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Nice. Yeah, that's more understanding than about 75% of my students (It's distributive). And I wouldn't/don't have any problem with the "old school" way if kids are understanding that it's the distributive property as they do it. And how you could use that distributive property for binomial multiplication. Or what is has to do with this figure:

But the vast majority of my students put the zeros there "because my second grade teacher told me to."

Well getting a piece of paper 30+ years ago that says I am a ChemE does help with the math stuff. Granted I could not do a differential equation now if my life depended on it. I fit into the use it or lose it category.

Distributive - yes that rings a bell. Teacher - can you check my work from your example problem???

(x+1) * (2x+3) = 2x^2 + 2x + 3x +3 = 2x^2 + 5x + 3

I struggled in the beginning with those types of problems, but then something clicked and they were really easy after that.

ApolloAndy:

But the vast majority of my students put the zeros there "because my second grade teacher told me to."

I'm expanding this beyond the current discussion, but I think this is 90% of the problem with everything.

We go through the motions and do things without understanding why we do them. This is becoming especially prevalent in the workplace where everything is boiled down to checklists or task sheets or whatever. Doing A, B, and C is not the same as understanding the job or the company or your role in the process.

It applies to almost everything though. As I get older, I'm amazed at how many people go through life not understanding why they do what they do every day.

</old man mode>

2+2= cat.
My math ain’t good.
My English ain’t either.

Neither are your haikus.

Naw, it’s just a pome.

Shades:

(x+1) * (2x+3) = 2x^2 + 2x + 3x +3 = 2x^2 + 5x + 3

Yep. Sorry I switched to a different diagram after you solved it, but yes, that's the idea. The real take home is the similarity between what you did here, the two digit multiplication, and the geometrical representation. If a student can get that, then I feel like I've done my job and I hope/assume/believe they will have a better experience of math on the whole.

I used to try to do those problems that someone would post on my FB page. Inevitably they would say that I was doing it wrong, and my answer was incorrect. I reply is that 's the way I was taught 55 years ago. Get over it. I then permanently hide the page......