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.

Friday, 24 June 2016

Grade 10 Applied Math, February - June 2016

At some point during this semester I said I would blog about what I did with my grade 10 applied class. I have spiralled this course for a number of years now and felt that it was time to change things up a bit. Here is my attempt to remember what I did and why. Firstly though, here is the Google sheet that I used to plan and record what I would be doing during each cycle. There are pictures of the cycles included below.

The first big change I made this time around was to start with trig. I have found that starting with 26-squares did not pose enough of a challenge to some students who decided right away that they didn't need to do any work in this course and behaviour issues ensued. Trig, being new to all students, was a great way to see how they learned and dealt with new content. I also introduced sine, cosine and tangent right away. I used to wait until cycle 3, but found that students tried to pick it up and then forgot how to use the tables. I taught with both trig tables and SOH-CAH-TOA the entire way through the course, and students generally chose one way and stuck with it.

Another change I made had to do with warm-ups. I still did them daily, but I created them as we worked through the course so that they would connect more with the content, either as practice for what we were doing or as lagged practice for a topic that we hadn't looked at for a bit. Here are this semester's warmups. There are no warmups for weeks 6 and 12 because I was testing those weeks.

Here is a look at cycle 2:

Here is a look at cycle 3:

And end-of-year stuff:

I didn't make any other major changes, but did try to spend a bit longer on each topic in cycle 2. I am sure I added a few new things but when you wait several days or weeks to blog about them, you forget. Well, at least I do. If there is anything that you want to know more about, please get in touch!

Monday, 20 June 2016

Things to Remember

Teaching can be a tough job. But it is also the most rewarding. This blog post is something I hope to remember to read on the difficult days.

I find the end of the school year hard. I am so happy to see what amazing people my grade 12 students have become, and for them to go off and make their mark on the world. But this is always accompanied by a sense of loss. I have spent at least 5 months in their company, and for some it has been 4 years, on and off. The connections built over that time are meaningful to me. This is why I write cards to each of my grade 12 students each year - I say something good about each of them and how they have impressed me along with good wishes for the future. Each message is different, as is each student. They also each get a lollipop which is a symbol of achievement in my class - a great result or a marked improvement earns a lollipop on a returned test. So it always fills my heart to receive some thank you cards from my students - I hope they have learned that a small gesture can mean a lot. Their words make me proud of who they are and what we have accomplished together. Here are a few excerpts:

"Thank you for actually making me like math and giving me back some confidence I lost a long time ago about my capabilities in math. You are the nicest teacher I have ever had and I really do appreciate everything you have done for me this year. Thanks for letting me know it's ok to make mistakes and giving me the time and all the effort you have put in to let me grow. I hope no matter what I accomplish in the future, I achieve it by hard work, and with all the kindness, empathy and dignity you have taught me."

"Your style of teaching and attitude towards math is so inspiring to me. It feels like you genuinely care about getting your students interested and getting them to succeed. I am honestly amazed you are able to keep up such a positive attitude and it honestly inspires me so much. ... If I grow up to become half as passionate as you about something, I would be so happy."

"I know I didn't get the greatest marks, however you not giving up on me and continuing to demand my best work will always be something I'm grateful for."

"I want to sincerely thank you for everything you've done for our class this semester. Thank you for being more than a calculus & vectors teacher, thank you for being patient, kind, supportive and always having a smile on your face."

"You are one of those teachers who truly cares about her students, and I'm so thankful for that."

"You have been incredibly inspiring to me and have greatly affected me. I admire your passion for math and especially learning. Your love for math is nerdy and contagious and you are the best teacher I've ever had because of it. ... I wish one day I can be half as lovely and caring as you."

Knowing your content is important. Being passionate about it is far more so. And the relationships we foster in our classrooms help inspire students to be their best selves. We do make a difference and our students notice what is important to us. And every so often we will hear a gem like "Your love for math is nerdy and contagious" - it doesn't get much better than that!

Wednesday, 1 June 2016

What's Your Best Question?

Yesterday, before class, I tweeted this out:

And, as usual, the #MTBoS came through. Here are just some of the replies I received:

I answered the question thanks to the great replies I got. But it was not until a student asked me how to solve the question that I realized that, despite knowing that many students would struggle with this question, I did not plan out what I would say when they asked for help. My answers ended up being just like those given to me on Twitter - "this is how you start it" which really took some of the fun out of solving this "puzzle". So I am now wondering what a good question would be to help move my students' thinking forward without giving away the solution. I should have at least asked "What do you notice?", but am not sure that would have been enough to get them going. Please tell me if I am wrong! This is the question I came up with in the van ride to take my kids to Jiu-Jitsu:

I am wondering what you would ask - what would your best question be? Please let me know in the comments!

Friday, 13 May 2016

OAME 2016

You know that feeling when you are trying really hard to get caught up on everything, but the "finish line" keeps moving? That's been me this week. I'm woefully behind on replying to emails, apparently have 9 comments awaiting moderation on my blog that I don't even remember getting notifications about (?!?) and I arrived home yesterday to a flooded upstairs bathroom which, of course, had spread to the downstairs bathroom, hallway and bedrooms while I was at school. When it rains, it pours! (sorry - couldn't help myself) 

However, I am attempting to work my way through my to-do list and owe David Petro my slide decks from OAME 2016. So, here they are:

WODB Session

Spiralling Panel Session
(this was with Bruce McLaurin, Jon Orr, Alex Overwijk & Sheri Walker)

Rethinking Grade 10 Math Double Session