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2011-02-01T00:00:01
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When it's the end of January and we need to turn over a page of our
calendar, the next page may
look a bit like this: <br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="195" width="264" src="Picture%202.jpg" alt="pic2"></mdo:image></div>
<br></br>
<br></br>
Of course, it probably has a good picture on the page as well,
perhaps of a scene from a T.V. show or an animal from your
favourite story book. <br></br>
<br></br>
If we ignore the edge squares which tell you what day of the week
it is, then we have a rectangle with twenty eight squares in it
like this:-
<div style="text-align: center;"><mdo:image height="159" width="267" alt="pic3" src="Picture%203.jpg"></mdo:image></div>
<br></br>
<p>Well, how about going a bit wild and designing a new shape for
these twenty eight squares in which the numbers follow on in some
way that is not too difficult to follow? For example, you could put
the twenty eight squares into a triangle and decide to go in an
up-and-down order like this:-</p>
<div style="text-align: center;"><mdo:image height="273" width="275" alt="pic1" src="Picture%201.jpg"></mdo:image></div>
<br></br>
What other wonderful designs can you come up with? <br></br>
<p>You can then go further by looking at the way the numbers are
arranged and see if you can spot any patterns. For example in the
first usual way:-</p>
<div style="text-align: center;"><mdo:image height="159" width="266" src="Picture%204.jpg" alt="pic4"></mdo:image></div>
<br></br>
you may notice things like:
<p>1. If you take any two by two square (like the one in at the top
left-hand corner with $1$, $2$, $8$ and $9$ in it), then the
opposite corners of that square add up to $10$. Then, if you take
the next two by two square along [with $3$, $4$, $10$ and $11$ in]
then the opposite corners add up to $14$ and so on.</p>
<p>2. Going down any column the numbers increase by $7$.</p>
<p>Do let us know what you discover in the designs you have
created. Can you explain why the patterns occur?</p>
<p>Then ask "I wonder what would happen if ...?''</p>
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<p class="editorial">John from St. Edburg's wrote;</p>
TOP LEFT<br></br>
I made the numbers into a triangle, starting in the top corner.
When I got to the bottom I started again at the top. I noticed that
the middle row and the middle column both added up to $75$.<br></br>
TOP RIGHT<br></br>
I wondered what would happen if I went down, then up, then down
again in a zigzag instead of just filling in the numbers downwards.
I had no idea that the middle row and the middle column were going
to add up to $75$ again!<br></br>
BOTTOM LEFT and BOTTOM RIGHT<br></br>
I tried putting the numbers in a different order again, starting in
the bottom left hand corner. In the bottom left picture I did the
same zigzag as in the top right, and in the bottom right picture
the numbers are filled in diagonally from top to bottom. I wondered
if the middle row and middle column would add up to $75$ again, but
they didn't. They all added up to $70$!<br></br>
<mdo:image height="564" width="599" alt="Sol" src="MonthmaniaSol.jpg"></mdo:image><br></br>
When I put the numbers into this grid, I spotted that on diagonal
rows the units repeat. This is because each number down the
diagonals is $5$ higher than the one before it ($1, 6,
11, 16$ and $13, 18, 23, 28$).<br></br>
<div style="text-align: center;"><mdo:image height="296" width="183" src="Monthmanis%202.jpg" alt="Sol2"></mdo:image></div>
<br></br>
<p class="editorial">Well done this was really a good investigation
for you - keep going !</p>
<br></br>
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<h2>Month Mania</h2>
<br></br>
When it's the end of January and we need to turn over a page of our calendar, the next page may look a bit like this:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="pic2" height="195" src="Picture%202.jpg" width="264"></mdo:image></div>
<br></br>
<br></br>
Of course, it probably has a good picture on the page as well, perhaps of a scene from a T.V. show or an animal from your favourite story book.<br></br>
<br></br>
If we ignore the edge squares which tell you what day of the week it is, then we have a rectangle with twenty eight squares in it like this:-
<div style="text-align: center;"><mdo:image alt="pic3" height="159" src="Picture%203.jpg" width="267"></mdo:image></div>
<br></br>
<p>Well, how about going a bit wild and designing a new shape for these twenty eight squares in which the numbers follow on in some way that is not too difficult to follow? For example, you could put the twenty eight squares into a triangle and decide to go in an up-and-down order like this:-</p>
<div style="text-align: center;"><mdo:image alt="pic1" height="273" src="Picture%201.jpg" width="275"></mdo:image></div>
<br></br>
What other wonderful designs can you come up with?<br></br>
<p>You can then go further by looking at the way the numbers are arranged and see if you can spot any patterns. For example in the first usual way:-</p>
<div style="text-align: center;"><mdo:image alt="pic4" height="159" src="Picture%204.jpg" width="266"></mdo:image></div>
<br></br>
you may notice things like:
<p>1. If you take any two by two square (like the one in at the top left-hand corner with $1$, $2$, $8$ and $9$ in it), then the opposite corners of that square add up to $10$. Then, if you take the next two by two square along [with $3$, $4$, $10$ and $11$ in] then the opposite corners add up to $14$ and so on.</p>
<p>2. Going down any column the numbers increase by $7$.</p>
<p>Do let us know what you discover in the designs you have created. Can you explain why the patterns occur?</p>
<p>Then ask "I wonder what would happen if ...?''</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=60&part=">This activity</a> is useful for helping learners to identify and explain number patterns. It is one that can help the more reluctant see that numbers and mathematics can be absorbing. It also has strong design and art links.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could start by showing the group some examples of various styles of calendars and encourage them to talk about the ways the months are laid out. Then show the class the picture of the calendar in the problem and invite them to talk in pairs about any number patterns they see in it. Can they offer any explanations? Next, show the second arrangement of the twenty eight squares. What number
patterns can they see now?</div>
<br></br>
You could then give them all a piece of $1$ cm squared paper and suggest they explore different arrangements of the twenty eight squares with the numbers arranged in some logical order. In the same way, they can look for any patterns of numbers that appear. If they work in pairs they will be able to talk through their ideas with a partner.<br></br>
<br></br>
<div>At the end of the lesson the group should come together again to show their various designs and discuss the number patterns they have found. Are there number patterns that keep on appearing? Are there some that have only been found once? Can they suggest why this should be? Can they explain why the patterns occur in each case?</div>
<br></br>
<h3>Key questions</h3>
<div>How else could you arrange the squares?</div>
<div>How will you write in the sequence of numbers?</div>
<div>Can you explain why this pattern of numbers occurs?</div>
<br></br>
<h3>Possible extension</h3>
Learners could move on from the calendar situation and arrange other numbers of squares in an interesting way. They could be asked "What would be a good new number to try?" and "Why?".<br></br>
<h3>Possible support</h3>
The help needed will depend on whether the learner is finding the numbers or alternative arrangements of the squares difficult. You could suggest arranging the numbers in the original calendar rectangle starting in different corners before trying other arrangements of the squares.<br></br>
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Try using squared paper to explore the possible arrangements of the
squares and numbers.<br></br></mdoxml>
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<p><span class="editorial">Natasha, Jill, Chris and Robert from
West Flegg School sent in a collection of solutions for February.
First of all there was</span> <em class="editorial">Month
Mania</em> <span class="editorial">and these most wonderful results
came from them that could be investigated further by you as another
challenge for this month.</span></p>
<div style="text-align: center;"></div>
<div style="text-align: center;"><mdo:image width="428" height="151" alt="60 2" src="ID%2060%202.jpg"></mdo:image></div>
<div style="text-align: center;"><mdo:image width="314" height="381" alt="60 3" src="ID%2060%203.jpg"></mdo:image></div>
<div style="text-align: center;"><mdo:image width="514" height="171" alt="60 5" src="ID%2060%205.jpg"></mdo:image></div>
<div style="text-align: center;"></div>
<div style="text-align: center;"></div>
<div style="text-align: center;"></div>
<p style="text-align: center;"></p>
<br></br>
<p class="editorial">I'm just so pleased to see these examples of
really good thinking. Well done all of you!</p>
<p class="editorial">Some more examples of good thinking came from
Yinkhe from Friar's School in Colchester. Lots of these I have
never seen before.</p>
<div style="text-align: center;"></div>
<div style="text-align: center;"><mdo:image width="427" height="195" src="ID%2060%201.jpg" alt="60 1"></mdo:image></div>
<div style="text-align: center;"></div>
<br></br>
<p class="editorial">There are so many patterns and relationships
in these collections of solutions.</p>
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Month Mania
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
Using, Applying and Reasoning about Mathematics
Investigations
Numbers and the Number System
Patterned numbers
Numbers and the Number System
Counting
Calculations and Numerical Methods
Addition & subtraction
Calculations and Numerical Methods
Multiplication & division