Representing fractions

What is it about?

A fraction $\frac{z}{n}$ describes a part of a whole:

The denominator $n$ tells you into how many equal parts the whole is cut.
The numerator $z$ tells you how many of these parts you take.

So $\frac{3}{4}$ of a pizza means: cut the pizza into 4 equal slices and take 3 of them.

Expanding means cutting every piece again into $k$ equal parts. Then there are $k$ times as many pieces, and you take $k$ times as many – the amount stays the same:

\frac{3}{4} = \frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8}

Reducing is the reverse: divide numerator and denominator by the same number, $\frac{6}{8} = \frac{3}{4}$ .

Every fraction can also be written as a decimal: $\frac{3}{4} = 0.75 = 75\,\%$ . Some decimals are recurring, for example $\frac{1}{3} = 0.\overline{3}$ .

If the numerator is greater than the denominator, you need more than one whole: $\frac{7}{4} = 1\frac{3}{4}$ .

A fraction can also refer to a quantity. $\frac{3}{4}$ of €80 means: split €80 into 4 equal parts ( $80 : 4 = 20$ ) and take 3 of them ( $3 \cdot 20 = 60$ ). So $\frac{3}{4}$ of €80 is exactly €60.

Try it

Tap the pizza slices to select them. Which fraction do you get?
Expand with $k = 2$ : What changes in the pizza, what in the fraction? Expand
Choose the sheet cake and set $\frac{1}{3}$ . What does the decimal look like? One third
Set $\frac{7}{4}$ . How many pizzas do you need? Seven quarters
Compare with the line segment. Line segment

Exercises

Exercise 1: Read the fraction

Load exercise Which fraction is shown? Also give it as a decimal and as a percentage.

Show solution

$\frac{5}{8} = 0.625 = 62.5\,\%$

Exercise 2: Expand

Expand $\frac{2}{3}$ so that the denominator is $12$ . Check with the simulation.

Show solution

$\frac{2}{3} = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}$ – every third is cut into 4 pieces. Show in the simulation

Exercise 3: Compare

Which is more: $\frac{2}{3}$ or $\frac{3}{4}$ of a pizza? Explain using a common denominator.

Show solution

$\frac{2}{3} = \frac{8}{12}$ and $\frac{3}{4} = \frac{9}{12}$ . So $\frac{3}{4}$ is more – by $\frac{1}{12}$ of a pizza.

Exercise 4: Recurring decimals

Write $\frac{1}{6}$ and $\frac{5}{6}$ as decimals.

Show solution

$\frac{1}{6} = 0.1\overline{6}$ and $\frac{5}{6} = 0.8\overline{3}$ – here one digit that does not repeat comes before the recurring part. Show 5/6

Exercise 5: Fractions of quantities

Work out: a) $\frac{3}{4}$ of €80, b) $\frac{2}{5}$ of $35$ students, c) $\frac{2}{3}$ of $1$ hour. Picture the whole in the simulation.

Show solution

a) $80 : 4 = 20$ , $3 \cdot 20 = 60$ , so €60. Pizza in quarters

b) $35 : 5 = 7$ , $2 \cdot 7 = 14$ , so $14$ students. Line segment in fifths

c) $1$ h $= 60$ min, $60 : 3 = 20$ , $2 \cdot 20 = 40$ , so $40$ min – like two thirds of a clock face. Pizza in thirds

Exercise 6: Special fractions

What do $\frac{0}{4}$ , $\frac{4}{4}$ and $\frac{3}{1}$ mean? Set them and write them without a fraction bar.

Show solution

$\frac{0}{4} = 0$ : no piece taken. 0/4 $\frac{4}{4} = 1$ : all pieces, so exactly one whole. 4/4 $\frac{3}{1} = 3$ : each whole stays uncut, you take 3 wholes. 3/1

Ideas for the classroom

Use: About grade 6 (fractions and parts of a whole, expanding and reducing, decimals).
Operation: Pieces can be tapped directly – good on an interactive whiteboard. The selected pieces are pulled out.
Common stumbling blocks:
- The parts must be equal in size – otherwise it is not a fraction of the whole.
- “ $\frac{6}{8}$ is more than $\frac{3}{4}$ because the numbers are bigger.” – Expanding shows that it is the same amount.
- “A bigger denominator means a bigger piece.”
Fractions of quantities: The simulation shows fractions of a whole. For quantities (money, time, numbers of people), let the whole stand for the quantity, for example the pizza for €80 or for 1 hour (exercise 5).