In this lesson (from the Connect Geometry and Algebra unit) the Dilate and Translate functions move from Lineland to Numberland.
Numberland is like Lineland, but the line has numbers so you can tell where you are.
1 Dilate and Translate in Numberland
Only the Dilate and Translate functions are visiting Numberland. The other functions stayed home because, as you may remember, they didn’t have as much fun there.
The Dilate function and Translate function want to do their first rides solo, to see what they can discover about their numerical behavior.
- Construct a number line and put an independent variable `x` on it.
- Dilate `x` and change the function to `D_(0, 2.0)` by adjusting scale factor `s` to 2.0.
- Vary `x` to give this Dilate function a ride on the number line. Pay attention to who’s ahead and who’s behind, and whether `x` or `D_(0, 2.0)(x)` moves faster.
- Record three different value pairs for `x` and `D_(0, 2.0)(x)`. What do you notice about the numbers?
- Adjust `s` to change the Dilate function to `D_(0, 0.5).`
- How is the motion of `D_(0, 0.5)(x)` different from the motion of `D_(0, 2.0)(x)`?
- Record three different value pairs for `x` and `D_(0, 0.5)(x)`. What do you notice about the numbers?
- Set `s` to a negative number. Before dragging, predict three value pairs that you think `x` and `D(x)` can make using your chosen value of `s.` Then test your predictions by dragging. What happened?
Go to page 2 and give the Translate function a ride.
- Construct a Translate function on the number line.
- Change the function to `T_(0→3.0)` by adjusting the vector slider `v` to 3.0.
- Vary `x` to give this Translate function a ride. Pay attention to who’s ahead and who’s behind, and to the relative speed of `x` and `T_(0→v)(x).`
- How is this function's behavior different from the Dilate function?
- Adjust `v` to make the Translate function `T_(0→7.0).`
- Record three different value pairs for `x` and `T_(0→7.0).` Use at least one negative value for `x.`
- Choose a negative value for `v,` and write down three predictions for value pairs you can make. Include a negative value for `x.`
- What happened when you tested your predictions?
Go to page 3 to experiment on your own.
Use this page to conduct your own experiments to make interesting rides for the Dilate and Translate functions.
For instance, you could have both of them ride on the same number line, or you could make several number lines with a variety of rides.
Summary Questions
From your work on page 1 explain, for any value of `s,` what the Dilate function does to independent variable `x` to produce dependent variable `D_(0, s)(x).` What number operation does this function perform?
Write an equation that connects `x,` `s,` and `D_(0, s)(x).`
From your work on page 2 explain, for any value of `v,` what the Translate function does to independent variable `x` to produce dependent variable `T_(0→v)(x).` What number operation does this function perform?
Write an equation that connects `x,` `v,` and `T_(0→v)(x).`
2 Compose Dilate and Translate Functions
The Dilate and Translate function have each explored their own behavior on a number line. Now they want to team up and explore a number line together.
Their idea is to attach the Translate function’s independent variable to the Dilate function’s dependent variable.
They need your help to make this happen.
- Put a Dilate function on a number line, decide what scale factor `s` to use, and vary `x` to see the function in action.
- Now put a Translate function on the number line, attaching the independent variable to `D(x).`
- Decide what translation vector `v` to use, and then drag `x` to see the composed function in action.
- Write down `s, v,` and three value pairs you found for `x` and `T(D(x)).`
(The full name of the dependent variable would be `T_(0→v)(D_(0, s)(x)),` but the two functions decided to use only their nicknames, and call it `T(D(x))` for short.)
On pages 2 and 3 try out different settings of `s` and `v.`
Use this page to compose and test different rides.
- As on page 1, construct the composition of `D` and `T.`
- Adjust `s` and `v` to make an interesting ride.
- Record your values of `s` and `v,` and write down three values pairs of `x` and `T(D(x))` to keep track of different places along the ride.
- What number operations do you suspect might connect `x` and `T(D(x))?`
- As before, construct the composition of `D` and `T,` and adjust `s` and `v` to make a different ride.
- Record your values of `s` and `v,` and write down three values pairs of `x` and `T(D(x))` to keep track of different places along the ride.
- Try to write an equation that connects `x` and `T(D(x)).` Use `s` and `v` in your equation.
Summary Questions
From pages 2 and 3, which composition made the most interesting ride? Explain why its values of `s` and `v` made the ride so interesting.
How can you read the abbreviation `T(D(x))` as words instead of just letters?
Explain why your equation connecting `x` and `T(D(x))` makes sense, based on what you know about the Dilate and Translate functions.
3 Construct a Dynagraph
The variables of the two functions were getting tired of bumping into each other.
They decided it would be even more fun if `x` and `T(D(x))` could use different number lines, separate from each other but with the variables still connected.
They need your help to construct this arrangement, which is called a dynagraph.
- Construct a number line with an independent variable `x` and a Dilate function. Drag `x` back and forth to check your construction.
- Use the Transfer tool to transfer both the origin point and `D(x)` downward. Construct a new number line using the transferred origin point.
- Construct a Translate function on the new number line, attaching its independent variable to `D(x).`
- Vary `x` to see the combined function in action.
- To see just the original variable `x` and the final variable `T(D(x)),`use the Visibility Widget to hide the transfer arrows and the `D(x)` variables.
- Connect `x` and `T(D(x)),`turn on tracing for the connecting line, and drag `x` to see how the connecting line shows the motion of these variables.
- Try out different different Dilate and Translate functions by adjusting the values of `s` and `v.` (Every time you change the functions, erase the traces and use Vary x to make smoothly spaced new traces.)
- Write down at least two things you noticed as you changed the functions (by adjusting `s` and `v`).
- Use pages 2 and 3 to make functions with particularly interesting traces.
On pages 2 and 3, construct a dynagraph and change the two functions to make make interesting or surprising traces when you vary `x.`
Draw a picture of your traces. Show the limits and origins of the number lines and the values of `s` and `v.`
Explain how you can use the shape of the traces to describe the motion of the two variables.
Why is it important to erase your traces every time you change `s` and `v?`
4 Dynagraph Challenges
Every page of this sketch has ten challenges. Can you get a perfect score of 100 on Level 5 (page 5)?
Levels 1, 2, 3 and 4 will help you develop the skills you need to crush the Level 5 challenges.
Directions: Adjust scale factor `s` and vector `v` to match the function description. Then press Check to earn 10 points and move to the next challenge.
Good luck!
Level 1: First adjust `s` and `v,` and drag `x` to test your answer. Then press Check. If you’re wrong, you get a second chance (worth 5 points).
Level 2: You can’t drag `x.` Press Vary x to get a hint (but it will cost you 2 points).
Level 3: Now `D(x)` is hiding, so you have to imagine where it is.
Level 4: Now all the variables are hiding. Press Vary x to show the traces—but now it costs 3 points.
Level 5: Every challenge is all or nothing, the variables are hiding, and you only get one chance to adjust `s` and `v.`
Summary Questions
Which of the function descriptions have multiple solutions? Explain how you figured out your answer.
Write a description of your own that has multiple solutions.
Write a description of your own that has a single solution.
Describe one thing you figured out as you worked on these challenges.
Which challenge page did you like the best, and why did you like it?
5 Dynagraph Slope-Intercept Challenges
The previous section described its functions as transformations, with a dilation followed by a translation: `T(D(x)).` This section describes its functions in slope-intercept terms: `y = m·x + b.`
Every page of this sketch has ten challenges. Can you get a perfect score of 100 on Level 5 (page 5)?
Levels 1, 2, 3 and 4 will help you develop the skills you need to crush the Level 5 challenges.
Directions: Adjust the slope `m` and intercept `b` to match the function description. Then press Check to earn 10 points and move to the next challenge.
Good luck!
Level 1: First adjust `m` and `b,` and drag `x` to test your answer. Then press Check. If you’re wrong, you get a second chance (worth 5 points).
Level 2: You can’t drag `x.` Press Vary x to get a hint (but it will cost you 2 points).
Level 3: Now `m·x` is hiding, so you have to imagine where it is.
Level 4: Now all the variables are hiding. Press Vary x to show the traces—but now it costs 3 points.
Level 5: Every challenge is all or nothing, the variables are hiding, and you only get one chance to adjust `m` and `b.`
Summary Questions
Which of the function descriptions have multiple solutions? Explain how you figured out your answer.
Write a description of your own that has multiple solutions.
Write a description of your own that has a single solution.
Describe one thing you figured out as you worked on these challenges.
What does slope mean on a dynagraph? How is this different from a Cartesian graph, and how is it similar?
What does intercept mean on a dynagraph? How is this different from a Cartesian graph, and how is it similar?
Which challenge page did you like the best, and why did you like it?
6 Play the Dynagraph Game
Can You Find the Mystery Function?
In this game you control `s` and `v.`
There is a mystery function (named “??”) connected to `x,` with its dependent variable labeled ??`(x).`
Your job is to discover the mystery function by adjusting `s` and `v` so that dependent variable `T_(0→v)(D_(0, s)(x))` matches ??`(x)`.
Page 1 is a practice page. Pages 2, 3, and 4 contain increasingly difficult games.
Good luck!
Page 1 is for practice. Drag `x` back and forth to figure out how the mystery function works.
Then adjust `s` and `v` to make your dependent variable `T_(0→v)(D_(0, s)(x))` match the mystery function’s dependent variable ??`(x).`
Page 2 is the first game page. On this page you can’t drag the independent variable; instead `x` is always in motion!
Adjust `s` and `v` to make your dependent variable `T_(0→v)(D_(0, s)(x))` match the mystery function’s dependent variable ??`(x).` Once you have the two dependent variables matching, press Check.
If you’ve successfully identified the mystery function, you get a new mystery.
Record the highest level of play at which you scored 8 of 10.
Describe something you noticed that helped you figure out the mystery function’s scale factor s.
On this page ??`(x)` is hidden; you can see only the traces of the mystery function.
Adjust `s` and `v` to make your connector from `x` to `T_(0→v)(D_(0, s)(x))` match the mystery traces.
When you successfully identify the mystery function, you get a new mystery.
Record the highest level of play at which you scored 8 of 10.
Describe a strategy you invented to help you find the mystery function.
Oh no! Even your own variables are hidden!
Can you use the mystery function’s traces to figure out `s` and `v?`
(Warning: Even Level 1 is hard; Levels 3 and 4 are exceedingly difficult!)
Record the highest level of play at which you scored 8 of 10.
Describe a strategy you invented for analyzing the traces.
Summary Questions
Write down two things you figured out as you played the Dynagraph Game.
Which game did you like the best, and why did you like it?
6 Accomplishments
To prepare for a summary class discussion, use the prompts below to record your accomplishments.
- For the dilate function, what did you discover about the relationship between scale factor `s` and the motion of the variables `x` and `D_(0, s)(x)?`
- For the translate function, what did you discover about the relationship between the length `v` of the translation vector and the values of the variables `x` and `T_(0→v)(x)?`
- How can your discoveries be written as mathematical equations using `s, v, x, D_(0,s)(x),`and `T_(0→v)(x)?`
- Describe something you noticed when you composed the Dilate and Translate functions.
- Describe something you wondered about composing two functions.
- What connections did you figure out between the shape of a Dynagraph trace and the motion of the two variables?
- What was your favorite discovery as you learned to play the Dynagraph game?
- Which was your favorite part of this lesson? Why?
- How could this lesson have been more interesting or more fun?
The term dynagraph was coined by Paul Goldenberg, Philip Lewis, and James O’Keefe in their study “Dynamic Representation and the Development of a Process Understanding of Functions” published by Education Development Center, Inc., and supported in part by a grant from the National Science Foundation.
4 The Fine Print
Requirements:
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Release Information
“You may freely use Web Sketchpad in your own Web pages, provided you do so for non-commercial use only” [per kcpt.github.io].
Update History:
21 Oct 2015: created an early version of the sketch
This web site is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through December 2020) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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