In PreCalculus today we began a short unit on sequence and series. Today students played around with about twelve different sequences to discover various patterns. Next we looked at a graph of the sequence 20, 17, 14, 11...in Desmos. When I asked students what they noticed, they said it followed a line. When I asked what the slope of the line would be, they could quickly see that it was -3 and that the slope made sense relative to the pattern of subtract 3 from the previous number. However when we graphed the line y = -3x, it didn't go through the points. How could we get the correct y-intercept? Natalie said that the y-intercept would be 23, because she used the pattern backwards - adding 3 to 20. Without formally talking about arithmetic sequences and the formal notation of an arithmetic sequence, students were able to create a formula to generate the nth term.
Next students worked in groups to generate formulas for 1, 5, 25, 125... and 1, 3, 7, 15, 31,... After graphing them, the students realized that the pattern was curved and that an exponential equation would work better than a linear equation. With a little bit of trial and error in Desmos, students discovered the equation y = 5
x-1 would work for the first sequence and y = 2
x-1 would work for the second sequence. We didn't formally talk about geometric sequences today, but the fact that students are already thinking about a pattern found by multiplying by the same number again and again, means that they will be ready for the formal introduction of a common ratio next week.
Tomorrow we will begin class by looking at some patterns found at
Visual Patterns by Fawn Nguyen. Here are two that I plan to use. Can you figure out how many stars and footballs are in step 43?
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