## Great Classroom Action

Mitzi Hasegawa links up a clever game called Entrapment that helps student understand these reflections, translations, and rotations that (I'm told) now constitute the entirety of K-12 mathematics:

One table debated the transformation highlighted in today’s picture. Is the figure on the grid a reflection of the figure or a rotation?

Federico Chialvo asks his students "How safe is a Tesla?"

Jonathan Claydon poses the deceptively simple challenge to spin a measuring tape handle at exactly one mile per hour:

Most students experience miles per hour in a car. The challenge would imply "I should spin pretty slow." Yet, the handle isn't very long. And there's the curiosity. What does 1 mph look like at this small scale? So they spun.

Mary Bourassa repurposes a classic party game for the sake of learning features of quadratic equations:

They would ask "Is my h value positive?" but then either interpret the answer incorrectly or not be sure whether the person they had asked truly understood what a positive h value meant. They all figured out their equations and had fun doing so. And they want to play again next week when the whole class is there. I'll be happy to oblige.

## Great Classroom Action

Cathy Yenca gives Graphing Stories a go and the going gets tough (and interesting) when she runs into Christopher Danielson's step-function:

The last video we tried today was Ponies in Frame. I heard the most awesome muttering as soon as the video began. “Oh! I get it. This one’s discrete.” [..] It wasn’t all lollipops and rainbows. A comment laced with negativity that resonated with Lauren and me was an outburst that “graphing used to be so easy, and this just made it hard.” How would you take a comment like that? What does that comment say about the student’s true level of understanding?

Jonathan Newman has his students analyze parametric motion by creating stop-motion videos.

Nicora Placa reminds us that the one of the best ways to assess a student's understanding of direct proportions is to give her an indirect proportion and see if she treats it directly.

At a workshop last week, the following task caused a bit of confusion. "If a small gear has 8 teeth and the big gear has 12 teeth and the small gear turns 96 times, how many times will the big gear turn?" Several participants were convinced it was 144.

Megan Schmidt uses one of the Visual Pattern tasks and surprises us (me, at least) with all the different interesting equivalent ways there are to express the pattern algebraically:

They came up with the following pre-simplified expressions for the nth step:

2n(n+1) + 3
1 + (n2 +n2) + (n+1) + (n+1)
2[n(n+1)] + 3
2n(n+1) + 3
3 + [(n+1)n] + [(n+1)n]
3+2(n+1) + 2[(n+1)(n-1)]
2n2 + 2n + 3

For each of these, I had the student put the expression on the board. I then had different students explain the thinking of the student who came up with the expression and relate it to the pictured pattern. I saw a real improvement here from when I had them do this activity the first time last week. I had many more students volunteer to explain the thinking of their cohorts and much less hesitation to work out what the terms in the expressions represented.

## Great Classroom Action

Nathaniel Highstein engineers a counterintuitive moment about graphing, one that subverts his students' expectations and creates intellectual need for new knowledge:

I love this problem because the answer becomes totally clear when you make a time vs. elevation graph – and the answer violates nearly everyone’s expectations and leads to a surprise! Many students got stuck in their initial guess, and even when we went over together what the intersection of the two lines implied, they tried desperately to draw a version of the graph where the two lines didn’t intersect. When they figured out that even skydiving down wouldn’t work, some resorted to teleportation.

Option 1. Explain how to use place value.
Option 2. Explain how to use place value while first asserting its usefulness to humanity.
Option 3. Explain how to use place value while first putting students in a position to experience life without any kind of place value.

Anna Weltman took option three:

Three fingers is Na Na Na. But four fingers – now, that’s a lot of fingers. Na Na Na Na is quite a mouthful and it’s getting hard to tell the numbers apart. Here is where the cavemen bring in a new word. Na Na Na Na is Ba.

We continue counting. Ba Na, Ba Na Na (giggles), Ba Na Na Na – now what? The kids think until – Ba Ba, of course!

I remember Hung-Hsi Wu's frustration with incomplete pattern problems. Paraphrasing him: "You can't find the next term in the sequence '4, 10, 16, … ,' because it could be anything." He's right, of course, but you can find a first term and Chris Hunter turns that fact into an icebreaker and a robust exercise in justification. He asks students to "Extend the pattern 'Ann, Brad, Carol, … ,' in as many ways as you can."

Not mathy enough for you? Remember, not all teachers will have a positive attitude towards mathematics. This is a safe icebreaker. You can always follow it up with the mathier "Extend the pattern 5, 10, 15, … , in as many ways as you can."

The first week of Exploring the Math Twitter Blogosphere asked teachers for their favorite tasks. Lots of people mentioned Four Fours. Megan Schmidt offers us an interesting cousin to that task and a useful description of what makes it effective for students.

The students also developed some interesting strategies, like grouping pairs that totaled 16 and 25. By the end of the 30 minutes, every single student had arrived at the correct solution. I’m not sure if it was the physical manipulative or the puzzle-like feel of the task, but I was so proud of this group of kids.

My guess: it's the puzzle-like feel. (Extra credit: What makes a math task feel puzzling?)

## Great Classroom Action

Bob Lochel has his statistics students take a "Rock, Paper, Scissors" prediction robot down:

Anyway, the NY Times online Science section has shared an online game of “Rock, Paper, Scissors”, where you can play against a choice of computer opponents. The “Novice” opponent has no understanding of your previous moves or stratgey. But, the “Veteran” option has gathered data on over 200,00 moves, and will try to use its database to crush your spirit Here's what we did.

Sarah Hagan links up the definition of a function to dating advice:

After this short conversation, I think I saw some light-bulbs come on. There was laughter, and I heard several girls discussing how they were going to ask the next guy they were interested in if he was a function or non-function. They decided this was problematic, though, because he wouldn't know what they were talking about if he hadn't taken Algebra 2.

Campaigns for "Literacy / Numeracy Across The Curriculum" always seem to wind up a joyless exercise in box-checking for teachers outside those disciplines, but Bruce Ferrington's school approached schoolwide numeracy with some whimsy:

The boss has decided that teachers are going to wear a number around their necks. No, we are not all convicts doing time. It's a cunning plan to get the kids to look at numbers in new ways. The whole idea is that the students are not allowed to use the names of the teachers. They need to call them by a number combination or calculation that equals their number.

Bryan Meyer takes a rote numerical calculation (multiplying numbers in scientific notation) and adds several mathematical practices with an extremely canny, extremely simple makeover:

We decided that if we wanted kids to talk, they needed to have something rich and complex to talk about and make sense of. After brainstorming some different options, we turned the computational question into a conceptual one.

## Great Classroom Action

James Key uses a function monster to illustrate transformations:

f(x) is a function monster, and it can only *eat* numbers between -2 and 4. Now we define g(x) = f(x-3). We know that f eats numbers from -2 to 4. What numbers can g eat?

Cathy Yenca uses a number talk to draw out the distributive property:

I love how this scenario never fails me. Inevitably when I ask – not for the final answer – but the process and thinking that students used to find the answer, someone shares that they thought of “outfits” … 3(20 + 25) … and someone else shares that they thought of shirts and jeans separately … 3 • 20 + 3 • 25.

In the middle of a lengthy and fun post describing his first day of school, Andrew Knauft asks his students which number in the set {9, 16, 25, 43} doesn't belong and why:

Here was a student, on the first real day of class, evaluating an argument independently of her own person bias, without forgetting that bias! (She believed her reason, for a different number, was more convincing, so the argument she read, although good, wasn't good enough to sway her off her choice.)

Also in the vein of constructing and critiquing arguments, Andrew Shauver asks which image-preimage reflection is "best" out of a set of imperfect reflections rather than which one is "right":

That question opens up a lot of potential thought-trails to wander down. As I did this activity with students today, the class settled on three criteria for rating these reflection attempts. The first is that the image and pre-image should be congruent. The next thought was that the image and pre-image should be the same distance away from the line of reflection. Finally, the students thought that the segment connecting the image/pre-image pairs would should be just about perpendicular to the line of reflection.

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