Monday 31 October 2016

Student-Created WODB

Once your students have done a number of Which One Doesn't Belong, they should be able to create their own. I described what I do with my calculus classes here, but thought I should take the time to outline my creating process.

Thanks to Chris Hunter, both when I am vetting or creating a WODB, I use a table like this where there are 4 criteria/characteristics labelled along the top.

I consider it a strong WODB if 3 of the items share a characteristic that is not present in the 4th. For example:

The criteria used for Shape 5 could be:

I find that working through one like this with students is a really important step for them to understand that "It's the only one that's pink" is not the depth that we are looking for (unless you are looking at colours!).

I also strongly encourage you to do some Incomplete Sets before your students create their own. Here is an example:

I love that there are many different options for the missing number here. And students (may) quickly see how helpful the table is to ensure a strong WODB.

What have I missed? Let me know in the comments!

Tuesday 25 October 2016

The Box Method for Factoring Trinomials

I love using the box method (area model without appropriately sized side lengths) to help students learn how to multiply polynomials. I love, love, love to use the box method to divide polynomials. But I only started using it to factor non-monic trinomials last year and I did a horrible job of it. Really horrible. I'm sure none of my students understood it because I didn't really get it. I am happy to report that I now LOVE using the box method for factoring non-monic trinomials. I shared this with the other math teachers at my school (I send out a weekly "Math Minute" - a link to a cool activity, a blog post, an idea that is worth sharing... and this was what I sent this week). I tried to colour code it to make it easier to follow and made a second box to better show the steps.

Here is an attempt at an explanation in case the example is not clear. Start by finding two numbers that multiply to the product of 'a' and 'c' (here -120) and add to 'b' (here -2). In this case the number are 10 and -12. The box represents the area (trinomial) and we are looking for the length and width (binomials). I always put the x^2-term top left, the constant term bottom right and the x-terms along the remaining diagonal. The number we found are used as the coefficients of x so 10x and -12x go in the boxes along the diagonal. Then I common factor the first row and the first column (that's where the 2x and 4x come from). This would normally all happen with one box, but now jump down to the second box. Figure out what multiplied by 2x will produce 10x and what multiplied by 4x will produce -12x. Those complete the factors and you can check that it all works out with the constant term (does -3 times 5 equal -15?). There you go. I love this because it is not a trick - it makes sense and has built-in error checking. I find it really fast, too.

I should note that I also show my students how to factor using decomposition and give them the choice of which method to use. So far more are choosing to use the box method. I can't wait to show this crew how to divide polynomials in a couple of years!

Rethinking Factoring Special Quadratics

Have you ever had a day when a lesson you took the time to rethink actually worked noticeably better? Let's be honest - I don't have the time (and sometimes not the motivation either) to rethink all my lessons. "What worked well enough last year is good enough for this year" happens far more frequently than I'd like to admit. I try to make notes if something really doesn't work or if I have a brilliant idea after the fact. And I do my best to act on those notes to my future self. Occasionally, if I teach more than one section of a course, I will make changes on the fly as I teach the second class. But the reality of teaching full-time and raising a family is that every lesson may not be as good as it could be. This is a difficult reality for me.

The change I made to yesterday's lesson was a simple one - I did the opposite of what has been done in the past. Let me back up for a minute (and I apologize if you've heard this all before)... The math teachers at my school all share lessons for all courses. I am the renegade who sometimes does things differently. I have been spiralling my grade 10 applied classes for several years and I spiralled my grade 10 academic class for the first time last year. I put a lot of thought into the order of topics and how each would be approached and blogged daily. This year I am tweaking what I did last year - the biggest change being that I am introducing more quadratics concepts earlier in the course. I am trying to be intentional when I look at past lessons and ask myself whether this is the best way to approach the topic. I looked at the "department lesson" on factoring special quadratics (at this point I have no idea who created it - it could have been me???) and just wasn't happy with it.

The old:

... followed by exclusively difference of squares practice questions. Then:

... followed by exclusively perfect square trinomial practice questions.

The new:

As I wrote above, I got students to do the opposite of the old lesson. Instead of expanding, they factored. This was good practice for them and they could see that there was a shortcut within the patterns. We had a whole-class discussion, talk with the people at your table, test your conjecture(s), come back to whole-class discussion kind of thing going, but we got there. They came up with the patterns (I didn't tell them) and they saw the value in what we were doing (I think). I think the old lesson tended to fall flat because they didn't see a need for more ways to factor - it was just confusing. They didn't see these special cases as being helpful. I hope this year's students do. They also know that they can also successfully factor them as they would any other trinomial if they don't notice that they are dealing with a special case. (Confession: Until I started teaching grade 10 applied, I did not think of a difference of squares being a trinomial where the x-term has a coefficient of 0. Factoring these with algebra tiles was a revelation!)

One of the things I love about spiralling is that it freed me from common test days. When my students need more time on a topic, I give them that time. So tomorrow we are factoring a little more. A few students are really solid with all types of factoring, but most have a more tenuous grasp of what to do when. My room is currently all set up for some factoring speed dating. Tomorrow should be a fun-filled day of factoring!

Monday 3 October 2016

Distance Between Two Points with Tacos & Zombies

I thought it was about time to dust off my blog. I am spiralling my two grade 10 academic classes this semester (I also have a section of grade 12 Advanced Functions). It is really nice to have last year's plan and homework sets to work from and tweak. I recently saw what Nathan Kraft did to a cool Desmos Activity Builder activity created by Andrew Stadel and knew that I wanted to adapt it for my class. Their activity focused on horizontal and vertical distances between two points. I needed to include the distance between any two points. Due to the size of the activity (so many images!) Desmos struggled to keep up with my changes and I ended up having to hard code the points I used rather than having them referenced with variables. Using variables would have made it easy to move the points, but I will have to save that for another time. I think I'm getting ahead of myself here though. I should back up for a minute and start at the beginning of today's class. We started with Dan Meyer's Taco Cart 3-Act found here

After setting the scene with the Act 1 video, I asked what information they needed to know. They requested distances and speeds so they got this:

Next, they worked on big whiteboards in their table groups to determine whether Dan or Ben would get to the taco cart first. We talked a little about the Pythagorean theorem along the way.

Then, I played the Act 3 video to confirm their answers (all groups said that Dan would arrive at the taco cart first).

It was now time for the zombies! I used my Popsicle sticks to select random pairs and handed out Chromebooks. The link to the activity is here.

Despite the activity not running as smoothly as I would have liked (points were not showing up for some students and overall it was very slow), students seemed to learn what I wanted them to. After working out the distance between four sets of points on this screen, they understood the process and were ready to generalize. (Screen 13 was planned as an extension - I really wanted to ensure that everyone completed screen 8.)

We consolidated to close the lesson and did one quick practice question.