Multiplication: Finding the Greatest Product

From a set of 1 through 9 playing cards, I draw five cards and get cards showing 8, 4, 2, 7, and 5. I ask my 6th graders to make a 3-digit number and a 2-digit number that would yield the greatest product. I add, "But do not complete the multiplication — meaning do not figure out the answer. I just want you to think about place value and multiplication."

I ask for volunteers who feel confident about their two numbers to share. This question brings out more than a few confident thinkers — each was so confident that he/she had the greatest product. (I'm noting here that I wasn't entirely sure what what the largest product would be. After this lesson, I asked some math teachers this question, and I appreciate the three teachers who shared. None of them gave the correct answer.)

I say, "Well, this is quite lovely, but y'all can't be right." I ask everyone to look at the seven "confident" submissions and see if they could reason that one yields a greater product than another, then perhaps we might narrow this list down a bit.

Someone sees "easily" that #7 is greater than #6. The class agrees.

Someone says #7 is greater than #1 because of "doubling." She says, "I know this from our math talk. Doubling and halving. Look at #1. If I take half of 875, I get about 430. If I double 42, I get 84. Both of these numbers [430 and 84] are smaller than what are in #7. So I'm confident #7 is greater than #1."

Someone else says #5 is greater than #4 because of rounding, "Eight hundred something times 70 is greater than eight hundred something times 50. The effect of multiplying by 800 is much more."

Someone says, "Number 2 is also greater than #1 because of place value. I mean the top numbers are almost the same, but #2 has twelve more groups of 872."

But the only one that the class unanimously agrees on to eliminate is #6. Then I ask them to take 30 seconds to quietly examine the remaining six and put a star next to the one that they believe yield the greatest product. These are their votes.

I tell them that clearly this is a tough thing to think about because we've had a lot of discussion yet many possibilities still remain. And that's okay -- that's why we're doing this. We've been doing enough multiplication of 2-digit by 2-digit during math talks that it's time we tackle something more challenging. So #3 gets the most votes.

I then punch the numbers into the calculator, and the kids are very excited to see what comes up after each time that I hit the ENTER key. Cheers and groans can be heard from around the room. Turns out #3 does has the greatest product (63,150) out of the ones shown.

Ah, but then someone suggests 752 times 84. I punch it into the calculator and everyone gasps. It has a product of 63,168.

Their little heads are exploding.

I give them a new set of five for homework: 2, 3, 5, 6, and 9. They are to go home and figure out the largest product from 3-digit by 2-digit multiplication. They come back with 652 times 93.

The next day, we try another set: 3, 4, 5, 8, and 9. We get the greatest product by doing 853 times 94. There is a lot — as much if not more than the day before — of sharing and arguing and reasoning about multiplication and place value.

Many of them see a pattern in the arrangement of the digits and are eager to share. They've agreed on this placement.

Then we talk about making sure we know we've looked at all the possible configurations. They agree that the greatest digit has to either be in the hundreds place of the 3-digit number or in the tens place of the 2-digit number. We try a simple set of numbers 1 through 5, and we agree that there are just 9 possible candidates that we need to test. The same placement holds.

Then we draw generic rectangles to remind us that we've just been looking for two dimensions that would give us the largest area.

I remember saying to the class, more than once, that this is tough to think about. To which Harley, sitting in the front row, says, "But it's like we're playing a game. It's fun."