Primary School fractions question

Question

My daughter asked me for help with the following question. While it's fairly easy to work out with a bit of common sense and trial and error, I was wondering if there is a simple way to work mathematically that you'd expect an 8 year old to understand?

Answer 1

We want to calculate $\frac{2}{3}$ of something. To make it easy, that "something" should be a multiple of 3 because it's easy to divide those into thirds. If that something is 3, 6 or 9 (and so on), we get 2, 4 and 6 (and so on) as results (so all the even numbers)

At the same time, we want to calculate $\frac{3}{10}$ of something. For simplicity that something should be a multiple of 10. If that something is 10, 20 or 30 (and so on), we get 3, 6 and 9 (and so on) as results. (so all the multiples of 3)

We now see that 6 appeared as a result in both: Two thirds of 6 is the same as three tenths of 20. Because 6 is a multiple of both 2 and 3.

Answer 2

Perhaps something like the following:

$$ \frac{2}{3} \times \frac{\text{top}}{\text{bottom}} \;\; = \;\; \frac{3}{10} \times \frac{\text{top}}{\text{bottom}} $$

If the left side "top" is $3$ and the right side "top" is $2,$ then the numerators on both sides will be the same, because $2 \times 3 = 3 \times 2.$ And if the left side "bottom" is $10$ and the right side "bottom" is $3,$ then the denominators on both sides will be the same (because $3 \times 10 = 10 \times 3),$ and we'll have equality. Note that this solution is a sight-recognition task.

However, in showing this to the student you'd want to use an empty box for "top" and an empty triangle for "bottom", or something along these lines. The fact that the two boxes (top's here) can represent different numbers should not be made an issue of. For the student, the boxes simply represent places to put numbers and not variables representing numbers (the latter of which they don't know about, so shouldn't be brought up).

Answer 3

I might try drawing some blank lines to represent the situation:

$\underline{\hspace{.35 in}} \hspace{.2in} \underline{\hspace{.35 in}} \hspace{.2in}\underline{\hspace{.35 in}} \hspace{.3 in}$

$\underline{\hspace{.35 in}} \hspace{.2in} \underline{\hspace{.35 in}} \hspace{.2in}\underline{\hspace{.35 in}} \hspace{.2in} \underline{\hspace{.35 in}}\hspace{.2in} \underline{\hspace{.35 in}}\hspace{.2in} \underline{\hspace{.35 in}}\hspace{.2in} \underline{\hspace{.35 in}}\hspace{.2in} \underline{\hspace{.35 in}}\hspace{.2in} \underline{\hspace{.35 in}}\hspace{.2in} \underline{\hspace{.35 in}}\hspace{.3 in}$

And then give a hint/explanation like this: The line of three boxes each contain the same amount. And the line of ten boxes each contain the same amount. The first two boxes of the three-box line contain the same total amount as the first three boxes of the ten-box line. We are interested in the grand total of each line (two-thirds of the first grand total is equal to three-tenths of the second grand total).