We began today with a funny video on how mot to use questioning in the classroom. It elicited a lot of good discussion about funneling versus focusing and how we use that in our classroom. We also talked about Standard 4 and looked at an example video.
Charles began with the material on functions. We spent a lot of time understanding what makes something a function. We learned the difference between range and co-domain. We connected the idea of function to the previous concept of process and then learned function notation. There was a lot of good discussion about what makes a function one-to-one, onto and finally what one-to-one correspondence is. Before lunch we began with the idea of functions relating to counting but got a bit hung up on the idea of what the domain and co-domain were and whether the domain changed when we skipped a stone when counting.
After lunch Marina began a discussion about measurement relating to functions but many found the idea a little too abstract for them to understand. When we started the pattern problems many participants realized they related to last night’s homework. The discussions were great and the presentations showed many ways to think about the pattern. We revisited the idea of (n/2)(n+1) from the Days of Christmas problem and reinforced the reason that the formula makes sense.
The perimeter problem brought three very different and powerful approaches to y=4n and reinforced the idea that it could be repeated addition of n four times or repeated addition of 4 n times. There was also a very interesting question about why the candle burning problem was written as y = 12 -2x instead of in slope intercept-form of y = -2x + 12. We ended with a good discussion of how to think about the f(x) notation.
We ended with a recap of our thoughts about questioning listed below:
- A focusing question doesn’t lead to a one word answer.
- A focusing question doesn’t just relate to this particular problem.
- Ask a student to rephrase their thinking because that’s the basis for our questioning.
- Let the students defend their positions (Socratic method)– issues and errors may just fall out.
- Keep in mind the offense and defense model of adjusting on the fly.
- It is important to know where the kids are going mathematically beyond what we are teaching them so we can question effectively.
- It is also important to have thought about questions ahead of time so that we don’t have to come up with them on the spot. Keeping notes from year to year is helpful.
We also talked about the think-pair-share idea: How do we deal with the fact that we can’t attend to every group at the same time, ways to ensure students remain on task and ways to reassemble them after. We watched a video of a first grade class and Standard #5 – we saw that the paper plate activity modeled the idea of algebra but discussed other ways the teacher could have set it up and had the students more involved. We ended with a gallery walk putting definitions, floating capacities, pictures, questions, etc. on the words – slope, y-intercept, x-intercept, proportion, and linear relationship.
By request I am listing some of the ideas here but please continue the gallery walk tomorrow morning in case things were added after you saw it.
slope: y = kx,(underline k) y = mx+b (underline m), rise/run, change in y/change in x, + or -, horizontal line has zero slope, constant rate of change, think of a straight line, daredevil compare the steepness of a hill for sledding or comparing ski slopes with diamonds.
Linear relationship: predict where a point will be on a graph, constant rate, graphs like a straight line, one-to-one, not quadratic, + or – slope, comparing fees for phone companies, y = mx+b, each x has only one y, pairs and how they relate to form a line.
x-intercept: where the line crosses the x axis, picture drawn, phone bill minimum with no extra minutes, reserve a kayak(0 hours rented), ?? posed – what is its relation to domain??
y-intercept: Deposit on kayak, y = mx+b (circle the b), picture drawn, f(x)-range, point where x=0, monthly charges for phone bill,
Proportion: a:b, a/b, a to b, 2 equivalent ratios, line going through the origin, “parts of”, “k value”, one ration = to another, graphing lines
Math HW – finish page 4.18
Pedagogy – read P to A pgs 59-69 and in your journal use your textbook or curriculum guide to list the floating capacity for : doing and undoing process, rate, proportional reasoning, linear relationships.