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Juice Mixing

You have two glasses with an equal amount of juice in them. One has apple juice, the other: orange juice. You take some amount from the apple juice and put it into the
orange juice glass, then mix it up real nice. Then you take the same amount from the orange juice glass and put it into the apple juice glass, and mix it up.

This would be great with a video… I think the question you ask is which glass has more of it’s original juice now? Answer to follow.

Workflow: Worksheet design for Factoring

I’ve been still trying to wrap my head around factoring, and I think I’m approaching a much reasoned presentation. The work I gave out today in both Algebra classes was more successful than it’s been before. In one class I gave them a very much scaffolded set of problems and explanations to help them tie together much of what we’ve learned. I haven’t made an answer key.

Ws Factoring Overview All Techniques

Hoping for a similar effect I used this sheet to introduce factoring ax^2 + bx + c‘s by splitting the middle in a different period. The justification I’m giving for splitting the middle is one I think extends beyond some of the simple (2x + 3)(3x - 5) type expressions. Anyway, this isn’t a magical topic, but I think my students this year will have a more solid ability to reason about factoring than before. That’s a good thing, I suppose.

This starts on the 3rd page ‘cuz that’s where I kicked off the page with the class.


I’ve got one more to share, which may get used later.


In other news, I thought I’d post a picture of how I get this generated. I write the worksheets in LaTeX using Emacs to write the code, and I use Maxima to help simplify and factor expressions so that I know they’ll be correct on the worksheet. Here’s a snapshot of how that all works.

Art Appreciation: A Geometry Project

My Geometry class is entering the 3-d part of the year. But instead of watching Avatar, we’re studying Area, Surface Area and Volume. Or at least that’s what the last three chapters in the book are. Thing is, in some ways the book throws a ton of easy pitches to the kids:

Find the surface area of this shape, given only the relevant dimensions and nothing else.

Same thing for volume and everything else. I thought I’d kick them off with a little detour to get them thinking about why we care about things like surface area and some of the problems that arise from surface area calculations as applied to real problems. I told the class that we were going to spend a day doing a little “art appreciation” and with no other introduction started the following slide show of images happily lifted from Christo and Jeanne-Claude’s site.

Christo Project

The slides naturally spark lots of “is this really art??? I could cover this place with stuff!” type sentiment. I took some straw polls like “How many of you think it would be easy to wrap our school building?” a split favoring those who voted “not easy” and then made the easy votes describe how they’d go about getting the school covered. We continued through the projects, discussing issues that seem to come up naturally whether or not they seemed especially mathematical. We stopped to read the press-releases as they were in the order. And by the time we reach the “WWCAJCD?” pyramid slide, they have lots of ideas about a project that might be done and the issues involved in making it happen.

We’ll be working on the project for two more days. Part of my hope is that good questions arise, and that students feel a little more confidence in working out solutions because there isn’t necessarily an answer they can check in the back of their book. This problem has not already been solved.

Learning a lot outside of school

I haven’t taken the time to write, and I’ve barely been keeping up with all the great posts being written these days. Very big thanks to Dan for summarizing the many sessions at NCTM and NCSM.

I’m enrolled in a class titled Problem Solving for the MS Math Teacher. The class is great. We get three problems for each HW assignment, and they really forced us to justify everything about our solutions. If you set up an equation or a formula it needs to be justified, if you perform a calculation the theory behind it should be clearly explained. The emphasis on reasoning has been a great push, and it’s really valuable to see how so many others reason through the same problems. Here’s a sample of the problems we’ve been assigned

  • Saved by Zero. How many zeroes occur at the end of the expanded numeral 999!?
  • The Last Straw. Two piles of straws are on a table. A player can remove a straw from either pile, or a straw from both piles. The player who takes the last straw loses. If there are two players how should you play?
  • The Case of the Grouchy Customers. Every morning at local cafes sleepy customers stumble in for their morning cup of coffee. One such cafe has a row of 10 seats at the counter. Typically, morning customers do not like to engage in conversation. How many different ways can three customers sit in those 10 seats so that no two customers are sitting adjacent to one another?
  • Put Down the Ducky. A man selling ducks sold half his flock and half a duck to Amy. He then sold one third of what was left and 1/3 of a duck to Beth; then 1/4 of the remaining flock and 3/4 of a duck to Cathy, and finally he sold 1/5 of the remaining flock and 1/5 of a duck to Dina. He now has 19 ducks and he never cut a single duck (whew!) What was the size of the man’s original flock?

It’s been a fun challenge to figure out how best to explain myself as I approach these, and some of them have been good problems to have on hand for students.

I’m also taking Intro to Computer Science and Programming for no credit through the MIT OpenCourseWare offerings. As a model for online learning I love that MIT is doing this. I have already found myself applying python to multiple problems beyond this class in a more sophisticated manner than prior to beginning the lectures / readings / problem sets. It’d be great to be able to get credit for this, but you can’t beat the cost.

The entire offering is fantastic and well curated. A lecture I recently watched was entirely re-taped because technical glitches ruined the video of the original class session. The professor entirely re-did his near hour long lecture for the sake of OpenCourseWare. It is lecture based, so not for younger folk, but if you have the patience to really work through the materials provided you really could educate yourself with little more than internet access and time. Take a look at the offerings if you are interested!

Exponent Rules, Polynomials, and Adaptations

It’s the end of Spring Break, and I’m just finishing up my Algebra plans for the coming week. The backstory to this post is the week before break I began exponent rules with the Algebra classes and successfully confused lotsa kids. Woosh. I’ll take responsibility for some of it, and I think I know what went wrong.

First, I tried to use Smart Slides to guide the classes through the material. For those of you who pre-prepare your slides for class, you’re stronger planners than I. My pre-prepared Smart notes have some significant flaws. For example, I might introduce a new concept like negative exponents with 2^{-2}, get some head nods and then run the train straight into a brick wall with the next slide: \left(\dfrac{2x^3}{3x^3y^5}\right)^{-2}. I’m not kidding. And the best part? I won’t know it’s coming either! Kids will just look at the slide and think: “I had it. Then I lost it.”

This sounds gnarly. My problem is not that I can’t plan, and normally the gaps aren’t that gross, but I’m trying to make a point here. What you think will flow smoothly on Sunday at the Coffee Shop doesn’t always flow smoothly on Monday in class. And you’ll have a much better sense of what should come next when you’re in class. So, wouldn’t it be nice to not be tied to the next slide?

Solution: Return to the whiteboard. I realize that lecturing is not the most progressive education strategy available, but many of us still present content this way. So we should at least do it well. There’s a reason that many professors who lecture for a living, even in Computer Science, still use a chalkboard. It puts a subtle break on the amount you write, it gives the audience time to follow you, and allows you to adjust on the fly without being tied to your next slide. I connect more with the class when the material is coming from me not from the surprise next slide I created a few days ago. I’m also giving students a little more processing time because I can’t reveal paragraphs with the click of a button. The white board also doesn’t change with a click so choosing carefully what you put up there helps students because they’ll have the reference for the rest of class. This is very hard to do well with any brand of Powerpoint/Smart/Keynote presentation.

Not that I want to present everything by lecture, but Exponent Rules and some brand new skills are worth presenting this way. There are many rules, they’re not so tricky, but they deserve clear names and a multitude of examples. So I outlined my notes, the examples I want to give, and I’ll have a printed copy to work off in class this week as we revisit the confusing stuff from last week, and jump into the void of new stuff.

Here are my notes.

Plan Multdiv Rules Lecture Notes

Once kids have a decent grasp of the rules I’m going to give them a shot at two problems I like. There’s slight WCYDWT bend to the first problem.

What’s the largest open box you can fold from a sheet of paper?

Plan Boxfold

If you want to run calculations on the fly for this problem, you might want to download Maxima (a great Computer Algebra System). I’ve made a little file that can crunch numbers for this problem given arbitrarily sized pieces of paper, and fold lengths.

Here are the resources for the boxfold problem.

  1. The problem guide (with solutions) / student handout
  2. The maxima calculator file.1

The rest of the files I have for this unit are in the box.


By way of David Cox if you want to do the Box Folding problem, consider incorporating this virtual manipulative it’s perfect.

  1. Note the second option in the “Cell” menu in Maxima is “Evaluate All Cells” this will crunch the numbers initially, and you should do it again whenever you change any of the input…

Right Strategy, Wrong Time.

A few weeks ago I asked students in my Algebra class a problem akin to the typical

A car leaves Philly for DC at 3 going 60mph and another leaves DC for Philly at 4 going 65mph, if the two cities are 138 miles apart how long until they cross paths and how far from Philly will they be when they do?

Nobody had any idea how to think about motion like this. Despite having done a few problems over the course of the unit, they got lost trying to think this through. It didn’t help that up until I asked them on the test I had been making them translate this stuff into a system of equations.

What went wrong? Well, students lacked a clear understanding of the problem in its context. Other problems in which they needed to set up a system of equations were fine.

7sodas + 8slices = \$15 \\ 8sodas + 9slices = \$17

This stuck; they get the situation and could probably talk like an expert about how they set out to solve the problem. With the motion problems they don’t have this familiarity and easily accessible framework.

To try to put a point on it, the problem was: my students lacked a framework for understanding a problem and consequently the ability to translate the problem from situation to mathematical calculations. To solve this involved some backpedaling on my part. First, I was arbitrarily forcing the tool on my students to solve this problem, and, in this case, I was missing the point. Kids didn’t understand how to think about objects moving together and apart, especially with respect to their velocity.

We needed some good visual fodder and time for students to get their hands dirty. I found an interested student to volunteer to bring a Flip in and shoot a bunch of scenes of kids walking back and forth. We spent a day taking these shots. We came back in and watched them, watched them and drew on them. Kind of like Dan’s “Graphing Stories” except the goal of graphing was to get kids to see that in different situations it is appropriate to add or subtract the speeds of two objects. We also noticed that if two objects cross at a given point, the sum of their distance traveled is each to the total distance between them originally.

Lights started to come on for them, and our class discussions improved dramatically. Kids started to explain why if two objects are approaching each other you can add their velocity, and that if you know the distance between them it becomes easy to think about when they’ll cross and exactly where on the route that will take place.

We worked through this problem set and students had some good discussions. But I dragged it out too long, and didn’t have an answer key ready. That has been fixed, and the final page has answers to the activities.


We spent a few more days talking about different problems. They worked through one which I collected and scanned, through I apparently didn’t get a double-sided scan so quite a few of these are cut off, but I still think it helps to see what they were doing. Edit: I’m going to check and see if it’s cool if I post work before it shows up here.

A few take aways. First, I shouldn’t be forcing a tool or strategy, say systems of equations, for solving problems onto students as a general rule. Instead of viewing problems through the lens of the current unit, it’s important to constantly be assessing whether or not students have a general familiarity, comfort and ability to work within the environment of a given situation. Before you get to any of the heavy lifting of solving problems in a tough context, create a solid understanding. Role play, demonstrate, enable them to perform the “back-of-the-napkin” calculation that demonstrates an ability to translate the context from a situation into calculation. Despite that I was teaching a unit on systems I shouldn’t force students to apply them blindly to every problem they see in the unit, and when past knowledge is sufficient for the task at hand, I should be praising them for choosing a successful strategy and make sure that others notice the connection as well. If I want to teach systems of equations I need to choose problems and situations that make them necessary and useful. In many cases, this is a place where our textbook fails us. The problems are often contrived, shallow, and/or easy to solve by much simpler strategies. In a few cases they get it right, this is where you want to spend your class time if you want your kids to appreciate the fancy new mathematical tricks they’re learning and not yearn for the old days.

Finally, this comes back to WCYDWT. We shot those videos, and I’ve been working on editing them. I think there will be a general wow effect when students see the final product, but for this class I don’t think that will translate into more meaningful work, we’ve spent our time in this context, they’re familiar now. As I’ve been editing them I can’t shake the feeling of being frequently rewarded when a calculation adds a meaningful piece of info into the scene or turning pixels roughly into meters and determining a subjects’ speed by calculating the difference in location over time. The math feels really useful. It’s hard to get that by having them watch the video. I had it by making it.

The magic of WCYDWT is when students are What-Can-You-Do-With-This-ing. Taking a situation and opening up a bag of math on it. Finding meaningful calculations, bringing ideas out of their repetoire and putting them to work to do something meaningful, interesting, and rewarding. That’s a hard thing to engineer. But it sure seems like a good way to spend your class time.

Shawn Cornally, Think Thank Thunk

Having spent a good chunk of my evening reading over Shawn Cornally’s fantastic blog “Think Thank Thunk” I’m engaged. I enjoy so much having a window into how teachers make decisions around their curriculum and class design. Knowing how someone chooses to introduce concepts and motivate students is a big bonus, and something Cornally communicates deftly. Read two posts and see if you agree. I highly recommend it.

Doug Lemov’s Taxonomy

Having had the luck to attend two of his seminars, I eagerly read through the NYT article on Building a Better Teacher. At his presentations he had a list of simple behavioral choices that improved teachers’ effect. I walked away from both presentations with lots of ideas of concrete changes to make in class, and recommend the article, and forthcoming book on the strength of his presentations.

Update I just checked out Uncommon Schools and found they have a page devoted to Lemov’s Taxonomy. I noticed there are a few more videos than in the Times’ article. That might be a good place to start if you’re looking for info on it. Though they don’t offer access to all their 700 clips. Too bad for everybody else.

Equation Challenge

I took Maria’s MathType vs. LateX equation challenge. Partially out of curiosity for how long it would take and also to put out an example of how I use emacs and LaTeX on a regular basis in my planning and in the creation of materials. If you’re having trouble falling asleep, let me recommend this video to you.

Here’s the original document. And here’s the pdf I generated.

Equation Challenge

Download the LaTeX equationchallenge.tex file if you want.

How To Teach: Will It Hit The Corner?

Here’s what I did. I taught this lesson today, and I had mixed results. For the record I’d say that kids in the second group that got the lesson where really into it, whereas the first group gave it a “so-what-else-did-you-cook-up-for-us?” rating. I started this post thinking I’d tell you what I did, but I kept thinking things like “if I’d only done this it would have gone better,” so here’s the if-I-would-have-done-this-then-it-would-have-gone-better version of what I did today. I know, too much hyphenation for one day.

At the end of this lesson, students will be able to

  • Work backwards to solve a complex problem
  • Use patterns and inductive reasoning to extrapolate data (logo positions on a large grid)
  • Apply software to investigate (potentially) algebraic relationships
  • Construct a rule for the corners problem

Setup: Show kids office clip #1 (see post). Then open up the 5 min. gridded version and play that in the background while acting like you’re trying to give directions. Get the kids to make guesses about whether or not it will hit a corner. Then after they’re done shouting about the epic moment at 1:43 (.333) when it does, get them to predict the next corner. It doesn’t happen. At least, not in the 5 minutes that Dan provided.

Distribute copies of the warm up below

Ws Office Boxcorner Warmups

As kids graduate from this, ask them if they’d like a bigger version (the desired answer being: “YES!”). Then hand them the first page of this sheet.

Ws Office Boxcorner

The answer here being that it doesn’t matter all starting squares result in a corner hit after varying numbers of steps. This can be shown with the software or through a pattern that many kids will notice after they have traced out sufficient numbers of steps. Talk about the patterns, ask them if they have any theories about how this might work on grids of varying dimension.

If you have access to a set of Mac’s you’re in luck. Ask the students if they have developed any general theories about these types of problems. How does the path to the corner depend on the starting location and grid dimensions? Allow students to work with the software to investigate and develop their theories. Use this sheet and have students download, the simulator I designed.1

Ws Office Boxcorner Pythonlab

I leave off here at this incomplete point, because I’m not sure of a good way to finish this off.

Any ideas? How would you finish this lesson? Would you do anything differently? I’d love to hear, as I might be doing this again with different classes in a week or two.

  1. If you intend to run the program, see the prior post for how-to info.