I can’t create a story as I think I’d hurt myself. I’d love to hear from the student about their rationale though. It’s fascinating and yet the student is in strong need of basic number sense. Pick any of those fractions… say one-half. IOne-half isn’t even in the middle of 0 and 1. The one thing positive I would be able to build from is the fact the student was able to somehow identify the five increments. Good luck Chris!
Here’s a possible logical thought train: it’s behind the decimal, so it’s a fraction. You told me decimals were another way of doing fractions. So I’ll put the numbers on the line like fractions. Still, I know how to count! And you told me fractions were less than one, so it makes sense that they’re all before the one. THere ya go… Now… what I don’t know is how sure I am of this. I could be following the time-honored “even if you don’t think it’s right, put something down so that they know you tried and maybe you’ll get partial credit,” or even, as I have done: “I know this is wrong but it’s the only ***ing thing that I can think of…”
Good one Megan! Once again the epic failure of number sense. Maybe if more time were spent with younger students on what numbers actually mean (and how they relate) rather than how to apply algorithms, we (I) wouldn’t be seeing mistakes like this quite so often.
Megan’s got this. Once you’ve got that .n = 1/n, it explains the sequencing, .1,.2,.3,.4. Good thinking guys – I was mystified. I looked at the fractions, and couldn’t think why the student would order them that way.
I’d do more representation work with decimals with this student before getting to the fractions. Even though most curricula do fractions first, I’m not sure that’s the right approach for most students.
Here’s another question: Why are they plotting fractions at all? Most students I know prefer decimals. The question just says plot 0.4, not turn it into a fraction. Labeling 0.1, 0.2, 0.3 seems more natural. Is it the word “fractions” in the header? Was it the fact that the number line was divided into sections?
Seems they completely ignored the labels {0, 1} on the number line anyway. (re: “where they plotted 1/4″… they just seemed to be labeling each dash… if there were more dashes, would they not have kept going to 1/6, 1/8?) Either way, seems to be a disconnect between the idea that Larger Denominator -> Smaller Number while Larger Decimal -> Larger Number. Which stems from Megan’s remark above.
l hodge
A researcher did some really interesting interviews with a student in a self paced program from the 70s. The student was bright, but developed some misconceptions similar to what Megan suggested.
If we want to be creative with the student’s thinking, I think the student may have read the decimal as “point 4” so from there labeled the four points on the number line. The directions say “Fractions and Decimals” and since the decimal was given, it was time for some fractions! So the fourth point, 0.4 = 1/4.
Would love to hear the student explain this one!
-Kristin
Julie Campos
The student plotted “.4” on the number line as 1/4 because they probably are still unfamiliar with what “.4” is as a fraction. Also, if you notice, in general, they are unfamiliar with the equivalence of fractions to decimals. For example, if you see they put 1/1 on the first dash. The students here probably thought the marks on the number line went in order of 1/1, 1/2, 1/3, 1/4. For someone who is unfamiliar with fractions and with the equivalence they have to decimals, this mistake is understandable because 4 is bigger than 3, 2, 1 but when you put a number like one on top of each of these numbers, 4 turns into 1/4 and it is now the smallest fraction on the number line. If this was a third or fourth grader who made this mistake I understand because they are still becoming familiar with the importance in comparing fractions to make sure that each fraction refers to the same whole.