1 Mystery 1: Warmup
Reflect `triABC.` Make its reflected image fit `triDEF` exactly.
What did you notice while adjusting the mirror?
What did you wonder?
Rotate `triABC` so it exactly fits `triDEF.`
What did you notice while adjusting the angle or center?
What did you wonder?
Dilate `triABC` so it exactly fits `triDEF.`
What did you notice while adjusting the scale or center?
What did you wonder?
Translate `triABC` so it exactly fits `triDEF.`
What did you notice while adjusting the vector?
What did you wonder?
Glide-reflect `triABC` so it exactly fits `triDEF.`
What did you notice while adjusting the mirror vector?
What did you wonder?
Press the New Case button if you want a new challenge.
Is this transformation an isometry or a similarity?
When you’ve done all five pages...
- For which function families were you able to get an exact fit?
- Of your successful efforts, which was hardest? How did you overcome the difficulty?
2 Mystery 2: Invent Shortcuts
These five pages are similar to the warmup, but you have several extra tools available. You can use these tools to invent shortcuts that make it easy to solve superposition problems. For instance, you might invent a shortcut for finding the mirror for a reflect function, or a shortcut for finding the center point for a dilate function.
Reflect `triABC` so it exactly fits `triDEF.` Then use one or more of the other three tools to figure out a shortcut you could have used to find the mirror.
You may get some ideas for shortcuts by using the Animate button to of the vertices.
If you can figure out a shortcut for constructing the mirror, describe your shortcut.
Dilate `triABC` so it exactly fits `triDEF.` Then use one or more of the other three tools to figure out a shortcut for finding the center for dilation.
Rotate `triABC` so it exactly fits `triDEF.` Then use one or more of the other three tools to figure out a shortcut for finding the center for rotation.
Translate `triABC` so it exactly fits `triDEF.` Then figure out a shortcut for finding the vector for translation.
Glide-reflect `triABC` so it exactly fits `triDEF.` Then use one or more of the other three tools to figure out a shortcut for finding the mirror vector.
Reset the page and press New Case to make sure your shortcut works with a different reflectdilate function.
Is this transformation an isometry or a similarity?
When you’ve done all pages...
- For which functions were you able to invent a shortcut?
- Which shortcut did you like the most? Why?
- Draw a picture showing that shortcut, along with the two triangles that it works for.
3 Mystery 3: Choose the Function
Each page shows a different function family—but which one?
You can use handedness to help you decide.
When you think you know the function family, use the shortcut you invented invented for that family, either before or after actually transforming `triABC.`
- Figure out which function connects the triangles.
- Use that function to superpose an image of `triABC` on `triDEF`.
- Drag points `A, B,`and `C` to make sure the triangles remain superposed.
Draw a picture showing the function you used. Your picture should show both triangles along with all the parts of the function rule (center point, angle, mirror, vector, and/or scale factor).
Did you use a shortcut to help adjust your function rule? If so, describe your shortcut.
When you’ve done all pages...
- For one of your functions, draw a picture showing the triangles, the function rule, and the shortcut. Beneath the picture, write a description explaining how the shortcut connects to the function rule.
4 Mystery 4: Exploit Your Shortcuts
As before, decide which function family connects the triangles on each page.
But on these pages you’ll do your shortcut construction first, and use the shortcut to help you transform `triABC`.
- Figure out which function connects the triangles.
- Before you transform the triangles, construct your shortcut for the function.
- Use your shortcut construction to help you construct the transformation function.
Did you use a shortcut to help adjust your function rule? If so, describe your shortcut.
When you’ve done all pages...
- For one of your functions, draw a picture showing the triangles and the function rule, and the shortcut.
- Beneath the picture, write a description explaining how the shortcut actually helped you construct the function rule.
- Did the shortcut contruct all parts of the function rule, or only some parts of the rule? What parts (if any) did you still have to adjust?
5 Mystery 5: Match Domain and Range
- Choose a function tool and restrict `x` to `triABC` as its domain.
- Then adjust the transformation so that `triDEF` is the range.
Draw a picture showing the function you used, including any center point, angle, mirror, vector, and/or scale factor.
If you used a shortcut to make it easy to adjust your function rule, show the shortcut in your picture.
When you vary `x,` does the image point move faster, slower, or at the same speed as `x?`
When you vary `x,` does the image point with the same handedness, or opposite handedness, as `x?`
6 Accomplishments
To prepare for the summary discussion, use the prompts below to record your accomplishments.
Activity 1 (Warmup): Each page has a different function-family tool with which you superposed an image of `triABC` on `triDEF.`
- For which function families did you achieve the goal?
- Which success was hardest? How did you overcome the difficulty?
Activity 2 (Discover Shortcuts): Each page has a function-family tool and three construction tools. With the construction tools you invented an easier way to superpose an image of `triABC` on `triDEF.`
- For which functions did you invent a shortcut?
- Which shortcut did you like the most? Why?
Activity 3 (Choose the Function): Each page has five function-family tools and three construction tools. By identifying the right family and using a shortcut, you superposed an image of `triABC` on `triDEF.`
- Which family was easiest to identify? Why?
- Which family was hardest to identify? Why?
- How did your shortcut discoveries help you identify the right family?
Activity 4 (Match the Domain and Range): Each page has all eight tools, but the function tools only transform points. You identified the family, restricted `x` to `triABC,` and superposed the range on `triDEF.`
- Can you construct the same superposition with point tools as with triangle tools? Which is easier? Why?
- For which family did the variables move at different speeds?
- How is variable speed connected to whether a function is an isometry or a similarity?
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
[an error occurred while processing this directive]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.
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