Quadratic Visual Patterns

You all know how much I love Fawn Nguyen's Visual Patterns site. I use them a LOT. They have been part of my warm-ups for years now and have been some of the best moments of my class each week. I have been recreating my warm-ups for my grade 10 applied class this semester (no, I can't leave things alone). I decided to do this so they align more with the curriculum expectations we are working on or provide lagged practice for other expectations. The warm-ups have included quadratic visual patterns for a few weeks now and I decided to step it up a little this past week with with a couple of patterns from Michael Fenton. If you haven't tried these ones before, I encourage you to do so before you scroll down.

We didn't actually work with the colour-coding, instead looked at the squares that overlapped by 1 each time. We worked with the number of circles first, established that this is a quadratic relationship and then found the rule by comparing the "side length" of each square to the step number.

I really also wanted to look at this pattern using the colours as a guide so we started over and found that we ended up with the same simplified rule.

I am totally impressed that some of my students can do these as they are not easy, especially for students who have struggled a lot with math and have trouble making connections. They have shown incredible progress and I love how willing they are to try.

Here is the next one we did:

 There is a lot going on with this pattern, but the colours really help show the squares emerging.

I should note that these "warm-ups" took about 45 minutes to work through. It was definitely time well spent. 

MPM2D - Day 5: Finishing Quadratic Visual Patterns & Multiplying Binomials

We started the week by looking at the remaining visual patterns, starting with this one:

Here is the next one:

I have to say that my students are rocking this! They really seem to understand how to relate the length and width of a rectangle back to the step number. We looked at different ways of seeing the shape grow for pattern #10:

And this is how we came up with the rule:

Then it was time for pattern #11:

I told my class that I had been unable to figure out a rule for this one algebraically so I had reached out to my math friends and Dave Lanovaz showed me a brilliant way of doing it. Here is the way I showed my class:

The original pattern was in blue. I repeated the pattern in yellow, then put them together thus creating rectangles! <insert angels singing> I asked if they could find the rule for this pattern and they all said yes. When I asked what they had to do to get the rule for the original pattern, they said we'd have to divide by 2. 

Here is what we did:

With that, our visual patterns came to an end. At least for now.

I decided to spend the rest of the class multiplying binomials in a somewhat more formal way than what we had done last week. We started by multiplying numbers, beginning with 23 x 51. It was eye-opening to watch them attempt this without a calculator. So many place value issues and *magical* answers with no work and no ability to explain what had been done. Then I showed them how to break up the numbers like this:

One student actually said that this was like a miracle! She has been relying on a calculator for as long as she has been allowed to and I don't think she understood how to multiply properly until today. At the end of class she even said that this was fun. (O.M.G. day made!)

I did explain that the area models should be representative of the size of each number, like this (which I stole from Tina Dittrich):

We went on to repeat the process with variables in the mix:

Then we went over the distributive property again, which I also demonstrated with a trinomial multiplied by some crazy quartic function. The point being that it works regardless of how many terms are in each set of parentheses.

My students now all have a tool they can use to successfully multiply binomials. I then put up four more examples and wrote a student's name next to each. I explained to them that whenever they go to put work up on the board, they can choose to insert mistakes along the way. So we will never know if someone actually made a mistake or if they planted one for the class to find. My goal in doing this is to ensure that no one ever feels dumb for making a mistake in front of the class. They will (hopefully) come to appreciate all the mistakes and learn from them so they won't make them come test day.

MFM2P - Day 78: Quadratic Visual Pattern & Linear Systems

I have been reading 5 Practices for Orchestrating Productive Mathematics Discussions and participating the in the book chat moderated by Jeff Lay of #OKMath (Tuesdays at 9 pm EDT). We are about half way through the book and there are already so many good strategies around giving a task. The huge amount of preparation suggested in the book pays dividends in terms of knowing what to say to students who are on the wrong track and guiding discussions. I planned on trying to incorporate some of the ideas in my MFM2P class today, but changed plans half way through class. I hope to share that with you tomorrow. Instead, rather ironically, I was under prepared when it came to today's visual pattern. I had found one way of expressing the pattern, but did not give myself time to look for more ways (which is precisely what I should be doing).

Here is today's visual pattern:

The version the students were working with was in grayscale so the patterns formed by the different coloured circles were not evident. They recognized that this was a quadratic pattern, but when I asked how they knew many said that it was because the number of circles was not increasing by the same amount. So I threw some numbers (something like (1,5), (2,15), (3,23), (4,63)) on the board and asked if they represented a quadratic pattern and again, got lots of "Yes, they don't go up by the same amount.". Yikes! I pushed further and asked them to tell me what they knew about this "pattern". They found the first differences and were even more convinced. Someone said something about second differences so we found those. Someone said that the second differences needed to not be constant and I decided to just correct that misconception. We could have looked at other known quadratic patterns and verified one way or the other, but I wanted to keep moving forward. We looked at the number of circles in each step, adding the 4th step to get more data, and showed that the 2nd differences were, in fact, constant.

Then we looked at different ways of seeing the pattern growing and found a rule.

This all took quite a bit of time which is why I decided to keep the ropes for tomorrow. I explained the goals for the ropes. Firstly, to find a relationship between thickness and diameter so that we could extrapolate to find the length of a thicker rope needed for a given number of knots. Secondly, given two different ropes, determine how to make them the same length with the same number of knots. The latter will require students to solve a system of linear equations which we have not done for a little while. So the rest of today's class was spent practicing solving systems with this handout that my colleague put together (thanks, Michelle!). Many students could not remember how to solve these, but with a few questions from me (What's the same in the two orders? What's different? What does that mean?) they did quite well.

MFM2P - Day 69: Quadratic Visual Pattern & Trig

As we missed doing the visual pattern warm-up yesterday, we did it today. This is what they started with:

Most showed me the pattern growing along the diagonal but were able to see it growing in other ways. They figured out that you needed to add 3 blocks, then 4 blocks, then 5 blocks... I asked if this was a linear pattern and some said yes while others said no. I asked what a graph of the number of the blocks vs. step number would look like. 

After they thought about that for a while I pulled up Desmos and plotted the points.

They could see that the points did not form a line and someone said that this was quadratic (yay!). I asked how we have found the equation to represent a quadratic pattern so far, and they said that they have used graphing calculators. I then told them that we could figure out the rule without using graphing calculators. We talked a little about how rearranging blocks to form a square plus some blocks can be helpful with quadratic patterns, but that it didn't work here. Admittedly, I was stuck on this yesterday, but thanks to Dave Lanovaz, I was able to share a fantastic strategy with my class today. He suggested doubling the pattern like this:

We now have rectangles whose side lengths we can relate to the step number:

Now that we have a rule for this pattern, we can divide by 2 to get the rule for the original pattern:

I love this! I love that I learned a new way of finding a rule for a quadratic pattern and that it is accessible to all of my students.

We spent the rest of today's class working on the trig matching activity. There was an error in my original fine - the corrected version is here. They are very slowly working through these problems and will continue tomorrow.