Puzzling Problems Preferred

Yesterday morning I determined that I would find something from my day to blog about, and hey! I did! :)

My geometry kids are learning about quadrilaterals. The prior day’s lesson was about properties of rectangles, rhombi, and squares. Here’s the warm-up problem I gave them:

In rhombus QRST, the diagonals QS and RT meet at point U. If QS=12 and RT=16, find the perimeter of the rhombus.

I think it’s a strong problem, because they have to recognize all of these things:

1. that since the rhombus is a parallelogram, the diagonals bisect one another;
   
2. that the diagonals are perpendicular;
   
3. that the four small triangles are right triangles;
   
4. that the Pythagorean theorem will allow them to find the length of one side of the rhombus; and
   
5. that all four sides of the rhombus are congruent.

Almost all of my students struggled with the problem. I gave some hints – “What do you know about the diagonals of a rhombus? Okay, now think about triangles.” One student asked me if it was a “trick question;” I told her that it’s not a trick, just a puzzle.

But once they solved the problem, whether they needed hints or not, they felt proud of their work. That was awesome to me. Some of them told me that they really liked the problem. It was a challenge to them, so they had a sense of accomplishment once they had solved it. One student asked if I could give them more problems like that for their homework; the book’s problems are pretty simplistic and generally don’t require them to make connections across multiple concepts. I really need to get better about giving them more challenging problems for practice.

In other news, I am ordering sample texts for almost every mathematics course we offer for grades 6-12. The math department (all 3 of us) will be meeting at the end of March for a day of articulating who we are, and then using those decisions about curriculum. I’m really excited about this. :)

As part of that process, I’m reviewing the calculus materials available from The Worldwide Center of Math, and so far I am loving their model for the modern textbook. Check it out.

Trig Functions – a w00t and a help request

First off, a “hey this went well,” and then a “help me out.” :) These are both about my trig class.

Hey, This Went Well
I started today talking about amplitude and period of sine functions. They’d already understood that the period of sin(x) is 2pi, and they got pretty easily that the amount it goes above and below the x-axis is 1; I just had to add the word “amplitude” to their vocabulary. Check.

So then I put a GeoGebra window up on the projector and entered f(x)=sin(x), and I asked them what I should change about the equation in order to make the amplitude different. They weren’t quite sure what I meant, so I said, “Should I multiply sine x by something? Should I multiply the x by something before I take the sine? Should I divide instead? Should I add something at the end? Should I add something to the x before I take the sine? Should I subtract instead?”

Then they started conjecturing. I made them explain why they thought their particular change would effect the desired transformation, made them convince each other before I would type it in so we could see what it actually did. After we saw each change, I went back to f(x)=sin(x) before making another change, so we were always comparing our changes with the plain ol’ sine function. They were really pleased with being able to see the changes immediately.

It was also cool because we’d looked at what happened when we used f(x)=2sin(x) and f(x)=0.5sin(x), and then somebody said, “What if we used a hundred?” So I changed the coefficient to 100, and a couple of them said, “Whoa! It’s just straight lines!” Others said, “No, it just looks like that because it goes so far up and down.” The magic of the zoom feature allowed us to see that it was, in fact, still curving like we expected it to, just with really skinny humps. :)

So I think that using GeoGebra like that was a good way of looking at these transformations – they can make a guess, and we can see whether it’s right immediately. We can also use crazy big (or crazy small) numbers if we want to.

Help Me Out
I’ve got an idea, but it’s only partially formed right now. I don’t want it to flop like the last one (which involved kids spinning in circles and then releasing a stuffed animal – it was supposed to be about linear velocity, but it was a Fail, though they did enjoy going outside). So…here goes.

You know how when you walk, your body moves up and down? Walk across the room now. When your legs are side by side, you’re at your full height, but when one leg is in front of the other, the distance from the ground to the top of your head is less than your height. If you hold an arm straight out with a marker held to a long piece of paper, and you walk normally, the mark you make on the paper will be sinusoidal. (I remember my instructor showing us this in the health class I took in college…I don’t have a clue what his point was, but I thought it was cool.)

So I want to let my students do this. I want them to ask questions about what will be different for the curves formed by the tallest person versus the shortest person; what will be different between a girl and a guy; what will be different if a person is walking versus running; what changes if a person is on crutches (okay, I’m not sure if we could pull that one off or not, but it’s an interesting question, right?); what will be the same in each of these cases as well.

But I don’t want to get out there (I have an idea for where we can do this) and then just be wasting time with questions we won’t be able to answer. So I need to figure out how to structure this thing. Suggestions?

Unexpected Misconceptions

My trig students are now learning about the graphs of trig functions. (Yes, I’m behind.) I started off by talking about periodicity, and then I talked about how the values of sine and cosine repeat once we get all the way around the unit circle and start going around again. I then presented the idea of f(x)=sin(x) and started getting them to tell me what values I would plot.

They were confused. I figured out two major reasons for their confusion. The first one, I think I’ve addressed, but the second is, I think, still causing befuddlement.

Reason #1: I put a graph up there and used crazy things like pi/2, pi, 3pi/2, and 2pi as my notch marks on the x-axis. They’re used to integers there, and to think that I could divide my axis so that each notch represented pi/2…well, that was just insane to them. I think it relates to the way they don’t really like to think of pi as a constant, since they can’t express its value exactly in any way other than a weird Greek symbol. But I think I dealt with this one okay, though I might mention it again today just to make sure they caught on.

Reason #2: This is best expressed by giving a quote from a student.

“Why are you saying ‘sin x’ when sine is y?”

The student was confused because, on the unit circle, she had memorized that the y-value is the sine of the angle (which we’ve been calling theta). All that relating of the values on the unit circle to right triangles? Yeah, that didn’t work for her. She just memorized that “sine is y.” I tried to show her that we had been saying that sin(theta)=y on the unit circle, and that now instead of theta, x is the angle of which we’re taking the sine for our function…but I don’t think it clicked for her. Well, I thought it had, but then when I said that we could also do a function f(x)=cos(x), she said, “But then cos(x) would be y, and I thought sine was y?”

So I’m still working on that one. But I’m making myself notes, so that next year I’ll be able to anticipate these misconceptions and hopefully prevent them by changing my earlier instruction. I’m sure new ones will pop up to surprise me then. ;)

Apples and Oranges

We compared apples and oranges in statistics today.

The teacher’s resource manual for the textbook comments that standard deviation lets us compare apples and oranges, because we can use it as a “ruler” to figure out how a particular data value compares with the mean. I took that idea and ran with it – to the grocery store, during my 4th period prep. I got 9 apples and a bag of clementines (there were 20 of them in the bag).

We put each piece of fruit on a paper plate with a letter on it (we had to use three Greek letters). We borrowed a balance from a science teacher. Before weighing the fruit, the class made conjectures about which apples would be biggest and smallest, and which oranges would be biggest and smallest. (They did know that our method of measurement would be using the balance.) I told them that in making those decisions, they were comparing apples with apples and oranges with oranges, but what if they wanted to determine which was relatively bigger, the biggest apple or the biggest orange? That, I said, was what they were going to figure out how to do today.

After they made their conjectures, they wrote the name of each piece of fruit on the board. (They decided the fruits needed names, so instead of boring old A, B, and C, we had Amy, Billy, and Caroline as our first three apples. I’ve studied Greek, so I was able to come up with some Greek names for those extra letters.) Then they recorded the mass of each piece of fruit.

The next part was easy: For each type of fruit, which piece was the biggest, and which the smallest? It turned out that the biggest apple was Francis (212.1 g), which had been their guess, but that was the only one they guessed correctly. Caroline was the smallest apple (144.4 g), Wilhelm the biggest orange (84.6 g), and Violet the smallest orange (52.1 g).

Then I told them to figure out the mean and standard deviation for each type of fruit. Doing this REALLY helped some of the students to better understand what standard deviation is (a measure of spread); I was able to point out that the standard deviation of apple masses (19.72) was more than twice that of orange masses (8.26), and the students were able to look at the actual fruits and see that yeah, there’s a lot more variation in size in the apples than in the oranges. They had been pretty confused by it before, so I was really glad to have that visual for them.

Once we had that concept a little more firmly understood, I said that we should look at Francis (biggest apple) and Wilhelm (biggest orange). I asked how we could figure out how many standard deviations above the mean Francis is, and they knew right away how we could do it. We did the same for Wilhelm, and we discovered that Francis was 0.981 standard deviations above, but Wilhelm was 1.746 standard deviations above. So even though Wilhelm is SMALLER than Francis when we look at the measurements themselves, Wilhelm is bigger as a big orange than Francis is as a big apple. We did the same thing with the smallest ones, and I pointed out that since we were subtracting the mean from smaller data values, we were ending up with negative answers…which just show us that we’re that many standard deviations below the mean. They got it.

And then I told them that these “how many standard deviations away” things they were coming up with are called z-scores. They were excited that “it has a cool name.” :) And I’m excited that I think they will actually remember it because of how we got there – they figured it out rather than having me throw a formula at them.

And then? We had a healthy snack to conclude the class. :) (My homeroom may or may not have juggled the leftover clementines right after the statistics class left.)

Updated link for quilt solution

Thanks to Elizabeth S for informing me that I had a bad link in my post about my solution to the quilt problem. The correct link for my solution is this one. I’ve updated the original post with the right link as well.

Historical Intro to the Tangent Line

I mentioned when I started this blog that this is my first year teaching math, but that I spent the last seven years teaching history. So it’s probably not a surprise that I find the history of mathematics interesting, though I’ll admit I have a LOT to learn in that area. I’ve pulled a few lessons from this CD-ROM, called Historical Modules for the Teaching and Learning of Mathematics, which I highly recommend. And then I developed a lesson for my calculus students as an introduction to The Derivative.

For full disclosure, you guys are actually getting the edited version. What I gave the students was overly ambitious and overlooked the fact that ellipses and hyperbolas have more complicated derivatives than parabolas, and I was really just going for something simple to ease them into things. I made some on-the-spot changes during the presentation of the lesson, and now I’ve come back to the lesson and changed it to what I really ended up doing.

All that said, here’s the lesson.

Some things to note about the execution of it…

• I didn’t actually say the word “derivative” until they got to the very end. Their textbook doesn’t give them the word until the next chapter, but hey, they found it! Why not tell them that’s what they were doing?
  
• Your students need to be familiar with GeoGebra in order to make this thing work. Mine ended up having to use the Van Schooten worksheet to find their tangent line slopes because they couldn’t actually do what I’d asked them to – this was the first time I’d told them to do something other than graph a function on GeoGebra, and it was over their heads. At the same time, they did learn a lot about what it can do and what doesn’t work! :) I’m hoping to have next year’s group prepared by working with GeoGebra a good bit while I have them in trig/analyt this year.
  
• Don’t tell your students you just ordered a book on the history of calculus unless you want them to look at you like you’re insane.
  
• My students didn’t know what “analytic” meant or how it was different from “geometric.” My own understanding of geometric solutions to problems has been greatly increased as I’ve done my thesis (on Descartes’ La Geometrie), so hopefully I was able to explain it to them well. I am much more of an analytic thinker than a geometric one.

My goal in using this was to have the historical setting provide a context for studying the mathematical concept. I think it worked pretty well. In any case, staring at the board where they’d done a lot of the work together and where I’d circled an expression and written “derivative,” one student said,

You know, I really do think I get this. It looks complicated, but it’s really not that hard – it’s just slope.

Water Tower Exploration

There’s a water tower right next to the building I teach in. Naturally, I had my trig students figure out how tall it is.

They were lying on the ground and measuring angles.

They were borrowing tools from other teachers (the science teacher has something she uses to see how high her 7th graders’ rockets go; the PE coach has a long tape measure for when he makes lines on the field).

They were mad that I wouldn’t let them climb the fence so they could get to the base of the water tower. (Dude, that’s not our property!)

They were making estimates before they took measurements.

They were recognizing when an answer they came up with wasn’t reasonable, working to figure out what went wrong, then trying again to correct it.

They were enjoying the nice weather.

They were noting that the ground isn’t completely level and trying to compensate for that in their measurements.

They were drawing pictures to represent their work.

After they’d gathered measurements and performed their calculations, I let a student call the city to find out the actual height of the water tower. Most of the students were within 8 feet of the right answer (which was 216 ft). One group was way off, but they realized that they hadn’t done a good job of determining the angles of elevation, so we got to see how much accuracy matters.

I love doing things like this. It seems to me like the students really feel like they own the mathematics when they tackle a problem like this and reach a solution.

However, I’m thinking about Dan Meyer’s recent post on pseudocontext. If we can just call the city to find out how tall the water tower is, what’s the point?

Well, it’s fun. It’s a chance to go outside. I think it’s significantly more engaging than the example in the textbook where you have to figure out how long the rope is that’s holding the tent up (the example is labeled as “Real World Application: Entertainment” – really? entertainment, because it’s a tent? yeesh).

But is the water tower activity flawed because there was an easier way to get the answer? My gut tells me no, but I’m still working on why.

Quilt problem: My solution

I finally got my solution for yesterday’s quilt problem typed up and published as a pdf. It’s here. I used both composition and multiplication of functions. Would you do it differently?

I had to go and ask another teacher a question right before that class came in today, so they were already in the room when I got back. And they were at the board, putting up what they’d done on the problem last night, or looking over each other’s work. I hadn’t instructed them to do that; they just did. YAY! :)

I think they probably could have continued working on it all period today, but I did have other things to teach them as well. So first I let them share their thoughts, but I gave more telling feedback. “Hmm…I think I’d run out of fabric if I only got that much. Can you figure out why?” (That kid was very close; he just didn’t think about how I need to cut rectangles of a particular size, so I can’t just divide the area of all the rectangles by the width of the fabric – in other words, the necessity of the greatest integer function for this problem.) “So for a quilt that’ll only be 48 by 64 inches–” (she was giving me an example of a specific block size rather than a function for any block size) “–you want me to get thirty-four YARDS of fabric? It’ll probably cost $3.99 a yard…do I need to spend THAT much money?” (I think she was trying to make me cut one super-long strip of fabric, leaving about 37.5 inches of the 40-inch fabric width untouched.)

So I took a few of their ideas, and really, most of them were focusing on figuring it out for a particular size block. One pair of girls who tried to make it a function didn’t recognize that the width and the length of a fabric strip in a block can both be expressed in terms of the same variable, so they were working with x and y. But you know what? When I was organizing my thinking to start off, *I* was going through a specific example in my mind. So I don’t think there’s a problem with using a specific example to help orient yourself to a problem.

I think the key is that in my mind, I always knew that I’d identified x and was working to figure out what I wanted to do to it by rehearsing what I do when x=6. I’m not sure whether they did that or not; in fact, I’d be willing to bet that most of them were just planning to figure out what x should be after they’d solved the problem with their particular example, or else that they forgot they were supposed to be looking for a function and would just consider themselves done when they reached the solution for their example. So the next time I use this activity, I need to make that point more clear at the outset, and I need to emphasize it over and over while they’re working as well.

I just realized that I’m going to be late for our church supper tonight if I don’t get out of here, but I think that was pretty much what I wanted to say. Oh, and after letting them share, I walked them through my solution, asking them questions to get them to come up with the functions I had. I don’t know if any of them drew out the fabric with rectangles cut out of it…importance of drawing a picture to help you solve something!

A great day in trig

My trig students are reviewing some algebra concepts before we jump into trigonometry, and the most recent thing was composition of functions. So today, I put this picture up on the screen:

And I asked them to find a set of functions, the composition of which would show how much fabric I need to buy of each color. I told them to work with one another, and that I would answer any questions they have about quilting or fabric.

At first, they were just sitting there at their desks, making uncertain pencil marks on their papers. I said, “Work together. Talk to someone near you. Get up and talk to someone far away from you. Use one another to figure it out.”

Within a couple of minutes, they were all sitting on the floor at the front of the room “so we can powwow.” One girl didn’t want to sit on the floor, so she started writing on the board. Most of them ended up getting up and crowding around the board (or the pull-down graph, or the other board); a few of them continued to huddle on the floor and work on paper. But they were all engaged and trying to figure this thing out.

They first asked me how many blocks there are in the quilt, and I pointed to the image and counted. They asked me how big the blocks were. I said that for the one I made in the picture, I used blocks that were 6″ squares, but that the block size was what needed to vary. I did have to point out to them that quilting uses 1/4″ seam allowances, but one student who does some sewing knew what I meant and explained once I said the phrase, catching on immediately to why that was important.

They asked if they could just write a function of how much it would cost to buy a quilt from Wal-Mart instead. I denied that request.

There was a lot of argument over whether the size of the block or the size of the strips within a block was the most important. Once they remembered that they were looking at how much fabric per color to buy, they focused in on the strips instead of the block.

Then they pretty much all got hung up on the area of the block.

“If it’s gonna be six inches finished,” they told me, “then the strips will be six-and-a-half by two-and-a-half inches –”

Wait, I said. Where’d you get two and a half?

“We added the quarter inch seam allowance four times, once for each side.”

Didn’t you already give it a seam allowance on the top and bottom? Why are you using those quarter inches again? (This part actually caused more confusion than this summary indicates.)

“Oh! Right, so they’re six-and-a-half by two inches.”

Okay, got it.

“So then, 6.5 times 2 is…thirteen, and then we need 48 of those, so we need 624 inches.”

So…you multiplied a length times a width times a count, and you ended up with a length?

“Ugh! I knew there was something about that that didn’t quite work! (to friend) Come on, let’s try again.”

At one point a group asked me how long a bolt of fabric was. Hiding my surprise that a 16-year-old boy knew that fabric comes on bolts, I clarified, “Do you want to know how long or how wide?” He changed the question to how wide, and I told him, normally 40 to 44 inches, and I usually use 40 when I’m calculating how much to get. A girl drew on the board to show a classmate how the fabric is wrapped around the bolt and where the 40 would be. A boy asked if I would want to buy any extra fabric; I said that normally, yes, I would, but for the sake of our problem they could assume that I was buying exactly what I needed. (The amount of extra I get varies, so it’s not like a constant they could add.)

They weren’t done at the end of class. I told them their homework is to continue to struggle with it – it’s okay if they don’t come up with an answer, but I want them to work on it some more. One girl said, “But my work is on the board!” So I let her use her phone to take pictures of her work, and the rest of the class asked her to send the pictures to them.

All of them were engaged. All of them were thinking and working to figure out what information was important and what wasn’t. It was awesome, and I hope I can come up with more problems like it. :)

Day One, Sort Of

Today was the first day of school, but our high schoolers (it’s a K-12 school) don’t have regular classes for the first two days, so it was a little odd. They have Orientation, which is basically a series of seminars. I did lead two math seminars today, one for the juniors & seniors and one for the freshmen & sophomores.

I had a little more than an hour and a half with each group, which was way too long; I’m going to request that the times be reduced for next year. I split the session into two parts.

The first thing we did was based on George Woodbury’s post on doing a study skills inventory. I did a lot of asking, “How do you do that?” For example, a few kids listed “organization” as something that’s characteristic of a successful math student. We talked about what it means to be organized as a math student, and how one can make that happen. It was a pretty good discussion, but I think a lot of the kids were zoning out. That may have been related to their schedule for the day (lots of seminars, as I said), but I think even so I should work on making sure ALL students take part if I do this again in the future.

For the second part of the seminar, I took an idea from @Mythagon and decided to have them investigate spirolaterals. I chose this because it’s accessible to all of them, regardless of math course level, and because it was something I could do to get them thinking mathematically, looking for patterns and using mathematical terms to describe what they saw. Here is the worksheet I developed. (Buddy the Bunny is one of the stuffed animals who lives in my classroom, just as an fyi.) The kids really got into doing the spirolaterals, and they were engaged and working hard to find the patterns.

With the younger group, I didn’t get into questions 4 and 5, except to point them to this website where you can make those changes and generate more spirolaterals. But the students in that group were asking fabulous questions as they tried to articulate the rules they were developing about the kinds of patterns they were seeing – a lot of “what ifs” came from them. It was really awesome to have them so into what we were looking at, and it was great to say, “That’s a great question. Here’s another sheet of graph paper – why don’t you try to figure it out?” We didn’t have time during class to explore whether palindromes in spirolaterals make any particularly cool patterns, but I think it will be something to investigate!

Tomorrow I don’t have any classes, and then Wednesday will begin the real deal. I’m excited. :)