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Here's another use-case that came up in my college remedial algebra class tonight (and again, this boils down to translations to mixed numbers): Finding fractions in a graph. So the specific example that presented itself tonight was a book exercise: "Graph the equation: $y = - \frac 5 3$". At that point, my students could tell me that this would be ...

0

For me it is simply a difference between diving 1 unit vs. dividing a group(more than 1 unit). I teach very small kids and sometimes I explain them a difficult topic for example fractions and try to find out WHAT EXACTLY are they not understanding, why they don't understand. So I have found that difference of diving/splitting 1 vs dividing/splitting group is ...

2

The appearance of "mixed numbers" is inherent to the Euclidean algorithm. For example, \eqalign{\scriptsize{\frac{355}{113}}\rightarrow& 335=\color{red}{3}\cdot113+16,\ \scriptsize{\frac{355}{113}}=3\scriptsize{\frac{16}{113}}\\ \scriptsize{\frac{113}{16}}\rightarrow&113=\color{red}{7}\cdot16+1,\phantom{xx}\ \scriptsize{\frac{113}{16}}=7\...

3

I did some research, which you can follow here, that explains the why of the nomenclature we use. Basically, something is proper if it is contained in something else, and improper otherwise. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6. This set excludes 12, because that represents the entire portion, e.g. no fraction occurs. Similarly, in ...

6

Interestingly enough, we do not have such a distinction in France. From the French Wikipedia article about fractions (emphasis mine): Dans l'enseignement français depuis la fin du xixe siècle, la fraction est définie comme le quotient de deux nombres entiers sans contrainte sur la taille du numérateur et du dénominateur (...) In French education, since the ...

6

The use of rulers with fractional inches is the first thing that springs to mind. Like this: The four keys have a width of $2 \frac{11}{16}$ inches at the tops of the key caps. If I calculated a length, and got $\frac{43}{16}$ inches, I'd have to convert it to $2 \frac{11}{16}$ to actually measure it. The "improper fraction" $\frac{43}{16}$ is ...

7

Student are introduced to fractions as part of a whole. They are then taught that improper fractions can be more than a whole - this is not ideal terminology or helpful for understanding. Improper fraction is a terrible name since it implies that there is something wrong with the fraction. Once student start to do calculations with fractions greater than one,...

8

I do not know of any relevant research. Here are my own not-research-informed ideas. Most people refer to fractions as parts of a whole. If someone says "I lost a fraction of a pound on my diet", you can be fairly certain that they didn't lose $\frac{23}{1}$ pounds. Since the common usage of the word and the mathematical usage differ, it is useful ...

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added Oct 6 The reason mixed numbers are found in US education is that mixed numbers are found outside of school in the US, so the children need to learn to understand them. Mixed numbers are found in road signs, cooking recipes, length measurements, and so on. (Denis Nardin commented that mixed numbers are never seen in Italy. Meters, centimeters, and ...

2

The Kahn Academy is quite extensive, and in my experience of high quality. It is partitioned by grades (kindergarten, 1st, .2nd, .., 8th), and by topic (arithmetic, geometry, pre-algebra, etc.) It has a nice interactive interface, practice questions and levels. A student can have a personalized dashboard. They've now branched out to Economics and AP-Biology ...

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Beast Academy is wonderful. It goes from a level 2 to a level 5, but those are higher than U.S. grade levels. Also, I have some of my favorite books listed at my blog.

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