 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum  MENU        A B C D  Notes for Session 7, Part A Note 2

Groups: Work in pairs at the computer.

Start your spreadsheet program and create a worksheet. Remember the following tips:

 • Click on a cell to see the function it contains displayed in the edit line above the worksheet. • To edit the entry in a cell, click on the cell, highlight any characters that were already there, and type over them. • To "fill down" (when you have a function used throughout a column of cells), highlight the cell you want to use as a starting point, then drag down to the cell you want to use as your ending point. Choose the Fill Down command to fill down. • To create the graph of a function, highlight the input and output cells and click on the "chart" button. Go through the menus, using the first column as the inputs.

For the functions in this session, you may choose to connect the points with line segments or with smooth curves. Consider the benefits of each type of graph. A smooth curve might not make sense if the function is only defined by integer values, but it might help to see the graph more clearly. Note 3 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. Note 4

Create a new worksheet, and then work on Problem A8. As you make your own functions, experiment with interesting or strange cases: What if you use 1x? What about using a number close to 1? Or using very large or very small numbers? In some cases, the numbers get so big that the graphs are distorted between the input points. For example, this graph was created with the rule 100x. It does not look like a smooth, constantly increasing function. Surely it shouldn't dip below the x-axis! Groups: Discuss why this is a mistake, and why the software might do that.

Before moving on, think about exponential functions and describe two different kinds. Some exponential functions -- those with a base that is greater than 1 -- are increasing (bigger inputs always produce bigger outputs, and the graph never slopes down), and others -- those with a base less than 1 -- are decreasing (bigger inputs always produce smaller outputs). The exception is 1, which produces a constant function that graphs as a line. Using numbers close to 1 as bases produce graphs that look like lines, but actually are not. Extending the graphs to more inputs better shows the behavior of the functions.

All exponential functions (except with a base of 1) have graphs like these:     Session 7: Index | Notes | Solutions | Video

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