Geometric line: constructing fractions

Question

I am interested in teaching maths visually. in page 36 of Growing ideas of number (by John N Crossley) the following image appears, yet I cannot fully grasp how to interpreted it.

Answer 1

In Growing Ideas of Number, Crossley provides this diagram, Figure 3.4, in a discussion of the notion of the "geometric line." From the perspective of modern mathematics, all of the points on the continuous "geometric line" correspond to real numbers, given some choice of 0 and 1. Many early mathematicians did not think of numbers this way. They saw numbers as corresponding to measurements taken using some unit length. We start with a unit length, and all other numbers must be constructed from this unit length. Bass (2018) calls these two perspectives the "occupation narrative" and the "construction narrative."

In the construction narrative, the integers and the rational numbers are legitimate numbers because they can be constructed. This is what Crossley is talking about in the following paragraph.

It is easy to take some measure as a unit, and then to measure off all integer multiples of that unit. I shall ignore the question of where negative numbers come from... Then it is easy to construct fractions of these multiples by drawing similar triangles. See Figure 3.4.

The purpose of Figure 3.4 is to show how, from a segment of length 1, we can construct segments of length $\frac{1}{n}$ and $\frac{m}{n}$. Here is a construction procedure that you can try to follow while looking at the diagram.

1. On a horizontal line, mark a segment of length 1. Call it OC.
2. Construct a vertical line passing through O.
3. On the vertical line, mark a segment OU. (U corresponds to the "1" on the vertical axis.)
4. Copy segment OU end-to-end along the vertical line to produce segments OM and ON whose lengths are $m$ times OU and $n$ times OU respectively.
5. Draw line CN.
6. Construct a line parallel to CN passing through U.
7. Mark the intersection of the line just constructed and the horizontal line as A. Triangle AOU is similar to triangle CON, and OA has length $\frac{1}{n}$.
8. Construct a line parallel to CN passing through M.
9. Mark the intersection of the line just constructed and the horizontal line as B. Triangle BOM is similar to triangle CON, and OB has length $\frac{m}{n}$.

This is related to the construction procedure of "dividing a line segment into equal parts." Here's a step-by-step animation and proof of that procedure. Instead of similar right triangles, it uses similar acute triangles. The underlying principle in both constructions is that the side lengths of similar triangles are in proportion.

As Sue mentions in her answer, Figure 3.4 is a bit confusing because the axes don't use the same scale. Perhaps this is because the scale of the vertical axis doesn't matter for the construction. We just need similar triangles whose corresponding sides are in the ratios $1:n$ and $m:n$. We can interpret the "1" on the vertical axis as an artifact of this part of the construction. The "1" on the horizontal axis marks the unit length that we want to divide into $\frac{1}{n}$ and $\frac{m}{n}$. Or perhaps the “1” on the vertical axis is meant to illustrate that, from a unit length, lengths $n$ and $m$ are also constructible.

This construction gives us the "rational number line," but early mathematicians knew that not all lengths could be represented in this way, and so this is not the "geometric line." But should the geometric line be discrete, like the rational number line, or continuous? According to Crossley, both are logically possible. Crossley attributes the resolution of this problem to Dedekind, referring to his discussion in Section 5.1 of Dedekind's construction of the real numbers.

In the occupation narrative, the continuous nature of the geometric line is assumed from the beginning. As Bass says, "In contrast with the construction narrative, wherein more and more points are installed to build the line, all points are present from the start in the occupation narrative, and more of them acquire numerical names across the curriculum."

Answer 2

I don't think this diagram would help kids understand fractions. But I do like how it makes me think.

Your problem might be that the 1's are not to scale.

If we are given that the 3 lines (or line segments) are parallel (which I hope was mentioned in the book), then the slopes are all the same. If we are given the numbers on the y-axis and the 1 on the x-axis, we can use those equal slopes to find the other numbers/points on the x-axis. Call those points a and b. Then looking at the common slope gives us n/1 = 1/a, so a = 1/n. And n/1 = m/b, so nb = m and b = m/n.

Perhaps constructing diagrams like this in geogebra yourself would help you. I started to do that here.