Equivalent Transformations and Symmetry

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SWBAT find rotations and reflections that map a figure onto itself, and find single transformations that are equivalent to a composition of transformations.

Big Idea

Symmetry in terms of reflections and rotations. Students unpack the standard and solidify their knowledge of compositions of transformations.

Do Now and Mini-Whiteboard Practice

15 minutes

Class begins with the Do Now.  In this particular assignment, I ask the students to perform a composition of transformations, and I intentionally chose the ones that they seem to find the most difficult.  Anything with fractions seems to be scary to my students!

I also include finding equations of lines again.  I continue to revisit this topic because students seem to struggle with it.  The fourth line requires that they recall slope intercept form, y=mx + b from their previous math courses; this is the first time that I have asked them to apply this skill.

I display the Do Now on my SMARTboard and ask that the students do the work on mini-whiteboards. It can, however, also be distributed as paper copies.

After going over the Do Now, I give the students a couple more problems in which I give them two points and ask them to find a composition of transformations that maps one onto the other. I ask that student volunteers display their compositions on the board and have the rest of the class check their work. I also walk around the room as students are working on this, looking for particularly interesting compositions.  We pay close attention to the students' notations and to the accuracy of their transformations (MP 6).

Going Over the Homework

10 minutes

This homework assignment, Composition of Transformations Worksheet, is intended to give students practice with compositions and the composition notation. I ask the students for the answers, we go over any problems on which the students do not agree, and then move on.

Symmetry: Having the Students Unpack the Standard

15 minutes

I begin with a class discussion on Common Core Standard HSG-CO.A-3: (see Adjustments to Practice reflection above)

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 

This year I also provided my students with an example that I told them we had just used in my precalculus class.  I drew the graph of y=x³ on the board, and asked if this graph has symmetry. After a few minutes of silence, one student said, "I got it - it's got symmetry across the origin! It's like a rotation of 180 degrees or a reflection across the origin."  We discussed her response, and I gave a few more examples of symmetry about a point.

Then I handed out Alphabet Symmetry.  I asked the students to determine for each letter of the alphabet whether it has horizontal line symmetry, vertical line symmetry, symmetry about any other lines, and/or point symmetry. We discussed their conclusions when all were finished.

Equivalent Transformations

45 minutes

Students work together in their groups on Composition and Equivalent Transformations.  In this exercise, the students perform a composition of transformations, and then find a single equivalent transformation.  

Students compare and debate their answers in their groups (MP 3), so little whole class discussion is needed until the final question regarding a dilation and its effect on area and perimeter.  This question uses a good deal of prior knowledge (the Pythagorean Theorem, work with radicals, and ratios) and students' memories may require some refreshing. Students will ask what kind of an answer I am expecting (whether exact or approximate) and we discuss the appropriateness of exact answers in this case (MP 6).

When we complete this question, I use it to allude to a future unit, asking:

Are these figures congruent? If not, what are they?

Teacher's Note: This year a student asked if the relationship we had observed between the areas of similar figures is always going to be true - I was psyched to see my students thinking in these terms, attempting to generalize patterns (MP 8) on their own!

Closing and Homework

5 minutes

I hand out the homework, Practice with Transformations, and explain that these are typical multiple-choice questions on a statewide geometry assessment. My students know that I am not a big fan of standardized tests, but they are also aware of the importance of these assessments so they appreciate the practice.