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.

11-22-2015 Image001

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?

11-22-2015 Image009

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?

11-22-2015 Image010

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.

movable line


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.

Leave a comment

Filed under teaching

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:

box plot histogram

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.


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!


Filed under Baxter, teaching

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.

Leave a comment

Filed under problem solving, teaching

Another year begins

It seems like school ended just yesterday, or maybe the day before. And yet the new school year is already up and running.

Fort Williams 2015We had two days of orientation last week – organized by students, with adult assistance – and they were really great. The first day was once again at Fort Williams Park and the weather, though quite hot, was sunny. Three years in a row. How can we be so lucky? The difference this year was that 9th and 10th graders were organized by paired advisory groups. Each group had a couple of juniors with them, as mentors, and their adult advisors, of course. The team-building activities were facilitated by juniors with adult assistance. The senior class was off on their own, team building at a ropes course nearby. They joined us at the end of the day, so we had most of the school together, out there under the trees, to debrief the activities of the day.

On Thursday, we were in Portland, at Baxter. Since we have all four grades for the first time this year, we are squished into a building that is too small. So, we needed a satellite campus. This was not a surprise. In fact, several people have been working on this problem for a couple of years. Every time they thought that they had a solution, something got in the way. Late in the summer, the building search team was able to secure a second space in Portland, a short walk from our main building. The space is large enough to house a few (like three) classes, but small enough so that the 9th graders, who will be spending their mornings there, will have a cozy space to make their own.

Once again our faculty has expanded. We’ve added art, engineering, and computer programming teachers, along with math, science, humanities, and special education. We have an outstanding team that shows a great commitment to our mission and purpose, even as we continue to refine what that is.

My year looks to be very interesting and challenging. I have a whole bunch of new courses to create, starting this term with Introduction to Logic and Transformational Geometry. Were those really my ideas? What was I thinking? The first time through is always the most difficult. It will be nice when this year is over. I think that we will have some pretty solid classes.

We’ve emerged from the terrible two’s to begin our third year. We’re toddlers. We’re ready.

Leave a comment

Filed under Baxter


What a week: Ending classes, field trips to Funtown and Boston, and Flex Friday celebration. That last thing was today.

Flex Friday is this thing we do at Baxter where kids work on long-term projects. They have to submit a proposal that goes through this pretty rigorous approval process. But then they get to work on their project. All year. Every Friday.

The concept is based on Google’s 20% time idea, where Google employees get to spend 20% of their time working on their own projects. Actual or perceived at Google makes no difference at Baxter. We have it. And at the end of the year, they get to show us all what they’ve accomplished with their 20%.

So, today, our students presented and exhibited their results. They shared what they have built or created. They talked about their struggles and successes. They focused on what they had learned (time management, organization, and communication being the big three). And then when I get home, this note from our principal is in my inbox:

Dear Baxter students,
I am so proud of you. Days like today reveal just how much you have accomplished this year. You are so passionate and articulate with adults and with each other. You are professionals, in dress and demeanor. You are honest when you share your successes and when you talk about your failures, and all of you seem to know that both are part of learning. I love it when you are innovative and your projects this year sought to reach the boundaries of what you know, what any of us know: what we can do with a mechanism or ingredient or measuring device; what we can make with paint or pencil or editing program; what we can build in the lab, on the street, or in cyberspace; what we can grow; how to bring compassion to a community of innovators. You are curious and creative and you are already changing the shape of the world around you. Thank you for changing mine.


And thank you, Michele, for making Baxter Academy the place where I want to be.

Leave a comment

Filed under Baxter

A Visit from the Charter Commission

Baxter Academy is a charter school in Maine. We are independent, meaning that we are not “owned” by some education company and we are not “aligned” to any particular program that already exists. We create our curriculum, our courses, and our materials. We answer to our students & their parents, to each other, to our Board of Directors, and to the Charter Commission.

Each year, at the end of the year, we get a visit from the Charter Commission. Well, it’s happened for the past two years – I imagine that they intend to continue this practice. The visit is kind of like a cross between an IRS audit and a NEASC accreditation visit. This year the Commission decided to take about 45 minutes in the middle of the day to split up and have conversations with different groups: students, teachers, parents, and community members. At other times they met with administration and the Board. At the end of the day, they gave some feedback to our Head of School:

They reported excited students and satisfied parents. They shared with us, at the end of the day, that teachers reported a strong sense of community, that Flex Friday is great, that special ed students are thriving, but that we need more time to communicate with parents around proficiency based diplomas. Teachers also liked working in start up mode, even though it is hard. Students liked best: the community, being able to connect inside school learning with outside the school, their control of their own learning, choosing their own curriculum, that teachers are guides, that they can challenge teachers respectfully and get answers. They report very little bullying and a culture of kids sticking up for one another.  If they could change one thing it would be to have more space, to add sports, more balanced m/f ratio, more clarity around grading.

“You can tell after even 2 minutes in the building that this is a great school environment.”

Clearly there is more work to do, but getting good feedback at the end of the year makes the rest of the hard work all worth it.

Leave a comment

Filed under Baxter

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:

baby weights

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.


Filed under problem solving, teaching