Number balance
Can you hang weights in the right place to make the the number balance balanced?
Problem
Number Balance printable sheet
This is an interactive number balance:
There are blue weights on the table, which you can hang onto the numbers. Have a play with it!
Here is a picture of the number balance with some weights on it:
Where would you need to hang a weight to make it balance?
What about this picture below? Where would you hang a weight to make it balance?
If you had to use two weights and make the picture below balance, where could you put them?
How many different ways can you make it balance with two weights?
Getting Started
Try experimenting with the interactivity!
How can you tell from the pictures which side of the number balance is lighter?
Student Solutions
Thank you to everyone who sent in their ideas about how to make the scales balance.
Amy from Stoke by Nayland Middle School sent us her work on this problem:
On the first question you would need to put a weight on number 2 on the right side.
On the second question you would need to put a weight on number 7 on the left side.
On the third question you could put one weight on 1 on the left side and the other on 2 on the right side.
Well done Amy!
Sam from Shepherd School noticed that there were several answers for the third question:
You could put them on point 6 on the right and point 5 on the left, for example. There are 9 solutions to this, they could go anywhere where the weight on the right is 1 more than the weight on the left.
Thank you for that clear explanation, Sam.
Saif from Pierrepont Gamston Primary School in England also found several solutions to the third question:
To balance the third set of scales you can have 9 possibilities which add one weight to the following columns:
1 left and 2 right
2 left and 3 right
3 left and 4 right
4 left and 5 right
5 left and 6 right
6 left and 7 right
7 left and 8 right
8 left and 9 right
9 left and 10 right
Well done Saif! How has Saif made sure that they have found all the possible solutions?
Teachers' Resources
Number Balance
Number Balance printable sheet
This is an interactive number balance:
Why do this problem?
This problem requires children to utilise their knowledge of number bonds in an unusual context, and offers the opportunity for them to explain and reason through their solutions.
Possible approach
This problem featured in an NRICH Primary webinar in January 2021.
Before tackling Number Balance, pupils will need practical experience of working with balances, if possible.
To introduce the problem, show the class the interactive balance on the board and, without saying anything, hang weights to make it balance (perhaps one weight on one side and two on the other to begin with). 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 two examples in the problem as a whole group, asking the children to talk to each other about what they might do before sharing ideas amongst the whole group.
Ideally, for the main challenge, children would have access to the interactivity in pairs on a tablet or computer. You can also encourage them to record their ideas on paper or mini whiteboards.
After leaving time for them to work on this, bring them together and discuss their solutions. You could ask pairs to write up some solutions on individual strips of paper which can be stuck on the board. You could follow on by looking for patterns in the solutions, which may lead to the children suggesting other possibilities. Alternatively (or as well) you could focus on the number sentences that can be written from the pupils' solutions.
Key questions
Can you tell me how a balance works?
How can you tell from the pictures which side we need to add a weight to?
What is the total of the weights on this side?
What is the total of the weights on the other side?
Possible extension
You could encourage children to work systematically to find all the possible solutions to the final part of the problem. They can also create their own challenge for another pair to work on.
Possible support
Having lots of practical experience of real balances will help children access this task.