Patans.

Comparing fractions

Cut them both the same

20 minutes

Start the lesson

Before you read anything

$\frac{2}{3}$ against $\frac{3}{5}$ — the pair module 9 left you with.

Thirds do not cut into fifths and fifths do not cut into thirds. Say what you could do to both of them instead.

Make the slices match

Two thirds and three fifths, both cut into fifteenths. The shading never moves; ten against nine is all that is left to do.

Cut every third into five, and cut every fifth into three. Both bars are now in fifteenths, and module 9 says that changed nothing about how much is shaded.

$\frac{2}{3}$ is $\frac{10}{15}$. $\frac{3}{5}$ is $\frac{9}{15}$.

Ten against nine. Once the slices match you are back in lesson 10.2's easy case, and you are simply counting.

Where the fifteen comes from

Three fives, or five threes. Any number both will cut into would do — $30$ works, $60$ works — but $15$ is the smallest and makes the smallest numbers to count.

Otim

compared $\frac{2}{3}$ and $\frac{3}{5}$ by comparing 2 with 3

Otim compared the counts while the slices were different sizes. Two big pieces can easily beat three small ones, which is exactly why the slices have to be made to match first.

The order to do things in

  1. Same slices? Count them.
  2. Same count? Compare the slices.
  3. Neither? Cut them both until the slices match, then count.

Only the third one is work.

Check yourself

Six to try

Two of these ask only for the re-cutting. The rest want the answer, and the re-cutting is how you get it.

1

To compare $\frac{2}{3}$ with something in $5$ths, both are cut into $15$ths. $\frac{2}{3}$ becomes how many $15$ths?

2

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

3

To compare $\frac{5}{6}$ with something in $5$ths, both are cut into $30$ths. $\frac{5}{6}$ becomes how many $30$ths?

4

Which is biggest: $\frac{1}{4}$, $\frac{2}{5}$ or $\frac{1}{10}$?

5

Which is biggest: $\frac{3}{5}$, $\frac{2}{4}$ or $\frac{1}{10}$?

6

Put these in order, smallest first.

Away from the screen

Make the slices match Write both fractions in the same slices before you decide.
Make the slices match, page 1 Make the slices match, page 2 Make the slices match, page 3

Open the PDF Save it to print

The question this leaves you with

$\frac{1}{2} + \frac{1}{3}$.

You now know how to make those slices match. So you can already do this — but first say why $\frac{2}{5}$ is wrong, because it is the answer almost everybody writes.

Show that you have it

Neither of the easy cases

The first needs the slices made to match before anything can be compared. The second is four fractions to put in order at once.

1

Which is biggest: $\frac{3}{8}$, $\frac{2}{5}$ or $\frac{1}{16}$?

2

Put these in order, smallest first.