Illuminations: Algebraic Transformations

# Algebraic Transformations

This applet allows you to transform a plus sign in eight different ways: rotate it ¼, ½, ¾, or a full turn, or flip it vertically, horizontally, or diagonally.

### Instructions

 In free mode, you can investigate the results each time you perform a transformation. Click on any of the eight transformation buttons to rotate the shape. Which letter appears at the top after each transformation? The Reset button will reset the figure so that the A returns to the top. In double transformation mode, you can investigate what happens to the shape when two movements are performed one after the other. Click on any transformation along the left side of the table, and then click on any transformation along the top. After both transformations, which letter appears at the top? The Clear button will remove all results from the table. The Show All button will not be active until results have been deteremined for at least 5 squares. After that, the button can be pressed to reveal the results for the entire chart. (But it is recommended that you attempt to complete the chart on your own and identify any patterns before using this button.)

### Exploration

 Is the act of double tranformations commutative? That is, does the order in which you perform two actions influence the result? Perform a ¾ turn, followed by a ½ turn. Which letter appears at the top? Then, reverse the order of the moves. That is, perform a ½ turn, followed by a ¾ turn. Now which letter appears at the top? Is it the same letter? Now, try this. Perform a vertical reflection, followed by a diagonal reflection. Which letter appears at the top? Then, reverse the order of the moves. That is, perform a diagonal reflection, followed by a vertical reflection. Now which letter appears at the top? Is it the same letter? Investigate other double transformations. When the order of two transformations is reversed, is the result the same or different?

### Lessons

 More and Better Mathematics for All Students
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