We began our week in Algebra I with Function Families.
We eventually end up here as a wrap up. Students come to the board and share their sorts.
The following day we summarize their findings on a foldable…descriptions of the equations and graph shapes from their groups. The inside of the foldable contains an example of each type of function, table of values and a graph.
I began with quadratic because I see the most mistakes here. Students will use their calculators and jot a number down without pausing to ask if it’s reasonable. We had 10, -8 and -27 for the first table value. Hmmm? How’d they get those? I actually used an entire set of wrong calculations and graphed, then asked, Is that what you expected it to look like? No. So we need to check our work and find the mistake.
We completed the first table and they were asked to write about what they noticed in the numbers. And we shared.
Next, we looked at the first differences. They wrote about their noticing again. “Oh,” a girl says. “That let’s me see what’s happening in the graph!”
And we finished with the second differences.
I went to the absolute value next.
One student claimed, it’s doing the same thing as the first but with different numbers. Another student disagreed because the numbers were constant and not changing like the first. But the directions were the same. I explained that different operations would cause the graphs to look differently and we were creating a guide to help us sort through the patterns and learn to recognize them.
In both cases, I heard students mention reflection, symmetry, matched – up referring to numbers in table, not the graph.
We continued with linear and the exponential.
I began with 4^1 on this table and asked, can I write this 4*1 and it’s still 4? Yes. So, 4^2 would be 1*4*4 and 4^3 1*4*4*4.
Which means 4^0 would be 1* (zero 4s)…or just 1.
We had done simple function inverses prior to fall break. I had used the -1 exponent to represent inverse. So our discussion went back to 4^-1. Student ask, “well, if exponents are repeated multiplication, would an inverse exponent be dividing?” And we continue with that discussion.
We ended the day with some reflection on our learning. They were asked to tell which 2 functions were most alike and why. Which 2 functions were most different and why. Very eye opening to read some of their thoughts.
At the end of one class, a couple of students we still discussing something. He shared, “I was wondering what I’d get if I graphed y=x^-1” and he showed me the graph. Why does it graph that way he asked. Why does it graph that, I asked him back.
His group mate shared, well, I graphed y=x^-2 and instead of reflecting into the 3rd quadrant, it’s like it reflected across the y-axis. Why did it do that? I replied, why do you think it did that?
I told them both, that was my goal…to let them start asking their own questions…and to keep pondering their graphs, we would talk more about them next week. It was a good way to end the week.