DEVILISH DIVISIONS
aHa Gotcha Ha Insight Page 47
Blake has a problem besides being in a class with eight women. He has become a surveyor and specializes in dividing curious lots into congruent parts. He was asked to take a lot shaped like an L and cut it into four congruent parts. How did he do this? Here is a couple for you to try on. There are extra papers with the shapes if you need them.
HINT: There is only one way this can be done.
If you have time, can you divide it into any other number of congruent shapes?
Blake’s next job is to cut this piece of land and cut it into four congruent parts. How did he do this? Here are a couple shapes for you to try on.
Like before, can you divide this trapezoid into any other number of congruent shapes?
Divide this next shape into four congruent parts.
Not too tough was it?
But, how can you divide the same shape into five congruent parts? Here’s a couple to try with.
Are you struggling? There is only one way this can be done.
Think obvious. Do not make it too hard. How many other number of congruent shapes can this square be divided into?
Now, find the paper with the two identical shapes on it. Figures 11 and 12 on page 49).Take the top shape and divide it into four congruent shapes.
Now, can you divide it into five congruent shapes?
When you cut a shape into pieces which are the same as the original, we call there REP-TILES. (cute, huh?)
Find the paper with the three unusual shapes. (Figures 13 on page 49.) These are also rep-tiles. Can you cut them into congruent shapes that replicate the original rep-tile?
How many congruent shapes can be made for each?
What happens if you have an infinite number of rep-tiles and start connecting them?
What about cutting rep-tiles into smaller rep-tiles?
What we know about rep-tiles is they tile a plane by translation, without rotation or reflection.
Therefore, don’t we have a _tesselation______?
Now for some fun. Find figure 14 – the T (see page 49). Is it a rep-tile?(No it is not). Can you cut the T into smaller congruent Ts?
Can you divide it into four smaller shapes?(There are no answers for these. Maybe someone will find one and we can still share.)
Now, look at the three figures labeled 15 (page 49). These are not rep-tiles, but have fun dividing them into at least two congruent pieces. Can you divide them into more than two congruent pieces?(The answers are on page 177)