### This comes from the Common Core

*Develop understanding of fractions as numbers.*

*1. ……*

*2. Understand a fraction as a number on the number line; represent*

*fractions on a number line diagram.*

*a. Represent a fraction 1/b on a number line diagram by defining the*

*interval from 0 to 1 as the whole and partitioning it into b equal*

*parts. Recognize that each part has size 1/b and that the endpoint*

*of the part based at 0 locates the number 1/b on the number line.*

*b. Represent a fraction a/b on a number line diagram by marking off*

*a lengths 1/b from 0. …….*

### ….but how do you do it ?????

The mystery is solved…….

Here is the line, with 0 and 1 marked. /You chose it already !

Here is a numbered line, any size, equally spaced, at intervals of one unit.

It only has to start from zero.

Now construct the line from point 5 to the “fraction” line at point 1, and a parallel line from point 1 on the numbered line.

The point of intersection of the parallel line and the “fraction” line is then 1/5 of the distance from 0 to 1 on the “fraction” line.

1/5, 2/5, 3/5, 4/5 and 1 are equally spaced on the fraction line.

L cannot be moved in the static picture.

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This is so cool! I love it and I think I understand! This is a grade 3 standard, and normally we would do what you did at line “a”: start at 0 and mark off 5 evenly spaced (well, as even as possible) intervals. Then we’d label each 1/5, 2/5, etc up to 5/5 or 1 whole. So we’re starting with the partitions first. But the standard says to start with the whole first and then do the partitions. I guess my question would be: Does the difference have a significant impact later on that caused the standard to be written that way?

Joe: The standard is conceptually wrong. It makes no sense at all.

I spoke to my wife on this and suggested a set of equally spaced (parallel) lines, separation immaterial, and a brick or a piece of ribbon. Set up the ribbon so that its length fits between the parallel lines, at an angle, to give five (or whatever) equal bits – and there you have one fifth of the ribbon in each section. My wife explained that she learned that at her mother’s knee in the 1960’s.

I will do you a diagram if it helps.

Done on the next post of mine.

Thanks Howard! So I see how you did it on your latest post. It’s so simple really. But now I want to understand what makes the standard conceptually wrong.

My thoughts are that yes you can set up a line segment and call it one unit, but you cannot assert that the point 1/5 of the way along has any real existence without a construction (a Euclidean construction). Perhaps “conceptually wrong” is a bit harsh!

I was thinking also of angles and constructions, and it would appear that very few angles at all can be constructed. Therefore most angles have got to be measured, or deduced from measurements. These measurements are inevitably approximate

You can tell me to stop at any time, but I am genuinely curious because I’ve spent so much time living in this standard. So I would want to know: how would you rewrite the standard? Do you feel it’s placed appropriately?