Recall in activity, Exponent 1, we investigated constructing doubles. From that investigation we produced a chart
like the one below.
Number of doubles |
Number of segments in exponential form |
Number of segments |
-2 |
|
|
-1 |
|
|
0 |
|
1 |
1 |
|
2 |
2 |
|
4 |
3 |
|
8 |
We have already discussed the different arithmetic operations that you use to move up and down the columns.
As you go down the columns which set of entries is getting larger faster?
How are you measuring the speed of increase?
What are the similarities among the entries of the doubles column and the entries of the number of segments column? What are the differences?
Construct a correspondence between the entries of the number of segments column and the doubles column using two number lines like the ones below or click HERE to construct your correspondence on a GSP sketch.
Is there a number on the upper number line that corresponds to 
doubles? Based on the correspondence you constructed what do you think
that number might be?
Where is the number 
on the upper number line?
Approximately what number does it correspond to on the lower number line?
Does every number on the upper number line correspond to some number on the lower number line? If you think so then explain why. If not, then why not? What numbers do not have a corresponding number on the lower number line?
Is there a number doubles that corresponds to 3 segments? Based on the correspondence you constructed what do you think that number might be?
Where is the number on the lower number line?
Approximately what number does it correspond to on the upper number line?
Does every number on the lower number line correspond to some number on the upper number line? If you think so then explain why. If not then why not? What numbers do not have a corresponding number on the upper number line?
Suppose now that we graph our points in the coordinate plane.
What is the geometric relationship between the points C and I? Is is different than the relationship between the points E and G or D and H?
If you need some help, click HERE to investigate the relationship with GSP.
Reflecting on patterns
Explain what you think the correspondences in this activity are? Keep in mind there are correspondences between terms in a sequence such as {..., -2, -1, 0, 1, 2, ...} as well as correspondences between sequences, and correspondences between points in the coordinate plane.
Explain what the number 2 had to do with this activity? Does it have any importance in terms of the points plotted in the coordinate plane?
If the graph of the points in the coordinate plane what x-values can
not be mapped to y-values? What y-values can not be mapped to x-values?
Can x = 7 be mapped to a y-value? Approximately which one? Can 
be mapped to a y-value? Approximately which one?
