Milo's teacher gave him a square and asked him to cut it into pieces to create a puzzle. After experimenting with different ways to cut his square into pieces, Milo said to Sarah: "Look, Sarah, I cut the square into 8 pieces and then rearranged those pieces to form a nonsquare rectangle!"
1. Sarah offers a challenge to Milo. "Okay," she says, "you were able to reassemble your 8-piece square puzzle to fit this rectangle exactly ...
. . . but, what if I take that rectangle and scale it up so that it has 1.5 times the width and 1. 5 times the height ...
"Can you dissect this rectangle into 8 puzzle pieces and reassemble all of them to make a square?"
a. Help Milo meet Sarah's challenge by cutting this rectangle (there is a copy of the rectangle in the Shapes Template handout) into 8 pieces and rearranging the pieces to form a square. Draw the square you formed below:
b. Without measuring,
i. state the relationship between the side length of Milo's original square and the side length of the square you just created.
ii. state the relationship between the area of Milo's original square and the area of the square you just created.
2. Now Milo wants to pose a challenge to you! Your challenge is to cut the square in the Shapes Template handout into only 6 pieces that can be rearranged to
form the nonsquare rectangle below.
a. Can you combine the shapes in Milo's 8-piece puzzle to form a 6-piece puzzle out of the square in the Shapes Template handout that can be rearranged to form the nonsquare rectangle, below?
b. Can you make a different 6-piece puzzle out of the square that can be rearranged to form the nonsquare rectangle shown below?
c. Choose one of the two 6-piece puzzles you created in parts a and b and show how your square puzzle could be reassembled to make a nonsquare rectangle.
Draw your puzzle pieces in the square and nonsquare rectangles below.
d. For the same puzzle, describe in words how your square could be reassembled to make a nonsquare rectangle.
e. What do you know about each of your puzzle pieces for the puzzle you described in parts c and d? Complete the table.
f. Sarah says to you, "I know all rectangles have 4 right angles, 4 straight sides, and opposite sides that are the same length. It may look like your final rectangle
has those properties but I need you to convince me so that I know for sure." (You might want to use what you know about the properties of your puzzle piece shapes (part c) to help convince Sarah.)
i. Convince Sarah your nonsquare rectangle has 4 right angles.
ii. Convince Sarah your nonsquare rectangle has 4 straight sides.
iii. Convince Sarah your nonsquare rectangle has opposite sides that are the same length.
Extension: Use another square from the Shapes Template handout and make an 8-piece square puzzle that, like Milo's, also can be rearranged to form a nonsquare rectangle. Your puzzle should not look exactly like Milo's, though. Create your puzzle so that the pieces are symmetrical about one of the square's diagonals.
Here's an example of a square with 2 puzzle pieces symmetrical about its diagonal.
a. Draw a picture of your square, showing how you divided it into puzzle pieces that were symmetrical about one of its diagonals. Then draw a picture of how
you rearranged the puzzle pieces to form a nonsquare rectangle.
b. Name all of the angles in your pieces without measuring.
c. Convince Sarah that your final shape is a nonsquare rectangle.