One of the most difficult (and fun!) lessons to teach in Middle School math is Transformations. I found this out, (after many years of teaching 7th grade math), when I was asked to teach two sections of 8th grade. I’ve taught 8th grade before, but never transformations. Over the summer, I looked for resources to teach this geometry concept. One week before school started, due to Covid 19, I was forced to stay home a while longer with my disabled son. So now, I had to focus on writing detailed lesson plans for a potential sub, who might or might not have a strong math background. As a teacher, I know one of the best ways to learn a concept, is to teach it to someone else. So I began to delve into the subject in depth and write detailed lesson plans.
REFLECTION
Reflections looked like fun. Fun to teach hands-on with graph paper, or digitally with Google Slides. I looked through my resources to see how to reflect across an axis, reflect across a line, and reflect a figure when part of your figure crosses the line of symmetry. You can teach reflections by referring to a mirror image. Ask the students why the word “AMBULANCE” is written as a mirror image on an emergency vehicle. Show students a great Youtube video on reflections such as “Mashup Math”.
TRANSLATION
Translation is also fun, and fairly easy to teach. “Mashup Math” has a Youtube video on translations, and compares this transformation to sliding on a skateboard. With translations, students move a figure on the coordinate plane, up, down, left, or right, without changing the orientation of the figure. This can also be done digitally on Google Slides. (This year I planned to mix it up between hands-on and digital assignments. Both are so important.) Students take the coordinates of the point at each vertex and move the point a required direction. Once they move each vertex and connect the new points, they will have created a new translated image. You can extend this lesson by having students both translate a figure, and then reflect it over a line.
DILATION
Dilation is just an extension of 7th grade math on Scale Drawings. Before teaching dilations, I review Scale Drawings, and re-introduce the vocabulary of scale and scale factor. I show students the blueprints to a house I built with my son-in-law. I explain that the city would not give us approval to build the house until we had these scaled down plans where every single door, window, and room size had to be drawn exactly to scale, or in proportion to the actual house. I remind students that a dilated, or scaled figure will be the the same shape but a different size. The sides of the pre-image and the new image will be in proportion. The new size is determined by the scale factor. A scale factor of 2 means the new image will be two times the size of the pre-image. A scale factor with a fraction less than one, means the new image will be smaller than the original pre-image. A scale factor of 1/3 means the new image will be 3 times smaller than the original pre-image.
Dilations are figures drawn to scale on a coordinate plane. Students will enlarge or shrink their figure by multiplying by the scale factor. If the scale factor is 4, each coordinate is multiplied by 4. So, if you have a point at (2, 3), multiply both 2 and 3 by 4. Your dilated point will be at (8, 12). If this is done with all vertices, you will have a new shape that is larger, but in proportion to your pre-image. If your scale factor is a fraction less than 1, you will multiply each coordinate by that fraction. If the scale factor is 1/2, the coordinates of point (8, 10) will be multiplied by 1/2, or divided by 2. The new point will be at (4, 5). Again, I recommend “Mashup Math”. This Youtube video introduces the topic by showing the dilation of the human eye.
ROTATION
Rotation is the transformation that gives teachers the most headaches. It’s not easy to teach. It took me quite some time to gain an understanding of how this works. A rotation is a circular movement of a figure on the coordinate plane. Sounds simple. I would start with a video. And again, Mash-up Math does a great job. You want students to have a conceptual understanding first, before just giving them the formula.
Rotate 180 degrees:
Have your students use patty paper to trace a figure on the coordinate plane. Add the coordinates of each point on the patty paper. Your figure will be rotated around the origin. On the original graph, draw a line from the origin to one of the vertices of your figure. Students will use this line to show they have moved 90 or 180 degrees.
Have them use the drawing on their patty paper to rotate the figure around the origin 180 degrees. It will be in the opposite quadrant. The point in which you connected with a line to the origin should be extended into the opposite quadrant, equal distance from the origin. (180 degrees forms a straight line.) If they draw a line from each vertex to the origin and then extend it as a straight line into the opposite quadrant, equal distance from the origin, they should have a figure rotated exactly 180 degrees. Have them use the traced figure to check for rotation. Ask students what they notice about the difference in the coordinates. This will help them understand why the formula (-x, -y) works in determining the coordinates of a point rotated 180 degrees.
Rotate 90 degrees:
With a new graph, have students again trace a figure and rotate 90 degrees around the origin. Trace the figure on patty paper and rotate to an adjoining quadrant, either clockwise or counterclockwise. This time, the line drawn from a point to the origin will form a right angle with the new point in the next quadrant. Use the traced figure on patty paper to help with this rotation. All vertices should form a right angle with the rotated point. Again, have the students look at the difference in points rotated 90 degrees. They can then come up with a formula to use to determine a point rotated 90 degrees. If they rotate the opposite direction 90 degrees, the formula will be different. See if the students can come up with a forumla for clockwise rotation 90 degrees (y, -x), and another formula for counterclockwise rotation 90 degrees (-y, x). Tricky! If the explanation seems confusing, go back to the video where you can see a visual of this transformation.
All Four Transformations
Once the students have a conceptual understanding, they can practice reflections, rotations, dilations, and translations on several different problems. The digital version is fun for students as they use the “fill color” feature of Google Slides to color in the squares for reflections and translations, or drag and drop points and lines to show dilations and rotations. They can also show a combination of different transformations. The digital version completes the lesson with a drag and drop vocabulary slide.
Best of luck with transformations! Click on the link for the digital Lesson: https://www.teacherspayteachers.com/Product/Transformations-Digital-Lesson-7220400
