Interactivate: Transformations (Elementary)

Abstract

This lesson is designed to offer an introductory look at transformations for elementary school students, with emphasis on congruency.

Objectives

Upon completion of this lesson, students will:

understand that shapes remain the same even as they are translated, rotated, or reflected
understand basic methods of evaluating translations, rotations, or reflections on graph paper

Standards Addressed:

Student Prerequisites

Geometric: Students should be able to:
- identify basic geometric figures

Teacher Preparation

access to a browser
access to a projector (recommended but not required)
access to pen and paper
physical models of squares, triangles, and other simple shapes
tracing paper and graph paper

Key Terms

congruent	Two figures are congruent to one another if they have the same size and shape
reflect	To perform a reflection
reflection	In the plane, a reflection is a rigid motion that keeps all the points on some line fixed, and flips the rest of the points to the opposite side of that line. In space, a reflection is a rigid motion that keeps all the points on one plane fixed, and flips all points to the opposite side of that plane. Note that if you perform any reflection twice, all points end up back where they started. When you reflect an object, you are creating a "mirror image" of that object, with the fixed line or plane being the mirror.
rigid motion	A rigid motion, of the plane or of space, is one that keeps the distances between all pairs of points unchanged. Rotations, reflections and translations are examples of rigid motions.
rotate	To perform a rotation
rotation	A rotation in the plane is a rigid motion keeping exactly one point fixed, called the "center" of the rotation. Since distances are unchanged, all the other points can be thought of as having moved on circles whose center is the center of the rotation. The "angle" of the rotation is how far around the circles the points travel. A rotation in three-dimensional space is a rigid motion that keeps the points on one line fixed, called the "axis" of the rotation, with the rest of the points moving some constant angle around circles centered on and perpendicular to the axis.
translate	To perform a translation
translation	A translation is a rigid motion that moves each point the same distance, in the same direction

Lesson Outline

Focus and Review

Review basic shapes and remind students of the definition of congruency:

What does "rotate" mean? What would it look like if you "rotated" this square/triangle?

Students should think of words like "turn", "swivel", "spin", etc.
What does "reflect" mean? Can you imagine what it would look like if you "reflected" this shape in a mirror?
What does "translate" mean? Help students understand that translating a word is basically just moving it from one language to another. Then ask students to consider how they might move or "translate" an object?

Objectives

Let the students know what it is they will be doing and learning today. Say something like this:

Today we are going to learn how to transform shapes by translating, reflecting, or rotating them.
We are going to use computers to learn more about transformations, but please do not turn your computers on until I ask you to do so. I want to show you a little about this activity first.

Teacher Input

Show students 3D Transmographer. Create an irregular shape, such as a trapezoid, to make your transformations easy to see.

Tell students that you are going rotate your shape by a quarter-circle. Ask students what they expect to happen. If you have access to a smart board/white board and projector, project the shape onto the board and then have students draw what they think will happen when the shape is rotated. Ask the following questions:
- What do you think will happen to the shape when it is rotated by a quarter-circle?
- Does anyone think something else will happen to the shape when it is rotated?
- How did you predict where the shape would rotate? Did you use a specific method or thought process?
Repeat the above activity, but with a translation of the shape instead.
Students are likely to find reflections to be the hardest to visualize, so demonstrate reflections by starting a 3D revolution at the slowest setting, and then pausing just as the shape is on the opposite side of the diagram (ten "steps"). By manipulating the graph view, show students that the shape remained the same even as it was reflected.
Using cut-outs of triangles, squares, and other basic shapes, physically demonstrate that shapes remain the same even as they are translated, rotated, or reflected. If possible, make your cut-outs the same size as the projection, so that you can show how the "before" and "after" images of the shape are actually the same.
Discuss with students why shapes remain the same even as they are transformed. Introduce the idea that a shape is congruent to its reflection, rotation, or translation. Ask the following questions:
- If I walk across the room, am I the same person even though I'm in a different place?
- What if I turn around? Am I still the same person then?
- When you look in the mirror, how do you know that the person you are seeing is yourself?
- Are there any things, like hair color, that you can look at to make sure that you're the same even as you move or turn around?
- Considering that, can you look at certain properties of shapes to determine if they are congruent?

Guided Practice

As a class, come up with algorithms for one or two of the following situations. With each situation is listed a sample algorithm, if students have trouble getting started.

Reflection across the x-axis.
- To reflect across the x-axis, fold the paper along the x-axis and then trace the figure.
Reflection across the y-axis.
- To reflect across the y-axis, fold the paper along the y-axis and then trace the figure.
Rotation by a quarter circle.
- To rotate by a quarter circle, place another piece of graph paper on top of the first piece, but rotated by 90 degrees, and then trace the figure.
Rotation by a half circle.
- To rotate by a half circle, place another piece of graph paper on top of the first piece, but upside down, and then trace the figure.
Translation up, down, left, or right.
- To translate up, down, left, or right, move each vertex of the figure as required, and then redraw the lines between the vertices.

Work through the problems on the Worksheet that correspond to the algorithms you've worked on as a class.

Independent Practice

Have students work individually or in groups to come up with algorithms for each of the other situations.

As they work, remind them to try to make sure the algorithms are efficient, so that they don't waste time, as well as such that they can check their work easily to avoid errors.

Have students complete the remainder of the worksheet using these algorithms.

Conclusion

Bring the class back together and discuss the results of the worksheet. Have students share and discuss their algorithms. Ask students the following questions:

How well did your algorithms work to complete the worksheet?
How can you check your work with these algorithms?
As you worked and heard your classmates present their algorithms, did you come up with any improvements to your algorithms that might make your work easier in the future?

Alternate Outline

This lesson can be rearranged in the following ways:

If students lack experience with creating algorithms, you can start by teaching students the various algorithms. Then, ask students to come up with other algorithms of their own invention that could also work for each situation.

Suggested Follow-Up

To build on the basic concepts used in this lesson, students can next go into more detail about the quantitative aspects of transformations with Translations, Reflections, and Rotations.

Interactivate

Transformations (Elementary)