Count them, or compare them

Before you read anything

$\frac{3}{8}$ against $\frac{5}{8}$. Which is bigger?

Now $\frac{2}{3}$ against $\frac{2}{5}$. Which is bigger? Neither of these needs any working.

Same slices: count them

Eighths in both bars, three shaded against five. The pieces are identical, so the answer is the overhang at the end.

Eighths are eighths. Five of them beats three of them, and there is nothing else to think about — the pieces are identical, so the count decides.

Same count: compare the slices

Two pieces shaded in each bar, so counting settles nothing. Thirds are longer than fifths, and that is the whole answer.

$\frac{2}{3}$ and $\frac{2}{5}$ are both two pieces. So the bigger one is whichever has the bigger pieces, and lesson 10.1 already told you which: thirds are bigger than fifths.

:x nutshell-the-comparison-bar

Two bars of the same length, one above the other, lined up at the left. The difference is the overhang — a length you can point at.

Draw them as :two bars, one under the other, starting at the same edge. The bigger fraction is simply the one that reaches further, and no arithmetic is involved at all.

Why this lesson comes before the method

Nakato

turned $\frac{2}{3}$ and $\frac{2}{5}$ into fifteenths

Nakato is right and she did four times the work. A learner who converts everything has a procedure where a picture would have done — and the picture is the thing that will still be there in three years.

Look first. Convert only when you have to.

Check yourself

Six to try

Before each one, say out loud which case it is — same slices, or same count. Then answer without writing anything down.

1

Which is biggest: $\frac{2}{10}$, $\frac{5}{10}$ or $\frac{8}{10}$?

2

Which is biggest: $\frac{4}{5}$, $\frac{4}{6}$ or $\frac{4}{8}$?

3

Which is biggest: $\frac{1}{12}$, $\frac{9}{12}$ or $\frac{11}{12}$?

4

Which is biggest: $\frac{2}{3}$, $\frac{2}{8}$ or $\frac{2}{12}$?

5

Which is biggest: $\frac{2}{12}$, $\frac{3}{12}$ or $\frac{6}{12}$?

6

Which is biggest: $\frac{3}{5}$, $\frac{3}{8}$ or $\frac{3}{12}$?

Same slices, or same count · page 1 of 3 Print it

Same slices, or same count · page 2 of 3 Print it

Same slices, or same count · page 3 of 3 Print it

Show that you have it

Which case is this

One of these is one of the easy cases and one is not. Saying which is the whole question.

1

Which is biggest: $\frac{2}{4}$, $\frac{2}{8}$ or $\frac{2}{12}$?

2

Which is biggest: $\frac{5}{6}$, $\frac{4}{5}$ or $\frac{1}{12}$?

Next: Cut them both the same The whole thing on one page