What’s going on here? What do you notice? What do you wonder?
✏️ How does this video relate to the rotate function you just constructed?
4 Rotate Games
This sketch includes five different Rotate Family games.
For each game, the level number (1, 2, and 3) determines how many hints are available, and the letter (a, b, and c) determines what angles are used. (For instance, multiples of 10° are harder to estimate than multiples of 45°.)
After you start a level, you must press Reset to change levels.
How well can you do at the higher levels of each game? (Don’t expect to get perfect scores at levels 2 and 3, but you will be able to get higher scores as you improve your ability to estimate angles and distances.)
Hit the Target: Where’s the missing dependent variable?
What’s the Angle: What’s the missing angle?
Where’s the Center: Where’s the missing center?
Inverse Hit the Target: Where’s the missing independent variable?
What’s the Moving Angle: Find the missing angle while `x` varies.
5 Trace a Star
6 Rotate Dances
6a Physical Dances
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.
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.
6b Virtual Dances
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.
Hint video: Trying the challenge at Level 4
7 Rotation Stars
7a Construct Stars
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.)
7b Describe Your Constructions
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`.
8 Where, Oh Where, Can the Center Be?
You are a detective, investigating a crime committed by a center of rotation, but you don't know where it is, nor do you know what angle it used.
Before you can solve the crime, you need to develop several methods for investigating rotations.
8a︎︎ Develop Your Methods
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`.
8b︎︎ Solve the Mystery
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.
9 Go Further
This activity is your opportunity to investigate rotation more deeply. Each page has a question that you can use to guide you.
This page supports the Rotate Family lesson from the Introducing Geometric Transformations as Functions unit. You will construct rotation functions, manipulate and observe them, and learn how this function family behaves.
By the end of this lesson, you’ll be able to perform these actions and answer these questions:
11︎︎ The Fine Print