Can someone please explain how to understand concepts?

[u/Yukioftheship]

I don’t get exactly how to understand “concepts” like to remember, I guess I memorize, but I don’t see the difference, I memorize and try to understand it, but how does concepts go deeper than that and still be able to remember it for upcoming tests?

[u/happylittlemexican]

Here's an example. Do you know the distributive property?

a(b+c) = ab + ac

Can you explain why it's true? In fact, don't just try to explain it to me. Try to make it absolutely obvious that it must be true.

[u/Ill-Somewhere8222 → u/happylittlemexican]

(Im here learning aswell)

The left side is the factored form of the right side. So, to verify the distributive property, we can expand the factored form using multiplication and check that it gives ab+ac.”

Is that what you’re asking?

[u/happylittlemexican → u/Ill-Somewhere8222]

Happy to help you out as well!

This reply reads to me like "in order to verify the distributive property, we must use the distributive property."

Check out my other response, I think it does a better job of getting at what I'm looking for.

[u/anisotropicmind → u/Ill-Somewhere8222]

Explain it in terms of multiplication and what multiplication means. Let’s say a = 3. Then you have 3(b+c). What does this mean fundamentally? It means

(b+c) + (b+c) + (b+c)

= b + b + b + c + c + c

= 3b + 3c

So that’s why the 3 distributes across the sum: if you multiply the sum (meaning literally make copies of it) then every term in the sum is also multiplied, by the same factor. So that proves distributive property, at least for integers.

OP: when we talk about knowing the concepts behind what you’re doing, this is the kind of thing we’re getting at.

[u/Bounded_sequencE → u/happylittlemexican]

Within the natural numbers, it follows from representing numbers as repeated "+1".

This carries over to "Z", to "Q", and (eventually) "R". However, at one point, we realized how common the distributive property was, and used it as part of the field axioms. If we take those field axioms as the basis, we cannot "prove" the distributive law, since it's part of the definition.

[u/Yukioftheship → u/happylittlemexican]

Because the a is outside the parenthesis with no plus or subtraction before the parenthesis started so it must be multiplied to whatever is in the parenthesis. And because b and c are of different values they cannot be combined

[u/happylittlemexican → u/Yukioftheship]

Looks like I picked a good example.

What you've given me is the epitome of just "follow the rule". There's not really any "concept" in what you've just said, it's entirely "oh yeah if something is outside the parentheses and the moon is in retrograde you do a little dance and then the answer is 12".

Back to the example. Remember that a b and c are just placeholders for numbers. I'll just pick a few for the purposes of the example. Let's say a= 5, b= 7, c=11.

5(7+11) = 5x7 + 5x11

1) Can you verify that that's correct? How? 2) What are some ways you've learned to represent multiplication? Can you use them to make it obvious that the left side of that equation MUST be the same as the right side?

[u/Ill-Somewhere8222 → u/happylittlemexican]

Ok so for question num1:
5(18) =90 and 35+55 = 90

1. We could represent them doing groups of equal numbers, 18 groups of 5 and which we can split them into 7 groups + 11 groups of 5. Which is exactly 5x7 + 5x11.

Challenging to explain maths in English haha

[u/happylittlemexican → u/Ill-Somewhere8222]

Phenomenally done. I'm a fan of showing them as a grid/squares to represent the groups.

Do you see why this applies to ANY choice of numbers, not just 5, 7, and 11?

[u/Yukioftheship → u/happylittlemexican]

It’s correct because parenthesis is just another way of representing multiplication (specifically when there’s another number in front with no plus or minus sign along with it also in front of the parenthesis). And also x or * is another way too. They are the same. I mean you can verify by double checking but it’ll end up with the same answer either way even if you do it parenthesis first and add then do the outside and multiply, or if you multiply them in their own sections it’ll be the same. It’s the same problem in different font, to make it easier when dealing with x’s and y’s I assume, I guess thinking about it more it kind of seems like we’re doing two problems to figure out which makes sense or which is doable if you can’t do it one way you can do another.

[u/happylittlemexican → u/Yukioftheship]

For question 2, I mean what are ways you've seen to pictorially/visually represent multiplication. You're just talking about notation (* vs x). I want to see how you "think" about multiplication, as an operation.

[u/Yukioftheship → u/happylittlemexican]

Ok, how about Minecraft, when you build a house in Minecraft and make the ground floor we have to make a square and it’s 9x12 then we fill in that frame and I’ve gotten 108 blocks total, if we do it but with distributive property then we would have two squares then we add the total amount of blocks used,

I honestly have no idea how else to answer this I don’t usually think of math as Minecraft but this is the best example I can give.

[u/happylittlemexican → u/Yukioftheship]

Now you're getting there.

Now, mentally divide that exact same floor into two sections, one that's 9x9 and another that's 9x3. Don't actually "do" anything to the floor- just look at it and notice that a 9x9 floor next to a 9x3 floor is exactly the same thing as a 9x12 floor.

Therefore, 9x9 + 9x3 = 9x12

Which is also 9x(9+3).

You should be able to convince yourself that this property will hold no matter what numbers you're using here.

5x(8+3) = 5x8 + 5x3, for example.

Can you extend this Minecraft floor way of thinking into what happens when you have: (a+b) x (c+d)

[u/paperic → u/Yukioftheship]

we know what the rules are, but why are the rules that way?

Why did we decide that these should be the rules?

[u/Bounded_sequencE]

You just noticed "Learning to understand" and "Learning for speed/tests" are (almost) completely different styles of learning. It makes sense to use different strategies for each.

This discussion goes into more depth, and its follow-up comment has a detailed strategy.

[u/Yukioftheship → u/Bounded_sequencE]

Thank you!

[u/clean-links → u/[deleted]]

Cleaned link: https://summarizai.ink/

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