Forging Connections
The Reflect Family

1 Warm-Up

After you master the challenge on page 1, you can try the slightly harder challenge on page 2.


✏️ What do you notice? What do you wonder?

2 Reflect Dance 1

Work with a partner to perform two short reflect dances, moving as if there is a mirror between you. To locate the mirror, put a piece of tape on the floor, with a dance guide paper in the middle of the mirror like this:

For the first dance, flip a coin to identify one person as the independent dancer and the other as the dependent dancer. You will switch roles for the second dance.

Independent Dancer: Always look at your partner, and take steps according to the directions below. As you dance, you can do any other body movement that’s both safe and simple enough that your partner can mirror it. Start one step back from the middle of the mirror.

Dependent Dancer: Mirror your partner’s movements, staying exactly across the mirror from your partner and the same distance away from the mirror.

First Dance:
  • 4 steps at 0°
  • 3 steps at 270°
  • 3 steps at 135°
  • 2 steps at 180°

✏️ Draw a diagram to show the path that each of you followed.

✏️ Did both of you move in the same direction?

Second Dance: (after switching roles)

Begin with the three-part dance routine below and finish by improvising your own dance pattern. If possible, ask a classmate to video your dance.

  • 3 steps at 315°
  • 2 steps at 90°
  • 3 steps at 225°
  • Improvise and record the rest of your dance routine

✏️ When did you both move in the same direction? When did you move in opposite directions?

3 Interlude

From I Love Lucy Season 4 Episode 27. Downloaded from

✏️ How does this video relate to the reflect dance you just did?


4 Construct Reflect Functions

  • Create an independent variable using the Point tool.
  • Create a function rule using the Mirror tool.
  • Create a dependent variable using the Reflect tool.

You and your partner should each take a turn making a design of your own, and you should each submit a drawing of your traces to your teacher.


Hint Video:    

5 Match the Traces

Each page of this sketch has a faint background image showing traces left by an independent variable x and its reflected image `r_j(x)`. But the mirror is hidden!

Construct your own reflect function and adjust it so you can drag x to make the same pattern as the one shown faintly in the background.

✏️ Try each of the two mirror tools. Which did you find more useful? Why?

✏️ What did you learn as you solved these challenges? Describe your method so another student could use it.

✏️ Why do you think there’s an extra tool at the bottom of the toolbox?


6 Reflect Games

Each of these three games has a different purpose:

  1. Given independent variable `x` and mirror `j`, how can you find `r_j(x)`?
  2. Given mirror `j` and `r_j(x)`, how can you find independent variable `x`?
  3. Given independent variable `x` and `r_j(x)`, how can you find mirror `j`?

For each game play a few times at easier levels, and work your way up to level 5. (Once you start a game at one level, you must press Reset to change to a different level.)

✏️ For each game, how many hits can you score in ten tries at level 5?

✏️ What methods did you invent to make it easier to get a high score?


7 Reflect Shapes

With this sketch, you can reflect a variety of shapes.

  1. Construct one of the shapes.
  2. Using the Reflect tool, attach `x` to your shape.
  3. Use the Animate tool to animate your independent variable`x` along the shape

Note that the Reflect tool in this sketch constructs three things: the independent variable `x`, the mirror, and the dependent variable `r_j(x)`.

  • Page 1: Experiment with reflecting shapes.
  • Page 2: Reflect to m,atch the given letter shape.
  • Page 3: Reflect to create other letter shapes.
  • Page 4: Experiment on your own.

8 Find the Mirror

This sketch challenges you to find the hidden mirror(s). On pages 1, 2, and 3, your job is to find the mirror so that if you animate `x` on the domain, `r_j(x)` will move around the range. Pages 4 and 5 povide slightly different challenges.

✏️ What did you notice, and what did you wonder, as you constructed these reflect functions?.


9 Reflect Dance 2

In a group of four or five students, tape a dance guide paper on the classroom floor to mark two perpendicular mirrors. Invent a reflect-family dance that uses both mirrors.

Dancer `x` is the independent variable and dancer `r_j(x)` is the reflected image, across mirror `j`, of `x`. Dancer `r_k(x)` is the reflected image across mirror `k` of `x`. Decide among yourselves how the fourth dancer should move. As you create and practice your dance, check to be sure the dancers are correctly positioned relative to the mirror.

As you choreograph and practice your dance, take turns so everyone in the group has a chance to be the independent dancer. When you’re ready to perform your dance, ask someone from another group to video your group dance.

✏️ As you created your dance, how did you decide the fourth dancer should move?

✏️ In what direction must each of the other dancers move when `x` moves at 0°? at 90°? at 180°? at 270°?

✏️ What else did you notice, and what did you wonder, about the reflect functions that connect your dancers to each other?

✏️ How could you use function notation to describe the fourth dancer?

NY Times 25 October 2016 Suzanne Farrell Ballet at Kennedy Center

10 Objectives

This page supports the Reflect Family lesson from the Introducing Geometric Transformations as Functions unit. You will construct reflect 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:

  • Construct independent variable x and reflect it across a mirror.
  • Label the dependent variable using meaningful function notation.
  • Drag independent variable x, trace both variables, and describe their relative motion.
  • Move the mirror to form and investigate a different member of the family.
  • Restrict the domain of x to a polygon and describe the resulting range.
  • Identify fixed points of the function and describe their relationship to the mirror.
  • Solve challenges that involve finding a hidden mirror.

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:

Record every major revision, in reverse chronological order

04 Mar 2018: Converted to new lesson format.
06 Mar 2017: Fixed a bug on page 5 where the x tool didn’t work.
16 May 2016: Updated websketch page 8 to eliminate tracing, which was confusing.
31 Oct 2015: Updated this page to the website format.