Category Archives: MTBoS Challenge

More 3D Geometry

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After my last post, Mike Lawler gave me all of these awesome ideas for my 3D geometry class. Considering that my class has been working on nets, I was most fascinated by the dodecahedron that folds into a cube, which came from Simon Gregg.

When I first watched the gif animation, I just couldn’t figure out what was going on. I thought, “I’ve got to show this to my students!” Thursday was that day. I tasked them with a build challenge. Of course 55 minutes wasn’t enough time to complete anything, but students had drawings (which gave us insight into the construction)

a CAD rendering (completed during a snow day)

and a previously constructed dodecahedron that had been re-purposed (completed during lunch).

So, thanks Mike, for the inspiration, and thanks #MTBoS for being there helping us to support each other.

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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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Stacking Pennies

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There are lots of things I love about my classes – my students are at the top, but I also am enjoying teaching a course for the 2nd or 3rd time. This activity is for a statistics course called “Designing Experiments and Studies.” It’s a course that lasts for one trimester (about 12 weeks) and we have just completed week 4. The Penny Stacking activity is an introduction to experiments. I ran this lesson last week (Jan 12 & 13).

The rules of penny stacking are simple:

  • You can only touch one penny at a time.
  • Once the penny is placed on the stack, you cannot move it.
  • Stack as many pennies as you can without having the stack fall over.
  • Each student stacks pennies once, with either their dominant hand or their non-dominant hand.
  • Students are randomly selected to stack pennies with either their dominant or non-dominant hand.

Before embarking on the experiment, we made the following predictions:

  • More pennies would be stacked with the dominant hand (although a few students disagreed and thought the results would be the same for both hands).
  • A few students thought the ratio of pennies stacked with dominant hand to non-dominant hand would be 3 to 1.
  • The range of pennies stacked with dominant hand would be 15-45 pennies, while the penny stacks from the non-dominant hand would range from 10-35 pennies.

Then it’s off to conduct the experiment. This is pretty tricky given that up to four students sit at a table and our building is so old that if someone walks across the floor upstairs, our floor will shake. Here are the results:

penny stacksHow would you interpret these results?

We calculated the means: dominant -> 30.9 pennies; non-dominant -> 26.25 pennies. It’s clear that, on average, the number of pennies stacked with the dominant hand is greater than the number of pennies stacked with the non-dominant hand. But is that difference in the means (of 4.65 pennies) significant? Is it unusually large? Is it more than what we might expect from randomizing the results?

To check this out, we randomize the results. Each pair of students received a stack of cards. Each card had a result. The partners shuffled up the cards and dealt them out in a stack of 10 (for the dominant hand) and a stack of 8 (for the non-dominant hand), calculated the means, and then subtracted (dominant – non-dominant). They each did this a couple of times and we made a histogram from the results of the randomization test.

pennies

(It’s a Google Sheets histogram – I don’t know how to get rid of the space between the bars)

If you look at our difference of 4.65 compared to these randomized results, it looks pretty common – not at all unusual – to get such a result. If you think that our randomization test was too small (with only 24 randomizations), then you can use the Randomization Distribution tool from Core Math Tools, a free suite of tools available from NCTM. And it’s the only tool that I know that runs this test effectively. Here are the results from 1000 runs, just like the card shuffling but faster.

more pennies

penny summaryAnd you can even get summary statistics that show that our result was within 1 standard deviation of the mean of the results that we got from randomizing the data. Not a very unusual result at all.

We followed this up on Thursday with an experiment inspired by an example from NCTM’s Focus in High School Mathematics: Reasoning and Sense Making – memorizing three letter “words.” Based on the experiment described in the book, I created random lists of three letter words and three letter “words.” The lists of words were meaningful, like cat, dog, act, tap, while the lists of “words” were nonsense, like nbg, rji, pxe, ghl. Students were randomly assigned to receive either a list of meaningful words or a list of nonsense words. They were then given 60 seconds to memorize as many words as possible. Like with the penny stacking, I made them predict what they thought the results might be. What would your predictions be?

 

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Working Together

onegoodthing

Like all schools, my school has committees. And, like all schools, my school has a student government. But, unlike other schools, our student government is a single body: the Student Senate, comprised of one representative from each advisory. This year, the Student Senate split into subcommittees to align themselves with the faculty standing committees. I’m on the Academics Committee – the group charged with looking at curriculum & standards and how they are aligned. On Monday, we met with the Senate subcommittee during lunch to discuss different ways that students can demonstrate what they’ve learned. We talked about the possibility of students creating portfolios and presenting their portfolios to teachers in a particular learning area to have them assess which standards the student has met, and to what level. So, while the teachers keep working on tightening up our standards & curriculum (we’re halfway through year three), our students are developing a portfolio review process.

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About Me

It would appear that I can’t resist a challenge from the #MTBoS. And so, mtbos blogthis January, I’ve been challenged to update my “About Me” page. Feel free to check it out. I’ve updated it with a bit about the school where I teach and what happens when I talk about teaching at a public charter high school.

By the way, we will be graduating our first class this year, and that’s pretty exciting. Even more exciting is the fact that they’re actually getting accepted to college! (Of course they are. But as the first group, this is still pretty exciting for us.)

I’m hoping to blog a bit more this year. A bit more than once a month (or less). But I’m not promising anything. I’m still building brand new classes – which takes a ton of time. I have a couple of drafts still in the pipeline that need some attention. Maybe I’ll get to them soon.

In the meantime. Thanks for the push #MTBoS (and Tina C). And Happy New Year!

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What Time Will the Sun Rise?

This week I begin Exploring the MathTwitterBlogosphere. I’m looking forward to these missions and challenges because I need someone pushing me to find the time to write in this blog. It’s good for me. Like spinach.

This week’s mission: What is one of your favorite open-ended/rich problems? How do you use it in your classroom?

One of my favorite open-ended/rich problems comes at the end of a unit on trigonometric functions. After exploring, transforming, and applying trig functions to Ferris wheels, tides, pendulums, sound waves, … I assess my students’ understanding by giving them some almanac data of  sunrise and sunset times for a specific location on Earth. Their job is to analyze the data and create a trig function to model either sunrise times, sunset times, or hours of daylight – their choice.

The data looks like this

and that makes it somewhat challenging for students to even begin. They are reminded that they should have “enough” data to know if the model they develop fits well. I point out that the times are given to them in hours and minutes, but that they probably want a single unit (hours or minutes after midnight). From there, they are on their own to solve the problem. Usually, they work with a partner.

In the classes that I’ve used this task with, we’ve modified the amplitude, period, and midline of the sine and cosine functions. We haven’t introduced phase shift, yet. So, there is also a reminder about selecting a convenient “Day 0” for the function they choose to model with.

What I love about this task:

  • Students are talking math, asking each other about the number of data points they should use: “Should we just pick the same day every month? Are 12 data points enough?” or “Do we just go every 20th day?” or “What should we use for the first day?”
  • Students are problem solving. They have to convert the times into a single unit. They have to make decisions about which variable to model, when to start, which type of model to use. Then, they can collect the relevant information to modify their chosen function.
  • Students are using technology. Although they don’t have to, it’s really easiest to have the kids making scatterplots on calculators or computers and then graphing their model on top of that. Then they have a built in way to check their work – they don’t have to ask me (the teacher) if they are correct. It shows up in the picture that they create.
  • Students think that working with trig models is really hard, so they feel very proud when they are able to complete this task without any help from the teacher.
  • It’s really easy to grade. Either the model fits or it doesn’t. Kids turn in their data tables and work showing how they calculated the necessary values for their model. This precludes anyone from using the old SinReg command.
  • Even though I’ve used this task for about ten years, it’s a perfect fit with the Common Core math standards (trigonometric functions) and practices. And since I live in a SBG world, this is a very good thing.

My favorite kind of assessment is one where students have to apply what they’ve learned to a different situation. Even though we create lots of different trig models in class, sunrise, sunset, and daylight hours represent a new application. And a new challenge.

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