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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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.

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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.

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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.

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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:

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.


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Order of Operations and Facebook

I received this text from my brother: Pam, who created the “Order of Operations” rule? Well, that’s a curious question. Why is he asking? I’d never thought about who “created” these rules. They just kind of made sense to me. Before I did any research, I thought for myself why these rules made sense to me. Here’s my response to my brother.

Suppose I needed to calculate 4 + 3 * 5. Without the order of operations, I would just make each calculation as I come to it. In this case 4 + 3 = 7 and 7 * 5 = 35. But, 4 + 3 * 5 is equivalent to 3 * 5 + 4, right? Because of the commutative property of addition, the order that we add numbers in doesn’t matter – we get the same result either way. So if that’s true that 4 + 3 * 5 = 3 * 5 + 4, then both calculations should give us the same result. But, if we don’t have conventions around this then the second expression, 3 * 5 + 4, would result in 19. Clearly that’s not equivalent to 35.

So why would we choose to multiply first instead of adding? We can think of the expression above (4 + 3 * 5) to mean that I am adding 4 onto 3 bunches of 5. I guess I think that it’s kind of implied that I would want to know what 3 bunches of 5 are before I add 4 onto that number. That’s why we would multiply before we would add.

There is a similar argument to be made with division. Suppose I come across 3 + 1/2. Without our order of operations, we might conclude that this was equivalent to 4/2 (or 2). But isn’t 3 + 1/2 the same as 3 and a half, or 7/2 (which is clearly not the same thing as 2)?

The PEMDAS mnemonic is a little misleading, too. It suggests that multiplication takes precedence over division and that addition takes precedence over subtraction. That’s not true. Multiplication and division are at the same level as are addition and subtraction. For example, if you want to calculate 6 * 3 / 2, you can first calculate 6 / 2 and then multiply that result by 3. Or, you can calculate 6 * 3 and then divide that result by 2. Either way, you end up with a result of 9. You can reason similarly with addition and subtraction.

Personally, I think that the real question is “Who invented parentheses?” I mean, to show grouping by using symbols is just genius. That would change the outcome of the original expression, right? (4 + 3) * 5 is very different from 4 + (3 * 5). At some point, mathematicians agreed that they didn’t need to write the parentheses around 3 * 5 (maybe for the reason that I stated above, maybe not), but that they would need to explicitly group (4 + 3) if that’s what they meant.

My Google search found this response to your question from Ask Dr Math.

Turns out there’s this Facebook post that asks you to calculate 6+1*0+2/2. Some people say the result is 7 and others say it is 1. Which is correct? That’s where the order of operations comes in. And that’s what was behind my brother’s question.


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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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