Birthday Cakes

'Birthday Cakes' printed from https://nrich.maths.org/

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Birthday Cakes

When Jack was one year old his mother bought a packet of $24$ candles for his birthday cake.

That year she put $1$ candle on Jack's cake. When he was two he had $2$ candles and when he was three he had $3$ candles, and so on.

candles

One day Jack's little sister Kate was born. She had $1$ candle on her first birthday cake, $2$ candles on her second birthday cake, and so on.

The candles were finished on one of Jack's birthdays with just enough left.

How old was Jack when Kate was born? And how old was each of them when the candles finally ran out?

Why do this problem?

This problem is a great context in which the merits of a trial and improvement approach can be highlighted. It could also give an opportunity to explore triangular numbers in the setting of a familiar-sounding situation.

Possible approach

You could introduce this problem by asking the class a few questions orally, using some pictures of candles on cakes to help. These could focus on simply adding the consecutive numbers on one child's cake at first. You could then introduce the idea of having two children, perhaps with twins, asking how many candles have been used altogether by various birthdays.

Then introduce the problem itself, again orally might be best, encouraging children to have a think on their own about what they might do. Then ask them to talk to a partner before sharing thoughts as a whole group. In this discussion, emphasise that we might have to try out some ideas and see what happens. It would be good to have rough paper and some sticks available for children to use (lolly sticks or even pencils are fine).

In a plenary, ask learners to explain what they did to solve the problem and share different strategies.

Key questions

How many candles will have been used for Jack's cakes by the time he has had his fourth birthday? Fifth birthday? Sixth birthday? Seventh birthday?

How does this help us to decide when Kate could have been born?

Why not try out some ideas using sticks or drawings?

Possible extension

Children could look purely at the numbers involved and explore what happens when consecutive numbers are added together (triangular numbers are formed). What patterns do they notice? They could investigate whether there are any differences between odd and even triangular numbers.

Possible support

Using practical equipment will make this problem more accessible for children.