Speedy Squares starts off by having you build a 2x2 square, 3x3 square, 4x4 square, etc. to help determine how long it would take to build a 26x26 square (see here for why 26x26). The participants in my class used linking cubes and a timer (on their phone for the most part) to collect data which they then graphed and modeled. They did a fantastic job with their data collection and worked together really well. Many had never worked with linking cubes before. As an aside, I showed them that if you took each square and made it into a tower, the heights showed the quadratic function (area vs. side length). They used their model to answer the original question. Jon Orr made a video answer which you can see here. I didn't actually show the video as I was concerned about time (so much fun stuff to do!) but I really should have. That was part I. Part II uses the data from part I to determine the relationship between the number of blocks and the building time which is then used to figure out how long it would take to build a house of your own design using Lego. That's the general idea, but there are more specific blog posts about this activity here and here.
Next up: Visual Patterns. I started by showing them Fawn's site: visualpatterns.org. I love this site so much. The (optional) homework I gave yesterday was a set of visual patterns.
Each of these patterns is linear and we talked about how we saw each one growing. It was really useful to have the linking cubes when we talked about the surface area of the second pattern. The third pattern is my favourite because there are so many ways of seeing it and you can show that they are equivalent using algebra. If you want to know more, I wrote about it here.
Then I gave them the second set of visual patterns:
It turns out (not by accident) that each of these patterns is quadratic. My big message was that I wanted them to see the pattern in the pictures, not go straight to the numbers. The first pattern has a square in the middle that has side length equal to the step number, so the number of tiles in the middle can be represented by n^2. Each also has 4 extra tiles on the corners so the rule here is n^2 + 4. We found two ways of finding the rule for the number of football helmet in the second pattern. There is one way shown here. See if you can find another. Although some participants did not want to stop working on the patterns, I decided we should move on and look at the other three patterns tomorrow.
I introduced 3-act math tasks with basketball shots from Andrew Stadel found on this page (he is also the creator of Estimation 180). I mentioned that Dan Meyer has many 3-act tasks and that Kyle Pearce has also curated a great collection.
We were running out of time, but I quickly described speed dating. This is a fun way of having students practice a skill that may not be that exciting. I chose factoring trinomials. I have blogged about it here.
I then quickly referenced Michael Fenton's fun Desmos activity: Match My Parabola to end today's class.
I also went to a couple of CWiC sessions today which were great. Philip Mallinson's talk, Solving Quadratic Equations with Origami, was about how to find the roots of a polynomial geometrically. It was great. I'm sure I have seen him give this talk before, but it still made me think and was brilliant.
Julie Graves did a talk entitled Quadratic Models without Quadratic Regression. It was really, really good. We saw how, given a partial data set, we could determine where the vertex of the parabola was and come up with an equation to model the data using only linear regressions. This was repeated for an exponential decay function. I generally love all sessions given by the North Carolina School of Science and Math teachers - they always make me think and I walk away with new tools and ideas.