Each page of this activity has an independent variable that can move along a restricted domain (a triangle). You’ll transform the variable, using a different function on each page, and investigate whether the dependent variable has the same handedness (both clockwise or both counter-clockwise) or a different orientation compared to the independent variable.
(The mathematical term handedness distinguishes objects like your hands or your feet, which are opposite from each other even though they have the same shape. You distinguish your hands by calling them right and left. We will call `triABC` clockwise or counter-clockwise based on the way `x` travels as it varies from `A` to `B` to `C`.)
To determine the handedness of `triABC,` vary `x` and observe its motion as it goes from `A` to `B` to `C.`
Does it travel clockwise or counter-clockwise on its domain? This is its handedness.
Reflect `x` and its domain (`triABC`). When you vary `x`, does `r(x)` move clockwise or counter-clockwise?
Is its handedness the same as `x,` or different from `x?`
Drag `A, B,` and `C` to experiment with different shapes.
Find a shape that makes `x` travel with the opposite handedness as before.
What effect does this have on the handedness of the movement of `r(x)?`
What is the handedness of `triDEF?`
Rotate `triDEF.` What is the handedness of `R(triDEF)?`
Do `triDEF` and `R(triDEF)` have the same handedness or opposite handedness?
Drag `D, E,` and `F` to change the shape of `triDEF.` Try different angles and center locations.
How did changing the shape affect the handedness of `triDEF` and `R(triDEF)?`
What is the handedness of `triABC?`
Glide reflect `triABC.` What is the handedness of `G(triABC)?`
Do `triABC` and `G(triABC)` have the same handedness or opposite handedness?
Drag `A, B,` and `C` to investigate different shapes. Also try different vectors.
How did your changes affect the handedness of `triABC` and `G(triABC)?`
What is the handedness of `triABC`?
Translate `triABC`. What is the handedness of `T(triABC)`?
Do `triABC` and `T(triABC)` have the same handedness or opposite handedness?
Drag `A, B,` and `C,` and change the translation vector.
What can you say about the effect of translation on handedness?
What is the handedness of `triPQR?`
Dilate `triPQR.` What is the handedness of `D(triPQR)?`
Do `triPQR` and `D(triPQR)` have the same handedness or opposite handedness?
Experiment with different triangle shapes.
Experiment with different locations of center `C` and different values of scale factor `s` (including negative values).
Describe your observations and conclusions.
2 Explore Multiple Reflections
On the pages of this activity you’ll reflect the independent variable (along with its domain) two or three times. In each case you’ll see if you can find a function from a different transformation family that produces the exact same result as the combination of reflect functions you used.
3 Construct Multiple Reflections
On each page of this activity you’ll try to construct multiple reflect functions to produce the exact same result as other transformations. The functions you'll try to construct this way include a rotate function (page 2), a translate function (page 3), a glide reflection (page 4), and a dilate function (page 5). (Hint: all of these except one are possible.)
The Fine Print