Category Archives: problem solving

Impressive

I spent today at Baxter Academy. Actually, I’m still here. See, I have a group of students working on Moody’s Mega Math Challenge. They have 14 hours to complete their solution to the problem. The clock starts ticking at the moment they download the problem. That was at 9:00 this morning.

I am impressed that this group, who in class is lucky to remain focused for 35 minutes (in a 55 minute class), has pushed through today with so much focus – I am assuming. You see, I’m not actually in the room with them. I make this assumption based on observations when I go and take some pictures or get a food order. I had to remind them about food, not the other way around.

Prior to today, they had done a bunch of work in class, on practice problems, getting organized, reviewing the modeling & problem solving process. One thing I learned from all of that is that we are definitely teaching these skills here at Baxter Academy. These students never once thought they wouldn’t be able to tackle any problem thrown at them. They would come up with a plan for what to do before the M3Challenge folks sent out their tips or hints.

Here they are, 8 hours into their day.

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That was four and a half hours ago. Now, with less than an hour and a half to go, it’s truly crunch time.

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Tidbits from the Week

I tried my #LessonClose a couple of times this week in my 3D geometry class. The first time I used the Collaboration poll and the second time I used a new Learning from Mistakes poll. Both polls were given while working on the question: “Which Platonic solids can tessellate space?” It was clear that cubes would work, but there was some disagreement about the tetrahedron.

Student comments about how they collaborated included:

  • I was part of the conversation when we were brainstorming answers
  • I participated but did not lead
  • I argued about shapes that would or wouldn’t work
  • Everyone’s voice was heard

Student comments about Learning from Mistakes included:

  • I assumed an angle stayed that same when rotated on the plane. This turned out to be false, and I had to later revise my answer.
  • I forgot Trig, and I may have messed up a little bit with the answer of #1
  • A few times, we started to follow a faulty train of logic, based on angle assumptions, that messed us up until we figured it out.
  • I wasn’t exactly wrong just wasn’t very sure. My group had made a prediction that was actually true.

I’m finding it difficult to decide on the specific poll to give. I might create a new poll that let’s the student select which aspect they would like to give me some information about. This is the heart of the PDSA cycle – learning quickly and making changes to improve.


Earlier this week, Dan Meyer wrote this post about explanations and mathematical zombies using z-scores as an example. In the comments I shared an activity that I’ve used, one that I posted about last year. It so happened, that it was time for that activity again this week. In both classes, students were able to develop the reasoning behind the formula through discussion. One student even described what we were doing as a transformation to the standard normal distribution. Never once did we write a formula.


Once again my Flex Friday work connects me with students who are new to Baxter Academy. This year we are teaching them skills through workshops. This Friday’s morning workshop employed design thinking: asking and answering “How Might We … ” (HMW) questions. (For more about this approach check out The Teachers Guild or the d.school at Stanford.) Once you have a bunch of HMW questions, you attempt to answer each one in as many ways as you can think of. My colleague called this “thinking around the question” and illustrated it with the question, “What’s half of thirteen (or 13)?” And here’s what we came up with.

img_20160916_120055631

 

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“That’s a Big Twinkie”

You know the quote. You can watch the clip.
twinkie

Yesterday I brought a box of Twinkies to class so my students could check Egon’s math. They measured twinkie dimensions and borrowed scales from the science lab. They made the classic error of not paying attention to units. And they argued – consulted – with one another. They didn’t quite finish during class yesterday. I predict that they will be somewhat surprised by the results.

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Dan Meyer, Girl Scout Cookies, and a Nissan in the driveway

A couple of days ago, Dan Meyer posted this new 3-Act problem about boxes of Girl Scout cookies being packed into the back of a Nissan Rogue. It came at exactly the right time for my 3D Geometry class. We’re entering the last couple of weeks, so I’ve been posing review problems each day to help them remember all of the topics we’ve tackled. As I was considering the plan for Wednesday, one dropped right into my inbox.

We watched the Act 1 video. I asked for questions:

  • How many boxes are there?
  • How many different shapes?
  • How many cookies?
  • How much do all those boxes weigh?
  • How much would that cost?
  • Could they have fit more?
  • Could they have packed them more efficiently?

I asked for estimates, including guesses that they thought were too low and too high: The too low & too high estimates ranged from 1 to 1,000,000 and the guesses ranged from 206-3000.

I asked for the information they would need:

  • How big are the boxes?
  • What is the cargo space?
  • How much do the boxes weigh?
  • What’s the maximum payload?

I knew that Dan would provide some of this information in Act 2, measuring the roguebut my students are very inquisitive and quite resourceful. They wanted to figure these things out for themselves. And as luck would have it, our principal drives a Nissan Rogue. We also had Girl Scout cookie experts who were quick to point out that not all cookie boxes are created equally. We sent a group out to measure the cargo space while two other groups worked on the problems of cookie box sizes, cookie box weights, and Rogue payload capacity.

In researching the payload, the group found that Nissan noted that the Rogue had a cargo capacity of 32 cubic feet, not the 39.3 cubic feet noted in the video. Guess we need to work on those research skills – clearly. The payload capacity was about 1,000 lbs.

Measured cargo space came out to be about 25.8 cubic feet (or approximately 44,600 cubic inches).

The group researching the cookie boxes decided to take a sample and find some average measurements. As a result, our boxes measured 2″ x 7.2″ x 3.5″ and weighed an average of 9.3 oz (or 0.58 lbs).

Final calculations showed that the Nissan we measured could fit about 885 boxes, which would weigh a little more than 500 lbs. But that was a pure volume calculation and the students knew from previous packing problems that we had done, that the reality would be less than that, and that there would be empty space.

Finally, we watched Act 3 (this is the Nissan version). We noted that the Rogue in the video was a different model year than the one that we measured. The measuring team also noted that the Rogue in the driveway had a floor at the level of the lift gate – it didn’t have the same depth as the model in the video.

So thanks, Dan, for the great set-up and giving my students the opportunity to revisit some of the work they’d done earlier in the term.

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“How Do We Know That?”

questions

I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?

I really felt that before we could go down the road of answering that question, my students needed a bit more knowledge and exploration around these shapes, and maybe some thinking around tiling the plane would help, too. So we spent some time trying to draw them, counting faces, edges, and vertices, visualizing what they might look like with vertices cut off, unfolding them into nets, and wondering why regular hexagons tiled a plane, but regular pentagons did not. We played around with the sides – a lot – and even talked about this thing called vertex angle defect. Then we returned to the question of why only five. Students were able to connect the need for some defect (angles totaling less than 360 degrees) and the ability to create a 3-dimensional figure. Through the investigation, they were able to see that the only combinations of regular polygons that worked (by sharing a vertex) would be 3, 4, and 5 equilateral triangles, 3 squares, and 3 regular pentagons. They could give solid reasons why 6 triangles, 4 squares, 4 pentagons, and any number of other regular polygons could not be used to create a new Platonic solid.

I had not anticipated this question, and had not included it in my plans. But, because it was asked, thankfully, by a student, it pushed us into thinking more deeply about these shapes (and their definition). And, ultimately, my students were able to answer the “why only five” question for themselves.

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Reasoning with Slopes

My geometry class has started investigating shapes on coordinate grids. Doing this reminded them that they knew how to use the Pythagorean Theorem and how to find the slope of a line segment. Here’s an example of what they worked on:

In looking at the slopes of side AB and side BC, which I wrote on the board as slope of AB = \frac{+3}{-3} and slope of BC = \frac{+3}{+5}.

Contemplating the situation, one of my students asked, “If you combine those two slopes, will you get the slope of the other side?” I’m thinking that by “combine” he means “add” and of course you cannot add these two slopes to get the slope of the other side. But instead of saying that I asked him, “What do you mean by ‘combine’?” He responded by explaining that if you add the numerators and denominators separately, it seemed like you would get the slope of the other side. He was thinking about this:

(+3) + (+3) = (+6) and (-3) + (+5) = (+2) which leads to a slope of \frac{+6}{+2}. But why would that be true? Vectors. It turns out that this student was learning about vectors in his physics class. I’m not sure that he consciously made the connection, but he did seem to be thinking about vectors. If you travel from A to B and then from B to C, you will have traveled 6 units up and two units to the right. That’s the same result you would get with vector addition.

Imagine what would have happened if I had asked the wrong question, or just replied without asking the question that I did.

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Why Standardize Normal Distributions

The new trimester started on Monday and I’m teaching a class called “Designing Experiments and Studies.” It’s a statistics class, so we’re starting with a bit about normal distributions. Most of the students in the class are juniors, but they’ve had very little instruction in statistics. They didn’t get it from me last year, so any knowledge that they might have is probably from middle school.

Today, I posed this question:

baby weights

And then I gave them some time to work it out. Here’s what happened in the class discussions (a bit condensed – the actual discussions took about 15 minutes in each class):

S1: The boy would weigh more compared to other boys because the boy is 0.25 pounds away from being one standard deviation above the mean, while the girl is 0.5 pounds away from being one standard deviation above the mean. Since the boy is closer to being one standard deviation above the mean, the boy weighs more, compared to other boys.

S2: But, 0.25 lbs for the boys is not really comparable to 0.5 lbs for girls because the standard deviations are different. I agree that the boy weighs more, but it’s because the boy is about 92% of the way to being one standard deviation above the mean, while the girl is only 75% of the way to being one standard deviation above the mean.

S1: What does that matter?

S3: It’s like if you’re getting close to leveling up (I know this sounds really geeky), but if you’re 10 points away from leveling up on a 1000 point scale, you’re a lot closer than if you’re 10 points away from leveling up on a 15 point scale. Even though you’re still ten points away, you’re a lot closer on that 1000 point scale.

S4: But you’re comparing boys to boys and girls to girls. You’re not comparing boys to girls.

S2: Yes, you actually do have to compare boys to girls, in the end, to know who weighs more for their own group.

Me: How did you figure out that the boy was 92% of the way to being one standard deviation above?

S2: Well, the boy is 2.75 lbs more than the mean weight and 2.75 / 3.0 is about .92. I did the same thing with the girl and got 75%.

At this point I showed them a table of z-scores, kind of like this one and we talked about percentiles. Looking at the table, they determined that the boy was at about the 82nd percentile, while the girl was at about the 77th percentile. Therefore, the boy weighed more, compared to other boys, than the girl weighed, compared to other girls.

I have two sections of this class, and this recreation of the conversation happened in both classes. I’m so happy when my students make sense of mathematics and reason through problems. I never had to tell them the formula to figure out a z-score, or why that might be useful or necessary. They came up with it.

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