MPM2D - Day 33: A Little Algebra with Quadratics

As my students get more comfortable working with quadratics, some have figured out how to calculate the 'a' value when determining the equation given a graph. I decided that it was time for everyone to see how to do this. My plan was to work through four examples together, which you can find here. I would like to say that I thought about where these would take us, but I didn't. I love how things turned out and need to make more time in my planning to ensure that I don't miss going down a path when it presents itself.

We started by writing down the three forms of a quadratic equation on the whiteboard (I wrote, but they told me what to write), and needed to clarify that the vertex was at (h,k). Then we worked on the first example, beginning with "Which form of the equation should you choose?":

This one was next:

At this point they had a fairly solid understanding of how to find the equation and they impressed me when they worked on example 3 - many students didn't need any prompting to find the x-value of the y-intercept, which is traditionally a common issue.

I asked them the do example 4 and, if they finished quickly, to do it a different way.

And then it just seemed natural to ask them to...

I love being able to connect to work that we have already done like this. I also know from checking homework that many students have had trouble correctly expanding and simplifying from vertex form. This gave me an opportunity to see what errors they were making and they all were able to experience success with this type of question.

I then asked what they knew about the graph from the equation in standard form. I wrote down all their responses without comment and then we went to Desmos to see the graph and adjust the answers.

I really like how it all came together. I am trying to ensure that my students don't simply learn a collection of skills, but rather that they understand how they fit together. I think that several pieces of the (quadratics) puzzle clicked today.

Here is today's homework set.

Quadratic Transformations

I thought I would write up a quick post to share the Desmos activities that I made over the past couple of days. I will be using them in my grade 10 academic class in a few weeks, when we get to quadratic transformations.

Creating an activity using Desmos Activity Builder is pretty straightforward and produces good results. I love that students can work through an activity at their own pace and that both students and teachers can get feedback throughout the activity. If you are interested in making your own, go to teacher.desmos.com, scroll to the bottom of the page and look for "Create with Activity Builder". There are great instructions once you have entered the title of your activity.

So - here are links to my quadratic transformation activities. Part I deals with horizontal and vertical translations and Part II works through reflections about the x-axis and vertical scaling before putting all the transformations together.

As always, I welcome feedback! If you are looking for more Desmos activities, check out the searchable Desmos Activity Bank. And if you create your own activities, be sure to submit them to the bank.

MPM2D - Day 4: Visual Patterns...going quadratic!

Today was awesome! My students were talking about math and wanted to share the patterns they were seeing and there was this great vibe going by the end of class. The bell rang and we all looked up, surprised. One student was not interested in leaving because she wanted to (needed to!) explain to me how she was seeing a pattern. This is how every class, every day, should be.

We jumped right in to our visual patterns, starting with two that we had already looked at. I wanted my students to start seeing them in a different way.

Instead of looking at how many blocks were being added each time, we discovered how the shapes related to the step number. They could see that the "extra" block on each side were constant, as was the width of the rectangle. Finding a pattern, based on the step number, for the length of the rectangle was more challenging. I wrote the values in a table to help them come up with a relationship. I had seen it as double the step number minus 1, which is also how at least one of my students saw it. But another said that it was the step number added to the previous step number. We figured out how to write that and simplified to show that both ways of seeing it produced the same result (2n - 1). It was great to have more than one way of seeing this. I asked how many blocks were in the rectangle at step 3. Although some counted the blocks to get 15, others said that they could multiply the length and width. We did this for our length and width at step n, simplified with a little help from the distributive property, and arrived at 6n -1, which was our result from yesterday.

Next, we took another look at the volume pattern from yesterday.

Again, we found the pattern without having to go back to step 0, instead relating what we could see to the step number.

Then, my favourite pattern and several ways of finding its rule.

There is so much richness in this pattern. The first three ways really reinforced the idea of relating the number of blocks to the step number. To work out the rule based on the area of the large square and the area of the inside square, we needed to step into the world of quadratics. My students have never really worked with quadratics before - the word itself was new to them. I loved that one student had created a table of values AND had calculated the first and second differences. We figured out a way of writing the area of the outside square for step n, but in order to show that the rule did still work out to 4n + 4, we needed to expand. They had never done this before, but I tied it back to the distributive property which seemed to make sense to them (I normally use some kind of area model to introduce multiplying binomials, but did not want to add that to the mix today).

And we kept going:

Then I let them work on the next few in their groups. It was so cool to have the same conversation over and over about the length of the rectangle for pattern 8.

They would tell me that you couldn't get it by adding something to the step number and as soon as I suggested multiplying the step number by something, the wheels started turning and almost immediately the light bulb went on. Visibly! Their faces lit and and they said an enthusiastic "Oh!". And I walked away.

I asked them to try to finish the visual patterns over the weekend (although #11 is hard) and gave them this linear review, as well. This is a modification of something I had created for my grade 10 applied class a few years ago. There is a worked example and then a practice question for each skill. As we haven't really done any formal taking of notes, I thought this would be a good approach.

I was planning on doing Solve Me mobiles next class, but now I think I will stick with quadratics for a while since we landed there today. My planning is always a work in progress!

Exeter Conference - Day 5

We started with a little 3-act fun from Andrew Stadel. Here is his blog post about Filing Cabinet with all the links to the videos. Here is my blog post from the latest time I have done it with my students. 

I also showed them some of Nathan Kraft's craziness (that's crazy in a good way, of course). Toothpick insanity and Starry Night.

Then we played Polygraph: Parabola from Desmos. It was a lot of fun and everyone saw how the game encouraged the use of correct vocabulary and helped you create better questions by showing you what others had asked. 

We also took a quick look at Central Park.

I probably sound like a broken record, but if you haven't tried Desmos Activities, you really need to check them out!

Time to get up and get moving! Tying Knots was next. The first part of this activity involves determining the relationship between the length of a rope and the number of knots in the rope. I really like this because, unlike most linear data collection activities, this has a negative rate of change. 

We skipped the part involving putting everyone's data together to be able to find the relationship between the diameter of the rope and the rate of change (but it's on the handout) and moved on to figuring out how to get the ropes to be the same length with the same number of knots. 

All but one of the groups got it to work which led to interesting discussions and to me adding what you see below for next time. 

I have blogged about the Tying Knots activity here, here and here and the handouts are here and here.

We took a quick look at some Always-Sometimes-Never statements and discussed how these can be really good warm-up activities that help students think beyond just their initial reaction to the statement.

I then gave a choice of matching activities.Quadratic (credit to the teachers at Sir Wil for that one), rational, right-angle trigonometry or combinations of functions.

A couple of participants then had a quick but lively game of log war. I think I originally got these from Kate Nowak, so I'll give her due credit.

And, finally, I showed off some of my students' parabolic art - art work created entirely with quadratic equations. Here is that activity's blog post and this is one of my favourites:

MFM2P - Day 37 (High 5s and Frogs)

We did two warm-ups today as we are off tomorrow and I didn't adjust the warm-up book accordingly. Extra fun! We started with this Would You Rather...

After I heard how many students don't like carrots, they did get to work. Some used their phones to help with the unit conversions. Here are some of the results:

I liked the different ways students went about showing which was a better deal mathematically.

Then we did the Daily Desmos which was perfect after the linear work we did yesterday. They were given the graph of the line through the two points shown below.

They found the slope (from the graph and from the points) and identified the y-intercept and put them together to form the equation:

We tested it out using Desmos and the line did go through the points. Yay!

We moved on to High 5s - here is the handout with several quadratic examples, including High 5s and Frogs. I explained the premise of players giving each other high 5s as they are introduced before a game. I asked for volunteers and the first one came up. How many high 5s so far? None. Next player...1 high 5. Next player...2 more high 5s for a total of 3. Player #4...add 3 more high 5s for a total of 6. Player #5...add 4 more high 5s for a total of 10. I didn't have any more volunteers and they seemed to have understood the concept so they continued to complete the table on their own. They also added a column for the pattern (1st differences) and for the pattern in the pattern (2nd differences).

I loved listening to them talking about how to create their scatter plots. They had to decide on an appropriate scale and make sure they chose independent and dependent variables correctly. They used graphing calculators to get the equation and used it to figure out the number of high 5s for the whole school.

Next up, frogs! They didn't have a lot of time left but they started playing the game and trying to figure out the pattern. Here is my blog post from last year which explains it a little more.

That was a good way to end the week. They were engaged and having fun.

MFM2P - Day 35 (Speedy Squares, Part II)

I chose an Easter themed Estimation 180 for today's warm up:

I have had students write their estimates on small whiteboards and hold them up, but this time I went around the classroom with a big whiteboard and recorded their estimates. This ensured that everyone was participating and let me see how close they were to the answer. Once I had everyone's estimate written down I asked for reasoning. One student said that there were "9-ish" layers and then we counted around 9 or 10 eggs per layer.

I know that some students just guess so I hope that showing them how to reason through to an answer will help them with future estimations. Two students had the exact answer and got chocolate Easter eggs as a reward!

We continued with Speedy Squares from yesterday. I have updated the handout (there were a few "oops" on my part). I collected the times that they calculated to build a 26 by 26 square:

My students did a really good job determining the relationship between time and number of blocks based on their data. I helped them by asking how many cubes they had in their 2 by 2 square and what the associated time was. They ran with it from there.

They needed a little guidance to "design" a house - really just a top view of the floor plan with approximate, but realistic, dimensions. I drew an example to help guide them. Most designs were simple but good. There were, of course, a few crazy ones that were ridiculously large or small.

Once they had calculated their square footage, they used the Lego My House site to determine how many blocks their house would need. 

Then they used their equation relating time and number of blocks to find the time it would take to build their house out of blocks. The time they calculated was in seconds so they had to finish by converting it to days so that it was a more meaningful answer.