Transformations is a big meaty unit in 8th grade math that students will revisit in Algebra 1 with quadratic functions. Let’s talk about all the ways you can help your students succeed with transformations!
CCSS and TEKS Standards
Let’s take a look at what the standards actually say regarding transformations:
Released STAAR Question from 2019
Backwards planning (looking at how a question has been assessed in the past) always helps me frame my instruction. This is a STAAR (Texas) question, but I immediately understand how important algebraic expressions are to showing mastery of transformations. If my kids can identify and explain transformations, but cannot show it algebraically, then I have not satisfied the totality of the standard.
Vocabulary is Key
Put transformations on the list of concepts that are vocabulary heavy! Not only are students going to need to internalize translation, reflection, and dilation, but there are plenty of prerequisite vocabulary terms that students will use: quadrant, y axis, x axis, coordinate, and orientation. Fortunately, students have been learning about the coordinate plane since elementary school. I would still provide a few minutes of class time to remind students what everything is called on the coordinate plane.
Image and pre-image can also seem basic, but prove tricky for students. I recommend reminding students that a preview is what you see at the movies first; therefore, a pre-image is first or the first image. The image is what happens after the pre-image has been transformed.
Luckily, a translation, a transformation that moves each point of a figure the same distance and direction, can be remembered that it involves a slide because sl exists in both translation and slide.
Similarly, a reflection, a transformation that flips a figure over a line of reflection, also shares that memory device with fl existing in both reflection and flip.
Lastly, dilation, a transformation that enlarges or reduces a figure by a given scale factor, can be connected to how our pupils dilate based on the amount of light in a given space.
With most vocabulary, I recommend using the words frequently during instruction, requiring students to use the words in their discourse, and making the words visible in your classroom.
Here is a little sneak peak of what our word wall looks like:
Algebraic Rules
Once students are familiar with what the transformations are and how they impact the orientation and congruence of a shape, then I would recommend spending one day per translation on their algebraic rules. You can see how I would teach transformations with the activities I would include.
CCSS Transformations Unit | TEKS Transformations Unit
Activities Included in Pacing Calendar:
- Translations Cut and Paste Activity
- Reflections Card Sort
- Rotation Maze
- Dilations He Said, She Said
- Scavenger Hunt Review
Note: If you have an 8th grade All Access membership, you have all of these activities already!
Scaffolding is your friend here! (Isn’t it always?) And I love how our curriculum team set this up in our student handouts. Let’s take a look at our Translation Student Handouts. You will notice that students will create a translation given the directions to move triangle ABC 5 units to the left and 2 units down. From there, students will record the vertices in the table and describe what is happening to the x and y-coordinates of the vertices. Lastly, students will summarize the effect of the transformation and record it algebraically.
(This is the same process that is used to introduce algebraic rules for both reflections and rotations.)
Remind your students that the purpose of the algebraic rules are to tell you what to do to the values in the ordered pair, or to evaluate what has already been done after a transformation. Verbalize: “(x-5, y+2) means left 5 and up 2.”
It’s also helpful to remind students that since the x-axis runs from left to right, translations to the left or the right will impact the x values. Similarly, since the y-axis runs up and down, translations up and down will affect the y-values.
Algebraic rules for rotations and reflections is where you will find the most misunderstandings. The negative sign indicates, “the opposite of.” (-x, y) does not mean that the x-coordinate will be negative. One way to combat this is to say “the opposite of” instead of “-x” whenever you or a student are reading algebraic rules aloud. I would also recommend annotating -x and -y’s with the word “opposite” on problems, multiple choice answers, etc.
When determining the coordinates for a figure that will be rotated, I try to simplify the rotation “rules” by having students remember to do the following:
- The x and y values only switch places when they have been rotated 90 and 270 degrees.
- Determine which quadrant the image will be in to determine the signs of your x and y values. For example, a point in the image in Quadrant IV must have a +x and a -y.
Technology is your friend
I think students can really benefit from understanding transformations when they can see the transformation in action. This is why I recommend using some of these websites to provide a visual as you introduce the transformations. Desmos has an entire unit devoted to Transforming Shapes which is divided into Describing Transformations Informally (which are more exploratory in nature) and Describing Transformations Precisely. I highly recommend assigning your students many of these activities. Here are my recommendations:
- Sketchy Dilations: Perfect activity to introduce dilations.
- Transformation Golf: Rigid Motion: There are lots of practice problems here, so you can pick and choose what is best. Essentially, students will use translations, reflections, and rotations to get the pre-image to the image while working around an obstacle. Students can watch the transformations they choose happen live and try again after learning what works and doesn’t work. For an activity that also includes dilation, try Transformations Golf: Non-Rigid Motion.
- Connecting the Dots: Watch and describe polygons rotating and practice determining the center of rotation.
This game is a great review of all transformations! It also offers hints if a student gets stuck.
Other Tips and Misconceptions to Watch For
- Symmetrical figures can be especially tricky. It might look like a translation, but students need to observe the order of the vertices on each figure to see if it’s actually been reflected rather than translated. Using colored pencils to circle and track the vertices can be extremely helpful!
- Students may confuse clockwise and counterclockwise in a rotation. If you have an analog clock in your classroom, I would point to that regularly.
- Students who are looking at dilations may incorrectly assume that any fraction would result in a reduction in size, forgetting that improper fractions will actually lead to an enlargement.
- Since reflecting over the wrong axis is a common error, have students label axes on each graph. Then have them highlight what axis they are supposed to reflect over or which axis the reflection had already been reflected over.
Transformations can be conquered! How do you teach transformations?
