Remember to use the Fill Down command after changing the output rules. The graphs and tables will change automatically with any changes you make to the columns.
Groups: After completing Problems A1-A7, discuss your responses to Problem A7.
The tendency may be to answer Problem A7 with particular numbers. For example, the first table will never contain the output 2, because the table starts at 3 and then increases. Think also about what types of numbers will never appear. For example, unless you use negative inputs, the first table will never contain negative numbers, numbers less than 3, fractions, or numbers that aren't multiples of 10 (except 3). Justify your claims. This will push you to think more deeply about exponential functions.
Groups: Before moving on, take a few minutes to read through the chart or put up an overhead of the chart. Take a moment to work a bit with the notation. For example, if you have a rule that is y = 10x, what would you get for x = 2? For x = 4?
Groups: You also may want to discuss zero as an exponent. Note that this is completely optional; knowledge of zero as an exponent is not assumed anywhere during the session. There are a couple of ways to explain the fact that any number (except zero) to the zero power is 1. First, discuss what 20 would mean and why. Then think about the following two explanations:
Multiplying by 1 doesn't change anything, so you can think of powers of 2 (for example) as 1 times some number of 2s.
23 = 1 x 2 x 2 x 2 (1 times three 2s)
22 = 1 x 2 x 2 (1 times two 2s)
21 = 1 x 2 (1 times one 2)
20 = 1 (1 times zero 2s)
Alternately, look at decreasing powers of 2:
23 = 8
22 = 4
21 = 2
Each power is half the previous one, so if the pattern is to continue, it must be the case that 20 = (1/2) x 2 = 1. Extending this pattern can help you find the meaning of negative exponents, as well. It should be the case that 2-1 = (1/2) x 1 = 1/2, and in fact this is what negative exponents mean.
Think about why a closed-form rule like y = 3x would give rise to a table with a constant ratio between successive outputs. A possible explanation is that if the input increases by 1, then the product is multiplied by another 3, so each output will be 3 times the previous one.
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