Punnet squares in math?

[LinRTX]

I had never heard of this before, but a child I was tutoring was talking about it.

[Clemsondana]

Some of the kids that I volunteer with are taught this way - if they have 62 x 27 then they make a square that looks something like 

         60       2

20

7

and then they fill it in like a punnet square.  I think it's a useful way for them to see what's happening, but they don't always seem to go back and show them how to do the algorithm and explain how it's the same thing, so kids continue to do this really inefficient method.  They can make a bigger square and do hundreds.  It is also a way of doing mental math, but the kids that I work with are often using a chart with the multiplication tables written on it to look up each step so definitely no mental math happening.  They do something similar for division - each problem takes 1/3 of a page because they draw these big boxes and do something in each box.  

[wendyroo]

Sure, Math Mammoth and many other programs call that the area model.
I introduce it as covering the top of a banquet table with 1x1 sticky notes...and wanting to know how many I will need to cover the whole thing.

I draw a rectangle to represent the table, and label the sides with the two numbers we are multiplying. We all agree we would need length * width post it notes to cover it...if only we knew how to multiply those big numbers. So then I suggest dividing it into pieces that we know how to multiply.

If my banquet table is 12 lights years by 34 light years big (the more ridiculous the visual, the more my kids like it and remember), then my area model looks like the 12 split into pieces of 10 and 2, and the 34 split into pieces of 30 and 4:

         10    |   2
-------------------
30 |   300  |  60   |
-------------------
4   |    40   |  8     |
-------------------

We can clearly see how many post its we will need in each section, and add them for a total of 408 post it notes each 1 light year by 1 light year big.

I like it because we revisit it in algebra to keep ourselves organized when multiplying polynomials like (x+4)(x^2+5x+3)

     x^2     5x       3
x   x^3    5x^2   3x

4   4x^2   20x    12

We can fill in each multiplication, combine like terms along the diagonals, and find an answer: x^3 + 9x^2 + 23x + 12

[City Mouse]

As a former 4th grade math teacher, I think that the area model works well for students (and parents) who do not understand why the traditional algorithm works. I never had a student who could not eventually transistion from area models to the standard algorithm, and seems to have a higher initial success rate over students who just memorize what I called a “recipe” and then struggle to remember where to put seeming random zeros (I know they are not random, but those students do not) and what to carry when. I never forced students who were successful with the traditional method to use area models outside of the initial introduction. 
 

While some parents balked at first, most were fine with it when I could verbalize why I felt area models were beneficial, and I included what Wendy mentioned above about the same method being used when multiplying polynomials.