Interaction trumps lecturing. While some students might have “mathematical minds” and quickly understand how to perform the many types of transformations from a vanilla lecture, other students benefit from interactive activities. A teacher, tutor or parent can engage in a simple hands-on activity using index cards and a whiteboard that helps students learn transformation methods by personal experience.

Shuffle Your Cards

Before you begin this activity, you’ll need to make some cards. Write the names of the translations you are teaching on index cards. For an introductory lesson on transformations, you might consider teaching reflections, translations and resizing. Write those words on the index cards and put them in one pile. For reflections, you’ll need a handful of lines with which to use, so write some lines, such as y = 0, on another handful of index cards. Stack these and mentally label them the “reflection pile.” Do the same for translations, but as translations are about movement, write statements such as “three to the left” or “three upward.” Put them in a pile you mentally label as the “translation pile.” For the resizing cards, write statements such as “double” or “half.”

Introduce Before You Produce

With the cards ready, teach quickly the basics of the methods of transformations you plan to teach. A quick introduction is enough; remind students of this, telling them, “If you feel this is confusing, don’t worry. We will work on this together in a second.” Just give them a brief overview of what translations are and how to perform each one. Use a simple shape for your examples, such as starting with the drawing of a square centered at the origin. Demonstrate each transformation, focusing on explaining how to perform each one. You might find it helps to list the steps. For example, a translation three to the left requires you to first find the new corners of the new square by subtracting three from each x-coordinate; then you must draw the new square using the new points.

Here's the Deal

Have your student or students choose one of the transformation cards at random. For example, if you are teaching translations, reflections and resizing, your student will have a one-in-three chance of getting a given translation type. After she chooses a card, have her choose at random from its corresponding pile. She will have chosen two cards, such as “reflection” and “x = 0,” for example. This is the transformation she will have to perform.

Evaluate and Dictate

Choose a shape according to your student’s level and draw it on the whiteboard. For less advanced students, consider squares and rectangles. For more advanced students, consider triangles, asymmetrical quadrilaterals and even more complex polygons. Have the student perform the translation, according to her card, on the shape you’ve drawn. Walk her through it if she needs help. For example, if she drew the cards that correspond to a reflection across x = 0, explain to her that she’s “flipping” the shape across the line x = 0. Give her the step-by-step procedure, if needed. This involves counting the distance between each corner point in the original shape and the line x = 0. Once she’s counted the distance from the corner point to the line, have her continue that exact distance across the line, in the direction opposite of the shape, plotting a new point when she lands. Once all the corner points are plotted, have her “connect the dots” to draw the new shape. If you are teaching multiple students, have them give feedback as to whether the transformation is correct. Review the steps of transformation and repeat them with another student and another combination of cards.