In this lesson you will figure out how to move two-dimensional transformation functions into one dimension.
In other words, instead of transforming points on the plane, you will transform points on a number line.
1 Warmup
The Reflect family decided to take a vacation by visiting Lineland.
But when `x` entered Lineland, her dependent variable `r_m(x)` had trouble getting into Lineland, and even more trouble staying in.
Can you help them?
- Press the Show Mirror button to show the function rule.
- Adjust the mirror to help `r_m(x)` stay in Lineland.
Describe how you adjusted the mirror to get the dependent variable to stay in Lineland.
Can you find a different way to keep `r_m(x)` in Lineland? If so, describe it, and explain how the two methods are similar, and how they’re different.
2 Introduce
Flatland the Movie is based on the classic novel Flatland by Edwin Abbott. The book is narrated by A. Square, a mathematician and resident of 2-dimensional Flatland who has had the extraordinary opportunity of visiting our 3-dimensional Spaceland after having an earlier chance to visit 1-dimensional Lineland.
The movie is available from Flat World Productions.
A. Square describes to his Spaceland readers the nature of Flatland society and politics, but is imprisoned after he returns to Flatland and attempts to explain to his fellow Flatlanders the existence of three dimensions.
(In addition to exploring the mathematics of dimensions, the book is also a satiric commentary on English society and politics of the time.)
✏️ How does this video relate to this lesson?
3 Reduce the Dimension
All of the function families decided to visit Lineland on vacation.
To get both their independent and dependent variables into Lineland at the same time, each family decided to bring the point(s) of their function rule into Lineland with them.
On each page, construct a different function family in Lineland, and experiment by varying `x` and changing the function rule.
Finally, compare the experiences that each family has in Lineland. Which families have the most interesting experiences? Which have the most boring experiences?
- Construct Lineland.
- Construct a reflect function, restricting `x` and `P` to Lineland. Put `Q` anywhere you want (but not on Lineland).
- Drag `P` and `Q` to different locations until you find a function rule that puts `r_m(x)` on Lineland.
- Make sure that `r_m(x)` stays on Lineland as you vary `x.`
Make a drawing to show how you got `r_m(x)` onto Lineland. Your drawing should show Lineland, the mirror rule, and all the points with labels.
Describe how `r_m(x)` moves in relation to `x.`
Can you find a different way to arrange the mirror to keep `r_m(x)` on Lineland? If so, make a drawing to show how you did it. If not, explain as best you can why it’s impossible.
- Construct Lineland.
- Construct a rotate function, restricting `x` and `C` to Lineland.
- If necessary, adjust `θ` to put `R_(C,θ)(x)` in Lineland.
Does `R_(C,θ)(x)` stay in Lineland as you vary `x?`
Describe what happens when you change the function rule by dragging `C` and `θ.`
Describe how `R_(C,θ)(x)` moves in relation to `x.`
- Construct Lineland.
- Construct a translate function, restricting `x` and both vector endpoints to Lineland.
- If necessary, adjust the vector to put `T_v(x)` in Lineland.
Does `T_v(x)` stay in Lineland as you vary `x?`
Describe what happens when you change the function rule by dragging the vector endpoints.
Describe how `T_v(x)` moves in relation to `x.`
- Construct Lineland.
- Construct a dilate function, restricting `x` and `C` to Lineland.
- If necessary, adjust `s` to put `D_(C,s)(x)` in Lineland.
Does `D_(C,s)(x)` stay in Lineland as you vary `x?`
Describe what happens when you change the function rule by dragging `C` and `s.`
Describe how `D_(C,s)(x)` moves in relation to `x.`
- Construct Lineland.
- Construct a glide reflect function, restricting `x` and both endpoints of the vector/mirror to Lineland.
- If necessary, adjust the vector/mirror to put `G_v(x)` in Lineland.
Does `G_v(x)` stay in Lineland as you vary `x?`
Describe what happens when you change the function rule by dragging the vector/mirror endpoints.
Describe how `G_v(x)` moves in relation to `x.`
Worksheet
4 Number the Domain
Numberland is like Lineland, but the line has numbers so you can tell where you are.
On each page, you will construct a number line and a function, and you’ll restrict the domain of the function to the number line.
You’ll measure the value of each variable so you can observe the values as you drag `x`, leading to some interesting and surprising discoveries!
- Construct a number line.
- Construct a reflect function, restricting `x` to the number line and `P` to the origin.
- If necessary, adjust `Q` to put `r_m(x)` on the number line.
- Measure both `x` and `r_m(x).`
- Vary `x` and observe these values as you drag.
Describe what you notice about the values in a sentence or two.
Use mathematical symbols to summarize your description.
Why do you think this happens? Explain.
- Construct a number line.
- Construct a translate function, restricting each point:
- Restrict `x` to the number line.
- Restrict `U` to the origin.
- Restrict `V` to the number line.
- Measure `x,` `T_v(x),` and `V.`
- Vary `x` and observe the measured values.
- Change the function by moving `V.` Then vary `x` and observe again.
- Try both positive and negative values for `V.`
Describe what you notice about the values in a sentence or two.
Use mathematical symbols to summarize your description.
Why do you think this happens? Explain.
- Construct a number line.
- Construct a rotate function, restricting `x` to the number line and `C` to the origin.
- If necessary, adjust `θ` to put `R_(C,θ)(x)` on the number line.
- Measure both `x` and `R_(C,θ)(x).`
- Vary `x` and observe these values as you drag.
Describe what you notice about the values in a sentence or two.
Use mathematical symbols to summarize your description.
Why do you think this happens? Explain.
- Construct a number line.
- Construct a dilate function, restricting `x` and `C` to the number line.
- If necessary, adjust `s` to put `D_(C,s)(x)` on the number line.
- Measure both `x` and `D_(C,s)(x).`
- Vary `x` and observe these values as you drag.
- Change the function by moving `s.` Then vary `x` and observe again.
- Try both positive and negative values for `s.`
Describe what you notice about the values in a sentence or two.
Use mathematical symbols to summarize your description.
Why do you think this happens? Explain.
- Construct a number line.
- Construct a glide reflect function, restricting each point:
- Restrict `x` to the number line.
- Restrict `U` to the origin.
- Restrict `V` to the number line.
- Measure `x,` `G_v(x),` and `V.`
- Vary `x` and observe the measured values.
- Change the function by moving `V.` Then vary `x` and observe again.
- Try both positive and negative values for `V.`
Describe what you notice about the values in a sentence or two.
Use mathematical symbols to summarize your description.
Why do you think this happens? Explain.
5 The Fine Print
Requirements:
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
“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:
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.
We are deeply grateful to McGraw-Hill Education
for making Web Sketchpad available for these lessons.
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