Grade 8 Geometry: Transformations & Similarity

Objective

Students will understand the concept of rigid motions (translations, reflections, and rotations) and how they relate to congruence. Students will be able to explain that two figures are congruent if a sequence of rigid motions can take one figure onto the other, and that rigid motions preserve the measures of line segments (lengths) and angles.

Materials

• The resource text/pages (Chapter 9: "Geometry: Transformations, Congruence and Similarity," especially Section 9.1 and Figure 1)
• Printer paper or student notebooks
• Pencils
• Rulers (straightedge) if available
• Tracing paper or transparencies (optional, if available in classroom, for demonstrating rigid motions)

Warm-up (~5 min)

1. Write these two words on the board: "Same Shape" and "Same Shape AND Same Size."
2. Ask students: "Can you think of two objects in this room that are the same shape and same size? What about same shape but different size?"
3. Take 2-3 quick verbal answers. No need to resolve — just get students thinking about the difference between size and shape.

Main Activity (~25 min)

Step 1: Introduce Figure 1 (10 min)

• Direct students to Figure 1 in the resource (Figures A, B, C, D showing sets of objects).
• Explain: In Figure A, all images are the same size and shape — you can move any one onto any other using a rigid motion (a slide, flip, or turn).
• In Figure B, the figures are the same shape but not the same size.
• In Figures C and D, the figures are neither the same size nor the same shape.
• Have students look at Figure A and try to describe (in words or by sketching arrows) how they would move the first object onto the others (sliding it, flipping it, or turning it).
• Ask: "Why can't we do this same kind of move for Figures B, C, and D?" Have students discuss in pairs (2 min) then share out.

Step 2: Define Rigid Motion and Congruence (8 min)

• Read aloud or paraphrase this key definition from the text:

"A rigid motion of the plane is a transformation of the plane that takes lines to lines, and preserves lengths of line segments and measures of angles."

• Write on the board the three basic kinds of rigid motions mentioned in the text:
• Translation (shift)
• Reflection (flip)
• Rotation (turn)
• Explain: "Two figures are congruent (same shape and size) if there is a sequence of rigid motions that takes one figure to the other."
• Emphasize the key idea from the text: rigid motions preserve both length and angle measure — nothing stretches, shrinks, or distorts.

Step 3: Apply the Basic Geometric Facts (7 min)

• Have students copy down (or read together) these basic facts listed in the resource:
• A line is determined by any two different points on it.
• Two lines coincide, intersect in exactly one point, or never intersect (parallel).
• Two circles don't intersect, intersect in one point, or intersect in two points (or coincide).
• Two lines that never intersect are parallel; if they intersect and all angles at the intersection are equal, the lines are perpendicular.
• The sum of the lengths of any two sides of a triangle is greater than the third side.
• Have students, in pairs, pick one of these five facts and draw a simple sketch on paper illustrating it (e.g., draw two intersecting circles, or a triangle demonstrating fact #5).

Wrap-up / Exit Ticket (~10 min)

Have students answer the following on a half-sheet of paper to turn in:

1. In your own words, what does it mean for two figures to be congruent? (Look for: same shape and size; can be moved onto each other using rigid motions.)
2. Name the three basic types of rigid motions discussed today. (Translation, reflection, rotation)
3. True or False: A rigid motion can change the length of a line segment. (False — rigid motions preserve length and angle measure.)
4. Look at Figure B in the resource. Are the figures in Figure B congruent? Why or why not? (No — they are the same shape but not the same size, so no rigid motion alone can take one to the other.)

Collect exit tickets as students leave or at the end of class.

If Time Remains

Have students revisit Figure 1 and try sketching their own simple example of two shapes that would belong in "Figure A" (same shape and size) and two shapes that would belong in "Figure B" (same shape, different size). They can draw these freehand or with a ruler, labeling which set shows congruence and which does not.