So it can be difficult for people to do long multiplication in their heads. Luckily, the distributive property of multiplication and division can make it easier. Distributive property: a(b + c)=ab + ac. This can help, because basically, if you have two numbers multiplied by each other, you can split one or both of the numbers into smaller, more manageable digits–and you multiply the groups together. This can be easier than trying to carry digits in your head like you carry digits when doing long division on paper.
So say you have 17*15. You can split that into (10 + 7)(10 + 5). That equals, somewhat in the order you would do it doing long multiplication on paper (you can do it in any order you want)–(5 * 7) + (5 * 10) + (10 * 7) + (10 * 10). That’s 35 + 50 + 70 + 100, which is pretty easy to remember when you’re doing it in your head, as opposed to trying to carry digits etc like you do when figuring out long multiplication on paper. (I didn’t do it in the “FOIL (first-outer-inner-last)” order, I wanted to keep the order more like when you do long multiplication on paper.)
This is similar for division. For example, say you have 500/3. First, 3*goes into 5 1 times, but we have 500 not five, so 3 goes into 500 100 times, so 3*100=300, with 500-300=200. Then 3 goes into 20 6 times, but we have 200, so 3 goes into 200 60 times. So 3*60=180, and 200-180=20. We know that 3 goes into 20 6 times, 20-18=2. Then 2/3= .6, you keep getting a variation of that over and over. So add it all up: 100+60+6+.6 etc = 166.6666 and on.
Just read about this JUMP system which is supposed to make math easier for kids (and adults) to learn–I’m going to look into, that’s what I want to do too, make a system like that!
Other very important algebra for breaking down math problems:
(a + b)/c = a/c + b/c. This is really because of the distributive property: dividing by c is the same as multiplying by 1/c. So (a + b)/c is the same as (1/c)(a+b)=(1/c)a + (1/c)b=a/c + b/c.
(ab)^c=a^c * b^c
ln (a^ b) = b* ln a
Image from Katey Nicosia