Hundred Square
A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?
Age
5 to 11
Challenge level
A hundred square has been printed on both sides of a piece of paper. One square is directly behind the other.
What is on the back of $100$? $58$? $23$? $19$?
Can you see a pattern?
Image
Click here for a poster of this problem.
This problem is also available in French: Grille de 100
Can you imagine the square on the back? Where will its $1$ be? Where will its $10$ be?
So what will be on the back of the $20$ in the $100$ square in the picture? The $30$? Does this help with the $100$?
You could print off the $100$ square, or draw your own, and write just, for example, $1-10$ on the back of it to get you started. Can you see where $11-20$ would be? Perhaps this helps to predict the other numbers.
We received many good solutions for this challenge.
Mikey from Archbishop of York CE Junior School wrote:
Having printed the page out I realised that if you turned it over you could see through the page. Looking where $100, 58, 23, 19$ would be meant you could read off the answers as $91, 53, 28, 12$. Or is this cheating?
I don't think this is cheating at all Mikey! Mikey then realised something else which was also spotted by "N" (he or she didn't give us a full Christian name):
If you draw a mirror line down the middle of the square you will be able to work out what number will be behind each number. You choose a number, then find its mirror on the other side of the line, this 'mirror' number will be the number on the reverse!This is also a very handy method - well noticed. "N" sent an image with the "mirror line" drawn in:
Image
Devonshire Maths Club, Devonshire Primary School have found a pattern which they describe:
The tens in each pair don't change ie $58 - 53$, both $5$ tens.
The units in each pair add up to $11$.
$100$ & $91$ are different. $100$ has a nought in the tens column, and $90$ has a nine. In the units, $1 + 0$ doesn't = $11$.
$100 = 9$ tens & $10$ units. $91 = 9$ tens & $1$ unit. Now the tens stay the same, and the units add up to $11$.
St Non's Class at St Mary's in Wales wrote:
We found this problem interesting and challenging.Behind 100 would be 91. Behind 58 would be 52. Behind 23 is 28 and lastly behind 19 would be 12.
We counted how many space from the left it was, if it was 4 from the left it would also be 4 spaces in from the right.
We also knew that 5 was in middle and it would be matched with a 5 on the other side.
We tried adding the numbers on opposite sides together. We noticed that 1+10 = 11, 11+20 = 31, 21+30 = 51 and so on. These numbers went up in 20s.
We then tried the next line in 2+9 = 11, 12+19 = 31, 22+29 = 51.
These numbers also went up in 20s. We thought this was interesting.
We went to try the third row in 3+8 =11.... it was the same!!!
These are the patterns we noticed.
Thank you for information about the patterns that you found. I wonder whether you can explain why these patterns appear?
Hundred Square
A hundred square has been printed on both sides of a piece of paper. One square is directly behind the other.
What is on the back of $100$? $58$? $23$? $19$?
Can you see a pattern?
Image
Click here for a poster of this problem.
This problem is also available in French: Grille de 100
Why do this problem?
This problem is an unusual way to explore number patterns in a well-known context. The activity will reinforce the construction of the hundred square, and increase children's familiarity with the sequences contained within it. Using a common resource, such as a hundred square, is a good way for children to begin to use visualisation, which they may find quite difficult at first. The act of visualising in this problem tests children's understanding of how the number square is created.
Possible approach
You could start by asking the group to close their eyes and try to see a hundred square in their heads. Ask some questions such as "What number comes first and what is below it?"; "What number is below $10$?"; "What number is two places to the right of $34$?". Encourage them to justify their responses before using a real hundred square to check.
You could then pose the question in the problem and encourage the group to work in pairs. Give them time to talk to each other about possible solutions without making a hundred square available at first. [This sheet has two hundred squares on it which may be useful for some
children.] Asking pupils to work in pairs on this task will encourage them to begin to argue mathematically. Listening to their explanations and justifications of which numbers are where would be an excellent assessment exercise for you.
When it comes to articulating their method, it is a chance for you to see how well they can put into words what they notice and how they use mathematical understanding and vocabulary to support this. Once the whole group has shared some ways of coming to a solution, pose some further questions for pairs to work on. It is interesting to find out whether some children adopt ways of working on
the problem that they learnt from their classmates rather than sticking to their original method. There are many different ways of 'seeing' a solution.
Key questions
If you turned over the hundred square, where would you write the $1$ of the new hundred square?
So, which number would this $1$ be behind?
Where would the $2$ be? And the $3$?
What number is below $1$ on the hundred square?
Can you imagine the square on the back?
Where will its $1$ be? Where will its $10$ be?What is on the back of $100$? $58$? $23$? $19$?