Problem Solving: Folding

Folding

Fold a strip of paper. Some of the creases point vertex up, some down. Predict the sequence of ups and down for any numbers of times you carry out the folding operation.

Plan: We decided that one of us would fold the paper until we could not clearly see the pattern and the other would record, then predict the sequence of the pattern based on the patterns we have.

The Pattern we have from the first fold to the fifth.

 U

UUD

UUDUUDD

UUDUUDDUUUDDUDD

UUDUUDDUUUDDUDDUUUDUUDDDUUDDUDD

At first, we thought about creating an equation for the pattern, but we did not succeed on that. Then we thought since we folded the paper in half each time, it might have something to do with symmetric. So we started to stare at the pattern we have. As I compared the first letter to the last one, the second to the second last, and so on, I found out the pattern!

We divided the pattern into the first half, the middle and the second half.

U

UUD

UUDUUDD

UUDUUDDUUUDDUDD

UUDUUDDUUUDDUDDUUUDUUDDDUUDDUDD

The first pattern depends on the way you hold the paper, and in this exercise the first pattern would be U since we folded the right end first then unfolded the entire strip.

For the first half, just simply copy the whole previous pattern.

The middle part is always the same as the first pattern.

For the second half, reverse of the first half and change each letter to the other.

Anther way:

U

UUD

UUDUUDD

UUDUUDDUUUDDUDD

UUDUUDDUUUDDUDDUUUDUUDDDUUDDUDD

The middle part is always the same.

The letters in red are symmetrical. They copy the letter in red in the previous pattern as the first half of the pattern (only for red letter), and the second half is the reverse of the first half and it changes each letter to the other (only for red letter).

The rest in black are all in the pattern of UD.