Climbing conundrum
Five children are taking part in a climbing competition with three parts, where their score for each part will be multiplied together. Can you see how the leaderboard will change depending on what happens in the final climb of the competition?
Problem
There are three climbing walls in the playground at Winterbrook School. Five children are having a competition to see how far they can climb up each one.
They have decided that whoever climbs the furthest on each wall should get a score of 1, then the person who comes second should get a score of 2, and so on. The three scores for each child will then be multiplied together, and the person with the lowest score will win.
Here are their scores for the first two climbing walls:
| Wall 1 | Wall 2 | Multiplied scores | |
|---|---|---|---|
| Anu | 3 | 4 | |
| Bryn | 1 | 5 | |
| Clare | 2 | 3 | |
| Duncan | 4 | 1 | |
| Elise | 5 | 2 |
At the moment, who is in the lead?
If the scores were added together, rather than multiplied, would this change the order? How?
On the third wall, after the first four children have climbed, the scores look like this:
| Wall 1 | Wall 2 | Wall 3 | Multiplied scores | |
|---|---|---|---|---|
| Anu | 3 | 4 | 1 | |
| Bryn | 1 | 5 | 3 | |
| Clare | 2 | 3 | 2 | |
| Duncan | 4 | 1 | 4 | |
| Elise | 5 | 2 |
Have a go at calculating the multiplied scores at this point in the competition. (Remember, these scores will only be accurate if Elise comes fifth on Wall 3, because if she overtakes anybody else then their score for Wall 3 will change.)
Elise starts her climb. Anu says to the others, “I’m joint first out of the four of us, so if Elise overtakes me then I’ll still finish in the top three.”
Do you think that Anu is probably right? Can you explain your thinking?
Have a go at working out what everybody's scores will be if Elise climbs the highest on Wall 3. Compare this to what would happen if Elise came fifth on Wall 3. (You might also like to work out what would happen if Elise came second, third or fourth, to compare all the possible outcomes.)
How would the outcome have been different if the scoring is reversed (5 for the best, 1 for the worst)?
How would the outcome have been different if the scores were added rather than multiplied?
Which scoring system do you think is fairest? Why?
Teachers' Resources
Why do this problem?
This problem provides an engaging context for exploring multiplication, and particularly for noticing that the product of two numbers can get bigger very quickly as the numbers increase. The fact that Elise's score on the final wall can drastically change the positions of the other climbers will spark learners' curiosity, and this will encourage them to notice facts such as how whoever comes first on a wall is at a huge advantage due to the nature of multiplying by 1.
Possible approach
As a whole class, talk through the first table and make sure that all children understand how the scoring system works. The idea of multiplying the scores for each wall to get the final scores will need to be explained, but learners might also be confused by the fact that a lower score is better. You might like to give some scoring methods for different sports as a reference for this, such as how the winner of a race will be the athlete with the shortest time.
Give children some time to calculate the multiplied scores after the first two walls have been climbed, and encourage them to compare these to the scores that would be produced by adding the numbers together instead of multiplying them. Learners can then begin to consider the differences between these scoring systems - what sorts of scores are generally better if the numbers are being multiplied? Or if the numbers are being added? As you walk around the room, listen out for children who have noticed that a score of 1 on any wall will be heavily rewarded in the multiplication scoring system.
Bring the whole class back together to discuss what the children have found out, before explaining the next part of the task. For the second table, learners can fill in the multiplied scores as they stand, but they will need to find a way of recording different possible scores based on Elise's position on Wall 3. Some children might draw their own tables for this, whereas others might prefer to make notes in a different way. You could give learners the freedom to choose their own way of recording their findings, or you could give them a sheet of pre-made tables if you'd prefer them to just focus on the multiplication. Every child should be given time to complete at least two tables - one where Elise comes first, and one where she doesn't - in order to give them the 'aha!' moment of seeing just how much the scores of the other children can change depending on Elise's score on Wall 3.
At the end of the activity, bring the whole class back together to discuss what they've noticed. What would their strategy be if they were climbing? Is it better to do equally well on all three walls or better on one and worse on the others? You might then like to share some information about the 2020 Olympic climbing, which is at the bottom of this page.
Key questions
Who is in the lead now? What order are the five climbers in?
What would happen if Elise came first on Wall 3? How would everybody else's scores for Wall 3 change?
If everybody's scores for Wall 3 (apart from Elise's) increased by 1 (because they were overtaken by Elise), would their order be the same on the leaderboard? Why/why not?
If gold, silver and bronze medals were given out, what do you notice about who would get the medals depending on how well Elise climbs? (Interestingly, Duncan doesn't get a medal if Elise comes fifth, but if Elise comes third (overtaking him) then Duncan gets a medal...!)
Is it best to be first on one wall and then not do very well on the other two walls, or to do quite well on each wall? Why?
How would the scores change if we used a different scoring system? What if we added the numbers instead? Or what if...? (Children will have their own ideas about different possible scoring systems, and might like to look at the pros and cons of each.)
Possible extension
A lot of time could be spent on critiquing this scoring system and trying out alternative scoring systems. Children might like to write to the IOC with their suggestions!
Climbing Complexity is a version of this task using the actual scores from the 2020 Olympics, which might be suitable as an extension for some learners.
Possible support
Some children will need support with understanding this task, and will benefit from focusing on creating just two final tables - one where Elise comes first on the final wall and one where she comes last. Multiplication tables (or calculators) can also be provided, as the focus in this task isn't about answering the multiplication questions but is instead about being able to notice patterns such as how multiplying by 1 gives a very small answer, and how multiplying slightly bigger numbers can give a much bigger answer.
More information about this scoring system:
This task uses similar scoring rules to the ones used in sport climbing in the 2020 Olympic Games, where there were three different disciplines – speed climbing, bouldering and lead climbing – but only one set of medals, so an athlete's scores for each of the three disciplines were multiplied together. In the men's final of those Olympics, this scoring system led to some very interesting things happening. If you'd like to investigate this, take a look at the problem Climbing Complexity.
In the 2024 Olympics, the scoring system changed. There were two separate sets of medals awarded – one for speed climbing and one for lead climbing and bouldering combined. For the lead and bouldering medal, climbers’ scores for each of the two disciplines were added together rather than multiplied. Take a look at the Sport Climbing at the Summer Olympics Wikipedia page for more information.
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