Trying to find math inside everything else

Posts tagged ‘math education’

Lab –> Lecture –> Assessment

Next year, the weekly schedule at my school is going to be 2 double periods for a particular class (alternating sections on an A/B day schedule) with a single period for every section on Wednesday. Because of the new schedule, I wanted to make a new structure for my class, which is the title of this post: Lab –> Lecture –> Assessment.

There are roughly 30 proper weeks of learning in the year, so I figured I would have 30 Learning Goals to cover, and do one each week. I would introduce each learning goal with a “math lab,” which may be an actual lab (like the popular M&M Lab for exponential growth/decay) or a 3 Act problem or something else that the students can really engage in before getting down to the nitty-gritty and symbolic way mathematicians deal with the problem.

The next double wouldn’t necessarily just be lecture, but it would be the abstraction of what we did the lesson before, including lecturing on technique and practicing what we’ve learned. Then assessment could be any number of things, but will almost certainly involve a targeted quiz.

Seems like a good structure, right? Problem is, while I have a lot of good labs and problems for most of the topics (and will keep improving), not all of them do. Particularly:

  1. Radicals – Simplifying & Arithmetic
  2. Unit Conversion
  3. Solving in Terms Of
  4. Box-and-Whisker Plots / Percentiles
  5. Scientific Notation
  6. Statistics Vocabulary (univariate/bivariate, etc.)

 

So my major goal this summer will be to develop something for each of those. The rest I can fall back on what I have, even if I don’t come up with something new/better. But these have nothing. My first task/idea is to develop a board game about radicals. That’s still under development. Any other suggestions would be appreciated.

Category:

labs, math

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Which Is Bigger?

I was going through some old stuff and dug out this gem:

It caused some disagreement when I first posted it, and my students jumped right in, arguing with each other and demanding to know who was right.

It’s a good way to show how mathematical language is precise, and it’s important to choose your words carefully.

Category:

ela, math

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3 Acts – Potatoes

I tried this out today in class (and will repeat tomorrow), and it worked out quite well. So now I want to share, my first 3 Acts problem.

Act 1

Potatoes – Act 1 from James Cleveland on Vimeo.

The question I intended to be asked was “How many of each potato do I need for the recipe?” or variants such as “What does he do now that the scale is broken?” or “Did he buy enough potatoes?” Those were all asked, along with some others.

Act 2

The video shows some things (how many potatoes I bought the first time, and the cashier says the totals), but it’s easier to lay that out when the students ask.

After that, they also wanted to know how much the potatoes cost, so I provided that.

But that’s all the information I can give: my scale is broken and I didn’t take the receipts from the cashier. Luckily, this is enough.

Act 3

After we calculated the weight, we compared when I weighed them in my “new” scale.

I wish I had my digital scale for a better Act 3, but it’s actually broken (and the calculations I had to do when it was inspired this problem) and the analog was cheaper. The solutions you calculate (.36 lb and .43 lb) are pretty close to the values on the scale (which I peg at .37 lb and .5 lb).

The problem itself, in terms of the system of equations involved, is not that complicated (because I am using it to introduce the concept of elimination), but for the students who solved it quickly, I had a trickier problem up my sleeve:

Extension

What if I had bought 3 Idaho potatoes on the second trip? How can I figure out how much each one weighs now?

The Complete Problem

Potatoes.zip

Category:

anyqs, lessons, math

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Steepest Stairs and Wacky Measurements

After reading Dan Meyer’s post mentioning a Steepest/Shallowest stairs contest, I decided to go for it. But Dan had them do it for homework after they knew what slope was. I decided that I thought steepness of stairs would be a great way to introduce the concept, and then we can have the contest after. So Ms. Barnett (my co-teacher) and I went around the area and took lots of pictures of stairs we can find. Then I put them up as a warm-up and asked them which were steepest and which were shallowest.

In every class, there was near-universal agreement on which stairs were the shallowest (the top-right), but lots of different votes for the steepest. So then I asked them, “How can you know? What does it mean to be steep?” I got a lot of good, intuitive answers from that (My favorite was that something is steeper when it is closer to being vertical). I asked them what they needed to know to find out which was steeper, and they said we should measure it.

But what exactly should we measure? That took a little cajoling and probing, until we eventually decided on the height of the step and how deep it was. So I gave it to them:

Alright, now we have these measurements, what can we do with them? I lead them on a discussion on how best to use these numbers (a ratio), and we looked at another example. This is a pretty clear example (1/2), but not all of them are. So we used our estimating skills.

 

 

 

 

 

 

 

And my personal favorite…

(They really asked if I had 11 cell phones. I guess my Photoshop skills are better than I thought.)

The best part of these pictures is that they so naturally prompted them to question the units of measurement. “That one is cell phones, but the other one is hands. How can we compare them?” And so it’s natural to talk about slope as a ratio with no units. I didn’t have to artificially insert it. I even had a picture of a curved slide at the end, so we could theorize about the steepness of that.

Finally at the end I mentioned the contest. Unfortunately, I’m afraid Ms. Barnett and I did too well finding stairs. I’ve had students say they’ve been looking, and some say they found some (but don’t have pictures yet, though they have one more week). I hope someone can knock us off our thrones:

SLIDES:

Steep Stairs (PDF)

Steep Stairs (PPT)

Category:

lessons, math

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No Better Feeling

At the beginning of yesterday’s lesson, I threw up this monster of a problem:

I told my students that, by the end of the lesson, they would be able to solve it. They flipped and freaked out. “No way, Mr. Cleveland, not going to happen.”

In all 4 sections, 1 hour later, every student correctly solved the problem. And they were all so proud of themselves for doing so. There’s no better feeling than that.

Category:

joy, student response

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How to Lie with Statistics – Stations

Cover of "How to Lie with Statistics"

I base my statistics unit on the book How to Lie with Statistics, by Darrell Huff, because I think it contains a lot of really important information that everyone should know to live their lives and to stop themselves being taken advantage of. (Which is one of the majors benefits of learning math, as it’s so easy for people who know math to swindle people who don’t.)

The problem is, though, all that important information that I think is so critical is not that important in the minds of the NY Board of Regents. Last year I went ahead and taught the whole unit around it anyway, but I’ve learned the errors of my ways when I found that my students were lacking in, say, the ability to make a box-and-whisker plot.

To compensate, I (along with my fabulous co-teacher) did a one-hour lesson that I broke up into stations, with each station representing a different “Lying Technique” that I wanted the students to learn about. (I had already covered The Sample with the Built-in Bias and The Well-Chosen Average as a normal lesson prior, because those are still relevant topics.) Each station lasted about 10 minutes, with some time for wrap-up and transit.

First page of Station 1

I also did this to support the project I had them working on, which is the topic of another post.

Stations in pdf form.

Category:

lessons, math

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They Don’t All Go So Well

I’ve known what my next post was going to be about for some time, which is why it’s been so long between posts, as I’ve been putting it off. The failures are less fun to write about, but it’s just as important when your lesson is a bust. Now I have lots of other things I want to write about, so more posts in the next few days.

Shortly after my successful Egyptian Fractions lesson, I wanted to tie a lesson into another ancient society they were learning about, so I decided to teach the Mesopotamian Number System. The idea was that we’d reinforce some ideas about exponents, place value, and scientific notation by working with another base.

The problem: working with another base is hard, especially if you’ve never done it before, and sexagesimal is not a great place to start, even with the boost I got with the fraction lesson. Introducing the idea with binary probably would have worked, but I didn’t have the time to do both and also teach the cuneiform and do the activity. Unfortunately, to save the activity, the basis of the understanding got cut. Which left me with a fun but useless activity.

I used hours:minutes:seconds as an analog to help understand base-60, but because they got that they couldn’t move past it. I gave them numbers to translate and had them carve cuneiform tax tablets (and they learned about Babylonian taxes), but that didn’t work out too well.

And then I didn’t even get nice product to display for too long, because they were too brittle.


As I said. A bust. Or rather, busted.

Category:

history

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What’s My Set?

For the past two lessons I’ve taught about sets, including set notation, union, intersections, and complements. To practice what they’ve learned, I had them play a game called What’s My Set? I originally came up with the idea because I wanted the students to get out of their seats in the middle of the double period, and so organizing themselves into the sets seemed like the way to go.

I gave them all badges as they entered class with a number on it. They got them totally interested in what the numbers were for, but I just expressed the need for patience. When it was time to use them, they were interested.

We played it twice. First to practice their ability to read Set Builder notation, write it, and translate into roster notation. I would display the sets in Set Builder on the board, giving each set a location, and they would have to move to that part of the room. But it’s up to the people in each set to make sure they have everybody that belongs there, since I would check if a whole set was correct, and so the stronger students were forced to help the lost ones to get their points. I would give a point to the first set to complete itself. The interesting thing is, because the sets change, though the points are per team, really they are individual. I didn’t give a prize, but they didn’t seem to care.

In the second part, to practice unions, intersections, and complements, I just left 6 pre-defined sets on the board:

Then for each round, I would write on the board something like O ∩ P goes to the front of the room and (O ∩ P) complement in the back, so they had to think a little bit more for this round.

Category:

math

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