Representing numbers
Problem
Find as many ways as you can of representing the number of dots shown above.
Try to find at least five ways.
Now find ways of representing ten times as many dots. Can you still find at least five different ways?
Student Solutions
Thank you to everybody who sent us their thoughts about this challenge. William from Dronfield Junior School in England sent in these solutions, splitting the total number of dots into tens and ones:
Finley from Goodrich C E Primary School in the UK also used tens and ones to represent the number:
Polly from Gorse Ride Junior School in England sent in these number sentences to represent the dots:
20+3=23
10+10+3=23
15+5+3=23
10+5+8=23
5+5+10+3=23
Polly has used the same idea of splitting the total number into tens and ones, but has also found solutions where the number 5 is important. I wonder why the number 5 might stand out when someone looks at the dot picture?
If anybody has any different ideas about how to represent this number, please email us.
Teachers' Resources
Representing Numbers
Find as many ways as you can of representing the number of dots shown above.
Try to find at least five ways.
Now find ways of representing ten times as many dots. Can you still find at least five different ways?
Why do this problem?
This problem gives an opportunity for children to think about ways in which numbers can be represented and to be creative as they invent original ways of their own. The task may peak individuals' curiosity as they may explore different ways of grouping, lining up or gathering.
Possible approach
Show the image to the group and ask them how many dots there are. Take the image away and ask learners to talk to a partner about how they counted. (See the task How Would We Count?.) Put the image back up and encourage conversation about ways of counting, perhaps sharing some ways with the whole group. Highlight particular strategies that have been used, such as
grouping, seeing lines of dots etc.
You can then set learners off on the challenge itself. You could ask them to record their representations on separate sheets of paper and then at an approprate time, invite everyone to walk around the room looking at the different ways.
A plenary could focus on discussing a few ways in particular, or you could ask if anyone has a question they would like to put to a pair about their representation.
Key questions
Tell me about these.
Will there be more ways?
Would someone else understand what you are representing here?
Possible extension
Replace the "ten times as many" with "nine times as many".
Possible support
It may be helpful to look at the article Children's Mathematical Graphics: Understanding the Key Concept which looks at the different ways children can record their thinking and understanding.