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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.

Celebrity Guessing Game

A few years ago, I came across this one on Dan Meyer’s blog. It’s from 2007, so it has been around for a while. I’ve been using it in Math 20-2 classes to adress mean, median, mode and standard deviation. Here’s my approach.

Create a PowerPoint or Notebook or Keynote file that had photos of celebrities of a wide range of ages. It works out nicely if you have one celebrity per student in your class, but if you have a class of 40, it would get a bit ridiculous. Give your kids a blank chart like this one, on which they can record their guesses as you run through the PowerPoint.

Run through the PowerPoint that has the celebrity names and photos, and ask the students to jot down the celebrity’s name and make a guess as to each celebrity’s current age. Make it competitive. Tell them there is a fabulous prize for the “Best Guesser”. Be vague if they ask you to define “Best Guesser”. This is my most recent PowerPoint.

Ask the class who has the most right, and act like you are going to give a prize to that kid. Inevitably, someone will stop you with a comment something like, “Wait! I only had 2 right, but I was really close on all the rest.” Let the class discuss how best to determine the winner. They will likely come up with a formula whereby they find the average difference each of them was from the actual answer. They will likely talk about whether or not it matters if they were 5 above or 5 below or just 5 away. Award the prize.

Let them calculate their average differences from the actual answer. Now you can define standard deviation for them. Standard deviation is a tough concept for Math 20-2 students, and this activity will help them picture it as an average difference from the mean.

Show them how to find standard deviation on their calculators (Math 20-2 students do not have to do this by hand).

Now ask them which celebrity they were best at guessing as a class. Assign each student a celebrity. That student must collect all the guesses that were made for that celebrity and calculate the mean, median, mode and standard deviation of the guesses for that celebrity. Put a giant chart on the board that has headings as shown below.

Get each student to fill in the calculations for his/her celebrity. Sit back as a class and look at all the data on the board, and decide which celebrity they were best at guessing. Have a class conversation around questions like: Is mean, median or mode the best one to use to determine which celebrity we were best at guessing? What does the standard deviation tell you? Does the one with the lowest standard deviation mean that was the best guessed age? When would that be true? When would it not be true?

Math 20-2 Research Project

When the revised curriculum came out with a Mathematics Research Project in Math 20-2 (Foundations of Mathematics 11 elsewhere in the WNCP), many of us were surprised by its inclusion. The exact wording of the outcome from the curriculum guide is shown below.

Mathematics Research Project

Specific Outcome

1. Research and give a presentation on a historical event or an area of interest that involves mathematics.

Achievement Indicators

1.1 Collect primary or secondary data (statistical or informational) related to the topic.

1.2 Asses the accuracy, reliability, and relevance of the primary or secondary data collected by:

• identifying examples of bias and points of view
• identifying and describing the data collection methods
• determining if the data is relevant
• determining if the data is consistent with information obtained from other sources on the same topic.

1.3 Interpret data, using statistical methods if applicable.

1.4 Identify controversial issues, if any, and present multiple sides of the issues with supporting data.

1.5 Organize and present the research project, with or without technology.

The Alberta 10-12 Mathematics Program of Studies with Achievement Indicators – Page 66

This outcome seemed really broad and wide open to me. I anxiously awaited the authorized resource so I could see how the approved publisher had treated this outcome. The approved resource is from Nelson (Principles of Mathematics 11). They introduce the research project on Pages 64 and 65. They outline a 9 step process to project completion, and then in each subsequent chapter, they have students complete one of the steps. Their process is:

1. Select the topic you would like to explore. (1 to 3 days)
2. Create the research question that you would like to answer. (1 to 3 days)
3. Collect the data. (5 to 10 days)
4. Analyze the data. (5 to 10 days)
5. Buffer Space. (3 to 7 days)
6. Create an outline for your presentation. (2 to 4 days)
7. Prepare a first draft. (3 to 10 days)
8. Revise, edit, and proofread. (3 to 5 days)
9. Prepare and practice your presentation. (3 to 5 days)
Nelson – Principles of Mathematics 11 – Page 65

In addition, on Page 64, they list “issues that may interfere with the completion of the project in a time-efficient manner.”

• part-time job
• after-school sports and activities
• regular homework
• assignments for other courses
• tests in other courses
• driving school
• time you spend with friends
• school dances and parties
• family commitments
Nelson – Principles of Mathematics 11 – Page 64

As I read this, I became quite disheartened. Remember, for this revised curriculum, there was one approved publisher. The WNCP worked closely with the publisher to ensure 100% curricular fit. That seemed to suggest that Nelson’s interpretation of the Research project outcome was 100% aligned with Alberta Education’s.  And what frightened me most was that Nelson’s interpretation seemed to be suggesting to students, early in Math 20-2, that they were going to have to do a project that would take between 26 and 57 days. Not only that, but this project was also going to interfere with after school activities. What kid, in his right mind, would want to commit to this? What teacher, in his right mind, would want to commit 26 to 57 classes (out of about 80, in total) to something like this?

I called Alberta Education and had a nice conversation with a person there. Based on my conversation, I believe I can provide you with examples of what this research project could look like. I’d hate to see teachers leave this outcome out, because I think students could have a lot of fun with it, if we present it properly.

What follows is my interpretation of this outcome. Hopefully I haven’t distorted Alberta Education’s intent too much. This is what I would do if I had a class of Math 20-2 students in front of me right now.

1. The research project should be something that interests the students. Ideally, I would give them several choices as to what to do, but I wouldn’t leave it as wide open as the outcome suggests. I would have some pre-packaged projects ready to hand out.
2. I would spend a maximum of 5 classes on this, and would not expect them to spend more than a few hours at home on it. 26 to 57 days is absolutely ridiculous.
3. I would need help coming up with project ideas for my students to choose from. I would ask for help on Twitter and this blog. Which is what I am doing right now. If you have other project ideas that you would be willing to share, please send them my way. I will post them here, and maybe we can develop a bank of these things so that we can give students lots of choice.
Below are three that I have that I think fit the criteria. Remember, there is only one specific outcome. The Achievement Indicators numbered 1.1 to 1.5 above are things the students may do, but do not have to do in meeting the outcome. I can’t think of a project that would hit all of those.

Will Women Soon Outrun Men?

David Petersen posted a link to this article in the comments section of another blog. I liked the article, so I turned it into a research project. You can download the project in Word format here. This project hits Achievement Indicators 1.1, 1.3, 1.4 and 1.5.
Edit (October 3, 2011)
Since I posted this, I have been shown a couple pretty interesting articles related to this subject.
Have we reached the limits of performance? Is the fastest human ever already alive? Article
Paula Radcliffe loses a world record marathon time because it occurred in a race that included men, giving her an unfair pacing advantage. Article

Is Farming the Root of All Evil?

At one of our UBD builds for Math 10C, the group developed a transfer task comparing height to historical period. I think this project fits nicely in the Math 20-2 research project outcome. It meets Achievement Indicators 1.1, 1.3, 1.4 and 1.5.

Survey of Classmates

I put this one together pretty quickly. It probably needs more thought and more work, but it will give you a starting point. It involves having students create a survey and then analyzing the results. It meets Achievement Indicators 1.1, 1.2, 1.3, and 1.5.

Other Ideas

I haven’t fleshed these out, but they could work.
• Analyze the scoring patterns of several hockey players over their careers. How many goals will they get this year?
• Analyze the earnings of several blockbuster movies over the past few decades. Based on that data, predict which current releases will earn \$100 million or more, and when.

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.

Dyatlov’s Pass

This is another activity that can be used in Math 20-2 to introduce the idea of reasoning. I got the idea from the Nelson textbook for this course. Their first activity is based on the ghost ship Mary Celeste. I discussed that one in an earlier post.

Based on the Learning Through Problem Solving steps, here’s how you can use this in a Math 20-2 class.

1. Play the first 2:08 of this video
2. Ask the students what they wonder about. They will likely begin to speculate what might have happened to the hikers.
3. Ask them to think of a theory on their own, and write it down. This is not a fleshed out theory, but just a quick first impression.
4. Ask them what further information they require. Much of the information in the video will have gone by pretty quickly, so you can give them this handout.
5. Have the students work with a partner to come up with a theory. Ask them to decide what information is important, and what is not as relevant.
6. Have students share their theories with each other.
7. Play the rest of the video.

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.

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.

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.