Birthday cakes
Problem
Birthday Cakes printable sheet
When Jack was one year old, his mother bought a pack of 24 candles. She put one candle on his birthday cake.
When Jack was two he had two candles on his cake, and when he was three he had three candles, and so on.
One day Jack's little sister Kate was born. She had one candle on her first birthday cake, two candles on her second birthday cake, and so on.
On one of Jack's birthdays, there were exactly enough candles left for Jack's mother to put the candles on his birthday cake. After that, the packet was empty.
How old was Jack when Kate was born? And how old was each of them when the candles finally ran out?
Getting Started
Draw the candles that are used for Jack's first few birthdays to get the idea of what is needed.
Real sticks are good if you have them.
Student Solutions
You had to be quite careful with this problem - Alice explained how she went about solving it:
First I got 24 lollipop sticks then I fiddled around and sorted out Jack's birthdays and put in Kate's birthdays where they fitted. I had 3 tries to get it right.
Alice sent in this very clear picture of the answer:
Amy and Nathan from Maadi British International School in Cairo also explained their reasoning:
We worked out it couldn`t be Jack's 7th birthday because that was too big. It couldn't be 5 because 5 was too small so it had to be 6 because it was in between the 5 and the 7. We had three more candles left so Kate had to be 2 years old. Kate was 1 when Jack was 4, so when Kate was 0, Kate was born and Jack had to be 3 years old.
Thank you to all of you for your very clear solutions.
Teachers' Resources
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.
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
Possible extension
Possible support
Using practical equipment will make this problem more accessible for children.