## What is Multiplication – part X

Yesterday, in the supermarket, a customer dumped a whole basket full of power bars on the counter.  The checkout counter person started to scan them in, one by one.  After seeing about seven of them scanned, with a lot more to go, the customer piped up: “they are all the same!”  The checkout person laughed and started to count them.

What just happened?

The customer apparently saw a faster way for the checkout person to handle the power bars, and the checkout person saw it too.  The critical ingredient was knowing that all power bars were the same.  How was this helpful?

I thought this was an interesting little puzzle.  Of course, given the title of the post, you may be sure that I see it as related to multiplication.   More specifically, multiplication viewed as a short cut.

I find this interesting since it is rarely the way kids in school relate to multiplication.  Kids in school often think of multiplication as hard and tedious, particularly as compared to addition.  In the context they usually encounter addition and multiplication, you might agree with them.  If you are given two numbers and you are supposed to add them, or you are given two numbers and you are supposed to multiply them, which of them is easier and less work?  Compared this way, you can see the preference for addition.

But comparing addition of two numbers and multiplication of two numbers may not be how it shows up in real life.  Let’s compare the situations that presented themselves before and after the customer’s comment: a pile of mixed power bars on the one hand, and a pile of identical power bars on the other hand.

mixed:       .34 + .36 + .45 + .29 + .34 + .34 + .35 + .40 + .60 + .20 + .29 + .34 + .36 + .36

identical:  .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34 + .34

On learning that all power bars were identical (and clearly trusting the customer on this) the checkout person knew she could simply count the remaining bars (14) and scan just one, and the cash register would handle it as a multiplication: $14 \times .34$ for faster service.  Multiplication as a short cut for repeated addition of identical numbers.

Note that even in the absence of a cash register that handles the multiplication for you, thinking of the problem as a multiplication gives you many different options for coming up with the result.  For example, you could split it in a group of ten and 4 separate ones, and know that the group of 10 cost $3.40, and the separate 4 could be added on one at a time. Or you could see it as seven pairs and compute the cost of a pair as .34+.34 =.68 and now only have to add .68 + .68 + .68 + .68 + .68 + .68 + .68, which is way fewer additions than you started out with. Or maybe you notice that 10 bars cost$3.40 and 5 bars cost half of that: $1.70, and so 15 bars add up to$ 5.10 and finally 14 bars cost $.34 less than that,$4.76!

So let’s see if we can state this result in an obvious and accurate way that yet sidesteps some big controversies that I don’t see as helpful at all.

repeated addition of identical numbers is multiplication!

What I found most interesting about the whole episode is how easily the customer and the checkout person understood each other, and how the notion of multiplication as a short cut for repeated addition seemed to underlie both their actions.