I learned this decades ago from my middle school math teacher for 6th and 8th grades. ** I still use the hack to this day.** Does anyone know of any other interesting and useful math hacks?

I learned this decades ago from my middle school math teacher for 6th and 8th grades.

**I still use the hack to this day.**Does anyone know of any other interesting and useful math hacks?Apparently, math enthusiasts on Twitter were interested this weekend in the math hack. So I thought I’d share a proof of why it works.

To prove in math that something doesn’t work, you only have to show one case of failure. To prove that something always works, you need to generalize using variables.

How could we generalize 65*65, 75*75, etc?

It looks like ten times a number plus five, then squared:

(10🦆 + 5)^2

So let’s do the math and hopefully we will decipher the magic.

(10🦆+5)(10🦆+5)

= 100🦆🦆 + 50🦆 + 50🦆 + 25)

= 100🦆🦆 + 100🦆 + 25

It took a moment to realize that regrouping will reveal the magic:

= 100(🦆🦆 + 🦆) + 25

Now to write this as a rule.

**Take a two digit number, last digit a 5, tens digit equal to****🦆**.I choose 35

**Multiply the 🦆 digit times itself, add 🦆 to the squared number.**3 times 3 is 9

9 + 3 = 12

**Multiply that number by 100.**1200

**Add 25**1225

********

In practice, with 35*35 I will mentally multiply 3*4 to get 12 and then tack on a 25 to the end of it.

Oh hmm, that’s a neat trick.

I wonder what the basis is behind it. Something provable?

My kids are learning from Beast Academy (in homeschool) and it's full of neat and useful ways to play with numbers.

Perhaps my favorite so far is a way to quickly determine if any number is divisible by 9, 6, and/or 3.

Say, this number:

1112172

(The photo below has my daughter looking at Latin, not math, but my husband's reaction is pretty much how I feel when I discover my kids have learned a shortcut I don't know. ;) )

I felt old as a teacher when high school students told me that they didn’t know the 3,6,9 rule. I didn’t use calculators in school until high school and remember learning that shortcut in middle school: I suppose it was more useful back then when you didn’t have a smartphone handy.

You know, I wasn’t sure of the age range of your homeschooler, but they might enjoy this game that @Mathgarden has used with his daughter. He says that knowing prime numbers is foundational, but I just think it’s a fun and addictive little game. Don’t try to beat his high score: last I heard, he got 56 correct in one minute(!).

Oh boy, you got me hooked on that game. I haven't managed to beat his score... yet.

I guess one math game leads to another because when I showed that one to my husband he turned me on to this one: Euclid the Game.