Are you well balanced?
Can you work out how to make each side of this balance equally balanced? You can put more than one weight on a hook.
Problem
You might like to have a look at the activity Number Balance before trying this problem.
Here is an interactive number balance:
You can hang weights on the hooks. All the weights are exactly the same and you can hang more than one weight from each hook.
Take some time to have a play with the balance.
I hang one weight on the 10. I want to use just one hook on the other side to make it balance.
- Where could I put weights and how many would I need to put on?
- Is there only one solution?
- How might you record your solutions so you don't forget them?
- Using only the 2 hook, how could I add weights on the other side to make it balance?
- Using only the 5 hook, how could I add weights on the other side to make it balance?
- How could I make it balance using both the 2 and the 5 hooks? Is there only one solution?
- Can you find all the solutions?
- How do you know you have found them all?
Getting Started
Try starting off by putting weights on the $1$ hook. How many will you need to balance with $10$?
If I then hang two weights on the other side, what would that balance with? So how many will I need on the $2$ hook to balance with the two on the $10$?
Student Solutions
Lottie and Adele sent us a great solution to this problem. They answer each part in turn and explain clearly as they go along:
1. To make it balance with 10 on one side, using only one hook, you can put two on the 5 or one on the 10 or five on number 2 or ten on 1.
2. Two weights on the 10=20. So put ten on number 2 hook to make it balance.
3. You could put four weights on the 5 to make it balance with 20 (2 10s).
4. You can put two on the 5 or put five weights on the 2. There is only one solution.
5. It balances by putting one on the 7 (with the 3) and the 10 on the other side.
Thank you to you both. I wonder if there are other ways of making it balance if you have one weight on the 10 hook on one side and can use the 3 hook and one other hook on the other side? Don't forget that you can have more than one weight on a hook.
Teachers' Resources
Why do this problem?
This problem can be used to introduce repeated addition and therefore concepts of multiplication. It could also be used to challenge children to work systematically to find all possible solutions.
Possible approach
Before tackling this problem, pupils will need practical experience of working with balances, if possible. It would also be a good idea to have a look at Number Balance which uses the same interactivity, but focuses on number bonds.
To introduce the problem, show the class the balance on the interactive whiteboard and, without saying anything, hang weights to make it balance, perhaps one on the left and two on the right. Take the weights off and put on a different combination of weights on each side to balance the equaliser. Repeat this a few times and invite learners to talk to each other in pairs about what they think is happening. You could then try the first question as a whole group, asking the children to talk to each other about what to do before sharing ideas and checking using the interactivity. During this discussion, you can specifically introduce language to help children talk about what they are doing, for example "2 add 2 add 2 add 2 add 2 equals 10 or five 2s equal 10, or 5 times 2 equals 10".
For the other parts of the problem, ideally learners would have access to the interactivity in pairs on a computer or tablet. You could encourage them to record their solutions on paper or mini whiteboards, and you could share some of these different ways of recording during a mini plenary.
After leaving time for them to work on this, bring them together and discuss their solutions. When it comes to considering how to balance a total of 20 on one side with weights on both the 2 hook and the 5 hook, you may want to ask pairs to write up some solutions on individual strips of paper which can be stuck on the board. You could look for patterns in the solutions and order them to help decide whether any have been missed out.
Key questions
Where have you tried hanging weights so far?
How will you know that you have got all the ways of making it balanced?
Possible extension
Children could investigate ways of balancing the number balance if, for example, you can only put weights on the 2 hook on one side and the 10 hook on the other.
Possible support
Having access to the interactivity, either on the whiteboard, or on individual computers, will help some children gain in confidence as they will be able to try out their ideas without the anxiety of getting things wrong.