After you master the challenge on page 1, you can try slightly harder challenges on pages 2 and 3.
✏️ What do you notice? What do you wonder?
✏️ Is this page easier or harder than page 1? What makes it different?
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.
Draw a diagram to show the path that each of you followed.
Describe the relative directions you moved in each part of the dance.
When did you both move in the same direction? When did you move in opposite directions?
3 Construct Reflect Functions
You and your partner should each create a design and submit your own drawing to your teacher.
On page 1, you used three tools:
These three tools emphasize the three parts of a function: the independent variable, the function rule, and the dependent variable.
On this page you’ll use the Reflect tool to create all three parts of the function. When you use it, keep in mind the three different things it makes.
Use a reflect function to trace a different letter.
The fixed points of a function are the places where the independent variable `x` and the dependent variable `r_j(x)` come together.
Construction Challenge 1
Construction Challenge 2
Construction Challenge 3
4 Find the Rule
These traces were left behind by an independent variable `x` and its reflected image `r_j(x)`. But both variables and the function rule are hidden!
Your job is to find the rule.
What did you notice, and wonder, as you solved these problems?
The cases on this page are harder, but your job is the same: find the function rule so that `r_j(x)` follows the blue trace while you vary `x` along the red trace.
Why was this page harder than page 1?
Make a drawing of your solution to a particularly interesting case. Show the traces, both variables, and the mirror line.
After your practice on pages 1 and 2, you’re ready to guess the location of the mirror ahead of time.
How close was your first guess? Did you get it exactly right?
To improve your skills, practice with several more cases. Follow the steps below for each new case:
How did more practice help you improve your accuracy?
Describe your strategy for locating the mirror. What did you pay attention to as you adjusted its two points?
On this page you have another new tool.
Experiment to see if you can use both the Midpoint tool and the Line tool to make a more accurate guess about where the mirror should be.
Construct the domain, restrict `x` to the domain, reflect in your line, and vary `x` to check your guess.
How close was your first guess? Did you get it exactly right?
Do these next steps at least two times. As you do so, think about how to use the Midpoint and Line tools together to make the best possible guess.
Describe your strategy for for using the Midpoint and Line tools to locate the mirror.
Why do you think this strategy works?
This page has a Midpoint tool and a Perpendicular tool.
As you did on previous pages, experiment to figure out how these two tools can help you guess the mirror correctly.
What strategy did you invent, using these two tools, to figure out the location of the mirror?
Explain in your own words why you think this strategy works.
How does this video relate to the reflect dance you did earlier?
6 Reflect Arcade
7 Reflect Dance 2
On the Floor
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, in mirror `j`, of `x`. Dancer `r_k(x)` is the reflected image in 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 north? south? east? west?
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?
On the Screen
Use this page to investigate reflections across perpendicular mirrors.
When Juanita and Donte did this, Donte said, `“r_k(r_j(x))` is the same as `r_j(r_k(x))`.” Juanita said, “No it’s not!” They both had good reasons for their answers.
Use this page to investigate reflections across angled mirrors that are not perpendicular.
What do you think Juanita and Donte said to each other when they did this construction?
How would you explain to a classmate what is special about perpendicular mirrors?
Use this page to investigate reflections across parallel mirrors.
Use this page to choreograph a dance using parallel mirrors.
Use this page to choreograph a dance using angled mirrors.
8 Dependent Dance Challenge
Until now, you’ve mostly been the independent variable (the penguin): either you were free to move around the plane, or you could move along a restricted domain. But in this challenge, you are the dependent variable (the frog), and your challenge is to follow the function rule as the penguin moves around her restricted domain.
To move correctly, you have to always be in the correct location across the mirror from the penguin. Fortunately you have a reflection segment to help you. When the penguin is in the center of the white circle, you are in the correct location for the function rule.
The hard part comes when the penguin starts moving: because you’re the dependent variable, you have to move to keep the white circle centered on the independent variable.
Before you start the game, practice moving the frog so the circle and cross-hairs move around the border of the polygon, because that's the path that the penguin will dance.
When 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 independent variable starts moving, you have to move the dependent variable to always be the reflected image.
Both the polygon and the trace will change color, from red to green, when you’re in the right place.
Once you master this dance, go to page 2 for a slightly harder challenge.
Page 2 has another challenge using a different polygon and mirror.
To prepare for the summary discussion, use the prompts below to record your accomplishments and “reflections.” (Sorry: bad pun ☹️)
The Fine Print
Record every major revision, in reverse chronological order