In the ten years that Scott Steketee and I have helmed the Sine of the Times blog, we’ve featured numerous Web Sketchpad activities that introduce mathematical topics through puzzles and games. The games featured below, all designed by Scott, find their inspiration in geometric transformations. If you find them as addictive and pedagogically rich as I do, you might wonder why most curricula have been oblivious to this playful approach to transformations.
The three Reflection games below (and here) are premised on a point, a mirror line, and the reflection of the point across the mirror. For each game, one object is hidden: either the point, the mirror line, or the reflected point. The goal is to identify the location of the hidden object as accurately as possible. (Use the page controls in the lower-right corner of the websketch to move from game to game.)
Note the labeling of the points: To ease the transition to algebraic functions, we label the preimage point as x. It behaves like an independent variable since it can be dragged anywhere. (In the actual games, we don’t allow you to vary x because that would enable “cheats” that make it trivial to solve the problem.) The image point—the dependent variable—is labeled rj(x), using meaningful function notation as an abbreviation to indicate that it is “the reflection in mirror j of point x.“
In each game, there are a progression of levels that adjust the difficulty and scaffolding provided to the player. In the first game, for example, the player eyeballs the location of point x’s reflection across a mirror line and drags circle y to that spot. Provided that the reflected point lies anywhere within the circle, the player’s estimate is considered a success. As the levels progress, circle y gets smaller, requiring the player to make better estimates. (Press Reset to switch from one level to the next.)
Some assistance arrives in level 4 to aid with these estimates. Circle y is now the endpoint of a draggable segment that includes a perpendicular bisector. Even if players begin with no familiarity of the mathematical properties of reflection, they soon discover the value of the perpendicular bisector in pinpointing the location of rj(x). Scaffolding of this sort appears throughout all the transformation games as way to position mathematical tools as an aid in achieving a higher score. In each instance, the tools appear with no explanation of how to use to them. It is left to the players to discover how the tools operate.
There is no set number of problems per round. A teacher has the flexibility to say, “To be a reflection apprentice, you must score 8 of 10 at Level 1; to be a reflection master, you must score 7 of 10 at Level 3; and to be a reflection superhero you must score 16 of 20 at Level 5.”
The five rotation games below (and here) are premised on a point, a center, an angle of rotation, and the rotated image of the point. For each game, three of the four pieces of information are provided, and the player must determine the location (or numerical value) of the missing piece.
In each game, the independent variable is x and the dependent variable is RC,θ(x) to indicate that this variable is “the rotation, about C by angle θ, of x.“
The five dilation games below (and here) are premised on a point, a center of dilation, a scale factor, and the dilated image of the point. For each game, three of the four pieces of information are provided, and the player must determine the location (or numerical value) of the missing piece.
In each game, the independent variable is x and the dependent variable is DC,s(x) to indicate that this variable is “the dilation, about C by scale factor s, of x.” In the text at the top of the game pages, the full name is used, though the point itself is labeled in shorthand: D(x) for “the dilation of x.“
Transformation Family Lessons
Each of these games was designed as part of a coherent series of activities in which students are introduced to the specific transformation family, construct it for themselves, analyze its behavior, and play the games above. Here are links to the games in their “native habitats:”
Reflection Family: https://geometricfunctions.org/fc/unit1/reflect-family/#6
Rotate Family: https://geometricfunctions.org/fc/unit1/rotate-family/#4
Dilate Family: https://geometricfunctions.org/fc/unit1/dilate-family/#4
(The “#6” or “#4” on each link opens the family’s web page to the particular section of the page that has the game. Omit them to open the page normally.)
While testing these games,, I found myself with a Wordle-like obsession to keep playing. My ability to estimate the hidden locations of reflected, rotated, and dilated points and their preimages was not always spot-on, but my visualization skills improved as I continued to play. Contrast this with the sanitized presentation of transformations where a student might be asked to reflect a point at (-2, 5) across the y-axis and name its location. So neat, so tidy,…and so boring. Students miss so much when given “nice” problems where the the heavy lifting is done for them!