MEL3E Day 20

Since it’s October I picked a candy theme estimate from Estimation180. We chatted for the first 5 minutes about our favourite Halloween candy.

After some Too high, too low, and best guesses we revealed the answer:

If you teach MEL3E you may know that regular attendance from some students is an issue. There always seems to be a few students you only see once or maybe twice a week. Years ago I used to give these kids hassle when they came to class. I would give them a lecture about attending regularly….and if you think of all the adult interactions that kid had that day most likely none of them were positive. And I contributed to that. Even though they made the decision to come to class that day. And you know what….most likely that kid wouldn’t be back for some time.

Now….with this group,  all interactions are positive. I want that student who comes only on Thursday to have at least one positive interaction with an adult that day. I want them walking out after the class thinking that my room is “good” place. I feel they will be more likely to come back to school even if it’s just for math class. If they are there some good will happen.

So, with different kids being absent on different days it becomes tough to get every kid the practice and learning they need. The mastery days and spiralling works well to address this. If a kid is away all week they won’t necessarily miss the whole banking unit. We’ll hit this again next cycle. Mastery days will allow kids to work on what they need.

Today wasn’t technically a mastery day, but the day was broken into a few tasks.

  1. Some students finished (or started) the transaction activity from yesterday.
  2. Some students practiced more with updating their account balances from transactions.
  3. Some students worked on past work (timezones, best deals, tax problems).

Tomorrow we’re on to credit cards.

Updating the MFM1P Spiral

“Have you taught for 25 years? Or have you taught one year 25 times?”

I don’t think I’ve taught the same course the same way ever. Why would we? We don’t have the same kids in front of us. And especially with the resources at our finger tips from our colleagues inside and outside of our schools. I’ve wrote before about the power of #mtbos and it changes the way you teach.

I started spiralling the MFM1P course a few years ago with Kyle Pearce. Since then I’ve taught that course 3 or 4 semesters in row…..and never the same way. New amazing lessons and tools are springing up. For past lessons I wasn’t completely happy with I’ve got to see if this new lesson or that lesson will help my students understand the concepts more deeply.

One change I wanted to make was to include solving equations earlier in the course. In my old plan I waited to introduce it after introducing linear relations. But, after teaching solving equations using the Double Clothesline and the puzzle nature of learning it that way….I can introduce it now and continually practice our skills through warm ups.

If you want to follow along as my day-to-day plan unfolds follow this link! If any of you have been spiralling MFM1P I would love compare notes, or see your plans.




Double Clothesline – Solving Equations

I have always taught solving 2-step linear equations by starting with a balance scale. Having students whittle their way down to see how many marbles were in each bag was always a win for me… most cases.

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I valued this approach. It’s easy to visualize and it strengthens the “whatever you do to one side of an equation you do to the other” mantra we tell students when solving . However, I’ve always been left wanting more especially when we introduce solving equations with negative coefficients or even when the solution is a negative value. The balance scale kinda loses it’s effectiveness.

Using algebra tiles help fill this hole. And now…. thanks to Andrew Stadel, double clotheslines.

I was lucky enough to attend Andrew’s NCTM Annual session on Error Analysis this year. In his session he demonstrated how to use a double clothesline to solve equations. I later found this resource on his site. Watch his videos on how to use the clotheslines….they helped me piece this lesson together. Stop now and go and watch Andrew’s video on solving two step equations.

I stared as Andrew did at the NCTM session:

I put 0 on the top line and 0x on the bottom line.

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I then held up the 3x card and asked where should this go? I asked if it should go on the left or the right of zero. The students overwhelming said it needed to go on the right. “3x is more than x, so it should go more to the right, just like a number line” (Always — Sometimes — Never was going through my head at this moment but i’ll wait to talk about this with the kids until a bit later in the lesson).  Screen Shot 2016-05-16 at 12.18.16 PM

I then said “I’m going to place this 15 right above the 3x and that means equivalence. 3x is the same as 15”

Where should 9x go? You could see the some students spacing out where 9x should go. This is what I love about this method. It’s so visual and we’re forced to always think about how terms relate to each other.

I want to know what number should be above 9x. I had them draw the number lines on their desks and let them work on determining the value of 9x.

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Going around the room there were a few different types of solutions. Some students said, “3 times 3x is 9x, so 3 times 15 is 45”

Some students said, “If 3x is 15 one x is 5, so 9x is 45.” Nice. We ensured the whole class understood both of these types.

Next puzzle: I asked where to place 3x + 4…then assigned it the value of 16.

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Where should 3x be placed? It was easy to see that 3x is less than 3x + 4 so it should go to the left. Now for the amazing moment! What should be the number above?

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from the class an overwhelmingly 12 was shouted. So now what must be the value of x?


Student: “The dividing is the easy part” We spent a few minutes here talking about why dividing 12 by 3 here makes sense.

Next Puzzle:

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Where should the 5x go? At first some students had some difficulty deciding if it should go to the left or right of 5x – 2.

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Once we settled to the right. They jumped to finishing it off to determine x.

Next Puzzle:

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Where 3x should go was a discussion. We all agreed it should be 14 down…..and where would that be? This is where the clothesline (number line) feels superior and the balance scale visual falls short. We can use the bi direction of the number line to continue working with negative values.

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What was awesome during this class was this wasn’t a big deal….the number lines seems natural!!


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Also watch Andrew’s example with negative coefficients.

I had students practice solving a variety of equations by drawing the cards on their handout.


They finally demonstrated their understanding by creating their own equation where x had to equal 4. They put their creations up around the room for the group to solve.


I feel that the number line (clothesline) method builds a lot of great number sense. We get to reinforce our inverse operations as we build from conceptual understanding to abstract. Students’ strengthen their understanding of algebraic expressions and how those expressions relate to others.

I’m now going to investigate how to to demonstrate solving multi-step equations…. 3x + 5 = 2x + 7 using the clothesline. I’m thinking this might be a difficult task. Any ideas????

[UPDATE] – Solving equations with expression on both sides.

Since this lesson my class used the double number line to solve equations like 4x + 10 = 6x + 2. It was great to keep some continuity here while we solved harder equations.

We placed each side of the equation on separate clotheslines just like before.

We didn’t want to re-invent a new strategy….we were great at solving equations when one line was used for numbers and the other for expressions… we wanted that. How can we get one line to be just numbers and one to have the expression? We subtracted 4x from both lines.  Which left us exactly where we were last class!!


and then we subtracted 2 from both to isolate the “x-term”


Finally dividing by 2



This will be our method too to solve a system of equations that are both in terms of y.

More clothesline:






Catch the Spiral! 

Last May I shared my day-to-day planning spreadsheet for my grade 9 applied course. On that sheet I recorded the topic, tasks, and resources for each day of the semester. I used that as a resource for myself when teaching 1P through a spiral this semester. I found that having that sheet to go back too was super helpful and a time saver. This semester I followed that timeline except with a few tweaks here and there.

Since that sheet was so handy to have I made one similar for my MPM2D class. It was my first time spiralling that course and I wouldn’t go back to teaching through units again.

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I heavily relied on Mary Bourassa’s blog….she is amazing. She shares her day-to-day plan as posts on her blog and also shares all of her resources and handouts. Thanks so much Mary!!!

Spiralling in Academic vs. Spiralling in Applied

I struggled initially with deciding to spiral the MPM2D course because of my experience with MFM1P. I had previously taught the 1P course through activities and 3 act math problems so it was a no brainer to just mix up the order of the problems and tasks. It was an easy transition since I had all the resources. For the 2D course though, it had been a while and I had not taught it with a task/activity approach.

What I found to work best in the academic class was to learn all new ideas/topics through activities and productive struggle with some direct instruction thrown in as a consolidation. Unlike the 1P course where I switched tasks/topics daily, I stuck to a topic/idea for a few days or a week in the 2D course. Once, for example, the class was comfortable with transformations of quadratics we would switch to trigonometry for a week, then analytic geometry for a week, etc.

I felt that through spiralling and teaching through productive struggle my students were better problem solvers. They were not just waiting to be told how to solve a problem. They were always actively thinking about which ideas they had learned could apply to solve a particular problem. That confidence I saw allowed us to go more deeply into the content than ever before. We just didn’t skim the surface of the processes, algorithms, and algebra needed, we solved problems!!

If you wanted to spiral the 2D course or a similar course I thought I would share out my plan to help out. Here is my day-to-day plan with links, resources, Desmos activities, 3 Act tasks, assignments, homework, etc from my spiralled MPM2D course. (It’s not fully complete for every day but you’ll get a sense of how the class ran).

Most files are either Smart Notebook, Apple’s Keynote, or PDF.

Get Apple’s Keynote on your Mac or on iOS.