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I take no credit for this idea at all. I was in a classroom this morning, and the teacher had the students play a game of Radical SNAP. The students were totally engaged, and were enthusiastically converting between mixed and entire radicals. It’s pretty simple to set up.

Materials: You need one deck of cards with the 10, J, Q and K removed for each pair of students, and one giant square root symbol per pair of students. This one should do the trick: Giant Root

Pair off the students in your class. Each pair gets a deck of cards, and should remove the 10, J, Q and K. Shuffle the remaining cards, and deal them so that each person has half the deck, face down.

Mixed to Entire

The students flip over their top cards. The student on the left puts his in front of the radical, and the student on the right puts hers under the radical. The first student to correctly convert the mixed radical to an entire radical wins the round.

Entire to Mixed

The students flip over their top cards, and put them both under the radical. The first student to correctly simplify the radical or to identify that it can’t be simplified wins the round.

I’ve been writing here about getting students to practice in class. Many of our students don’t (and probably shouldn’t) be doing practice math questions at home. We need to build opportunities into our lesson for them to do some questions. Kate Nowak has provided us with two great ways to get students to practice some questions in class in more compelling ways. This practice is formative assessment. I would classify both of her activities as activating students as instructional resources, using the language of Dylan Wiliam.

I have nothing to add to Kate’s work, other than to tell you I’ve used this stuff and it works. I’m just pointing you in its direction and making a link between what she has shared and the formative assessment I’ve been writing about in my recent posts.

Check out Kate’s Row Game and Speed Dating. She has several already prepared that fit our WNCP curriculum. It looks like she has been collecting other people’s Row Games here.

Both activities are easily differentiable, and allow kids to practice their math. They have built in accountability because students are responsible to each other. They are both fine examples of embedded formative assessment.

I take no credit for this one and I’ve never tried it. It’s all Rick Wormeli. He shared this strategy in a session I attended several years ago. It would certainly qualify as activating students as instructional resources for one another.

Rick works with one student a day ahead of the next lesson. He makes sure that student understands what is going to be taught the next day. That student becomes the “expert” on the material during the next day’s lesson. Students who have questions are expected to go to the expert first.

Simple. Shortest blog post I ever wrote. This strategy would be effective in showing students that support and help doesn’t always need to come from the teacher. Remember, formative assessment is about feedback. That feedback doesn’t always need to come from the teacher.

Rick has a great story about what he does when it is a weak student’s turn to be the daily expert. I won’t ruin it for those of you who haven’t seen him yet. And if you haven’t, I highly recommend him. He’s an engaging and compelling speaker.

Two by Four

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.