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Problems like

There is 30 students in the class. $\frac13$ of the students study biology, $\frac15$ of the students study physics and the rest study chemistry. What part of the class studies chemistry?

What I would explain is that first we calculate what part of the class doesn't study chemistry $$\frac13 + \frac15 = \frac{8}{15}$$

Then, for finding the fraction we need I would explain that we need to do $$\mathbf{1\,-\,}\frac{8}{15} = \frac{7}{15}$$

However, the student gets confused because there are $30$ students, no $1$.

Maybe there is an easier method for resolving these problems in a test.

Sorry if the question title isn't right, I'm not an English native speaker. Thanks for the help!

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    $\begingroup$ Just a tip: you wouldn't call sixth graders teenagers. That begins at ages 13-14, usually. $\endgroup$ – Javier Oct 31 '17 at 13:19
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You could try keeping 30 as the denominator throughout, that is, observing that $\frac{1}{3} = \frac{10}{30}$ and $\frac{1}{5} = \frac{6}{30}$, so the portion of the class that doesn't study chemistry is $$\frac{10}{30} + \frac{6}{30} = \frac{16}{30},$$ and the portion that does study chemistry is $$\frac{30}{30} - \frac{16}{30} = \frac{14}{30}.$$ Once they understand this line of reasoning, you can point out that these fractions can be expressed in terms of smaller denominators.

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I think this problem is much easier to handle if you refer throughout to the number of students who study each subject, rather than the fraction, only expressing the final answer as a fraction at the very end.

That is:

  • $\frac{1}{3}$ of the $30$ students study Biology — this is $10$ students, because $\frac{1}{3} \times 30 = 10$.
  • $\frac{1}{5}$ of the $30$ students study Physics — this is another $6$ students, because $\frac{1}{5} \times 30 = 6$.

Since 16 of the 30 students are accounted for, the remaining 14 must study Chemistry. Therefore the fraction that studies Chemistry is $\frac{14}{30}$, or (in lowest terms) $\frac{7}{15}$.

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  • $\begingroup$ +1 ... this avoids lots of fractions, compared to Daniel's solution. $\endgroup$ – Gerald Edgar Nov 1 '17 at 0:11
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Unfortunately, I think you may be asking the wrong question. When it comes to fractions, a huge share of students have no idea what they even mean. They have likely not internalized the idea that fractions are numbers, which makes all the fractional arithmetic you do completely rote (in their minds). In other words, it's prior knowledge gaps that are holding them back. Here's how I would assess for those prior knowledge gaps.

Ge them a list of numbers such as:

$\frac{11}{7}, \frac{8}{2}, 5, \frac{3}{4}, 1.6$

Then, with no hints, ask them to draw their own number line from scratch and place everything appropriately. A massive share of high school graduates can't do this, so it's very likely your grade 6 students can't either.

You could also ask them to do $3-\frac{4}{5}$ and see if they can answer that using "common sense" instead of "common denominators". Have them draw a picture to explain their intuition.

Lastly, ask them to determine if the answers to the following are bigger or smaller than 700 and how they know. See if they can explain their reasoning without doing the "standard algorithm" calculations. (All exercises involve 701.27 to discourage exactly that and to encourage thinking about meaning and estimation instead.)

$\frac{3}{8}\times701.27=\_\_\_\_$

$\frac{8}{3}\times701.27=\_\_\_\_$

$\frac{701.27}{3}\times8=\_\_\_\_$

$\frac{8}{3}\div701.27=\_\_\_\_$

$701.27\div\frac{8}{3}=\_\_\_\_$

$701.27\div\frac{3}{8}=\_\_\_\_$

$\frac{3}{8}\div701.27=\_\_\_\_$

$Two\ thirds\ of\ 701.27\ is\ \_\_\_\_$

$Two\ thirds\ times\ 701.27\ is\ \_\_\_\_$

$701.27\ groups\ of\ two-thirds\ is\ \_\_\_\_ $

Lastly, I'd suggest you draw a picture like this to represent the students, where P represents a physics students, B represents a biology student, and C represents a chemistry student.

P P P P P P

B B B B B B B B B B

C C C C C C C C C C C C C C

Given this picture, are they able to describe the class correctly using fractional vocabulary and equivalent fractions? Can they use those fractions to combine the fractions of physics and biology students? A shocking share of students can't. And if they can't, there's no way they're ready to tackle word problems in which they must determine and/or draw and/or visualize the situation.

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They do not grasp that $\frac{7}{15}$ means "seven out of every separate group of fifteen are being counted". Additional work on equivalent fractions is needed for them to understand the connection between your seven of fifteen and the fourteen of thirty that they are trying to determine.

They are also unclear about the meaning of $1$ as a fraction itself being $\frac{1}{1}$, that is, one of every item is being considered, i.e. when writing the $1-...$ we mean that we are starting with all of them and then taking away the ones we don't want.

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