Let’s have a number line. We can count up or down by moving to the right or the left, or actually up or down if the number line is drawn vertically.
But what else ?????
The following pictures show how points on the number line which represent fractions can be found exactly by simple geometrical construction, and then how results of multiplication and division of fractions can be found exactly as points on the number line (sorry, the numerical values are however not found).
This arose from a statement in the CCSS math document that the fraction 1/6 could be represented by a point one sixth of the way from zero to one, BUT NOWHERE DOES IT SAY HOW TO FIND THAT POINT.
So, you want to do a fraction subtraction. Here’s how, as a geometrical construction. You will need a piece of paper and a ruler.
Draw three number lines through a common point, which is the zero. Pick a nice point on the middle line to be the 1, say 6 inches away from the zero. Label the other two number lines 1,2,3,4,5,6,7 at equally spaced points, scale completely immaterial.
Now do what is shown in the picture below. (the pairs of lines are parallel)
Now measure the distance with the ruler, and divide by 6 (if you put the 1 at the 6 inch point).
A simpler version of this (2 number lines) can be used to locate the point on the number line corresponding to any (relatively simple) fraction.
While working on my software for 3D spline curves I needed to find a point between two others which was not halfway.
This turned into finding the ratio which an angle bisector of a triangle splits the opposite side. I worked it out with coordinate geometry and vectors, messy, messy, and then found out that this was a regular theorem in geometry. Here is the geometrical proof which I came up with. It sure ties a few things together: