First Dance

Use a group of three or four students to perform a 180° rotate dance: an independent dancer, a dependent dancer, and one or two choreographers. You will do at least four dances, so each group member should take a turn playing each role.

You can include specific arm, leg, and other body motions in your dance. The choreographer(s) should make sure the dependent dancer's motion is a rotation. (For instance, if the independent dancer raises her right arm, the choreographer(s) should check that the dependent dancer also raises her right arm (not her left arm).

For your first dance, follow the steps below.

1. Choose one person as the independent dancer and another as the dependent dancer.
2. Mark a center point on the floor.
3. The two dancers face each other, close to the center point.
4. The independent dancer moves in a similar pattern to the penguin in the sketch below, backing away from the center, moving around the center, and finally approaching the center.
5. The dependent dancer moves at the same speed in the opposite direction.
6. The choreographer(s) make sure the dependent dancer keeps the center half-way between the two dancers (as in the dance below).

More Dances

Change roles and do a 90° rotate dance, similar to the 90° dance the penguin and frog do below.

Do two more dances (first 180° and then 270°), changing roles each time. For these dances use a more challenging pattern. Page 2 of the sketch shows an example.

Up until now, you’ve mostly been the independent variable: you were either free to move around the plane, or you had to move around a polygon or other restricted domain. But in this challenge, you have to help the frog, who's the dependent variable. The penguin will move around a polygon, and your challenge is to help the frog follow the function rule.

On page 1 you have to drag the frog so you’re always the 180° rotation of the penguin as she moves around the polygon. Fortunately you have a 180° arc and cross-hairs to help you. As long as you can keep the penguin in the cross-hairs, you are in the correct location, so you just have to get the penguin in the cross-hairs and keep her there once she starts moving.

Before you start the game, practice moving the frog so the cross-hairs move around the border of the polygon, because that's the path that the penguin will dance.

When you think you’re ready to play, move the frog so the cross-hairs are over the penguin. Then press the Countdown button and get ready to move the frog. When the countdown ends and the penguin starts moving, and you have to move the frog to always be the 180° rotation of the penguin.

Page 2 has another challenge using a different polygon, center, and angle of rotation.

On page 3, you can adjust `C`, `θ`, and the vertices of the polygon and then hide them all to make an interesting challenge for a classmate.

Which is easier: being the independent variable or the dependent variable? Explain your thinking.

• What you notice, and wonder, as you practiced the dance on page 1?
• Did you get better as you practiced? Could you do the dance at a more difficult level and still keep the colors green?
• Are the dances on pages 2 and 3 easier (because of your practice on page 1) or harder? Explain.
• Having done physical dances and virtual dances, is it easier being the independent variable or the dependent variable? Why?

On page 1 the orange segment has already been moved to the correct location for you to use as your restricted domain. (You’ll have to do this yourself on the rest of the pages.)

On pages 2, 3, 4, and 5 you’ll construct several rotation stars. On each page you can press the New Design button to find a design you like. As you create your star designs, don’t do them all the same way. Instead, try to use as many different methods as you can.

To keep track of different constructions, draw a picture in your journal of each star, and use function notation to describe the function you used to make each part of the star.

Use pages 6 and 7 to create designs of your own. (You can make fancy stars, or other designs that you invent. On the right is a example that was made with just two different angles but a number of different centers.)

In the drawing below, this student shows that he used two different function rules. In the first line he writes `R_1: C_1, 60°` to say that rotation `R_1` rotates about center `C_1` by `60°`, and then he writes `x_1=R_1(x)` to say that dependent variable `x_1` was constructed by applying function rule `R_1` to independent variable `x`. Similarly, he writes another line to describe his function `R_2`, with a different center point and angle, which he used to construct dependent variable `x_2`.

In this second drawing, another student shows that she used a single function rule two times. In the first line she writes `R_1: C, 120°` to say that rotation `R_1` rotates about center `C` by `120°`, and then she writes `x_1=R_1(x)` to say that dependent variable `x_1` was constructed by applying function rule `R_1` to independent variable `x`. On her second line she uses the same function again, so she just writes `x_2=R_1(x_1)` to say that dependent variable `x_2` was constructed by applying function rule `R_1` to variable `x_1`.

Begin by learning to analzye the evidence:

Pages 1 & 2: Find two ways to locate the hidden dependent variable if you’re given center `C` and angle `theta`.

Pages 3 & 4: Estimate the hiding place for `C` if you can vary `x` and see the effect on `R_C(x).`

Pages 5 & 6: Aapproximate some possible hiding places for `C` even though you can’t vary `x`.

Pages 7 & 8: Precisely mark all possible hiding places for `C`.

You have a new perpendicular bisector tool to help you track down the center. This crime scene has three rooms, represented by the three pages. Use the clues in each room to learn more about the crime. When you reach the third room you will be able to construct the exact hiding place for the center.

• Page 1: In this room your only clues are points `D` and `E`. You’ll construct a rotate function with a center point that works even when point `D` moves around.
• Page 2: In this room the center committed two crimes, rotating `D` to `F` and `E` to `G`. You’ll find a single center point to solve both crimes—as long as the clues don’t move around.
• Page 3: In this room you’ll construct `overline(DE)` and `overline(FG)`, and then construct a rotate function to rotate the first segment to the second. You’ll also figures out what changes, and what stays the same, when the points move around.