2397
/www/nrich/html/content/id/2397/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>A hundred square has been printed on both sides of a piece of paper. One square is directly behind the other.</p>
<p>What is on the back of $100$? $58$? $23$? $19$?</p>
<p>Can you see a pattern?</p>
<p style="text-align: center;"><mdo:image alt="" height="341" src="100Sq.png" width="341"></mdo:image></p>
<p style="text-align: left;"> </p>
<p style="text-align: left;"><a href="http://nrich.maths.org/7202&part=">Click here for a poster of this problem</a>.</p>
<br></br>
This problem is also available in French: <a href="http://nrich.maths.org/7116">Grille de 100</a><br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">Thank you for your solutions to this problem.
You found several different ways of tackling it.</p>
<p class="editorial">Fergus, Nathan, Samuel and Jesse from
Rutherglen Primary School drew another hundred square with the
numbers on the opposite side of a printed hundred square. Jesse
says:</p>
On the back of $100$ was $91$, on the back of $58$ was $53$, on the
back of $23$ was $28$ and on the back of $19$ was $12$.
<p class="editorial">Mikey from Archbishop of York CE Junior School
wrote:</p>
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?<br></br>
<br></br>
<p class="editorial">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 first
name):</p>
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!<br></br>
<br></br>
<p class="editorial">This is also a very handy method - well
noticed. "N" sent an image with the "mirror line" drawn in:</p>
<br></br>
<br></br>
<p style="text-align: center;"><mdo:image height="303" width="300" alt="100 square with mirror line drawn in" src="mirrorline.gif"></mdo:image></p>
<p><span class="editorial">Well done to George from Bradon Forest,
Mary from St Swithun's, Miss Grewcock's Class from St James CEVA
Primary and Joshua from Queen Victoria Primary who also noticed
this mirror line.</span></p>
<p><span class="editorial">Devonshire Maths Club, Devonshire
Primary School have found a pattern which they describe:</span></p>
<div>The tens in each pair don't change ie $58 - 53$, both $5$
tens.</div>
<div>The units in each pair add up to $11$.</div>
<div>$100$ & $91$ are different. $100$ has a nought in the tens
column, and $90$ has a nine. In the units, $1 + 0$ doesn't =
$11$.</div>
<div>$100 = 9$ tens & $10$ units. $91 = 9$ tens & $1$ unit.
Now the tens stay the same, and the units add up to $11$.</div>
<br></br>
<p class="editorial">Very well noticed. Luke from Queen Victoria
Primary found another way:</p>
<br></br>
<div>If the number is $23$ then it is placed two places away from
the end of the line.</div>
<div>Next you go two places back from the other end of the same
line to get the correct answer of $28$. Like $19$ would be $12$ on
the other side of the sheet.</div>
<br></br>
<p class="editorial">Jake from the same school as Luke explained
this a bit more generally:</p>
<br></br>
<div>To find the solution you need to first find out whether it's
on the right or the left of the hundred square. Then see how many
squares it is away from the end of the square. Then use the number
of squares on the other side. Then you have the answer.</div>
<br></br>
<p class="editorial">Sohpie and Anna from St Swithun's and
Gabrielle from Hayesfield Girls School also tackled the problem in
this way.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2397&part=index">
This problem</a> 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 really tests children's
understanding of how the number square is created.<br></br>
<h3>Possible approach</h3>
<div>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.</div>
<br></br>
<div>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. [<a href="/content/id/2397/2397.pdf">This sheet</a> 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.</div>
<br></br>
<div>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.</div>
<br></br>
<h3>Key questions</h3>
<div>If you turned over the hundred square, where would you write
the $1$ of the new hundred square?</div>
<div>So, which number would this $1$ be behind?</div>
<div>Where would the $2$ be? And the $3$?</div>
What number is below $1$ on the hundred square?
<div>Can you imagine the square on the back?</div>
Where will its $1$ be? Where will its $10$ be?<br></br>
<div>What <span style="font-weight: bold;">is</span> on the back of
$100$? $58$? $23$? $19$?</div>
<br></br>
<h3>Possible extension</h3>
Learners could use a mirror on the hundred square and note all the
patterns they can find. What happens if you add each number on the
front to the number behind it? Using a different square such as a
[$64$] $8$ by $8$ square will produce a different, if related, set
of numbers.<br></br>
<h3>Possible support</h3>
Many children will find this task more accessible if they have a
real copy of a hundred square to use and annotate as they feel fit.
[<a href="/content/id/2397/2397.pdf">This sheet</a> has two hundred
squares on it.]<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Can you imagine the square on the back? Where will its $1$ be?
Where will its $10$ be?<br></br>
So what will be on the back of the $20$ in the $100$ square in the
picture? The $30$? Does this help with the $100$?<br></br>
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.<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">On the back of 100 is 99 On the back of 58 is 53 On the back of 23
is 28 On the back of 19 is 12 <br></br>
<br></br>
Patterns: Both numbers in the pair usually have the same tens digit
apart from 100/99 and 1/10 Units digit of sum of pairs is always
1<br></br>
<p class="editorial">Thank you for your solutions to this problem.
You found several different ways of tackling it.</p>
<p class="editorial">Mikey from Archbishop of York CE Junior School
wrote:</p>
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?<br></br>
<br></br>
<p class="editorial">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):</p>
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!<br></br>
<br></br>
<p class="editorial">This is also a very handy method - well
noticed. "N" sent an image with the "mirror line" drawn in:</p>
<br></br>
<br></br>
<p style="text-align: center;"><mdo:image height="303" width="300" alt="100 square with mirror line drawn in" src="mirrorline.gif"></mdo:image></p>
<p><span class="editorial">Devonshire Maths Club, Devonshire
Primary School have found a pattern which they describe:</span></p>
<div>The tens in each pair don't change ie $58 - 53$, both $5$
tens.</div>
<div>The units in each pair add up to $11$.</div>
<div>$100$ & $91$ are different. $100$ has a nought in the tens
column, and $90$ has a nine. In the units, $1 + 0$ doesn't =
$11$.</div>
<div>$100 = 9$ tens & $10$ units. $91 = 9$ tens & $1$ unit.
Now the tens stay the same, and the units add up to $11$.</div>
<p class="editorial">Very well noticed. Well done!</p>
<br></br>
<br></br></mdoxml>
2
4
1
0
0
0
0
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?
Numbers and the Number System
Patterned numbers
Using, Applying and Reasoning about Mathematics
Visualising
Mathematics Tools
100 square