Activity
Clare takes a piece of paper, cuts it in half, and then stacks the pieces. She takes the stack of two pieces and cuts it in half again to form four pieces, stacking them. She keeps repeating the process.
1. The original piece of paper has length 8 inches and width 10 inches. Complete the table.
number of cuts | number of pieces | area in square inches of each piece |
0 | ||
1 | ||
2 | ||
3 | ||
4 | ||
5 |
number of cuts | number of pieces | area in square inches of each piece |
0 | 1 | 80 |
1 | 2 | 40 |
2 | 4 | 20 |
3 | 8 | 10 |
4 | 16 | 5 |
5 | 32 | or 2.5 |
2. Describe in words how you can use the results after 5 cuts to find the results after 6 cuts.
Compare your answer:
Double the number of pieces, from 32 to 64. Halve the area of each piece, from to .
3.
a. Use the graphing tool above or technology outside the course, on the given axes, sketch a graph of the number of pieces as a function of the number of cuts.
b. How can you see on the graph how the number of pieces is changing with each cut?
Compare your answer:
The height of each plotted point is twice the height of the previous plotted point.
4.
a. Use the graphing tool above or technology outside the course, on the given axes, sketch a graph of the area of each piece as a function of the number of cuts.
b. How can you see how the area of each piece is changing with each cut?
Compare your answer:
The height of each plotted point is half the height of the previous plotted point.
Are you ready for more?
Extending Your Thinking
1. Clare has a piece of paper that is 8 inches by 10 inches.
a. How many pieces of paper will Clare have if she cuts the paper in half times?
Compare your answer:
pieces
1.
b. What will the area of each piece be?
Compare your answer:
Each piece has an area of square inches.
2. Why is the product of the number of pieces and the area of each piece always the same? Explain how you know.
Compare your answer:
It’s always the same because adding up the area of the little pieces of paper gives the area of the original sheet of paper.
Self Check
Additional Resources
Graphs of Sequences
Graph the sequence {18, 36, 72, 144, 288, . . .}.
First, make a table:
Term | 1 | 2 | 3 | 4 | 5 |
Value | 18 | 36 | 72 | 144 | 288 |
Now, plot the points:
Notice that each height is double the previous height.
Each term is also double the previous term.
Try it
Try It: Graphs of Sequences
Graph the following sequence and explain the pattern.
1, 3, 9, 27, . . .
Here is how to determine the pattern of a sequence with a graph:
First, make a table:
Term | 1 | 2 | 3 | 4 |
Value | 1 | 3 | 9 | 127 |
Next, plot the points:
Each point is 3 times the height of the previous point.
Notice that in the sequence, each term is 3 times the previous term.