Numbers for Words People

I have been reliably informed by students that fractions were concocted by a bunch of teachers who wanted to find the most effective way to torture children!

In this first instalment on fractions, I aim to explain what they are and how we determine that two fractions are equivalent. I will also give you a tip that makes children share equally without adult interference.

Birthday Bob eats half your cake!

Birthday Bob loves birthday parties and loves eating cake. His friends, Hadi, Freya, and Se-Ping all had birthday parties. They all invited Bob and here’s the best part: They each promised Bob that he could eat half of the cake. What’s more, the cake at every party would be rectangular to guarantee accurate cutting!

Hadi’s party was straightforward. There was only one invitee, Bob. The cake was cut in half and each kid ate one piece.

When halves are less straight forward

Four kids had to the share the cake at Freya’s party! Bob was unhappy: “You promised me half of your cake. Look, there are four pieces!” “Don’t worry, I’ll let you have two pieces. The rest of us will manage”, Freya replied.

At Se-Ping’s, the cake was cut into six equal pieces. Bob came to protest, but Se-Ping interrupted, “Relax! Three pieces are yours.”

Many ways to eat half a cake!

The figure below shows the cakes at all three parties. Bob always takes the pieces to the left of the dashed line. Can you see that his share is always one half of the cake?

Hadi’s cake	Freya’s cake	Se-Ping’s cake

Formal notation for expressing fractions

No matter how many equally-sized pieces there are, Bob always gets one out of each two. The conventional notation to express “one half” is:

The top number is known as the numerator and the bottom number as the denominator. The numerator tells us how many pieces make up our share, while the denominator tells us how many equal pieces there are in total.

Figure 3 below shows the same cakes as figure 1, except with Bob’s share expressed in fraction notation:

Bob’s share: $\displaystyle \frac{1}{2}$	$\displaystyle \frac{2}{4}$	$\displaystyle \frac{3}{6}$
Bob’s share: 1 piece out of 2	2 pieces out of 4	3 pieces out of 6

Why it all makes sense

We have seen that $\displaystyle \frac{1}{2}$ is the same as $\displaystyle \frac{2}{4}$ and the same as $\displaystyle \frac{3}{6}$. The reason this makes sense is this: Doubling the denominator leads to twice as many pieces. To keep your promise to me, you need to give me twice as many pieces, hence the doubled numerator. The same goes for tripling both.

Parenting Advice: Teach your children to share equally

Readers in Australia may recognise the legendary science populariser, Dr Karl Kruszelnicki. He describes a good way to have two children share anything fairly: One child cuts, then the other picks the piece they want. The first child knows that, if they cut the cake into two unequal parts, then the other child will pick the larger one.

Notice that, in my description, I didn’t say “the other child will pick the bigger half”. There is no such thing as the bigger half! As I used to tell my own children, “If the two pieces are not the same size, then they are not halves!”

Next steps

Let me know in the comments what you would like to see clarified about fractions, and I’ll make the next post about that. Also, let me know what you thought on my explanation. I’ll get better at explaining things if I listen to my readers’ feedback.

You may enjoy this previous post about the potato paradox, a puzzle that deals with percentages. Percentages are essentially fractions, with 100 in the denominator.