Session 5:
Homework

Problem H1

You can make your own tangram set from construction paper. Start with a large square of construction paper and follow the directions below:

 Step 1: Fold the square in half along the diagonal; unfold and cut along the crease. What observations can you make about the two pieces you have? How could you prove that your observations are correct? Step 2: Take one of the triangles you have, and find the midpoint of the longest side. Connect this to the opposite vertex and cut along this segment. Step 3: Take the remaining half and lightly crease to find the midpoint of the longest side. Fold so that the vertex of the right angle touches that midpoint, and cut along the crease. Continue to make observations. Step 4: Take the trapezoid and find the midpoints of each of the parallel sides. Fold to connect these midpoints, and cut along the fold. Step 5: Fold the acute base angle of one of the trapezoids to the adjacent right base angle and cut on the crease. What shapes are formed? How do these pieces relate to the other pieces? Step 6: Fold the right base of the other trapezoid to the opposite obtuse angle. Cut on the crease. You should now have seven tangram pieces. Are there any other observations you can make?

 Problem H2 When people work on the dissection problems, they often create a figure that looks like a rectangle, but they can't explain why the process works. Or sometimes they perform a cutting process that they think should work, but the result doesn't look quite right. Reasoning about the geometry of the process allows you to be sure. A cutting process is outlined below. Your job is to analyze if the cuts really work. Does this algorithm turn any parallelogram into a rectangle? If so, provide the justifications. If not, explain what goes wrong. First, cut out the parallelogram. Then fold along both diagonals. Cut along the folds, creating four triangles as shown. Slide the bottom triangle (number 4) straight up, aligning its bottom edge with the top edge of triangle 2. Slide the left triangle (number 1) to the right, aligning its left edge with the right edge of triangle 3.

 Try this with a very long and skinny parallelogram.   Close Tip Try this with a very long and skinny parallelogram.

 Here are some new cutting problems. When you find a process that you think works, justify the steps to be sure.

Problem H3

Start with a scalene triangle. Find a way to dissect it into pieces that you can rearrange to form a new triangle, but with three different angles. That is, no angles of the new triangle have the same measure as any angle of the original triangle. You should be able to demonstrate the following:

 a. The final figure is a triangle. (It has exactly three sides.) b. The two joined edges match. c. All three angles are different from those of the original triangle. d. As a challenge, can you solve this with just one cut?

 Form two angles that are smaller than the smallest one in the original triangle, and a third angle that is larger than the largest angle in the original.   Close Tip Form two angles that are smaller than the smallest one in the original triangle, and a third angle that is larger than the largest angle in the original.

Problem H4

Start with a scalene triangle. Find a way to dissect it into pieces that you can rearrange to form a new triangle, but with three different sides. That is, no sides of the new triangle have the same length as any side of the original triangle. You should be able to demonstrate the following:

 a. The final figure is a triangle. (It has exactly three sides.) b. The two joined edges match. c. All three sides are different from those of the original triangle.

 Problem H5 Start with an arbitrary quadrilateral. Find a way to dissect it into pieces that you can rearrange to form a rectangle. Test your method on quadrilaterals like these. Try to justify why it will always work.

Remember from Session 3 that you can divide any quadrilateral into two triangles.   Close Tip