She was concerned about fairness. Some students who would do well if she did straightforward “lecture->3 examples->homework” classes on this material will likely do worse. As she phrased it “I’m grading their innate abilities since I’m never teaching it. Is that fair?”

What does it mean, fair? If she gives this take-home assessment, it will *measure something different from other tests*. Most tests in a traditional environment measure ability to memorize, ability to process lecture and notes, ability to apply class-discussed examples to new (but usually similar) situations. Good ones test some ability to apply techniques in a larger context, or with a new wrinkle. These are fine and good things to measure. But ability to read and learn information from a text *without explicit guidance* is also a reasonable thing to measure. It isn’t the *same*, but that doesn’t make it less *fair. *I told her she should do it if she wants to.

And this made me think: what do we **want** to measure in a student? If grades are part of the process – and there’s certainly no way we will be removing them systemically any time soon – what would we ideally like those grades to reflect, and how are we doing at making that so?

There are many things I like about traditional tests, which I admit to using almost exclusively (along with homework completion and occasional classwork problems and exercises) at the moment. I think they are a good way to assess fluency in vocabulary and basic procedures, and a good traditional test will include problem-solving questions that call for “putting it all together.” But mathematical modeling is usually tested in a limited way or not at all. Same for explanations and derivations, which I emphasize heavily in class discussion then barely use at all on assessments. I think I often struggle putting in questions that can be viewed as “subjective,” as if my tests need to be unimpeachably numeric, a “numbers don’t lie” attitude that is specious but hard to shake. I also think I struggle with the idea of making a test too “hard,” both so as not to crush their spirits (because they don’t like feeling confused and it’s easier to cave to that than teach them to be fine with it) and because I feel weirdly tied to the arbitrary 70=D, 80=B, 90=A system we have created for ourselves. But by keeping things simple, I am overemphasizing the skills of the good memorizers and example-appliers and underemphasizing the skills of the problem-solvers and explorers, who would be able to demonstrate their brilliant attempts given more space to try. All with the goal of making it so that ones that freeze and give up under pressure can still squeeze out an arbitrarily-defined 70%.

It is also a hard truth that straightforward tests are easy to make, grade, and find time for, while problem solving and modeling can be hard and time consuming. And students *expect* tests. They expect a life of mostly-coasting with occasional nights of insane study. So it is simply the path of least resistance to take that path; no complaints, except in my own mind. And I definitely overemphasize a life of no conflict.

I don’t know the solution. I do not have the time or energy to do the sort of constant assessing of thinking that I think I would do in my ideal world. With two young children at home, there is not the window to look through exit slips every day and leave detailed feedback. It isn’t maintainable. And I haven’t yet found a way to keep my workload bearable and also do an honest assessment of the skills I actually find most important. And so, when I realize that I don’t have “enough grades” I write another test with a sigh and continue in the status quo.

]]>While grading my third assessment in the system – and my third assessment that tested only one standard – I realized that I needed to separate out algebra and arithmetic errors from the content itself. The standard was “Can move back and forth among area, perimeter, radius, and diameter of a circle.” And on the assessment, almost every student did that perfectly; equations were perfectly set up, understanding of the basic concecpts was clearly there. A few girls missed things under this standard – confused diameters and radii or circumferences with areas – but most of the mistakes on their papers were **procedural **not **conceptual**. Algebra, arithmetic, units, or simply following directions. So I added two learning goals to my list called “Units, directions, rounding, completeness” and “Algebra and arithmetic”. I gave each student a perfect score of 100 on those topics, and they can *only lose points*. 1 point here, 2 points there, 3 or 4 points for egregious issues. So instead of giving a student who left units off of everything a 90 on the Circle Area and Circumference standard, I can give them a 100 on it but take 4 points off of this rolling standard that is constantly important.

At the end of the unit, I will give a mastery badge to anybody who has a 90 or above in those areas, since they will have shown consistency in these areas.

]]>Here is the highlight version:

The spreadsheet has one tab, called “Master,” which controls most of the system. On this tab you define your standards or learning goals for the unit/course, put your roster, and assess each standard. Students can earn from 0-4 stars on each standard (though you enter it as a standard 1-100 grade). You can also award Mastery Badges to students when they have, in your estimation, mastered a goal. You can also add feedback, links to assessments or resources, and notes either to the class, a student, or to a student in response to a particular standard.

All of this is automatically imported by the students’ own personal tab. They can see their grade on every standard, which ones they have mastered, how many they’ve mastered, and what level this makes them. Their goal is to level up to the highest possible polygon! This can then be published to the web so they can see it as their own personal Mastery Portal, with links, feedback, and so forth that (should) automatically update. It looks like this:

All of this was awesome, but I immediately saw some things I could do to improve the sheet; my summer of working as a spreadsheet programming professional really came in handy here!

I made three major changes to Kyle’s sheet:

- I sped it up. The original student sheets relied extensively on HLookup and VLookup calls, which are amazing spreadsheet functions that also tend to be rather slow when used a lot. I was able to use some different commands (Index and Match) to speed up the calculation of student spreadsheets by limiting the number of times one sheet looks up data in another.
- I added some automation. Specifically, I added scripts to automatically create the student tabs from the roster, automatically get their URLs so you don’t have to copy and paste links one-by-one from a menu, and delete all of the student tabs if you need to start over. I also added a script to force the student tabs to update if for some reason they don’t change when you enter a score. Thanks to Alice Keeler for her TemplateTab script from which I started and got inspiration.
- I added a little bit of customization that was not in the original (though not as much as I’d like to add eventually)
- I added a tab with directions, so there’s no need to reference a blog post to remember how to work it. =)

I’m very excited to use this spreadsheet for this unit. Thanks so much to John Orr and Kyle Pearce and all of their inspirations for the brilliant idea and work – I think this could be a real game changer.

I’ve never mastered formative assessment. I have a hard time putting emphasis on it and time into it for the same reason that my students do – I’m a procrastinator and heart and work better with deadline and attached value. So we worked through a right triangle trig and area test in my geometry class, and some students *never really got it*, and I didn’t know that. Then there were some students who just screwed up on test day – as one student told me later, her test grade was collateral damage to a lab report. And there were just as many positive surprises – which is nice, but still tells me I didn’t know what I was doing.

I wrote this tweet:

And then, after finishing grading and writing necessary comments, I stayed up a little later, in a tired-but-annoyed fugue state. I stumbled upon this tweet by Kyle Pearce:

Go ahead and read his post. I’ll wait.

I followed the link, read the post, and realized that I needed to try it out. Immediately. And I couldn’t wait. I decided that I would try it starting the very next day, with the unit I had already been doing for two days with my geometry classes: circles.

The next morning, I had 80 minutes to prepare for my first geometry class. I was able to get their names entered on Kyle’s spreadsheet, create a sample web page to show them, get some preliminary standards written up, and make assessments for my first three standards – naming parts of a circle, sketching parts of a circle, and moving between area, circumference, and diameter of a circle. You can see the assessments I made here: **Circle Standard Assessments . **The assessments are not particularly clever or good – I made them fast – but it’s a start. I ended up doing standards 1 and 2 at the end of class with them, and assigned standard 3 as homework – they can either do it for practice and attempt it again later OR pledge not to use notes/books/others and do it for Mastery (we have an honor code that makes it reasonable for me to offer this option).

I’m really excited about this. I think it is going to be awesome. My students were excited as well.

If you want details on implementation, see this post on how exactly to use the spreadsheet to implement this system, with some modifications I added.

]]>I have since learned that StatsCrunch can output very similar data (and is nice enough that I may do subscriptions for it next year and use it more thoroughly with my students), but I did not know that tool this morning.

So what did I do? Made a google sheet that you can paste data into OR enter information for another google sheet (url, tab name, etc) and it will generate computer output for you. I may not ever use it again now that I have figured out StatsCrunch, but for data already in a google sheet this is quite possibly easier. I put it here for anybody who may find it useful.

]]>Here is the basic structure of MANY of the chapters of my text:

- Big Topic
- A bit about it generally
- How it applies to categorical data
- How it applies to quantitative data

As specific examples, we have:

- Chapter 1 – Exploring Data
- Analyzing categorical data
- Analyzing quantitative data
- Describing quantitative data

- Chapter 7 – Sampling distributions
- What is it?
- Sample proportions
- Sample means

- Chapter 8 – Confidence intervals
- The Basics
- Proportion intervals
- Mean intervals

- Chapter 9 – Hypothesis tests
- Basics
- Proportions
- Means

- Chapter 10 – two sample tests
- Comparing two proportions
- comparing two means

You get the picture.

I understand the appeal and purpose of setting things up this way; means and proportions are by far the most common things we do statistical studies and inference with, and the general process of, say, constructing a confidence interval is the same in both cases. But my students have struggled this entire semester with keeping straight the differences between them. This is partially my fault for failing to make the distinctions clear, but I have to wonder; would it be better to do everything to do with categorical data and proportions FIRST?

Here’s what I envision:

- Designing studies (chapter 4 in my book). Crucial to any longitudinal, semi-project-based approach, since I will want the students to design our at least have input on the design of our longitudinal projects
- A categorical data project. Not dissimilar from my Skittles activity but broken into pieces and interspersed with additional practice. This project, which will get touched on every single day, will require them to learn about:
- Graphing and representing categorical data, including discussion of frequency tables, marginal distributions, conditional distributions, etc. (Chapter 1 in my book)
- Some aspects of Probability; using our sample data as the “true” value, imagine other future sampling options. (Chapter 5 in my book)
- Sampling distributions of proportions (Chapter 7 in my book) – this will be our first quantitative data, so…
- Displaying quantitative data with graphs (dotplots and histograms) and describing them with numbers (mean and standard deviation) (more chapter 1)
- Normal curves (Chapter 2)
- and now we have neough for… Confidence intervals and hypothesis tests with proportions (chapters 8 and 9)
- Comparing two proportions (chapter 10)
- chi-square goodness of fit and independence tests (Chapter 11)

Do you see how one giant, connected series of investigations involving *exclusively categorical data and quantitative data about that categorical data*, could lead to all of these ideas?

Once that project is finished, we start fresh and do data that was quantitative *from the start*. Two quantitative variables that can be connected, so we analyze each variable separately, do inference on each variable separately, then combine them for regression and regression inference.

Finally, fill in any gaps: probability ideas that never came up seem like the most obvious ones.

The chapters felt very disconnected this year. A significant part of that was teaching (I was basically relearning the curriculum myself as I went, after all), but a big part is structural as well, and I wonder if this sort of linear, longitudinal structure would be helpful. What would I lose by doing this?

]]>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: https://drive.google.com/open?id=0B-C-lUvv4rQ4ZG9hcnh1azdHNjQ&authuser=0

]]>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.]

]]>Here’s a summary of topics I’ve covered so far:

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

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

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.

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…

We watched the Act 3 Video and clapped when the boat [spoilers!] made it through.

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*?

**TL;DR**

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).

]]>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.

**Changes?**

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 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.

]]>Having just finished up our giant soccer goal geometry project, I’ve given myself permission to work a little less hard on that class for a week or two – my students will actually enjoy a trip into traditional math land, I think, and that will free me up to focus on AP Statistics (well, and the grades and comments that are due on Wednesday).

AP Statistics has not always one well for me this year, and I spent some time over spring break strategizing. I need to teach a few more concepts, help many of them fill in the gaping conceptual holes that our year has left behind, and give them plenty of practical test-taking strategies, all in 8 weeks. To do that, I need to re-establish trust and fun. Rapport. It needs to be a class they, and I, look forward to, and right now it just isn’t.

So, thus: the candy strategy.

I know I’m not the first to realize that statistics and candy go hand and hand – there are pages and pages of candy activities (with a special focus on M&Ms) on the internet. I read a lot of those, today. I think that I can piece together those ideas, along with some of my own, and review *every single major concept we have learned* with candy. Specifically, for me, a combination of jelly beans, starbursts, skittles, and M&Ms.

Today I wrote a 10-page document (see link at the end) that reviews frequency tables, pie charts, bar graphs, two-way tables, sample design, proportion confidence intervals, 1-sample proportion z-tests, and introduces chi-square goodness of fit tests as a cap, using nothing but skittles. We will work through the activity alone and in groups for the next two class days. Along the way we will review vocabulary, conditions, graphing techniques on calculator and by hand, and learn something new. Plus, eat some skittles.

It feels a little cheap, but I hope that candy will improve the relationship with my students enough that we can tackle this final quarter together with positive attitudes. One more to go!

]]>On the whole, though, things are going well! All pictures below were created by students.

First, students calculated how long various pipes in our goal needed to be, given only “4 feet tall by 6 feet wide”

Then they calculated the angle they would need to rotate the diagonal beams:

The next day, we learned how to use the 3D modeling software 3DTin and built scale models of our goals. (click the image to go to the 3D model itself)

This 3D modeling software is a bit buggy (as they all are, apparently) and a couple of students have panicked or lost work, but most have handled it well. Some students didn’t put their pipes “inside” the fittings, so then their diagonals were too short and they had to troubleshoot. Others rotated the wrong angle – the student above, for example, rotated 53 degrees from the vertical rather than the horizontal at first and couldn’t figure out why the angles didn’t work. Good problem solving work to fix that.

The snow days came next, and students had to do what they could at home. Since the missed class day was going to be working on group projects, this was sadly little.

When we returned to class yesterday, though, we jumped right in!

Every student measured and cut at least one piece of pipe while another student managed and helped them out. While that was happening, the students who weren’t working on the goal itself were taking videos and pictures (which is where these came from) and working on figuring out a set of instructions for assembly to distribute with the goals.

By the end of class today, all of the pieces were cut and goals had been test-assembled in every class. I now have one more class day to help the girls finish their documentary videos and assembly instructions.

At this point, I think the project is a qualified success.

- The right triangle applications were practical and useful, and it was helpful for them to see a use for tangents before learning about them.
- 3D Modeling is in general a fabulous exercise in geometric thinking and coordinate systems, and when they ran into problems even more useful.
- They are learning to use saws safely, always a nice skill. Also, “measure twice cut once”.
- While making their videos, they have to work out the best way to present the mathematics of their goals, aiding mathematics communication.
- Making explanatory diagrams is
*always*a useful task in STEM communication and, I believe, quite mathy.

- Technically topical or not for a geometry class, I want to add budgeting for supplies to the project.
- The video product won’t really come together well, I fear, and the time devoted to it isn’t particularly intellectual for many of them. Need to adjust it a bit or a lot.
- If possible, it would be nice to open the design options a little wider to give them more ownership of the project design. Give them even less. If I pepper the project pieces over the whole quarter that could be more tenable.
- 3D Modeling is awesome, but would be better if we’d done something earlier with the software so less time needs to be spent on that. Find a way to introduce the software in first semester.

My conundrum is with the AP curriculum itself. Y’all, it is *so. packed.*I am sitting in the room right now watching my students taking a chapter test that covers 1-proportion Z-tests, 1-sample T-tests for the mean, and (most difficultly) the interpretation of all of the various points and numbers that come up in the process of taking those tests. This test will probably have an average (raw) score in the 65% range, though I do curve it (as the AP test itself is curved).

We had five class days to work on this chapter. 400 minutes of class. This isn’t nothing, of course, but it simply. isn’t. enough. to explore these in a way that helps them really *grok *what’s going on. And slowing down is not really a viable choice if we want them to have a chance of getting the questions that come up from chapters 10, 11, and 12.

So they muddle through. They learn trigger words that remind them which calculator command to use where, and they memorize phrases like “We are 95% confident that…[this thing] is not [that thing]” and they cram, cram, cram. I have excellent students who work very hard, and none of them are in danger of failing the class, but they *aren’t enjoying it. *And, perhaps worse, most of them *won’t remember it or apply it.*Part of that is me – this is my first year teaching it, so I muddle about quite a bit, usually don’t figure out the best way to explain things and frankly I don’t think I did a great job in the first couple of chapters setting up the overall meaning of statistics and really hammering the concepts when the numbers were easy. That would have made the interpretation parts simpler now. Next year will be better.

And part of it is the weird position AP stat takes at our school (and many others, I tihnk) – it’s the “easy” AP math, right? Well, sort of. It’s definitely not hard in the same *way* that AP Calculus is, but it is PLENTY hard. Doing well on the tests requires a **higher** level of logical thinking and explanation skills than calculus, but it doesn’t require the same level of comfort with numbers and formulas (though it requires some!) So it’s easier for some people, but not all. However, the rest of my department doesn’t necessarily understand that, and our recommendation process doesn’t really know how to screen for who will be ready for the class, so I have several who, simply, aren’t.

But a **big **part of it is the AP curriculum itself. We are required to submit a syllabus that includes projects, but mine was/is thoroughly shoehorned in and non-rigorous, because there simply isn’t TIME. We’re not going to get through the last chapter as it is! And that is sad. If there is any class that is ripe for project-based-learning, it is statistics, but there is simply no possible way to do it that way with any primacy inside the AP curriculum. There’s too much on the test.

AP Physics B used to have a similar problem, and the AP committee, bless them, eventually split it into two years to allow for more exploration and depth. I don’t think that’s necessary for AP Stat, I think we just need to cut some things. I’m not sure what we could cut – that’s not really my decision – but as it is, there is too much to do in one year, even with extremely intelligent, hardworking students with great mathematical backgrounds. I don’t want to go shallower, I want to *deeper* with *less*.

I think I will find a way to compromise, for this year. I can look through the last two chapters, write up a summary and sample questions that hits the absolute basics of them for the test – basically, intentionally go *very shallow* on those, expecting that students will miss some of the harder questions on those topics – and then spend time looking deeper at our earlier chapters. It might result in the same test scores, and may revitalize my faith in the curriculum.

Does anybody else have this problem? Suggestions on how to get some depth of understanding without compromising their test scores?

If you’re like me, you see White and Gold (in shadow!) If you’re like many others, you see Blue and Black. If you’re like some, you see Periwinkle and Gold. But the real dress has been confirmed as blue and black.

What’s also undeniable is the colors of the ACTUAL IMAGE are a light purply blue and a brownish orange – so arguably blue but definitely not black.

So there are three questions here:

- Why did the camera screw up so badly, as those colors are clearly NOT the darkish blue and black the dress is supposed to be? How is it possible that the camera did such a bad job?
- Why do my eyes/brain see the white and gold, even though it’s really blue and brown?
- Why do other eyes/brain see the blue and black it actually is even though the picture is terrible?

Here is my theory for question #1. The dress is indoors, in a store perhaps. The area behind the dress and camera is lit with bright incandescent lighting. The area in front of the camera and dress is lit by the sunlight we can see in the photo streaming in from the upper right.

When you take a photo with a camera in automatic mode, it has to make two big decision: what kind of light is it in and how much light should it “suck up”. These are called the “white balance” and the “exposure.”

This camera saw the sun and said “Oh! Sun! I’m in daylight!” and set the white balance to daylight. Since the sun is kind of bluish and incandescents are kind of reddish, this means that the image will be shifted a little bit redder under daylight than incandescent, so making the WRONG decision here means that all the colors in the dress shifted red: the dark gray (not black, really, because it’s shiny) of the trim shifted toward brown, and the blue of the main fabric shifted toward purple.

The camera ALSO realized that it needed to get a bunch of extra light, since otherwise we wouldn’t be able to see the dress in contrast with the bright sun, so it cranked the exposure up. This resulted in everything – the black and the blue – being brighter than they really should have been.

End result? A black and blue dress looks light purple and brown.

Here’s my proof:

I printed some blue and black stripes, taped it onto an opaque file folder, and put scotch tape all over it to make it a little shiny (the dress is shiny!).

Here is the original image and the printed out version taken with automatic options under good light.

Here is the same card, photo taken with bright incandescent lighting – however I manually set the camera to daylight AND overexposed it. In other words, I think THIS is what the camera did wrong to get the image we got.

I think that no matter WHO you are you can interpret that as potentially being white and gold! If anything I exposed it too much.

Okay, so that helps answer the question of “why does the picture look so bad?”

Second question – why do I see something even *worse?* After all, the actual photo is light blue and brown, and I see white and gold – even further away from the true value. This one I can’t prove, but my theory is that my brain is doing the same thing the camera did, and, just like the camera, getting it wrong.

When white things that are lit by daylight are in shadow, they actually look a little bit purple. My brain is smart. It says “Oh, look, it’s in shadow, so that means it will look a little bit purple even if it’s white. Therefore, I’m going to assume those purple bits are actually white, since the position of the sun makes me believe the dress is in shadow.” My brain also assumes the brown bits are shadowed, so it lightens them up and gives them some shininess, creating the illusion of gold.

In other words, I THINK the picture below is what I’m seeing. This picture is a white-and-gold card, in the daylight, but shaded.

But I’m not.

Third question – why do some see it the right way, and others flip back and forth? Well, this one is harder. I think this particular picture is amazing because it somehow managed to find that perfect middle zone of perception where things can go different ways. Some of our brains do what mine did and make the wrong call about lighting, making the bad picture even worse, Some go the other way, and make the right call, so their brain “fixes” the bad picture. This is what most of our brains do with most pictures – we are CONSTANTLY “fixing” bad pictures in our heads, and some people still get this one right. Others go either depending on the moment, light in the room, brightness of the display, or other factors. It’s in a gray zone. Personally, I’ve only managed to make the original picture look blue and black once – it was by projecting the image, zoomed in, on a white board, in a dark room. Suddenly it looked blue and black to me. Otherwise, I have always seen white and gold.

What makes this more frustrating ans simultaneously appealing than other optical illusions is simply that this one *was a complete accident!* Usually, to be this confusing, illusions have to be designed.

]]>The students will be broken into groups of 6 to 8. Each student will, individually:

- Decide what size pieces of PVC they will need using the Pythagorean Theorem for the diagonals
- Figure out the angle they need to rotate the diagonal pieces using tangent ratios
- Create a scale 3D model of the diagram using the online 3D modeling software 3DTin. I will provide them with this template for the fittings. This will require them to work and think in three dimensions, of course, as well as deal with scaling.
- Write a report explaining all of the above, with extra focus on the mathy bits.

The whole group will then break into subgroups to accomplish the big three group goals:

- Create the goal itself. Every student will measure and cut at least one piece, while one student manages the process.
- Make a video documenting the design, building, and assembling process. Several students will work on this as writers, editors, and videographers. (Mathematical communication)
- Make a document that contains instructions and diagrams for assembling the goal from the components, since some of our goals will be given to faculty members for their children in disassembled form.

We are lucky that we have a brand new maker space on campus where we can do the measuring and cutting with support from excellent staff members.

I’ve spent a TON of time trying to think through the organization and group job assignments that can make this effective, so I’m crossing my fingers that it goes well. It’s a big risk because in terms of “testable content” we are not doing much new in these five days – really, tangent ratios, a 15-minute mini lecture, is all we’re getting in that domain – but I think it will be an extremely valuable *general STEM* project. I’ll let you know in a couple of weeks.

You can access all of the support documents I’ve created for this project, including some instructions on using 3Dtin and job descriptions for the sub projects, in this shared folder on Google Drive. The .one file is the original – I created these documents using OneNote, which I use to distribute all notes and handouts to my students, but it is also available in Word (may have formatting issues) and PDF if you prefer.

]]>The problems are great, but I’m not comfortable (yet, I hope) grading them as high-value summative assessments, which means that I don’t have a “reasonable” distribution of grades. I could spend a day consolidating our knowledge from the problems and short-lectures – areas of various shapes, similarity theorems, scaling factors, area unit conversions – and throw together a quiz on it, but it just feels so… blah. This is always my issue when I get excited about process, is that grading process is *hard* and never feels fair, but of course it’s expected that I will have a reasonable semi-normal grade distribution, and the students who will eventually struggle with SAT/ACT/etc math should probably be at my low end. So I need a quiz.

I don’t want to do a quiz.

It’s probably what I’ll do, though.

]]>Here it is. Still stressing the idea of scaling area, but adding in a wrinkle of a regular polygon: we’ve never discussed their area, but they should be able to make rectangles out of them, I think, or maybe approximate them as a circle.

The following images are from Home Depot’s web site. The first kit allows you to create an octagon-shaped patio (a regular octagon) and the second kit allows you to expand it to make a larger octagon. The diameters of both octagons and the side length of the larger one are shown.

- Find the scale factor from the smaller octagon to the expanded octagon and use it to label the side length of the smaller octagon.
- Using what you’ve learned about scaling factors and areas, do you think it is reasonable for the company to charge the same price for the expansion as they do for the original kit? Why or why not? Your explanation should include mathematical work.
- Find a way to estimate the approximate (or exact) area of the octagons, then use that data to support (or confuse) your explanation in problem 2.
- After answering all of the questions above, either look in your book on page 441 for the formula for area of a regular polygon OR ask Mr. Griswold to give/show it to you. Use this formula to calculate the areas of each of the octagons. Did your method work? Is your answer to #2 the same?

Here was the first one, which I made up. Ensworth is another independent school in our area whose colors are orange and black.

This problem led to a very interesting and fruitful discussion both on the relationship between linear and area scale factors and the related (and commonly done wrong) idea of converting square units.

Darlene, a student and tennis player at Ensworth, has decided that she wants to pull a prank on Harpeth Hall before her big tennis match and paint their tennis courts orange. She needs to know how much paint to buy, so she uses Google Maps to print out a picture of the tennis court, as well as a picture of the Google Maps scale marker, as shown.

- Use a ruler to find the physical scale factor from the picture of the tennis courts to the ACTUAL tennis court. Your scale factor should NOT have any units in it – it should simply answer the question “How many times bigger is the real world than the picture world.”
**Hint:***You do not need to measure the tennis courts to answer this, only the scale.* - Measure the length and width of each of the two tennis courts to one decimal place (just the part inside the lines) then use your scale factor to determine the lengths and widths of the real tennis courts.
- What is the area of the picture of each tennis court (include units, round to 2 places)?
- What is the area of the actual tennis court (include units, round to 2 places)
- If each can of paint covers 50 square meters, how many cans will Darlene need to paint both of the courts orange?

Darlene went through the steps above and found the following information (these measurement numbers are made up and will probably not match yours):

Darlene first measured the scale and found it to be 2.1 cm long. To find the scale factor, she wanted 10 m / 2.1 cm. To get the units the same, she multiplied 10 m by 100 to get 1000 cm / 2.1 cm = 476.2 . Thus she knew the real world was 476.2 times larger than the picture. She measured the length of each court to be 4.3 cm and the width to be 1.9 cm

She then calculated the area of each court to be 4.3*1.9 = 8.17 sq cm

Next, Darlene wanted to calculate the area of the real tennis court, so she multiplied the area times the scaling factor: 8.17 * 476.2 = 3,890.5 sq. cm . Since that number was large, she then divided by 100 to get an area of 38.9 sq m per court. Thus for two courts she would need 78 sq. m of paint, so she bought two cans. She wanted to move fast, so she recruited her friend Lana to help her.

Lana wasn’t convinced that two cans was enough paint, so she tried the math again. She got the same measurements and scale factor as Darlene: 4.3 cm for the length and 1.9 cm for the width, with a scale factor of 476.2 She used the scale factor to find the length of the real court to be 4.3*476.2=2,047.6 cm, or 20.5 m, and the width of the real court to be 1.9*476.2=904.8 cm or 9.05 m. She then multiplied those numbers to get an area of 9.05*20.5=185.3 sq m, meaning she would actually need 4 paint cans per court, or 8 total.

- Who is right? Or are they both wrong? You should assume their measurements make sense, and only criticize the method. Explain.

I am back to teaching high school math this year, after a few years teaching middle school science, and I did a mediocre job first semester. My AP stat class was muddled and confusing and my geometry classes were boring and uninspired.

We’ve been back in class a week and a half, and I’m doing better. I’m feeling charged up by the idea of letting my students struggle with larger problems that don’t have every detail filled in – Problem Based Learning, I guess, as described on http://www.emergentmath.com . I don’t know how I’m going to have the time to do it much in AP Stat – due to my muddling first semester we have a TON of ground to cover before the exam – but geometry is just ripe for this.

I started simple, at the VERY beginning of a chapter on similar polygons. We had reviewed ratios and proportions before doing this problem, but had not actually discussed similar triangles or polygons. I basically improvised this problem, but it worked really well.

It was a sunny day, so I split my girls (I teach at an all-girls school) into teams of 3-4 and told them I was going to take them outside and they would need to measure the height of our athletic field’s flood lights. I gave them three hints:

- They would have a meter stick
- I could use my phone to put “pins” in a Google Map wherever they wanted me too (using satellite view) then measure the distance between pins.
^{1} - It was sunny outside.

This is a very traditional problem, setting up a proportion between objects and their shadows, but I gave them absolutely *no other hints*.

I gave them 5 minutes to strategize, then we walked to the field (it was cold, but nice!) and then I gave them 7 minutes to make their measurements and tell me where to drop pins on the map. Then we returned to the classroom. I gave them the distances they asked me to find, then they found their solutions and wrote it up on a Google Slides presentation that they presented to the class. Here is one example (student’s name was removed and replaced with “student”):

The whole process – setting up the problem, brainstorming, walking, measuring, computing, creating the presentation, and presenting – took about 60 minutes. I was able to give them a useful 10-point classwork grade based on my observations and their presentation. Everybody did well, and felt success. And there was good spring-board potential from the question “Why is the ratio between the object and its shadow the same in both cases?”, which we proved using the AA Similarity Postulate a couple of days later.

It could have been done better, if I had been more ahead of it. But I think it went really well, and it inspired me. We will be doing more things like this. Stay tuned.

** ^{1}**This is a pretty easy and fun trick. Long press on Google maps on your phone, and a pin will drop. Move it exactly where you want. Then you can save the pin to your account by clicking the star. When you log into that account on your computer ,the star will show up, and you can right-click and use “Measure Distance” to measure the distance between any two points on Google Maps. This saved time and effort involved in using the ginormous tape measures from the science labs.