If Not Methods: Scary Division

We’ve been exploring the many ways to help children reason about tough math problems, without giving them rules to follow.

As always, real math is not about the answers but the thinking.

But what about division with scary, big numbers? What if our kids get stumped on a calculation like 3840 ÷ 16?

When kids say, “I don’t know how”

We can teach without crippling children’s understanding if we follow the Notice-Wonder-Create cycle:

• Notice everything about the problem.
• Wonder about the possibilities.
• Create something new: perhaps a solution or a math journal entry, or perhaps just a deeper level of understanding.

“Notice, Wonder, Create” is not a three-step method for solving math problems. It’s the natural, spiraling cycle by which our minds learn anything.

Two ways to think about division

As you notice and wonder with your students, take the opportunity to remind them of the two models we use to think about division.

In partitive division, we imagine cutting our total amount into a given number of parts:

[total amount] ÷ [number of parts] = [size of part]

This is a fair-sharing situation, which makes it easy for children to imagine in their heads, especially if we move out of the abstract realm and set up a life situation.

$3840 shared 16 ways = How much per person?

In measurement division, we imagine measuring out chunks of a certain size:

[total amount] ÷ [size of chunk] = [number of chunks]

How many 16s are there in 3840?

Measurement division (also called quotative division) is the basis for the long division algorithm.

Partitive division: Fair sharing

Suppose our robotics team won a big competition. The prize money was $3,840, and we want to share it equally among the 16 members of the team. How much will each person get?

Because 16 people is an even number, we can make our calculation simpler by splitting the group in half. If all the team members get $3,840, how much will eight members get?

Half the team gets half the money:
$3,840 ÷ 2 = $1,920

Then how much do four members get?
$1,920 ÷ 2 = $960

Then what’s the share for two team members?
$960 ÷ 2 = $480

Finally, how much does one person get?
$480 ÷ 2 = $240

Measurement division: Convenient chunks

If we’re trying to find out how many 16s there are in a large number like 3,840, we don’t want to count the chunks one by one. That would take all day!

Instead, we can look for the largest, easiest chunks to count:

100 sixteens = 1,600
200 sixteens = 3,200

So we have at least 200 sixteens in our total. How close did we get?

3,840 – 3,200 = 640 to go.

Now we look for smaller chunks, but still avoid the slow process of counting one-by-one:

10 sixteens = 160
20 sixteens = 320
40 sixteens = 640

And that uses up our whole amount. How many sixteens did we measure out?

200 + 40 = 240

Long division: Running a cookie factory

If you insist on teaching the method of long division, at least build a foundation that gives the method sense. Don’t teach arbitrary rules and mnemonic tricks.

Imagine you are running a cookie factory that produces thousands of cookies every day and ships them out to warehouses around the country, which will then distribute the cookies to local stores.

You can ship the cookies as:

• single cookies (samples for the distributor’s marketing team)
• boxes (ten cookies each)
• cases (ten boxes each = 100 cookies)
• pallets (ten cases each = 1,000 cookies)
• truckloads (ten pallets each = 10,000 cookies)

Today your factory produced 3,840 cookies, and you need to ship them out to 16 warehouses. And you can’t play favorites — each warehouse must get the same number of cookies.

Your accounting department came up with a method to keep track of inventory, subtracting cookies as they are shipped.

Here’s how it works…

To keep delivery fees down, you want to ship out the largest packages possible. But today’s production was low. You can’t send a truckload to each warehouse, or even a pallet each.

But you can ship out cases.

16 warehouses × 1 case per warehouse = 1,600 cookies
16 warehouses × 2 cases per warehouse = 3,200 cookies

Send two cases per warehouse and subtract the cookies you’ve shipped. Make sure you record the two cases in the correct place-value column!

Now you can break open the six remaining cases and add the boxes inside to those extra forty boxes, making 64 boxes of cookies ready to ship.

How many can we send?

16 warehouses × 1 box per warehouse = 160 cookies
16 warehouses × 2 boxes per warehouse = 320 cookies
16 warehouses × 3 boxes per warehouse = 480 cookies
16 warehouses × 4 boxes per warehouse = 640 cookies

Send four more boxes of cookies per warehouse, and subtract what you’ve shipped.

In this case, our inventory has dropped all the way down to zero, so our job is done. We don’t have any single cookies left.

Each warehouse received 240 cookies in all.