## 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.

Iterate

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.

\$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?”)

Miscellaneous

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].

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Mr. K.

October 9, 2009 - 4:12 am -

why are we doing this digitally?

It may be because I teach middle schoolers, but my kids really do thrive with stuff they can grab.

I get why parts of this are done digitally – we want predictable answers, we want to be able to highlight the nickels only problem, and we want to avoid having the kids filch quarters to get themselves gatorade during lunch.

Still, It seems that there’s room in this lesson to let at least one kid shove his hand into a jar and pull out a handfull of change.

Oh, and I think it’s a wonderful follow on to the fish problem.

2. #### Jason Dyer

October 9, 2009 - 6:16 am -

Man, no pressure.

To me the issue is purely logistic. You may or may not be at a site where you feel comfortable having \$75 in change around.

Just to keep managing things simple if I ran this lesson I’d likely stick with pennies for the hands-on part. I’d feel uncomfortable running this with no hands-on at all, because to be honest I feel a little uncomfortable with just the pictures (I don’t have a good feel for what 5500 grams means) so I’m sure some of my students will be too.

Plus, y’know, kinesthetic learners.

There may also be logistics involved in how much time you have to allocate for the lesson. With all the topics a math teacher needs to handle digital may be the only way to squeeze it in here.

One last point is it’s possible to pull an estimate with only the original raw picture, with no weight and no extra scale. What is visible in the picture can be your random sampling, and you can pull out an actual nickel and measure it to get an idea of the scale; then do some geometry to get the volume. It’s a rougher estimate than the weight will give you but it works if you’re teaching Geometry rather than Algebra.

3. #### Claire

October 9, 2009 - 9:24 am -

I really like this activity; it hooks the students and follows a great progression.

The fact that you’ve shared what
“took a week of detailed planning and an afternoon of careful shooting” is fabulous! Hopefully most folks will be thinking “of course he shared it”, and not just because it’s Dan Meyer. When you put a bunch of effort into a really delicious unit the only thing better than it working really well with your own students is knowing that your effort is paying off for other teachers and their students.

1. The photo of the first jar of mixed coins, the mass of that one is 6,634 grams right?
2. In the shots where your hand is going in to scoop out coins and the mass of the jar is indicated below, a literal interpretation would be that the mass of all the coins is the printed mass plus the mass of coins in your hand. This isn’t the intent, but does contribute a bit of noise to the visuals. Not sure though that there would be a solution that wouldn’t create distractions of its own.

Cheers!

4. #### Dale Basler

October 9, 2009 - 1:37 pm -

Don’t forget about the pre-1982 pennies. That’s a story worth sharing.

@Claire Agree with your correction- it’s mass not weight. Surprised that the U.S. Mint has it wrong.

5. #### TheInfamousJ

October 9, 2009 - 2:07 pm -

Perhaps I am reading it wrong (very possible), but shouldn’t the coin weight be consistent as 5,500 grams or 6,634 grams? Why does it change?

Also, if you want to make science teacher’s lives easier by teaching weighing-by-difference what I’d do (’cause I’m a science teacher) is:

(1) Show picture of entire coin jar with mass total for coins at bottom.

(2) Hand out sample scoop images with mass of jar listed as what was left AFTER the scoop came out.

Students could then subtract to find the mass of the scoop.
Then on their own, they could set up their own masses to value ratios.

This would add time to the lesson, for sure, but would also really really work them, mathematically.