Multi-part area problem

I merged the study of triangle trigonometry and polygon area in my geometry class, since they go together very well. For their test, I created this multi-part area problem I like quite a bit. You can click the image to access the Geogebra sketch I used to make it on GeogebraTube if you’d like to download and modify it.

area problem
Click the image to see the Geogebra sketch I used to make it on GeogebraTube.

Correlation with candy is hard

I had a hard time coming up with a candy activity for bivariate data that was really good. I ended up kind of cheating, and wrote an activity that has students gather lots of data points, one of which is “how many jelly beans can you pick up in one no-palm pinch?” and then letting them check the correlation of that with any of a wide variety of data points (height, hair length, finger length, age in days, etc.).

I don’t think it is as good, or will allow for quite as deep a discussion, as the earlier activities, but it is still decent I think.

I still haven’t actually taught regression inference, so I’m using this activity right after a brief mini-lesson on interpreting and using computer output for regression inference. I definitely will not be tackling this topic at the depth our book does; will focus instead on practicality and the basic idea of extending our knowledge of T-tests and confidence intervals to a more uncharted area. Some of the data sets they decide to plot will probably NOT satisfy all of the conditions for regression inference, though, so that could lead to a good discussion.

You can find all of my AP Stat candy activities, including this one (forgive any typos, it was made very late last night) here:

How accurate is the official M&M data really? Add your data!

Yesterday I did a chi-square goodness-of-fit test with my class comparing a large sample of M&Ms – over 800 of them – to the data that is provided by Mars for the true count of M&Ms. We got a p-value of 0.0002, which seems crazy. So now I simply need to know how accurate their data actually is.

So here is my proposal: if you do an activity that involves counting the various colors of M&ms in any random sample, any year,  add your data to my collection using the form below. If you buy a bags of M&Ms and just feel like counting it, add data. If you want to put young children to work counting M&M colors, add their data too. If you have data from previous years, great! Add that too, and maybe I can add a time component to the analysis. If we get enough people on board, we should start to get an accurate picture of the true proportion of M&M colors, to see if Mars tells it true.

I have embedded the form and part of the analysis below, but you can also click here to access the full Google Sheet with the analysis of the total data, analysis by year (currently only 2015 makes sense obviously) submission-by-submission analysis, and original results.

[Note: I also had to make my own chi-square cumulative distribution function for Google Sheets, borrowing some source code from this online calculator at UCLA. If you want to know how to use it, or make your own custom Google Sheets functions, e-mail me and I can advise.]

AP Stat Candy Review so far – and a new one!

As of now, I have written 3 AP Statistics review activities centered around candy. You can access them all by clicking here. They are taking us around 2 class periods each, which is a full 160 minutes of class time; not a short commitment by any means, but I’m enjoying the long game connections a LOT.

Here’s a summary of topics I’ve covered so far:
Activity 1 – Skittles
This activity works itself up to chi-square goodness of fit tests (which, when I gave out this activity, we had not actually covered). Along the way we covered:
  • relative frequency tables
  • sample design, including
    • voluntary response sample
    • convenience sample
    • simple random sample
    • cluster sampling
    • stratified sampling
    • multistage sampling
  • collecting a sample and creating frequency and relative frequency tables
  • displaying categorical data with pie charts and bar graphs
  • two-way tables
  • marginal and conditional distributions
  • sampling distribution of sample proportions
  • confidence intervals for population proportions
  • 1 proportion z tests
  • Chi-square GOF tests
Activity 2 – Starbursts
This activity centered around a claim that the proportion of orange Starbursts is the same as the the proportion of orange skittles. We had never done 2-proportion z-tests, so the activity spends a lot of time walking through the logic of them and reviewing symbols and their meaning.
  • power of a test
  • calculating power (as an exercise in reviewing tests and CIs)
  • 1-proportion z tests (again)
  • 2-proportion z-tests
Activity 3 – M&Ms
In this one, we introduce quantitative data by measuring the mass of the M&Ms, while also reviewing some categorical data.
  • gathering data
  • review chi-square GOF with several possible distributions and sample sizes (exploring power, etc, along the way)
  • check percentage of actual rejected hypotheses against alpha
  • represent quantitative data visually and use SOCS
  • confidence interval for sample means
  • effect of sample size and alpha on power
  • 2-sample mean t-test
I honestly don’t know if this sort of long-range, vertical review is better than more traditional review or learning, but the students seem more engaged then I was getting them before, and I think they are grasping some of the big-picture conceptual things better than before, at least, which is satisfying for all of us. No idea if it will translate to AP scores.

3 Act Trial (and tribulations)

Inspired by the general enthusiasm for the process in the MtBOS, I tried my first 3 Act Math Lesson(s) today.

If this is somehow the first place you’ve heard that term, Dan Meyer has been the main force and originator of this structure, and I’ve also seen good explanations of the structure from Dane Ehler, Geoff Kraal, and many other. I think this series of blog posts by Dan Meyer is maybe the best place to really start understanding the structure.
We just finished basic triangle trigonometry and word problems, and are about to start applying trigonometry to area, so I was looking for problems to help us hone in on those, and I found a couple: Boat on the River by Andrew Stadel and the Big Nickel task by Kyle Pearce and others. 
I had never done this before, so I decided to try something new and NOT overprepare. Didn’t make a handout, or a structure, or a OneNote page, or a powerpoint, or anything. Just watched the videos myself, made sure I understood the basics of what happened, and wing it.
It kind of worked.

Boat on the River 

Act 1

I told the students “We will watch a short video. Ask whatever questions you can about what you see.”
We watched the act 1 video three or four times and I elicited and wrote down questions.
Result from one class:
Some of the questions were silly (9), some were answerable by watching the video we had more carefully (1, 2, 12, 13ish, 10ish), some were ineffable (16), some required outside knowledge (11, from a student who sails), and MANY were math. I was going for #3, so that’s good. To solve!

Act 2

I showed them the pictures on the PDF provided by the task. 

(There is another image that establishes a scale for these pictures of 1 in = 10 ft). Since we already knew how to work with sine ratios, this was easy for the girls to work with, and they found that the boat will make it… barely. I let them work on this in partnerships for 5 minutes, explain their work, and then…
Act 3
We watched the Act 3 Video and clapped when the boat [spoilers!] made it through.

Overall outcome

A success. I don’t feel like the students were hugely more invested in this than in more “normal” word problems, but if I increase the enthusiasm of my presentation I think we can get there. I do like the basic structure, and calling for questions brings me back to the first steps of the Engineering Design Process from my science teaching days; a useful way to get ideas.

Big Nickel

Since we had never done area problems that required trig, this was a harder, longer task for my students, but that wasn’t the only stumbling block I ran into here.

Act 1

After watching the short video at the top of the task,  my students thought the monument was funny, but their questions were silly, snarky, and didn’t really go where I was hoping. I eventually had to straight up give them the question I was hoping for, the last one.
I don’t know how to inspire the sort of sideways silliness needed to find that question; a perfectly designed 3 act may do a better job of getting them there simply from the video, and maybe this one isn’t perfect. Or maybe I just need a sillier group of students to get there.

Act 2

“What do we need to know?” was actually more interesting here. We needed to know dimensions of the big nickel and the real nickel (height and thickness) and my students pointed out that we also needed to know if it were hollow  or solid – basically is this a problem about surface area or volume! This part was fun, as we scoured the wikipedia entry for the information we needed:
  • It is 9.1 m tall, 0.61 m thick.
  • It is made of steel plates attached to a skeleton – basically hollow!
  • A real nickel is 2.12 cm tall, 1.27 mm thick
I told them this was all they needed and let them loose. We have studied the area and angles of regular polygons before; I told them they may want to remind themselves of those formulas and techniques, as well as look for right triangles to use trig. I would say about 2/3 of each class got very close to finding the area of both the big nickel (in square meters) and the little nickel (in square cm). Some needed more hints and guidance than others. This part was slow, and frustrating, but went well. We got to the end of class and I asked how many nickels it would take to tile the front of the big nickel, and then we had to remember that square meters and square cm are not the same unit – so we got to review square unit conversions again, which we tackled earlier in the year. The math of this was good.

Act 3

… there isn’t really one. We can find other people’s solutions, but unfortunately there is no way to get the kind of satisfactory resolution from this problem that we got from the boat. I think this may actually disqualify this as a true 3-act problem. It is a GOOD problem, but, frankly, nobody has ever built the Big Nickel out of nickels, so we don’t have an answer to the question outside the theoretical.
How do you create the resolution of Act 3 when the problem at hand can’t actually be done?


My first attempt at 3 act problems was fun, and led to good math, but needs work. I think I need to add some structure, and I need to think more about some of the components, specifically how to get the resolution of Act 3 when the problem does not allow for it and how to get the “right” question out of Act 1 when it’s not necessarily obvious (but IS creative and interesting).

Soccer Goals – final thoughts

The Great Geometry Soccer Goal project ended a couple of weeks ago just before spring break, and I’m finally ready to finally debrief. I already wrote many of my thoughts at the mid-point of the project, so this one will be brief.

If you’re interested in seeing the products, here are the videos they made. They vary in polish, but all show the scope of the project well This was not the only product, but it is currently the only publicly available one; need to do some name removing, etc, before I can share snippets from reports or instructions.

If you want the project materials, they are here.

Was it worth it?
Yes. I may streamline it a tad to take 4 class days instead of 5 next year (our class days are 80 minutes), but it’s worth it either way.

Did they learn anything that will help with their test?
Probably not. We learned tangent ratios, and practiced the Pythagorean theorem and some arithmetic, but especially with spring break right after they still had to be retaught tangent on our return.

So what DID they get out of it?
Visual thinking with 3D shapes. Recognition of the practical value of certain math topics. Mathematical communication skills. Learning to use a saw. Value of precision, but also seeing where there is room for a little error. Fun! Honestly, though, i think the mathematical communication and 3D manipulation skills are probably the most “measurable” for a math class.

I will probably use TinkerCAD next year instead of 3DTin, just because 3DTin seems to be a dying project, even though I prefer it a bit. I may skip the 3D modeling entirely, since it’s time consuming for the outcome, but I really like it so I’ll probably keep it. I will tweak the job assignments, but not much; my hard work in advance worked out well there. I will pray for no snow days. I may add other elements: have them calculate the cost of the goal using the Lowes website, for example.

I highly recommend this project, or one inspired by it, if you have the ability to buy and cut PVC with your students. It was a lot of fun.

The power of power

Today in AP Statistics we continued the Great Candy Review by comparing Starburst proportions to the skittles proportions; specifically, we started trying to decide if the proportion of orange starbursts could be equal to the proportion of orange skittles.

The activity covers both 1-sample proportion tests (by assuming that 20% of skittles are orange, as we surmised, and comparing our starburst sample proportion to 0.2) and then 2-sample proportion tests by dropping that assumption and comparing our actual samples, but before I dove into the tests I decided to spend some time dwelling on power.

This is my first time teaching this course, and I haven’t always figured out until too late what aspects to prioritize. Power is hard, it comes near the end of a chapter, and I skimmed it.

Big. Mistake.

Really thinking about the power of a test, even calculating it, turns out to be an extremely good way to really think about the underlying concepts of statistical inference. It took us 30-45 minutes to really get through the first two pages of the packet, which I didn’t expect, but I saw light bulbs going on all over the room as we slowly grasped the big picture. When students really understood the power of the test – when they realized that even if our friend is wrong there is a 75% chance we won’t be able to “prove” it with these techniques and understood why… well, they were obviously annoyed, but they also clearly understood the limitations and execution of inference tests better than they have all year.

It was a good moment.

Next class we will actually take a sample of starbursts and conduct the tests. I doubt we will be able to decide with high confidence that they proportions different (even though they really ARE) and now, hopefully, students will understand better why.

See this folder for the candy activities we’ve done so far.