Tag Archives: posing problems

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?

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. :)

Learning Targets and Brainstorming

Status on Learning Targets…

  • Calculus: done
  • Statistics: not all the way through the text, but I’ve gotten through everything we’ve done in the class I’m taking, and I can add to it later
  • Trigonometry & Analytic Geometry: done, unless I get through things and have time for an extra chapter, at which point I’ll add in those LTs
  • Geometry: just opened the book (I’ll spare you a link to an empty document ;))

If anyone feels like giving me feedback, please do so. This has been a bigger task than I anticipated!

Now for some brainstorming…I have an idea for how I want to structure my units. For each unit, I want to put one or more real problems (of the WCYDWT variety) up on the bulletin board. The idea is that I will choose these Big Problems (don’t I come up with creative names?) as things that can be found using what they will learn in the unit. We’ll keep looking at them as we go through the unit, evaluating how our new knowledge can help us reach new understandings about the Big Problems. (I totally need a better name than Big Problems.)

What I want my kids to recognize is that math can be useful. I think it will also push them in the right direction for completing each semester’s final assessment, assuming I stick with that idea.

So once I get my Learning Targets finished, I think I will tackle the first couple of units for each class and figure out…

  1. Big Problems
  2. Assessments
  3. Instruction
  4. Homework/practice sets

I think doing that will help me feel ready to start the year, though the more units I can prepare ahead of time, the better. Oh, and I also need to work on my Policies & Procedures sheet (to explain things like my crazy new grading idea).

Final assessment idea

I’m required to give a final assessment at the end of each semester. For my 9th and 10th graders, the final is worth 15% of the semester grade (with each of the quarters in that semester counting for 42.5%), and for 11th and 12th graders, the final is 20% (each quarter 40%). Not all that SBG-friendly, since it’s not something I can reassess.

BUT…we can choose to have the students do some sort of “exhibition” in lieu of an exam. This, I think, I can work with, and still stay true to the SBG philosophy.

So today I thought of something I could have the students do. It’s not totally processed in my mind, so there are probably pitfalls and things that don’t make sense. But here’s where I am with it so far.

  1. Students choose something they’re interested in. Ideally this is some sort of issue that is important to them.
  2. Students find numbers related to their issue. They may have to look online, they may have to go somewhere and take their own measurements, they may have to call someone in the field to ask for numbers.
  3. Students perform calculations on the numbers. At this point it will tie in with the Learning Targets for their course that semester, and they will have to demonstrate and explain how they have used a certain number of concepts. (Certain number from each unit, maybe? Not sure.)
  4. Students interpret their work. What does their work do to help us understand the issue better? Does it help us develop possible solutions? What action can they take now that they have done this work?
  5. Students develop a way to present their work – maybe an oral presentation, maybe a backboard for a “Math Fair,” I’m not sure yet.

Although the product (which would be a presentation of all the work) would determine their grade on the “final,” it’s something they would be working on over the course of the semester. I imagine setting due dates for the different parts of the project, requiring students to plan out what data they will look for before they go and get it, etc.; I’m not planning to throw this out there and say “do it.” I also expect that I would offer individual student conferences (formal and informal) so that they could get feedback on their work, make sure they really are demonstrating mastery of LTs through it (there’s that opportunity for reassessment), etc.

What do you guys think of this as a final assessment?

Reading too much into the question

More thoughts on my statistics class from today. We were going over some review questions, and I said that one of them didn’t have enough information for us to determine an answer. The instructor responded,

You’re reading way too much into the question.

Well, no, I really wasn’t. It was a poor question. I’m really not sure why she didn’t just acknowledge that it’s a poor question – it’s not even a pride thing, because these questions are from the textbook publisher, not from her. And even when I am the one who wrote questions, I’ve told kids who challenged them, “You know what? You’re right. I could be asking this question more precisely.” And I make a note to improve the question for the next time I come to that material.

So the first part of my point here is, it’s okay to recognize that a question/assignment/whatever is not the best and could be improved. I think my students like knowing that I will receive their critiques (delivered respectfully, of course) and consider their feedback. I know that I didn’t like feeling like I wasn’t being listened to, being told to just write down the “right answer” and move on.

But the second thing I wanted to say in this post is a question I thought of as a result of this experience. Is there a point at which it’s okay to tell a student “you’re reading too much into the question”? Obviously we want to stop kids from saying,

“Well, Johnny won’t have ANY apples left after he gives 3 to Suzie, because right at that moment Suzie gets turned into a ZOMBIE, and she doesn’t care about the apples anymore but just wants to eat Johnny’s BRAAAAAAINS, so Johnny figures life is more important than apples and he drops the apples so he can run away. Will he survive Zombie Suzie’s attack? Just wait until I turn in my next homework!”

Because that would just be silly.

But. There’s a lot of good thinking that students can do in between “just give the answer you know I’m looking for” and the zombie scenario. I’ve read several bloggers talking about WCYDWT (What Can You Do With This), which is all about posing new problems, digging deeper into what’s right in front of you instead of just relying on the (possibly/probably poorly written) questions in the textbook.

There probably are good occasions for telling a kid he’s reading too much into a problem. But I think we might jump there a little more quickly than we ought to sometimes.