Spiralled MPM2D Update

This semester I taught two sections of grade 10 academic math, both of which I spiralled. Although I have taught this course for many years, this was my second time spiralling it. I blogged my way though last year's journey, starting here. I thought it might be worth sharing what I did this time around as the changes resulted from my reflections the first time through the course.

The one big change was doing more quadratics earlier in the course. About half the course is quadratics so I tried to devote about half of each cycle to quadratics. We factored starting in cycle 1; the rationale being that factoring doesn't always stick so multiple exposures to it (and lots of practice) should help students better retain how to factor.

Cycle 1:

Cycle 2:

Cycle 3:

Not-really-a-cycle Cycle 4:

We managed to finish the content on December 22 so when we returned from the break there was a sufficient amount of time to review each strand and have an optional test for each strand. These tests covered 9 of the 10 curriculum expectations and each expectation was optional. Each student could choose which expectation they wanted to show - some did none while others did all 9.

I modified some of the homework sets from last year, but continued to spiral the homework as well. I tailored it to include questions that I know many students needed to practice, but always kept it to one page.

I'm sure there were other differences that I cannot recall at the moment. If you have questions I would be happy to answer them in the comments.

Median, Altitude, Perpendicular Bisector Warm-Up

My grade 10 academic students have spent the last couple of classes working on finding the equations of medians, altitudes and perpendicular bisectors. Some did a lot of work and some, well, let's just say that some did less. As we move into looking at properties of quadrilaterals, I wanted to ensure that everyone has at least a basic level of comfort with the prior work. So, I came up with a warm-up that I thought I would share.

It incorporates collaboration and finding mistakes (there is no way that all groups will do this correctly on the first try). I suspect some will struggle finding the group with the same question as them, but they will learn to follow instructions better (hopefully).

Here are the questions, linked here, that I will cut into strips for them.

I have 28 students, so 14 pairs which means that I needed 7 questions in order for each question to be repeated. Two groups will eventually find the centroid (via equations of medians), two will find an orthocentre and three will find a circumcentre.

If you can think of ways of improving this, please let me know in the comments!

Centroid of a Triangle

Yesterday, my grade 10 classes learned about triangle centres. They each had a triangle awaiting them on their desk as they walked in (I copy them onto thicker-than-normal paper). 

I gave instructions along these lines:
- cut out the triangle
- figure out how to balance it on your pencil
- make note of the balance point
- figure out how the balance point relates to the coordinates of the vertices of the triangle
- when you think you know, create your own coordinates for a triangle ABC, write them on the board along with their balance point

That last part was new. Its inspiration is Joel Bezaire's Variable Analysis Game which you can learn more about here. It helped some students see the pattern (especially the last line that I added to make it a little more obvious) and kept those who had found it early engaged as they were checking that other student's coordinates and balance points worked.

It was a fun way to start the class so I thought I would share... Thanks, Joel!

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!

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.

MPM2D - Day 79: Summative Day

Today was summative day. The tension level was high, but they seemed to jump right in and get going. I had told them that they should look at it as a conversation with me, as I will be the only one who sees their work, and they should explain what they are finding and why and reflect on their work as they go. From what I saw as my students were writing, I think they did a good job of it.

MPM2D - Day 78

Today my students worked on what they needed to work on to prepare for tomorrow's summative. For some that meant creating more good questions and answering them, for others it meant going over homework sets with which they had struggled. I gave one-on-one help, as needed. 

I don't know if it's just me, but I find it difficult at the end of the semester to give appropriate wait time when giving individual help. I will ask a question and want to jump in with the answer or will want to just tell them how to start a question rather than asking a question to get them to think about what they should be doing. I have to remember to keep myself in check and not rush through - even though the semester is almost over, my students are still learning.

MPM2D - Day 77: More Summative Prep

Not too much to report today.  I started today's class by looking at some of the work they did last night. They shared their work in their groups and then we looked at a couple of student's work with the document camera. We commented on strategies students used to come up with values/equations in questions and that the work done to create a good question may demonstrate a whole lot of the math I hope to see. I told them that I would really like them to do their summative in pen so that I can see everything they do. I tried to show them how the process expectations fit in with their work and how these could be improved, especially how they can reflect upon their work. We looked at sample solutions before they worked on part (b) of yesterday's scenario. We also talked about how they can make this type of open-ended evaluation work for them. They can choose numbers that will either increase or decrease the level of difficulty of their question. They can practice certain skills (finding a triangle's circumcentre, finding the shortest distance between a point and a line, converting between forms of a quadratic, etc.) and see if they can make those work with the scenarios they are presented with on Thursday. I think that they left feeling a little less anxious and like they may have a little more control over what they will be doing on Thursday.

MPM2D - Day 76: Summative Prep

I started today by going over what we will be doing for the remaining 11 classes before the exam. Then I had them continue the gallery walk from Friday, looking at each other's scenarios and questions. Next, we spent a little time recapping the curriculum expectations (although, because I spiralled the course, it was all pretty fresh) and talking about the process expectations. 

I then showed them a scenario and some questions that I came up with a few years ago and tasked them with completing the questions and writing up solutions.

While they started working on that, I discussed homework progress with individual students. Their homework is to complete question (a), above.

MPM2D - Day 75: Summative, Day 1

Unbeknownst to my students, today was day 1 of their summative. Our school board (district) requires us to have two final evaluations. For this class, that means a summative task and an exam. They are both board-wide evaluations so I can't really share them, but will try to give an idea of what we do. I didn't warm my class that this would be happening today as there was nothing they could do to prepare for today's work and I didn't want to raise anyone's anxiety level.

I organized my students into not-so-random groups of 3 and told them that their job today was to come up with scenarios and then good questions based on their scenario. They were not answering any of these questions. I gave an example of a scenario: Laura is flying a plane from Ottawa to Boston. We then came up some questions that would work for this scenario. We talked a little about trying to make connections between various curriculum expectations and that they should be working on creating rich questions.

They worked really well and had great conversations. I loved that they asked to use the whiteboards even though they knew that they would have to produce their final work on chart paper. Once they had worked through this process, they did a bit of a gallery walk to see what other groups had thought of. They were also encouraged to add feedback and their own questions to others' work.

I think that's all I can say about this. They were awesome - I love how easily they talk to each other about math.

MPM2D - Day 74: y = 2^x and Negative Exponents

We have what I think is a weird curriculum expectation lumped in with quadratics in grade 10:

I feel like it's a one-off lesson so I kept it until the end in the hopes that my students wouldn't forget it.

We started by looking at patterns.

They did well at seeing the pattern and answering the questions that follow. We jumped onto Desmos and looked at the two graphs as we zoomed further and further out. We also looked at what happens to the exponential as x becomes a very big negative number, and tied that into the pattern above.

Next, we watched James Tanton's video. I really like his explanation of 0 and negative exponents related to the piece of paper (around the 3 minute mark). Here is the link. My students didn't want to stop watching as they wanted to know about exponent ½, which is not in the curriculum for this course :) 

We looked at y = 3^x next to see that the same patterns hold true and then looked for a way of evaluating a power with a negative exponent without having to make a table and use the pattern.

We went through some examples together - I chose students to answer using my popsicle sticks. These were fine.

And the first row of the next set of examples went well, also. Then we got to (-2)^(-4) and my ability to take a quick question and turn it into a 10 minute class discussion hit. (Seriously, other teachers seem to finish a lesson with 20 minutes to spare and I don't get through the lesson.) Anyway, we had a lot of discussion about whether the answer should be positive or negative and whether the brackets were needed. I should say that the students were discussing it as I only asked what they thought of each others' answers without adding to the discussion. In the end, those who said yes to brackets convinced those who were saying that brackets did not matter.

I consolidated what they had figured out here:

And that was all we had time for. Here is today's homework.

MPM2D - Day 73: Cosine Law (day 2)

We jumped right back into cosine law today with this example:

They are really doing well with this. We went over what it means to solve a triangle before they worked on the next question.

As we had not practiced using the cosine law to solve for an angle very much, we did a couple of those types of questions.

They seemed to be getting the hang of seeing when they could use sine law and when to use cosine law. We talked a little about the upcoming summative where they will need to show multiple skills, not the same one over and over.

That example flowed nicely (pun intended) into answering "When do I use what?". Here is the link to the page with the following flowchart. I like its simplicity.

Here is today's homework.

MPM2D - Day 72: Cosine Law (day 1)

As with yesterday's class, I started today by asking them to find the side length of a triangle.

They tried various strategies but they could not determine the value of x. When I asked why they said they didn't have enough information, which we then turned into not having the right information for the tools they had. We then developed the cosine law.

I was surprised where the hurdles appeared in this process. This was the first one:

I assumed that they would all write the ratio without any difficulty, but student after student (that I chose using popsicle sticks) could not come up with this. There were strange combinations of trig and the Pythagorean theorem, proportions involving trig somewhere along the way, and other convoluted (incorrect) answers. I gave them a few minutes to discuss what was being asked after which we got to the equation we needed. 

The second hurdle came when I asked them to expand and simplify (a - x)². The errors didn't bother me so much as the lack of confidence working this out. We worked through it on the whiteboard and then moved on.

Seeing that you could substitute expressions from the left side into the right side proved to not be an issue which also surprised me.

They talked about the patterns they saw in the equations before we tackled our first example. I love that the student who wrote this solution made the mistake you see below. So many forget, or miss, the order of operations when they write out all the steps, which leads them to the wrong answer. I suggested that they should just put the entire expression in their calculator and let it do the work. I also reminded them that using the ANS key on their calculators can be really helpful.

I worked on one more example that was a little less straightforward and was their first attempt at solving for an angle using the cosine law.

We will do more work with the cosine law tomorrow and talk about when to use what in trig.

Here is today's homework.

MPM2D - Day 71: Sine Law

I started with this picture and explained that the surveyor was trying to determine the width of the river (if someone can give me the source of this question I would greatly appreciate being able to give credit):

They then worked in groups to find the width of the river using this diagram (it was so nice to see and hear them talking math!). Most groups started by finding the missing angle in the triangle and then got stuck. They knew they needed to use trigonometry but either tried with this triangle as-is, or realized that they didn't have a right angle. I circulated and asked them what they needed to use trig and each group replied that they needed a right triangle. So I asked how they could create one and walked away.

Many groups added altitudes that were not helpful as they didn't know a side length of the right triangle they had created. I asked if they could create a different right triangle (and walked away).

Here is the (edited) solution of the group that didn't need any hints from me:

I told them that they had solved for a side in a non-right triangle! I asked if they would like to not have to always draw the triangle and add the altitude and so on. This led us to...

We talked about when each form would be useful and what information you needed to use the sine law before working on some examples. Many had a really hard time with the first example. I tried to help guide them by writing the sine law on the whiteboard and putting check marks next the information we knew so they could see that we only needed the proportion using A & B.

The next example required them to notice that they didn't have an angle-side pair so needed to find the third angle before using the sine law.

In the final example they found the measure of an angle.

Here is today's homework.