (-4, 3), C (-2, 3), D (-1, 1). Demonstrate how to rotate the shape 180º around the origin, creating a congruent parallelogram at A' (3, -1), B' (4, -3), C' (2, -3), D' (1, -1). Show that the parallelogram moves from quadrant II to quadrant IV. Show how point A (-3, 1) when rotated around the origin ends up as point A' (3, -1). A is three units to the left of the origin and one unit above while A' is 3 units to the right of the origin and one unit below. If necessary, repeat by showing how each point of the parallelogram rotates 180º clockwise to quadrant IV. Determine congruence by measuring the sides and angles to show that they are equal.
Step 6: Distribute graph paper to students. Draw a right triangle with the following points and coordinates on the board: A (-5, 2), B (-2, 6), and C (-2, 2). Ask the class to pair up, complete the transformations below, and label the points:
- A translation four units to the right [Answer: Coordinates will be A' (-1, 2), B' (2, 6) and C' (2, 2)]
- A reflection across the horizontal axis [Answer: Coordinates will be A' (-5, -2), B' (-2, -6), and C' (-2, -2)]
- A rotation 180º clockwise around the origin. [Answer: Coordinates will be A' (5, -2), B' (2, -6) and C' (2, -2)]
Step 7: Checking for Understanding: Review answers as a class and respond to any questions.
Step 8: Assign the Turn Up the Music Student Worksheet printable for classwork or homework.
Step 9: Checking for Understanding: Review the answers to the Turn Up the Music Student Worksheet printable, which are provided on page 1 of the Answer Key: Designing With Geometry printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.
- Grade 8: Congruence and Transformations (CCSS: 8.G.1)
- Grades 6–8: Making Sense of Problems, Reasoning, and Attending to Precision (CCSS MP1, 2, and 6); NCTM Geometry
For more information, download the comprehensive Standards Chart: Geometry printable.