For each problem, start with a square sheet of paper and make folds in the sheet of paper to construct a new shape, then explain how you know the shape you constructed has the specified area.

- Construct a
**square**with exactly**¼ the area**of the original square.

Explain how you know it has ¼ the area: - Construct a
**triangle**with exactly**¼ the area**of the original square.

Explain how you know it has ¼ the area: - Construct
**another triangle**, also with**¼ the area**, which is**not congruent**to the first one you constructed.

Explain how you know it has ¼ the area: - Construct a
**square**with exactly**½ the area**of the original square.

Explain how you know it has ½ the area: - Construct
**another square**, also with**½ the area**, which is oriented differently than the one you constructed in #4.

Explain how you know it has ½ the area:

### Convincing Mathematical Explanations

Convincing mathematical explanations stand up to any challenge and can convince others of a mathematical result. Two conditions must be satisfied:

**Convincing explanations use facts, not opinions, to support claims. **

The figure below is a square.

[[{"fid":"5","view_mode":"full","type":"media","attributes":{"height":220,"width":206,"alt":"Square with corners marked A,B,C,D","title":"Square","class":"media-element file-full"}}]]

What is the measure of ∠ACD?

*Opinion*: ∠ACD is 90 degrees because it looks like an L.*Fact*: ∠ACD is 90 degrees because it is one of the 4 angles in a square and all 4 angles of a square are 90 degrees.

**Convincing explanations are complete and don’t leave any gaps or holes. **

The figure below is a square.

[[{"fid":"6","view_mode":"full","type":"media","attributes":{"height":220,"width":206,"alt":"isosceles right triangle ","title":"isosceles right triangle ","class":"media-element file-full"}}]]

What is the name of the figure in bold?

*Explanation with gaps*: This figure is an isosceles right triangle. I know this because two of the triangle’s sides are sides of the square, so they must be the exact same length.

*Complete explanation:* This figure is an isosceles right triangle. I know this because two of the triangle’s sides are sides of the square, so they must be the exact same length. And ∠ACD is a right angle because it is one of the 4 angles of the square, and all angles in a square are 90 degrees.

### An Introduction to Patty Paper

Patty paper can be used as a construction tool in geometry. You can use it to construct a variety of geometric figures. If you’re not familiar with patty paper, work through the following simple exercises and you will start to see how patty paper can support your geometric thinking.

- When patty paper is folded a crease, or line, appears. Try folding a piece of patty paper to make a line.
- Patty paper can be used to construct a line connecting two points. Draw 2 points anywhere on a sheet of patty paper. Now fold your paper so that the crease, or line, passes through both points.

[[{"fid":"8","view_mode":"full","type":"media","attributes":{"height":213,"width":220,"class":"media-element file-full"}}]] - You can place one point on top of another using patty paper. Draw 2 points anywhere on a sheet of patty paper and try it.

[[{"fid":"7","view_mode":"full","type":"media","attributes":{"height":107,"width":220,"style":"width: 220px; height: 107px;","alt":"Patty paper with fold","title":"Patty paper with one fold","class":"media-element file-full"}}]] - Like the points, you can place one line on top of another with patty paper. Draw 2 line segments on a sheet of patty paper and try it. Once you have folded so that one segment is placed on the other, you can check to see if two segments are congruent. Are your two segments congruent? How do you know?

[[{"fid":"9","view_mode":"full","type":"media","attributes":{"height":107,"width":220,"alt":"Patty paper with two lines","title":"Patty paper with two lines","class":"media-element file-full"}}]] - You can create congruent angles with patty paper. Draw an angle on a piece of patty paper. Now place a second piece of patty paper over your first and trace your angle onto this second sheet.
[[{"fid":"10","view_mode":"full","type":"media","attributes":{"height":220,"width":213,"alt":"Patty paper with angle ","title":"Patty paper with angle ","class":"media-element file-full"}}]]

[1] _{This is an adaptation of the Introduction to Patty Paper Geometry activity that originally appeared in Michael Serra’s Patty Paper Geometry. Emeryville, CA: Key Curriculum Press.}