7798 /www/nrich/html/content/id/7798/ 1 0000-00-00T00:00:00 <mdoxml version="1.0"><br></br> Here are some squares I have cut from centimetre-squared paper.<br></br>  <br></br>  <mdo:image width="245" height="193" src="Squares1.png" alt="squares"></mdo:image><br></br> If I cut squares of all sizes from one centimetre to ten centimetres arrange them in order they will look like this:<br></br>  <br></br>  <mdo:image width="456" height="73" src="Squares2.png" alt=""></mdo:image><br></br> <br></br>  I am going to take two squares which are next door neighbours and put the smaller one on top of the larger one, keeping one corner and two sides together. This bit is the difference between the two squares.<br></br> <br></br> To make it easier to see I have coloured the squares.<br></br>  <br></br>  <mdo:image width="161" height="162" src="Squares3.png" alt=""></mdo:image><br></br>  <br></br> Now I am going to cut off the extra piece to make the two the same size and this is what I get:<br></br>  <br></br>  <mdo:image width="160" height="158" src="Squares4.png" alt=""></mdo:image><br></br> <br></br> This can be arranged like this:<br></br> <br></br>  <mdo:image width="318" height="26" src="Squares5.png" alt=""></mdo:image><br></br>  <br></br> How many squares long is it?<br></br> <br></br> Now the job for you to do is to find the differences between all the next-door squares from one to ten and to arrange the differences in lines like the one above.<br></br> <br></br> You can use squared paper to do this or you can download <a href="/content/id/7798/SquaresB-W.pdf">this sheet</a> with all the squares on it. You will probably need two copies.<br></br> <br></br> If you would like the coloured version you can find that <a href="/content/id/7798/SquaresCol.pdf">here</a>.<br></br> <br></br> This means you need to find the differences between one and two squared, two squared and three squared all the way up to the difference between nine and ten squared.<br></br> <br></br> Now arrange the differences you have cut off neatly in order of size so you can compare them.<br></br> <br></br> Now look at the differences in order.<br></br> What can you say about them?<br></br> Write down the length of each one in order.<br></br> What do you notice?<br></br>  <br></br></mdoxml> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/7798&amp;part=">This problem</a>, although not difficult to start, can lead learners into generalising about the difference between successive squares. Learners will need plenty of $1$ cm squared paper, and you will need some larger squares if you want to demonstrate to a group. They will also need scissors and possibly coloured pencils and glue to make a permanent record.<br></br> <h3>Possible approach</h3> <div>You could start by showing some square numbers cut from squared paper and asking why they are called &quot;square numbers&quot;. You could either cut squares from squared paper with larger squares or download and enlarge this sheet. Can the group think of any other square numbers?</div> <div> </div> <div>Alternatively, if the group already knows something about square numbers you could go straight to placing one square on top of the next one and seeing the difference. Remember to put two sides and the corner between together. Then the difference can be cut off, laid out in a line, and compared with the differences between other 'next-door' squares.</div> <div> </div> <div>After this learners could work in pairs on the actual problem from a printed sheet or a computer, so that they are able to talk through their ideas with a partner. They will need $1$ cm squared paper or two copies of this sheet. This coloured version might be helpful for some learners.</div> <div> </div> <div>At the end of the session all could come together again and discuss their findings. Can they say what kind of numbers all the differences are? What do the differences look like if they are arranged in order?</div> <br></br> <h3>Key questions</h3> <div>What is the name of these numbers cut out of squared paper?</div> <div>Can you tell me why that is their name?</div> <div>What other square numbers can you think of?</div> <div>Did you put two sides and the corner between together?</div> <div>How long is the bit you have cut off if you arrange it in a line?</div> <div>What kind of numbers are these?</div> <div>What do the differences look like if they are arranged in order?</div> <div>Have you discovered anything else?</div> <br></br> <h3>Possible extension</h3> Learners could try to generalise from the results of this problem and possibly reach the formula $(a + 1)^2$ = $a^2$ + 2a +1.<br></br> <h3>Possible support</h3> Suggest just beginning the problem, arranging the squares in order, which is quite easy, or try this Stage 1 problem - <a href="http://nrich.maths.org/7785&amp;part=">Odd and even</a>. <br></br> <br></br></mdoxml> <mdoxml version="1.0"><br></br> Remember to put the corner and two sides of the top square exactly on top of the larger square underneath.<br></br></mdoxml> 2 4 0 1 0 0 0 Take two squares which are next-door neighbours and find the difference between the two squares. What kind of number is this? Using, Applying and Reasoning about Mathematics Making and testing hypotheses Using, Applying and Reasoning about Mathematics Generalising Numbers and the Number System Odd and even numbers Numbers and the Number System Square numbers