Archive for the ‘Math 20-1’ Category

The first strategy I’d like to share for embedding formative assessment (Remember, formative assessment isn’t always a thing. It’s feedback for learning) is referred to by the consultanty-types as “two by four”. While generally working in the field of consulting, I should point out that I’m not much of a consultanty-type. I’m not a fan of fancy processes and the equally silly names we give them. So, for classification purposes only, I present to you the two by four.

One of Dylan Wiliam’s 5 key strategies is to activate students as instructional resources for one another. This strategy is my go-to one, mostly because I’m trying hard to embrace the philosophy of the revised program of studies, which suggests I should orchestrate experiences from which my students can extract their own meaning. When I active them as instructional resources for one another, they learn from each other, rather than from me writing notes on the board.

This strategy is really simple, and can be used to teach just about anything I can think of in our curriculum. Two students learn something through some kind of structured activity. They check in with two more students who learned the same thing and compare strategies and results. When they agree, they move on.

Let me tell you how this worked in a class I tried recently. I was invited to model formative assessment in a Math 20-1 class (Pre-Calculus 11 elsewhere in the WNCP). The topic of the day was multiplying radicals. In my classroom days, I would have written some rules on the board, done some examples, and then given an assignment for practice. The students would not have been activated much at all.

For this lesson, I built a structured activity that I hoped the students would work through and learn what I needed them to learn about multiplying radicals. That activity is posted below.

Some notes about how it went:

  • First of all, I should point out that this does not need to be photocopied and handed out. It would work just fine as a set of questions that the teacher poses to the class. I ran it off, because I was going into a stranger’s classroom and I wanted to make it as simple as possible for the students.
  • As the students worked, I circulated and listened in. I helped when needed and as little as I could. I regrouped them at the appropriate times and with the appropriate people. By the end of class, almost all of the students taught themselves how to multiply radicals. I gave exit slips at the end (another formative assessment strategy) and the exit slips showed me that most students had met or exceeded my expectations. Two students didn’t manage to learn the material. This next part is huge. Wait for it…
  • It wasn’t my class. I didn’t know many of the student names. Despite that fact, when I saw the two exit slips that showed little or no understanding, I described to the classroom teacher exactly who I thought those two students were in terms of where they were sitting. I nailed it. Because I was circulating and working closely with the class, I knew exactly who was getting it and who wasn’t. That’s feedback, folks. That’s formative assessment, folks. The old me (remember, the one who stood at the board writing example after example) would have had no idea who was getting it and who wasn’t. Unit tests were frequently sources of great surprise for me.
  • I was fascinated by the strategies they were coming up with. On the first page of the activity, all students concluded that \sqrt{a}\times\sqrt{b} was equal to \sqrt{ab}. What they did when I introduced coefficients on the next page was really interesting. The room was split into two camps when they came across 3\sqrt{2}\times6\sqrt{3}. Many of the pairs of students got right to it and multiplied the coefficients, then multiplied the radicals, and came up with 18 \sqrt{6}. They confirmed their answers using their calculators, and moved on. The other camp wanted to rely on the only rule they knew at that point (\sqrt{a}\times\sqrt{b}= \sqrt{ab}). Their solutions looked like this: 3\sqrt{2}\times6\sqrt{3}= \sqrt{18}\times\sqrt{108}=\sqrt{1944}=18\sqrt{6}. They were converting to entire radicals, multiplying, and then simplifying. It worked, but it was highly inefficient. So when it became time to let the pairs meet with other pairs to see what they had learned, I made sure that each group of four contained one pair with the efficient strategy and one pair with the inefficient strategy. Students were taught the more efficient strategy by other students, rather than by me. That’s formative assessment.

I did something similar in a Math 30-1 (Pre-calculus 12) class earlier this year. The description and resources are available here.

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In my school days, I remember (and not fondly) problems like:

A train leaves Toronto for Montreal at the same time as another train leaves Montreal for Toronto. The cities are 500km apart. The trains pass each other 2h later. The train from Montreal is traveling 50km/h faster than the one from Toronto. At what distance away from Toronto do the trains pass each other?

I’m no whiz with a video camera or script writing. I’m not much of an actor (which you’ll see if you bother to watch the video below). But I think this is a more compelling way of presenting the same problem.

Act I

Act II





I need some help here. Any ideas? Is it worth bothering?

  • John was driving slower because he thought Darlene would drive farther to the meeting point. His plan is to drive 110 km/h all the way back, thinking that this would save him time overall. Would he have been better off driving 110 km/h the whole way?
  • What if Darlene left an hour later?

Production Notes: My wife says that there’s no way I’d be that calm if I had to drive her purse back towards home.

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I’m starting to see this stuff more places. This one could be fun. This one is easier mathematically than the Penny Pyramid that Dan Meyer describes here. In Dan’s 3 Act Format, here it is.

Act I


Ask the kids what they wonder about. They could go lots of interesting directions with this one.

Act II


This video will give them the information they need, assuming their perplexing questions require them to know how many watermelons are in the pyramid.


The Answer

The answer is in the form of a Word document with a photo and some calculations. I’d love a video. If anyone has the budget for 385 watermelons, the patience to stack them, and the video editing skills to insert a counter to the filmed stacking, I’d gladly take it.


What is the mass of the pyramid?

What is the cost of the pyramid?

How many watermelons could we stack in this room?

If the truck is 20 m from the pyramid, how long would it take to build?

Any other sequels? If so, throw them into the comments.


This zip file (29.7 MB) contains both videos.

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Graphing Inequalities

Last week, I mentioned to a group of teachers that I had never come up with a good way to teach kids where to shade when graphing an inequality. Vicky from one of our local high schools shared her method with me. It’s pretty nice.

Vicky gives her students an inequality like 2x-y \le 7

She asks them to each find two coordinates that satisfy the inequality, and then plot them on a giant grid at the front of the room. When 30 kids come up and plot points, it will look something like this.

From this graph, it becomes pretty obvious that there is a line involved, and which side of the line we should shade. It also becomes obvious that one kid made a mistake.

We could extend this method to quadratic inequalities. If the students were given the inequality y>x^2+3x-2 , we could ask students to find ordered pairs that satisfy the inequality, and plot them on a grid at the front. It might look like this.

Students could then have conversations about which of the shading should include the boundary, and which should not, and how to deal with that.

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RAFT in Math

Several years ago, I attended a session on differentiated instruction with a wonderful woman named Dr. Vera Blake. One of the suggestions she made was that we use the RAFT model as a check for understanding. For those of us more mathematical than Englishal,  RAFT is a writing tool typically used in English classes. RAFT stands for: Role, Audience, Format, and Topic. It helps students focus their writing by clarifying what their role is, who their audience is, what format is appropriate, and what topic needs to be covered. I may have just demonstrated a rudimentary understanding of the process, but I’m a math teacher…

I asked Dr. Blake to tell me how I might use it in a Math class. She showed me how to use it as a review, and a check for understanding. I was wrapping up a quadratics unit with an11th grade class, and she helped me write a set of RAFTs to use as a review with them.  I created as many as I could think of, and had pairs of students randomly select one RAFT.  For example, one pair was given the role of the discriminant. Their audience was a quadratic function, and the format was a letter from a stalker.  The topic was “I know all about you!”  A pair of quiet and shy girls wrote a really creepy letter from the discriminant to the quadratic function. Their letter clearly demonstrated understanding of what the discriminant indicated about the graph of the corresponding function.

The class had a lot of fun with it. We had songs, raps, poems, letters, posters, radio ads and many other things performed in class after one day of preparation. Some other examples included a quadratic formula writing a cover letter to a quadratic equation to apply for a job, a dating ad written by a quadratic formula who was looking for love and understanding, and a workout plan devised by a personal trainer aimed at making a specific quadratic function skinnier. What all of them had in common, was that they showed an understanding of the class material.

There were some bumps. The group that had to write the dating ad had no idea what a dating ad looked like, so they searched personal ads on my computer. That probably wasn’t a good career move. Another group had to design a twelve step process in the manner of AA to solve a problem. They also searched on my computer for addiction programs. Despite it all, I managed to keep my job.

I had also forgotten to consider assessment, so I forced a rubric on their presentations in the end. I should have left it as a formative assessment.

If you are interested in trying one of the two I created, feel free. I’d love to hear from you about how it went.

Quadratic Equations and Functions RAFT Topics – Math 20-1 and Math 20-2

Relations and Functions RAFT Topics – Math 10C

Here are some samples of student work from the 11th grade class on quadratics. They were a little better when seen performed live, in front of the class, but you will get the idea. The girls who wrote the first letter are clearly better students of English than I am.

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Thanks to the comments on my previous post, I learned that there is a much easier way to make some of the sets I described.  To make the \sqrt{3}set, I originally suggested measuring a 60 degree angle.  Here is a much better way.

Use the \sqrt{2} set that was built in the previous post, and place it on a different color of card stock grid paper.

Create a right triangle with legs 1 and \sqrt{2}.  This makes the hypotenuse \sqrt{3}.

Use the rest of the \sqrt{2} set to make right triangles with legs 2 and \sqrt{8}, 3 and \sqrt{18}, 4 and \sqrt{32}, 5 and \sqrt{50}, 6 and \sqrt{72}, 7 and \sqrt{98}, and 8 and \sqrt{128}.

Cut them out, and you have a much easier \sqrt{3} set to put on your radical ruler.

You could use similar patterns to create other sets.  For example, you can make a \sqrt{6} set by using your \sqrt{5} set.  Make triangles with legs 1 and \sqrt{5} and their multiples, and you have a \sqrt{6} set.

Thanks again for all the feedback on that last post.  You have helped make this activity even better.

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Last year I visited Shannon Sookochoff at Victoria School.  She showed me a neat activity on simplifying radicals on a Geoboard.  As I was thinking about how to introduce adding and subtracting radicals recently, that activity came back to me. I have created an extension of what Shannon Showed me.

Adding and Subtracting Radicals Using a Radical Ruler

Have students create “sets” of triangles.  The set I would start with is the \sqrt{2} set.

Use a 1 cm grid on a piece of colored card stock.  Have students draw right triangles with legs 1 and 1 (makes the hypotenuse \sqrt{2}) and then with legs 2 and 2 (makes the hypotenuse \sqrt{8}) and so on.  Continue all the way to one with legs 8 and 8 (makes the hypotenuse \sqrt{128}) .

Their grid paper will look something like this:

Notice I labeled the hypotenuse inside each triangle.  That’s because we are going to cut them out so we’ll have a set that looks like this:

Next, draw a line on a piece of paper.  This will be our \sqrt{2} number line.  Start by putting the \sqrt{128} triangle on the number line, and marking this point as \sqrt{128}.  Work your way in by putting the next one, which is \sqrt{98}, on the line.  Continue until you get down to the \sqrt{2}.

At this point, students should notice that all the gaps are \sqrt{2} in length.

This will allow them to label the bottom of the number line with \sqrt{2}2 \sqrt{2}3 \sqrt{2}, and so on. Notice that this gives a nice visual showing entire radicals and their equivalent simplified mixed radical.

Now we will extend the \sqrt{2} number line.  The quickest way to do this is to use the \sqrt{128} triangle and place it on \sqrt{128} as shown.

Then just mark the intersections along the hypotenuse.  These are each \sqrt{2} long, so we can extend our number line.  I took mine to 16 \sqrt{2}.

Now we can add radicals pictorially on our number line.  For example, if we wanted to calculate \sqrt{72}+\sqrt{50}, just put the two triangles on the number line as illustrated, and read the mixed radical below.

The answer is 11 \sqrt{2}.

Now you can have the students make other sets.  The easiest to make are ones that don’t require any angle measurements.  I would make a \sqrt{5} set in a different color, by using legs 1 and 2 cm long, and then just extending those by multiples of the original sides (your other legs are 2 and 4, 3 and 6, 4 and 8, 5 and 10, 6 and 12).

Cut them out, and add them to a second number line below the first. Line up the 0’s, so that students could compare and order mixed radicals using these number lines.

Other sets that are easy to make are \sqrt{10} (Legs 1 and 3, and then multiples of those), \sqrt{13} (Legs 2 and 3 and then multiples of those), and \sqrt{17} (Legs 1 and 4 and then multiples of those).  You could make as many of these as you want.

It would be nice to have a \sqrt{3} set, too, but this one will require some angle measurement.  Start with a base of 1, then measure an angle of 60 degrees. Where this meets the vertical will be a height of \sqrt{3}.

Continue this with bases of 2, 3, 4, 5, 6, and 7 to create a \sqrt{3} set. (Sorry about the stray line)

Cut out this set, and add it to your number line. Note that this set is much harder to build accurately.  I measured as carefully as I could, and my gaps are not consistent.

Edit: I have made improvements to the creation of the a \sqrt{3} set based on feedback from the comments section below.  You can see a better way to make the a \sqrt{3} set here.

Now that you’ve built these radical rulers, here are a few things you can do with them.

  1. Simplifying radicals.  This is a quick pictorial representation of what radicals become in simplified form.
  2. Comparing and ordering radicals.  Which is greater: 9 \sqrt{2} or 5 \sqrt{5}?  Typically, students would punch this in a calculator and compare decimals. It works, but has no understanding behind it. With their radical rulers, they can see which is bigger.
  3. Adding and subtracting radicals.  Put the triangles on the number line as shown above.  To illustrate adding only like radicals, ask them to add \sqrt{75} and \sqrt{45}.  They will discover that it can’t be done.  \sqrt{75}, in green, is not like \sqrt{45}, which is yellow.

I’d love some feedback on this one.  It’s not something I have ever tried in a classroom, but I would love it if someone tries it, and lets me know how it worked.

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