One wasn't square
Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.
Problem
Mrs Morgan, the class teacher, pinned numbers onto the backs of three children: Mona, Bob and Jamie.
"Now," she said, "Those three numbers add to a special kind of number. What is it?"
Michael put his hand up.
"It's a square number," he answered.
"Correct," smiled Mrs Morgan.
Mona leaned back to look at Bob and Jamie's numbers. "Oh!" she exclaimed. "The two numbers I can see also add to a square!"
"And me!" called out Bob. "The two numbers I can see add to a square too!"
"Oh dear," said Jamie disappointedly, "the two numbers I can see don't add to a square! It's either 5 too little or 6 too big!"
What numbers did the three children have on their backs?
Getting Started
What can you deduce from what Jamie says?
Making a list of square numbers might help.
You could try finding pairs of numbers that make squares.
All the numbers, including the squares, are less than $40$.
Student Solutions
Several people sent in the correct answer to this problem. Phoebe and Lea, both at Cottesmore School tell us how they worked through it:
We found this out by adding 6 and 5 together which makes 11.
We wrote out squared numbers to 100 and then we realised that the only squared numbers with a difference of 11 between them were 25 and 36. Therefore the numbers had to add up to 36.
We then did 20 + 5 and then 11 which all makes up to 36.
Lucy and Melissa who are at Woodfall Junior School explain which number is on each child's back:
Bob's number is 11, Mona's 20 and Jamie's 5.
Mona saw 11 + 5 = 16
Bob saw 20 + 5 = 25.
Jamie looked at Bob and Mona and got 31 which is 5 less than the square number 36 and 6 more than 25.
Kevin also sent in his work on this question:
We know that Mona's number plus Bob's number is 5 less than a square and 6 more than a square. So these squares must be 11 apart. I wrote out the first few squares and saw that they get further and further apart, and the only ones that are 11 apart are 25 and 36. So Mona's number plus Bob's number is 31. We know that when you add them all up you get a square, so Jamie's number plus 31 is a square. From the hint, all the numbers are less than 40, so Jamie's number plus 31 is 36. So Jamie's number is 5. Then I found that the only way we could make the rest of the problem work is to have Mona's number as 11 and Bob's as 20 (or the other way round).
Thanks for these solutions to what can be an interesting exploration.
Teachers' Resources
One Wasn't Square
Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children: Mona, Bob and Jamie.
"Now", she said, "Those three numbers add to a special kind of number. What is it?"
Michael put his hand up.
"It's a square number", he answered.
"Correct", smiled Mrs Morgan.
"Oh!" exclaimed Mona, "The two numbers I can see also add to a square!"
"And me!" called out Bob, "The two numbers I can see add to a square too!"
"Oh dear", said Jamie disappointedly, "the two numbers I can see don't add to a square! It's either $5$ too little or $6$ too big!"
What numbers did the three children have on their backs?
Why do this problem?
This problem invites learners to explore square numbers and the relationship between them. It can be used to talk about a trial and improvement approach and also acts as an informal introduction to algebra.
Possible approach
Key questions
Possible extension
Children could explore the gaps between square numbers up to $100$. What patterns can they discover?
Possible support
It might help for children to make a list of the square numbers, using a multiplication square if necessary. Finding some pairs of numbers that add to squares will get learners started on the problem.