Intuitive explanation of Addition, Multiplication, Division and Exponents

Hi,

Often times I get confused why two quantities in given equation or to derive a certain equation is multiplied or divided. e.g. to find a slop of a line we do (change in y) / (change in x) . so why division here?

Another example is, fraction 2/4 which we can think of that there are 4 slices of pizza and if we take 2 slices it represent 2/4 of whole. But how to understand fractions like 4/2, 10/2, 10.5/5, 25/1.5. may be like "there are 10 candies and we want to make batch of 2, how many batches we get? or we have a rod of length 10.5 meter and we want 5 equal parts, how long each part will be". T

I heard about following models:

Addition - repeated counting or sliding

Multiplication - repeated addition or scaling

Exponents - repeated multiplication or Growing for amount of time

I could understand some of them but not everything is very obvious mainly when it comes to fractions.

How to represent multiplication(scaling up) as addition(sliding up) for fractions. e.g. 3 * 1.5 as addition?

How to represent addition(sliding up) as multiplication (scaling up) for integers and fraction. e.g. how do you represent 3 + 5 as a multiplication?

How to represent division (scaling down) as subtraction ? It should be possible I think.

How to represent subtraction (sliding down) as division?

Multiplication and Addition are increasing effect but how to know if it's sliding up or scaling up so we can formulate appropriate formula, or it doesn't matter? because sliding up = scaling up ?

Division and subtraction are decreasing effects but how to know which ones to use for our formula? or it doesn't matter? because sliding down = scaling down?

Can you explain all this topics in details as possible with enough examples for each?

Intuitive explanation of Addition, Multiplication, Division and Exponents

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