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."

Two Pizzas and Five People

I'm thinking a lot about how my 6th graders responded to a pre-lesson task in "Interpreting Multiplication and Division" — a lesson from Mathematics Assessment Project (MAP) .

MAP lessons begin with a set structure:

Before the lesson, students work individually on a task designed to reveal their current levels of understanding. You review their scripts and write questions to help them improve their work.

I gave the students this pre-lesson task for homework, and 57 students completed the task.

I'm sharing students' responses to question 2 (of 4) only because there's already a lot here to process. I'm grouping the kids' calculations and answers based on their diagrams.

Each pizza is cut into fifths.

About 44% (25/57) of the kids split the pizzas into fifths. I think I would have done the same, and my hand-drawn fifths would only be slightly less sloppy than theirs. The answer of 2 must mean 2 slices, and that makes sense when we see 10 slices total. The answer 10 might be a reflection of the completed example in the first row. "P divided by 5 x 2 or 5 divided by P x 2" suggests that division is commutative, and P here must mean pizza.

Each pizza is cut into eighths.

Next to cutting a circle into fourths, cutting into eighths is pretty easy and straightforward.

Each pizza is cut into tenths.

I'm a little bit surprised to see tenths because it's tedious to sketch them in, but then ten is a nice round number. The answer 20, like before, might be a mimic of the completed example in the first row.

Each pizza is cut into fourths.

I'm thinking the student sketched the diagram to illustrate that the pizzas get cut into some number of pieces — the fourths are out of convenience.  The larger number 5 divided by the smaller number 2 is not surprising.

Each pizza is cut into sixths.

It's easier to divide a circle by hand into even sections, even though the calculations do not show 6 or 12.

Each pizza is cut into fifths, vertically.

Oy. I need to introduce these 3 students to rectangular pizzas. :)

Five people? Here, five slices.

Mom and Dad are bigger people, so they should get the larger slices. This seems fair. We just need to examine the commutative property more closely.

 Circles drawn, but uncut.

I'm wondering about the calculation of 5 divided by 2.

Only one pizza drawn, cut into fifths.

9

Twenty percent fits with the diagram, if each person is getting one of the five slices. The 100 in the calculation might be due to the student thinking about percentage.

Only one pizza drawn, cut into tenths, but like this.

10

I wonder if the student has forgotten what the question is asking for because his/her focus has now shifted to the diagram.

 Each rectangular pizza is cut into fifths.

Three kids after my own heart.

Five portions set out, each with pizza sticks.

I wonder where the 10 comes from in his calculation.

Five plates set out, each plate with pizza slices.

Kids don't always know what we mean by "draw a picture" or "sketch a diagram." This student has already portioned out the slices.

What diagrams and calculations would you expect to see for question 3?

There's important work ahead for us. The kids have been working on matching calculation, diagram, and problem cards. They're thinking and talking to one another. I have a lot of questions to ask them, and hopefully they'll come up with questions of their own as they try to make sense of it all. If I were just looking for the answer of 2/5 or 0.4, then only 12 of the 57 papers had this answer. But I saw more "correct" answers that may not necessarily match the key. We starve ourselves of kids' thinking and reasoning if we only give multiple-choice tests or seek only for the answer.

That's why Max Ray wants to remind us of why 2 > 4.