Forging Connections
The Rotate Family
Download lesson journal

1 Warm-Up


What’s going on here? What do you notice? What do you wonder?

2 Introduction

  • Create an independent variable using the Point tool.
  • Create a function rule using the Center & Angle tool.
  • Create a dependent variable using the Rotate tool.
  • Change `theta` to 180°. Drag `x` to trace a shape.
  • What do you notice about the relative speed and direction of `x` and `R_(C,θ)(x)?`
  • Erase the traces and draw a new shape.
  • Include parts where you drag x toward and away from the center, in an arc around the center, and in a straight line that’s not toward or away from the center.
  • On your worksheet draw a picture of the shape. Show `C` and `theta` in your picture.
  • What do you notice, and wonder, when you drag `x`?
  • Use this page to investigate the behavior of a `90°` rotation.
  • Create an independent variable, a rotate rule, and a dependent variable.
  • Adjust `theta` to `90°` and drag `x` to trace a shape.
  • Include parts where you vary `x` toward and away from the center, in an arc around the center, and in a straight line that’s not toward or away from the center.
  • Describe the relative speed and direction of `x` and `R_(C,θ)(x).`
  • On your worksheet draw a picture of the shape. Show `C` and `theta` in your picture.

  • Use this page to investigate the behavior of a `45°` rotation.
  • Trace a shape, and draw a picture on your worksheet.
  • Describe the relative speed and direction of x and `R_(C,θ)(x).`
  • Drag `x` to find fixed points, where `x` and `R_(C,theta)(x)` come together.
  • What do you notice, and wonder, about the fixed point(s)?

3 Interlude

✏️ 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

  • Page 1: Adjust the function rule to trace out the star.
  • Page 2: Can you find two ways to trace each arm?
  • Page 3: Trace another star of your own choosing.

6 Rotate Dances

6a Physical Dances

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.


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.

  • 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?

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.)


Page 1 hints for function #2:     One method Another method Still another method

7b Describe Your Constructions

The easiest way to describe your star constructions is to combine function notation with drawings, as described below.

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`.


Hint videos:     Page 1 Page 3 Page 5 Page 7

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.

  • 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.

Hint video:     Page 1

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.

  • Page 1: You will verify an important feature of rotate functions.
  • Pages 2 and 3: You will explore some interesting features of the rotate family.
  • Pages 4 and 5: You will compose and superpose rotations.
  • Pages 6 and 7: You will investigate connections between reflect functions and rotate functions.

Step 1 Step 3 Step 5 Step 6

10 Objectives

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:

  • Vary the independent variable. Vary the independent variable `x` by dragging it freely, and later by dragging it along a restricted domain.
  • Construct the dependent variable. Given an independent variable `x`, a center `C`, and an angle `theta`, construct the dependent variable.
  • Restrict the domain. Construct an independent variable on a restricted domain to limit where it can move.
  • Where’s the center? Find the center point given an independent variable `x` and its rotated image`R_(C,θ)(x)`.
  • What’s the angle? Determine the angle of rotation given an independent variable and its rotated image.
  • What’s the function? Construct the rotate function given an independent variable and its rotated image.
  • What’s the function? Construct the rotate function given a restricted domain and the corresponding range.
  • Use mathematical language. Use mathematical terms and function notation to describe an independent variable and its rotated image. Use both independent/dependent “variable” terminology and “preimage/image” terminology.
  • How do they move? Describe the relative motion (rate of change) of the variables for a given rotate function `R_(C,θ)`.
  • Draw the range. Given a rotate function (defined by its center point and angle of rotation) and a restricted domain, draw the shape of the range.
  • Do the rotate dance. Given a rotate function and an independent variable animated on a restricted domain, vary the dependent variable accordingly.
  • Construct a rotation star. Given a center point and positive integer n, create design with n-fold rotation symmetry.
  • Compose rotate functions. Apply one rotate function to the dependent variable of another. Consider special cases: a function is composed with itself, and a function is composed with another (the inverse) so that the composite is the identity function.
  • Construct the range. Given a rotate function RC,θ that operates on a point, construct the range given a polygon domain.
  • How is rotation different? Describe several differences between rotate functions and reflect functions.
  • Score! Use your understanding of the rotate function to put the balls in the net quickly and easily.

11︎︎ The Fine Print


These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.

Release Information

Update History:

27 Mar 2018: Updated to new organization; previous version is preserved temporarily at
02 May 2018: Rotation stars bjects lost due to resizing are now back on-screen.Rotate-games objects showing in error are now hidden.
27 Apr 2018: Resized Rotation Stars sketch to fit the widescreen format
11 Dec 2017: Added Rotation Stars section.
16 Oct 2017: Finished sketch, worksheet, and videos for the Restrict the Domain section
03 Oct 2017: Created Forging Connections banner
24 Jul 2017: Adjusted styles
26 Apr 2017: Added warmup game; converted to toggling section format.
31 Oct 2015: Updated this page to the website format.