Category: algebra

Total 38 Posts

Score One For The Forces Of Innumeracy

In these exponential times, I admit that even I find it easy to nod my head credulously at a passage like this:

When it comes to [Facebook’s] online chat function, 1.6 billion messages are sent every single day and 1.4 million photos are uploaded a second.

Not so Nat Friedman who crunched some numbers in an utterly classroom-appropriate exercise in unit conversion and calculated that this means everyone on Earth is uploading approximately 20 photos per day.

Which means I had better hurry up and get a Facebook account.

[via daring fireball]

[N.B. The Internet has been pretty generous today, right?]


What I Would Do With This: Pocket Change

[following up from here]

Appeal To Their Intuition

“How much cash is this?” Take guesses. The student risks nothing with a guess but that investment pays off huge for the teacher over the life of the exercise because the student wants to know who guessed the closest.

Build Slowly

Again, ask “how much cash?” but also ask “how heavy?” Show them the weight. (I zeroed out the jar from every weight measurement you’ll see here. Don’t worry about it.) Spitball some ideas for determining the value of those coins. You’re trying to motivate the idea that the weight of the coins ties directly to how much the coins are worth. Pull up the relevant Treasury website.

Then mix in some nickels. Scoop out a small sample. Play with that. Set up a proportion between value and weight.


Now you have pennies, dimes, nickels, and quarters. I took nine sample scoops, everything from small to big.

I formatted these at 4×6 so I could print them out at our local one-hour shop for a few bucks and put one in front of every student.

Throw A Curve Ball

Some will finish quickly. You tell them you have a jar of coins that weighs 5,500 grams. You reach in and pull out 14 nickels. How much is the jar of coins worth?

They’ll run these calculations and come up with an estimate of $55. You tell them it was really $34, which is huge error. Ask for sources of error. Then toss this up and talk about it.

Confirm The Answer

$84.00, if you were curious.

It’s essential to give some kind of visual confirmation of the answer, both so we can give credit to good initial guesses and so we can talk about sources of error. (ie. “who was off by the most? did sample size matter at all?”)


  1. Show them CoinCalc, the backend of which does exactly what we’ve done here.
  2. This activity follows-up nicely on the goldfish activity, where we used a small sample of fish to determine the total population of a lake.
  3. We yield the floor to Jason Dyer and anybody else who would like to debate the question, “why are we doing this digitally?”


Here’s the entire learning packet [62MB].

My Lesson Plan: The Door Lock

Michael Caratenuto:

Personally, I think that this particular image lacks opportunities for inquiry. Perhaps if it was presented with other kinds of door locks leading students to come up with and answer the question, “which is the most secure lock?” [emph. added]

This is exactly right. The latest WCYDWT? installment has provoked the usual litany of Really Interesting Bite-Sized Questions, the sort of prompts that will play great in the Applications & Extensions & Assorted Mindblowers section of your lesson plan but which, on their own, aren’t a lesson plan. Those questions don’t provoke the kind of iterated, increasingly difficult practice that students need for skill development.

Again, this image on its own is insufficient. With some creative modifications, however, it will carry you through permutations. Here is that lesson plan in its broadest strokes.

Start with the image.

Tell them the code is 1 digit long. Tell them the code is 2 digits long. Tell them it’s as long you want it to be. I respected the rule of least power here, which meant that when I took this photo I tried to stay out of the way of your lesson planning. Have them write down all the possible codes for n=1, n=2, n=3, etc. The increasing obnoxiousness of the task will motivate a formula for the general case. That’s arrangements.

Tell them the lock is a 4-digit lock. Now turn on the blue light.

Ask them to list the possible codes. You can iterate this a bunch of times until they have discovered on their own this tool that mathematicians call a factorial.

Remind them it’s a 4-digit lock. Then put up this image. It will be confusing, but only for a second. Ask them to list every possible code.

Iterate this with two and three buttons until they have generalized permutations. Then maybe you iterate the entire thing with another keypad lock.

Then maybe you dip into the comments of the original WCYDWT? post and help yourself to some very-interesting follow-up questions. I recommend Alex’s.

Let me close by saying how shocked I am at how little all of this costs.

[Update: Bruce Schneier has a good follow-up on information leakage. Two photos.]

[Update II: due to the peculiarities of many car door locks punching in “123456” tests both “12345” and “23456.” Consequently, there is a number string 3129 digits long that will test every five-number comination.]

[Update III: more information leakage.]

[Update IV: more information leakage.]

How Can We Break This?

I like this. The iPhone application RulerPhone will measure anything, in any photo, so long as the photo includes a credit card. It’s a great use of proportional reasoning, which, if pressed to name one, would be The Mathematical Skill I’d Most Like My Students To Retain After High School.

I added it to the What Can You Do With This? segment featuring The Bone Collector, which seemed like an obvious pair to me. In trying to find the best classroom entry point for this program, I can only think of the question, “How can we break this thing — trick it into giving an incorrect measurement?” I imagine someone can do better.

Asilomar #5: Michael Serra

Session Title

Games And Puzzles That Develop Sequential Reasoning

Better Title





A structure not dissimilar to Megan Taylor’s yesterday, where Serra debuted games and puzzles and gave us time to tease them out.

I sat with two former colleagues in the back — all of us now at different schools. One teacher enthused over Sudoku puzzles. They challenge kids. Kids like them. It gets them comfortable with numbers. The other enjoys Serra’s games and puzzles, like Lunar Lockout. Both cite improved student disposition toward math and improved deductive reasoning.

I disagreed with them. In general, I find it dangerous to put too much distance between “fun time” and “math time” preferring, instead, to have that cake and eat it too, creating as many challenges as I can that are both fun and mathematically rigorous. (Which Sudoko, to put it plainly, isn’t.) My task is harder, I think, and I know I fail at it more, but I’m more satisfied on balance.

It was a good conversation. Feel free to interrupt us.

Serra’s best offering for my money was Racetrack Math:

It’s like this:

  1. Draw a racetrack on graph paper, however crude.
  2. You and your opponent start anywhere on the starting line.
  3. You travel along vectors. You may increase or decrease either the x-value, the y-value, or both, but only by one unit per turn.
  4. First person to the finish line wins.
  5. (P.S. No crashing.)

This gets very interesting very quickly. You start out with tiny vectors which lengthen by one unit every turn. If you fail to notice the side of the track off in the distance, though, and fail to slow down in time, you crash. (Which I did in the example above.)

I hereby toss all of my battleship exercises in the recycling bin. This is a much more straightforward introduction to positive/negative coordinates since each new turn is relative to the last turn rather than relative to this strange coordinate axis thing.

Plus, your students can create racetracks of their own, of infinite complexity, within seconds. Serra cited some kids who created a pit lane, which you had to enter on your second lap, and oil slicks, on which you could not adjust your vector at all. I’m impressed.


PowerPoint. Which is tough when you’re asking people to solve a puzzle. If someone suggests an alternative route to the one you have programmed into your slide, you have to dodge their answer a bit.


Blank puzzles and games to draw on. Again, paper is not dead. How do you do this digitally? Load each picture one at a time into Skitch and pass a stylus back and forth? Moderation, please.


  • “There is no research that demonstrates these games improve outcomes in other mathematical procedures like two-column proofs,” Serra admitted reluctantly. “It has to be there. I know it is.