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At what age are most children able/taught to convert between rational fractions and decimals?

For example

  1. Convert 0.25 to a fraction consisting only of whole numbers.

  2. What is 3/4 expressed in decimal notation?

Notes

  1. Please answer with an age in years rather than a country-specific measure such as "Grade N"

  2. I'm interested in 'most' children rather than the exceptional prodigy.


Question

At what age (in relation to teaching an "average" class) is it realistic to expect young people to have the mathematical maturity to understand and carry out such computations?

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  • $\begingroup$ While I appreciate the green check, might I suggest that you hold off for a little while? It may be that someone else comes along with a better answer in the next couple of days (it is the weekend, after all). $\endgroup$ – Xander Henderson Jul 18 at 17:10
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    $\begingroup$ The title assumes that most children eventually can do this task, assuming facts not in evidence. The Q itself allows for "are taught," which is more answerable. $\endgroup$ – shoover Jul 18 at 23:33
  • $\begingroup$ @Xander Henderson - Okay! $\endgroup$ – chasly - reinstate Monica Jul 18 at 23:35
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    $\begingroup$ I know this is tagged as primary-education, but I assure you, the vast majority of high school graduates have not learned the nuances of fraction and decimal conversions. $\endgroup$ – WeCanLearnAnything Jul 19 at 8:01
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There is no single answer because there are several semi-overlapping methods to convert fractions and decimals. Depending on the numbers, they require drastically varying amounts of prior knowledge.

There are orders of magnitude more nuance here than meets the eye, making this an area where many educators, materials, and educational systems make curse of knowledge mistakes, both with and without awareness. So prepare for an epic post. It's gonna be loooooong.

But mastering the pedagogy here strikes right at the heart of why so many people hate math. If you need to cut back on time spent on surface area or bar graphs to do this, it's a worthy trade off. Frankly, f$%@ curriculum standards if you have to - mastering fraction, decimal, and percent conversions is far more important for the kids' futures.

Part 1: Converting Fractions to Decimals

Let's make the wildly optimistic assumption that the students already know that a fraction is a number and can intelligently compare and contrast fractional numbers with whole numbers. Only then should you bother with decimal conversions. Otherwise you will really pound into their head that math is just a bunch of weird rules to memorize. This assumption alone is worth many threads. :-)

But with that assumption in place:


Method 1: Apply Base 10 Conventions

Example: $\frac{3}{100}=0.03$ because the 3 is in the hundredths place. Though this is is trivial to memorize, it is actually quite difficult for students to internalize that this is just a logical extension of the base 10 number system, but, unlike their prior knowledge, we're scaling down here. Sadly, since many students, do not arrive in their fractions and decimal units understanding base numbers, all they can do is memorize this. In turn, that makes virtually all aspects of rational numbers extremely difficult to learn afterward.

Example: $\frac{36}{100}=\frac{30}{100}+\frac{6}{100}=\frac{3}{10}+\frac{6}{100}=0.36$ because the 3 is in the tenths place and the 6 is in the hundredths place. This requires an understanding of base 10 and the ability to see how tenths and hundredths correspond with number lines or 10x10 grids. They don't have to have conceptualized equivalent fractions yet, but that's ideal.

Method 2: Use Equivalent Decimal Fractions Then Base 10 Conventions

Example: $\frac{3}{5}=\frac{6}{10}=0.6$. Note that this requires all the previous knowledge mentioned above and then some.

Example: $\frac{7}{5}=\frac{14}{10}=1\frac{4}{10}=1.4$

Example: $\frac{7}{5}=1\frac{2}{5}=1\frac{4}{10}=1.4$

Example: $\frac{7}{250}=\frac{28}{1000}=\frac{20}{1000}+\frac{8}{100}=\frac{2}{100}+\frac{8}{1000}=0.028$. Notice the number of steps required to do this rigorously and correctly. Also since it's kinda hard to see visually how two-hundred-and-fiftieths and thousandths correspond, or even hundredths and thousandths, students must have conceptualized equivalent fractions to do this.

Method 3: Interpret Fractions as Quotients of Integers

Example: $\frac{8}{3}=8\div3=2.\bar{6}$

Example: $\frac{8}{3}=2\frac{2}{3}=2+2\div3=2+0.\bar{6}=2.\bar{6}$

Example: $\frac{12}{3}=12\div3=4$ [No decimal required! Remember to mix in plenty of whole numbers in ALL activities with students learning fractions and decimals as it is the only way to ensure integration of new knowledge (rational numbers) with prior knowledge (whole numbers).]

Example: $\frac{16}{32}=16\div32=0.5$

Notice that Method 3 assumes knowledge of fractions as a quotient. Tragically, this is usually just papered over in math classes and, for many students, becomes just another weird property to memorize. I'd say it's worth spending 3+ hours with students developing their intuition on why fractions can always be considered a quotient and relating that to their prior knowledge of division. A context to motivate this is also useful. "In a medical study, $\frac{379}{581}$ patients survived after taking a pill. $\frac{340}{521} $ patients survived after surgery. Which treatment has the higher survival rate? How do you know? Can you just use number sense and intuition to answer this? Or other methods of comparing fractions? Explain."


After learning the methods in that order, students should then be required to:

  1. Generate examples of fractions for which each method is most appropriate or to which multiple methods could apply, and carry out the conversions.

  2. Justify their reasoning with sketches of area models, number lines, and explicit references to base 10.


Part 2: Converting from Decimals to Fractions

If the students understand Method 1 above, then converting from terminating decimals to fractions is trivial and there's really only one method and its shortcut.

Method 4: Using Base 10 Conventions "Backward"

Example using base 10 conventions: $1.309 = 1 + \frac{3}{10} + \frac{0}{100} + \frac{9}{1000} = 1 + \frac{300}{1000} + \frac{0}{1000} + \frac{9}{1000} = 1\frac{309}{1000}$

Example using shortcut: $1.309=?$ "There are 3 digits after the decimal, so three zeroes in the denominator, so 1000." $1.309 = 1\frac{309}{1000}$

Students should then demonstrate further conceptualization of scaling down with base 10 by converting fractions such as, say, $23.00901460$ to fraction notation.

Method 5: "Easy" Repeating Decimals

Converting from repeating decimals to fractions is drastically harder and I'm unaware of any way to do this without either memorization of common repeating decimals OR the use of algebra.

Example: $0.\bar{6} = ?$ Let: $0.\bar{6}=x$ Then, we use 9 since there is only one digit that repeats and it's in the tenths column: $9x=6$, $x=\frac{6}{9}=\frac{2}{3}$.

Method 6: "Hard" Repeating Decimals

Example: $0.0\overline{142857} = ?$ Let: $0.0\overline{142857} = x$. Then $10x=0.\overline{142857}=\frac{1}{7}$ (as recalled from memory) so $x=\frac{1}{7}\div10=\frac{1}{70}$.

Example: $0.0\overline{142857} = ?$ Let: $0.0\overline{142857} = x$. Then, multiply $x$ by some number and... ugh, I hope kids aren't forced to do this too often. In my opinion, understanding - and not just execution of - the long division algorithm is enough to prove that all repeating decimals must have an exact simple fraction representation.

The strongest students will understand this from the age of maybe 12 onward, but many high school graduates won't really get this at all. Prerequisite knowledge of fraction arithmetic, base 10, long division, and algebra is so extremely varied that I don't see how anybody can know when an average class is ready to begin studying this.


So, perhaps a better question to ask is "At which age do average students have the prior knowledge in place so that they can learn each of those methods?"

My answer to that would be:

  • 8 or 9 years old for Methods 1 and 4
  • 10 or 11 for Method 2
  • 12+ for Method 3
  • 13+ and a very good teacher for remaining methods

That being said, in a truly spectacular school system, all of this should be mastered by all students by the age of 12.

One can only dream. :P

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  • $\begingroup$ Wow, this makes me wonder whether I understand fractions! Thanks for a very thorough and illuminating answer. I shall now read through again to check my own ability level. $\endgroup$ – chasly - reinstate Monica Jul 19 at 10:17
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    $\begingroup$ You're welcome. Many people were taught fractions horrendously when they were young, literally to the point of pseud-math. Hung Hsi Wu is an expert on this and how to fix it when it comes to fractions. Largely, his suggestions come down to thinking of fractions as a number. Check out his stuff. math.berkeley.edu/~wu $\endgroup$ – WeCanLearnAnything Jul 19 at 17:08
  • $\begingroup$ I find it amusing that you spend a paragraph at the top dismissing standards, then cite a source in a comment where one can find detailed descriptions of what the standards mean and how they apply, written by someone intimately familiar with (and in support of) the CC standards. For example, this document, from among those you cited. $\endgroup$ – Xander Henderson Jul 20 at 6:24
  • $\begingroup$ (1/2) @XanderHenderson Then you misunderstood what I meant and/or I wrote poorly! [Prob my bad]. "...many educators, materials, and educational systems make curse of knowledge mistakes". Hung-Hsi Wu is the person who I think is most famous for pointing this out for rational numbers education. He co-wrote the much better CC standards. I respect those standards and think teachers should largely follow them. What I would like to hear is the name of even one North American school system that takes such standards seriously as measured by how they teach and grade their students. $\endgroup$ – WeCanLearnAnything Jul 21 at 18:17
  • $\begingroup$ (2/2) @XanderHenderson - "... cut back on time spent on surface area or bar graphs... Frankly, f$%@ curriculum standards if you have to ...", I should have specified that many states have shifted away from CC standards, many ed publishers have almost literally put stickers that say "Common Core" on crappy old pseudo-math textbooks and tests, and, of course, there are plenty of weird math standards in other countries. Bad standards and areas of math that kids can catch up on easily later if they miss now (e.g. bar graphs) are worth sacrificing to help kids master rational numbers now. $\endgroup$ – WeCanLearnAnything Jul 21 at 18:24
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This is a US Common Core Standards based answer. The standards are grades based, because new knowledge builds on old, and that old knowledge is presumed to come from earlier grades. In the US, most states do not require children to enter until they are 6 years old (and many states don't require entrance until a child is 7), but children usually start kindergarten at age 5. Thus, in principle, any particular class of students might have an age range of 3–4 years. Hence pinning down ages is a more difficult task than pinning down grades. I've given approximate ages in my answer, under the assumption that a child starts kindergarten at age 5.

  • In the 3rd grade (approximately ages 8–9) the standard suggests that students should be able to

    Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

  • In 4th grade (approximately ages 9–10), the standard suggests that students should be able to

    Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

    and

    Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

  • In the 8th grade (approximately ages 13–14), the standard suggests that students should be able to

    Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

The standards don't seem to ever demand that students fluently convert back and forth between fraction and decimal representations. Instead, the standards mostly emphasize more general ideas of "number sense" (compare fractions, plot things on number lines, etc.). Given easy access to technology, I don't find this surprising.

In any event, based on these standards, it seems that an answer to your question is that students should be able to start familiarizing themselves with fractions and decimals around age 10, and that they should probably be maximally fluent (to the extent demanded by the standards) by around age 15.

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In my experience as a teacher, many children are asked to memorize certain equivalent fractions before they are taught the computation. Some of these include:

  1. $0.25$ and $1/4$
  2. $0.75$ and $3/4$
  3. $0.2$ and $1/5$
  4. $0.8$ and $4/5$

Student often memorize these and then memorize their percent equivalents. In this way they get away without ever really being able to do the calculations required. They may be taught how to do it, but for too many children, the computation never sticks.

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    $\begingroup$ It's true and sad that many students memorize this without ever understanding it. They should, though, be able to draw all of that on number lines and 10x10 grids, though. $\endgroup$ – WeCanLearnAnything Jul 19 at 17:05
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Translation: US 4th grade is ~10yo. US fifth is ~11yo. (not at the year start, but sometime within it). Again, that's averages and slightly to the high side. Few kids slightly younger (birthday in the summer, not hitting the mark until summer after the grade.)

My personal experience in US public school was that 5th grade was the standard spot for decimals. And I don't recall breaking it up by years (like learning some decimals in 4th and then the rest later as Common Core seems to say). Fractions are started earlier, although you don't get the whole topic (e.g. cross multiply to divide) in earlier grades.

FWIW, my experience teaching sports (with decimal timing of races) to children was that 4th graders almost always did not understand decimal representation and that kids 5th grade (say 50% through year) and above were fluent with them. It was uncanny how this played out, with even relatively sharp kids, and with all the decimal exposure on TV, watches, etc. But it was just an issue of them not being exposed to it, since decimals are traditionally a 5th grade topic.

As far as translation from fractions to decimals, like the easy examples you gave, that would be in fifth as well. Not saying average kid is invariably perfect. Or that there aren't laggards. But .25 to a quarter is completely reasonable expectation for an average kid halfway through 5th grade in USA.

P.s. I think this is actually reasonably close to CC standard (which just approximates normal practice anyhow). And I wouldn't get too wrapped up about slight differences of CC standard versus practice. The US really does not have a national curriculum like France. But sort of a loose defacto pattern of progression. [Why you will hear people talk about "Algebra 1" in 9th grade and the like.]

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    $\begingroup$ I don't see the CC "breaking things up" by year. Rather, the CC provides milestones: by year $x$, students are expected to be able to do, at minimum, $y$. I also wonder if you are misremembering your youth---my recollection is that many simple fractions were taught early (e.g. $1/2 = 0.5$, $1/10 = 0.1$), particularly in the context of money, while more general techniques were not introduced until later. This matches my daughter's experience (she just finished the second grade, and is not expected to know much more than $1/2 = 50¢ = \$0.50$). $\endgroup$ – Xander Henderson Jul 19 at 14:32

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