The denominator says how many divisions per region, numerator is the count.
Note inherently positive point of view.
Doubling both leads to a physically distinct situation, but same area
Adding the counts
Adding repeatedly -- leads to multiplying numerator
Dividing a rational by a natural number is dividing each divided region into more of its pieces. The numerator still just counts the total number of pieces. So the number gets smaller.
Putting those two things together, we multiply top and bottom.
Focus on dividing 1 region. Once that's done, we just multiply.
Scale the original 1 region to match the size of the denominator. Then count the regions that the 1 region occupies. So dividing by 1/2 leads to half the original region so there is a two count there. Dividing by 3/2 leads to a full region plus 1/2. So the original occupies 2/3 of that.
It comes out to multiplying by the reciprocal. Try to really sell this notion.
Common denominators
above was all positive rationals. Now we can employ the same trick as before.
show that doing all these things does not depend on the representatives. do a numerical example here; proof does it in general.
Each line through the origin represents a rational number. If it has slope m, then 1/m is the corresponding rational number (basically change in y over change in x and we represent the pairs as (numerator, denominator).
Addition represented by fixing the y-coordinate and then adding the horizontal line segment of the smaller past the larger one. Subtraction is the line segment in between, shifted to y-axis. Multiplication is projection to of coordinates and doing the multiplication there (what is multiplying line segment lengths)? then plotting the point and getting the line.