# Category Archives: teaching

## Stacking Pennies

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:

How 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.

(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.

And 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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## Intro to Statistics (Unit 2)

During the first few days of the new unit students explore relationships in the data we had collected about the class.

### Back to the Class Data

Looking at class data gives us the chance to ask questions about relationships among the variables. Here are some questions my students came up with:

• Is arm span really equal to height?

The easiest way to dig into this question is to look at a scatter plot of the data. So, we plotted the variables, along with the line height = arm span.

We noted that two people were on the line, two others were very close, and the rest were either above or below the line. What do those points above the line mean about height and arm span for those people? What about the points below the line?

• If my hand span is longer than my wrist circumference, then shouldn’t I be able to wrap my hand around my wrist and touch my pinky to my thumb?

One hundred percent of students had longer hand spans than wrist circumference, but only a couple of students could wrap their hands around their wrists.

• Is the age (in months) related to any other measure?

It would seem that none of the other variables is a good predictor for age in months. It also seems as if age vs height has a negative association. Huh?

### Digging a Little Deeper

If the line height = arm span doesn’t describe, or predict, that relationship well, then what would do a better job? We added a “movable line” and adjusted it until it looked about right.

Our line predicted that height = 0.85 * arm span + 26 cm. Wait, what? Height is 85% of arm span? And what is that +26 cm all about? It made for an interesting conversation, especially this question from a student: “How can a person who has an arm span of 0 cm be 26 cm tall?” Which prompted: “What does an arm span of 0 cm even mean?” I certainly don’t have definitive answers to these questions. What I can do is encourage the curiosity, the conversation, and point out that the relationship we discovered is for these measurements. Does it make much sense to use our calculated relationship to make predictions about heights for arm spans that are relatively far away from the data we collected?

### Correlation, Causation, Outliers, Influential Points

All of these topics follow from this initial discussion about the class data. Ultimately, students once again find their own variables of interest and complete an analysis demonstrating what they’ve learned. This time topics included unemployment rates, marriage rates, divorce rates, distances & temperatures of celestial objects, height & weight, obesity rate & life expectancy, and mean snowfall & mean low temperature.

Once again, the variety of topics that interested my students is greater than what I could have come up with. More importantly, because they chose their own variables, they were interested in analyzing the data and answering their own questions.

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## Intro to Statistics (Unit 1)

Statistics & probability in high school is often saved for 12th grade, though some progress has been made with integrating linear regression into algebra classes.
My school operates on trimesters, so each class is only 12 weeks long. We’ve created an Intro to Statistics class to focus on descriptive statistics during those 12 weeks. It’s really designed for students who are entering high school, not leaving it. I probably should have created this post a couple of months ago, since the term ends on Tuesday, but I’ve been a little busy.

### All About the Chips

Early in the term we investigated claims made by Keebler and Chips Ahoy about their chocolate chip cookies. Of course, in order to really investigate, we needed to dissect the cookies and count up the chips. Here are our results (from this term):

• Fifty percent of Keebler cookies have more chips than 100% of Chips Ahoy.
• Keebler has a mean of 34.4. chips per cookie. With 24 cookies per package, this means there are approximately 860 chips per package.
• Chips Ahoy has a mean of 25.9 chips per cookie. With 35 cookies per package, this means there are approximately 907 chips per package.
• Although Keebler has fewer chips per package, they have more than 25% more chips per cookie (on average) than Chips Ahoy. Keebler would need to have an average of 32.4 chips per cookie for their claim to be true. They had an average of 34.4 chips per cookie, which is more than 25% more chips per cookie.

Students were asked to write an introductory paragraph and a concluding paragraph. Here’s one introduction:

Are they lying? That’s the question we asked ourselves when we conducted tests to see if either Chip’s Ahoy or Keebler told the truth in their advertisements. Chip’s Ahoy promised 1000 chocolate chips in every bag, and Keebler promised 25% more. Our findings surprised us.

The findings followed, and then this conclusion:

We believe, based on our findings, that Chip’s Ahoy told the truth, while Keebler tried to get away with a misleading slogan. While Chip’s Ahoy had approximately 907 chips per package, which is 93 less than they promised, it would be unreasonable to expect our estimate to be exact, as some cookies may have more chips than others. Because of this, we must grant Chips Ahoy some leeway, as it could simply be our estimate was low. However, Keebler promised 25% more chips than Chips Ahoy. However, the total number of chips in Keebler was actually less than Chips Ahoy. However, we believe “25% more” may be referring to the number of chips per cookie, not per package. Because of this, Keebler may be technically telling the truth, but they are misleading consumers. Chips Ahoy was telling the truth all along.

### All About the Class

We also collected some data about the class, including height, arm span, and kneeling height. Students were asked to apply what they learned from the cookie activity to the this new data set. They represented the data graphically:

And then described what they saw:

The height is skewed to the left, whilst the kneeling height is symmetrical. Kneeling Height has a small interquartile range, and is less spread out than height. The minimum Height is larger than the maximum kneeling height. Kneeling Height and Standing Height do not share a single point.

They are similar because they are both a measure of distance/height. They are different because a person’s kneeling height will never be greater than their standing height, which leaves interesting data with you compare the two.

There is less variation in kneeling height than there is in standing height. No one in the class was so tall their kneeling height was greater than the minimum standing height recorded.

Or:

Height: The data for height are skewed to the left with a median of 170.5 cm and an interquartile range of 10 cm.

Armspan: The data for armspan are skewed to the right with a median of 166.3 cm and an interquartile range of 11 cm.

Comparison: The median of both sets of data have a difference of 4.2 cm and the interquartile range has a difference of 1 cm. The Height data are skewed to the left while the armspan data are Skewed to the right.

Conclusions: In conclusion, the rule of thumb that you are as tall as your arms are long is mostly true because the median of both data sets is only 4.2 cm off and the fact that the interquartile range is but one centimeter off proves this further.

### All About What They Learned

The first unit of the course ends with students finding and analyzing their own data. Data choices included movies, bass fishing, hours in space, world series appearances, touchdowns scored by the Giants and the Cowboys, wealth vs age, costliest hurricanes, and daily high temperatures for Portland, ME and Berlin, Germany. What I love the most about this assignment is that students are able to investigate something that interests them and show me what they’ve learned.

They always come up with topics that I would never think of!

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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:

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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## A few of my favorite things

Following the lead of the good folks over at One Good Thing, I’m sharing a couple of fun tidbits from my life at school this week.

Today a student came to ask for help with a trig problem he had. He’s part of a team building a 3D printer and stand and he needed to figure out how long to make a brace of some kind. (I don’t really know exactly what his team is working on, but that’s the gist.) He had everything set up properly, but it wasn’t making any sense to him. Turns out, his calculator was set to radian mode. Yay that he recognized that something was wrong. Yay that he asked for help. Yay that he knew what he was doing.

One of our math classes this term is called Euclidean Geometry & Introduction to Logic. The teacher (not me) has been focusing on precise communication of reasoning. The other day I observed a student in my advisory ask for some peer feedback on a proof. The first student he asked had been out sick for two days, so he very kindly declined. Then next student, also a member of my advisory, gave very solid and constructive feedback about how the proof could be improved. I love it when they talk math with each other.

My Introduction to Stats class was dealing with correlation vs causation this week. They were presented with these two variables: time in seconds spent draining a full bathtub, water depth in cm, and asked to identify the explanatory and response variables. Some students saw the draining time as the explanatory variable and others saw the size of the bathtub as the explanatory variable. The debate that ensued was engaging, animated, and enlightening. Plus, I was able to unleash the voice of a 9th grade girl who has been afraid to speak about math before that moment. Another student commented on her way out of class, “I’ve never had such an argument about bathtubs before!” I love it when we can respectfully disagree and have interesting conversations.

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## Baxter Academy Shadow Day

Yesterday was a “Shadow Day” at Baxter Academy. That means that most of our students were off on a job shadow of their choosing. I’m anxious to hear about the shadows that they were able to arrange during the snowiest week of the winter, so far. I would have checked in today, but we have another snow day – the third this week.

Anyway, while our students were off doing their shadows, we had about 120 prospective students, interested in attending Baxter Academy next year, join us for a “simulated day.” The students were placed into 16 different groups, each led by a couple of current Baxter students through a day of classes that included a math class or two, a science class or two, humanities, and an elective or two.

I co-taught our modeling class with one of our science teachers. This is the introductory math & science class at Baxter. It’s technically two sections, but they are integrated and teamed up so that the two teachers are working with the same groups of students. Sometimes we meet separately, as a math class and a science class, and sometimes we meet together. I’ve written about the class before, and the kinds of modeling we have made them do.

But what do you do with a bunch of 8th graders who are are with you for only an hour? Introduce them to problem solving through with this TED talk by Randall Munroe. And then take a page from Dan Meyer’s Three Act problems – a page from your own back yard: Neptune*. A brief launch of the problem and off they went. Not every group was able to answer both parts of the question: How big is the Earth model and where is it located? But most groups were able to come up with a solution to at least one part.

The point of the day was to provide a realistic experience of what it’s like to be a Baxter student. We grouped them together with others they didn’t know before walking into the building. We asked them to collaborate to solve a problem they’d never seen before. We asked them to do math without giving them directions for a specific procedure to follow. We asked them to share their results in front of strangers. We gave them an authentic Baxter experience.

*For more information about the Maine Solar System Model, visit their website. It’s really a rather amazing trip along this remote section of US Route 1. I’ve done it – I’ve driven through the solar system.

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## Modeling Projectiles

My Functions for Modeling classes are ending on Tuesday. These are the introductory math classes at Baxter Academy. This course is paired with Modeling in Science, which has a focus on science inquiry and the physics of motion – kinematics. The final assessment is a ballistics lab, where the student groups have to measure the launch velocity of their projectile launcher and, along with some other measurements and a few guesses, identify the best launch angle and launch point to fire the projectile at a vertical target. The students are not allowed to have test shots or simulated shots. They are expected to gather the necessary data and complete all of their calculations before testing their theory with one, single shot. The target is fairly forgiving, but students are still amazed when their predictions result in a projectile going through the target.

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## Problem Solving with Algebra

That’s the name of one of the classes I’m teaching this term. We have trimesters. So each term is 12 weeks long and we have a week of “intersession” in between the terms. Except that this first term is not quite 12 weeks. The expansion area of the building wasn’t quite finished when we started school, so we had some alternative programming called “Baxter Foundations.” It included stuff like my Intro to Spreadsheets workshop. Classes started this week. And one of my classes is called Problem Solving with Algebra. I came up with that name, and I honestly don’t know exactly what it means. I have a rough idea, but it could go in a lot of different directions. Mostly, I want my students (and all the students taking this course) to think and puzzle and use algebra and solve problems.

Then I got an email that Jo Boaler has published a short paper called The Mathematics of Hope. In it she discusses the capacity of the human brain to change, rewire, and grow in a really short time based on challenging learning experiences. We’re not talking about learning experiences that are so challenging that they’re not attainable, but productive struggle. Challenging learning experiences that produce some struggle, but are achievable. The ones that make you feel really good when you solve them. You know the ones I mean.

So I decided to start this class with a bunch of patterns from Fawn Ngyuen‘s website visualpatterns.org. The kids are amazing. They jumped right in. Okay, so I taught most of them last year and they know me and what to expect from me, but seriously. Come up with some kind of formula to represent this pattern. Kinda vague, don’t you think? And I’m pushing them to come up with as many different formulas as they can, and connect those formulas to the visual representation. For example, an observation that each stage adds two cubes to the previous stage would result in a recursive formula like: C(n) = C(n-1) + 2 when C(1) = 1 (which is a recursive formula for pattern #1).

On Tuesday, different groups of students were assigned different patterns. Wednesday, each group presented what they were able to figure out. Some had really great explicit formulas, while others had really great recursive formulas. A few had both. Most were stumped at creating an explicit formula for pattern #5, pattern #7, and pattern #8.

Tuesday night, I received this email from Sam, a student:

“After staring at the problem for 2 hours, (5:45 to 7:45) and scribbling across the paper as well as two of my notebook pages, I am still unable to find a explicit equation. Then, reading the directions, I realized that the way they are worded allows the possibility of no explicit equation, as well as the fact that I only had to come up with equations as I can find. So after 2 hours, several google searches, lots of experimentation and angry muttering, I decided I have all that I can muster, and must ask you in the morning.”

I left them with the challenge to find an explicit formula related to one of these patterns. Their choice. Just put some thought into it before we meet again on Monday. Wednesday night, Sam sent me this followup email:

“After another hour at work, I found the explicit formula. I realized that the equation was quadratic, not exponential, and youtubed a how-to for quadratic formulas from tables. I kid you not, the man said the word “rectangle” and from that, I solved the problem. Then I watched the video through and took quick notes for future reference.”

Then, Dan Meyer posts this: Real work vs. Real world. Makes me think – as always. What am I asking of my students? This is real work – they are engaged and they are thinking. Sam, and the others, were not going to be defeated by a visual pattern. The fact that they are working in a “fake world” doesn’t matter.

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## The Power of Interesting Questions

Today I led two groups of students through an introduction to spreadsheets as part of our Baxter Foundations workshops. Our framing question was, “How much is that Starbucks habit costing you?” Many students, of course, said $0, but we widened the question to include other vices, like Monster drinks, Red Bull, going across the street to Portland Pie every day, or down the street to Five Guys for lunch. And we broadened the question to, “What if you put your money into a retirement fund instead?” To make this real for my students, my friend Tracy admitted to her Starbucks habit and offered to be our real case study. Before we started creating anything, I asked the students to complete this quick survey to figure out what they knew and what they didn’t. Then we looked at the results as a group. Here’s what we found: Group 1: Mostly sophomores Group 2: All freshmen Group 1: Mostly sophomores Group 2: All freshmen Group 1: Mostly sophomores Group 2: all freshmen Clearly, the sophomores were bringing more to the table than the freshmen. After all, they had been instructed in spreadsheets in their engineering class last year, but they were still a bit unsure of what they knew. They thought they probably knew more than they had indicated, but didn’t know what I meant by “cell reference,” for example. And remember, I teach in Maine where 7th graders are given their own digital device. It used to be a laptop, but last year many districts changed to iPads. I would have expected the 9th graders to have had much more experience with spreadsheets, but I’m seeing that the switch to iPads is having an impact on that. Very sad. I began by explaining the situation: Tracy spends$x each day on her Grande Soy Chai at Starbucks. If we want to figure out how much she spending, and what she could be earning instead, what information do we need? And then I had them brainstorm for a couple of minutes.

Information needed: cost of the drink, how much spent each month, and interest rate for the investment.

We made a few assumptions:

• Tracy could find a mutual fund, or other investment, that earns an average of 7% annually
• that she is 25 years from retiring (I don’t actually know this)
• that the price of coffee would not change over the life of the investments (we knew this was unreasonable)
• that Tracy would invest the same monthly amount for the life of the investment (also unlikely)

But this is also part of problem solving. Take a few minutes to watch Randall Monroe’s TED Talk and you’ll understand what I mean.

So here’s the spreadsheet that we came up with.

#### So what did we learn?

• Tracy spends a lot of money on her Grande Soy Chai. But, it’s possible that the drink adds some value to her life and is worth the price.
• Investing early and for a long time really can pay off, even if the amount invested isn’t all that much each month.
• Learning about spreadsheets can be fun if you have an interesting question to answer.

Do I think the students in this 90-minute workshop will remember everything that we discussed? Of course not – I’ve been doing this job way too long to think that. But here’s the beauty of it all – they have their own model to reference, be it Google or Excel, they all created one and can take another look at any time. I heard from another teacher that a couple of his advisory kids started talking about making their own coffee instead. A couple of my advisory students commented on the experience at the end of the day. One said, “It was interesting to see how the numbers involved in the Starbucks added up if invested in a retirement fund. The actual application was nice.” Another said, “The spreadsheet exercise this morning was fun. I think it was the funnest way to learn how to do a spreadsheet I have ever done. So thank you.”

You’re welcome.

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