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2011-02-01T00:00:01
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<span style="font-style: italic;">Well I wonder how often you have noticed that there are numbers around the place that follow one after another</span> $1, 2, 3, \ldots$ <span style="font-style: italic;">etc.? Sometimes they appear in reverse order when a countdown is happening for a launch of a rocket. But usually they happen in an order going up, like when you read through a book and notice the
page numbers. These kinds of numbers are called consecutive numbers, you may have heard of the word before - it simply means that they are whole numbers that follow one after another.</span><br></br>
<br></br>
<br></br>
<span style="font-style: italic;">You can start anywhere [</span> $3, 4, 5, 6, \ldots$ <span style="font-style: italic;">etc. or</span> $165, 166, 167, 168, \ldots$ <span style="font-style: italic;">etc.] and they can be explored in a number of different ways, when they are not counting anything particular. This investigation is about using the idea of consecutive numbers and gives us other
numbers that we can explore much further and find out all kinds of things. You may very well discover things that NO ONE else has discovered or written about before, and that's GREAT!!</span><br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">So this is how it starts.</span> You need to choose any four consecutive numbers and place them in a row with a bit of a space between them, like this:<br></br>
<br></br>
<mdo:image alt="4, 5, 6, 7" height="31" src="b_1.gif" width="271"></mdo:image><br></br>
<br></br>
When you've chosen your consecutive numbers, stick with those same ones for quite a while, exploring ideas before you change them in any way. Now place $+$ and $-$ signs in between them, something like this :<br></br>
<br></br>
<span class="editorial" style="font-style: italic;">4 + 5 - 6 + 7</span><br></br>
<span class="editorial" style="font-style: italic;">4 - 5 + 6 + 7</span><br></br>
<br></br>
and so on until you have found all the possibilities. You should include one using all $+$'s and one that includes all $-$'s.<br></br>
<br></br>
<br></br>
Now work out the answers to all your calculations (e.g. <span class="editorial" style="font-style: italic;">4 - 5 + 6 + 7 = 12</span> and so on). Are you sure you've got them all?<br></br>
<br></br>
If so, try other sets of four consecutive numbers and look carefully at the sets of answers that you get each time. It is probably a good idea to write down what you notice. This can lead you to test some ideas out by starting with new sets of consecutive numbers and seeing if the same things happen in the same way.<br></br>
<br></br>
You might now be doing some predictions that you can test out...<br></br>
<br></br>
<br></br>
FINALLY, it is good to ask the question "I wonder what would happen if I ... ?"<br></br>
You may have thought up your own questions to explore further. Here are some we thought of:<br></br>
<br></br>
"What would happen if I took the consecutive numbers in an order going down, instead of up?"<br></br>
"What would happen if I only used sets of 3 consecutive numbers?"<br></br>
"What would happen if I used more consecutive numbers?"<br></br>
"What would happen if I changed the rule and allowed consecutive numbers to include fractions or decimals?"<br></br>
"What would happen if I allowed a $+$ or $-$ sign before the first number?"<br></br>
<br></br>
<br></br>
<span class="editorial">This problem was chosen as a favourite for the NRICH 10th Anniversary website by Bernard Bagnall. Find out why Bernard selected it in the</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=31&part=note">Notes</a><br></br></mdoxml>
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<p class="editorial">Andrew and Tom sent this in to us:</p>
<div>Numbers 14, 15, 16, 17:</div>
<br></br>
<div>
<div>1, 14 + 15 + 16 + 17 = 62</div>
<div>2, 14 - 15 - 16 - 17 = -4</div>
<div>3, 14 + 15 - 16 + 17 = 30</div>
<div>4, 14 - 15 + 16 - 17 = -2</div>
<div>5, 14 + 15 + 16 - 17 = 28</div>
<div>6, 14 - 15 - 16 + 17 = 0</div>
<div>7, 14 + 15 - 16 - 17 = -4</div>
<div>8, 14 - 15 + 16 + 17 = 32</div>
<br></br>
<div>The numbers are all even</div>
<br></br>
<div>Numbers 20, 21, 22, 23:</div>
<br></br>
<div>1, 20 + 21 + 22 + 23 = 86</div>
<div>2, 20 - 21 - 22 - 23 = -46</div>
<div>3, 20 + 21 - 22 + 23 = 42</div>
<div>4, 20 - 21 + 22 - 23 = -2</div>
<div>5, 20 + 21 + 22 - 23 = 40</div>
<div>6, 20 - 21 - 22 + 23 = 0</div>
<div>7, 20 + 21 - 22 - 23 = -4</div>
<div>8, 20 - 21 + 22 + 23 = 44</div>
<br></br>
<div>Compare the sets:</div>
<br></br>
<div>The numbers are all even in both sets. The sets of numbers
both have 3 negative numbers and 5 other numbers - one of which is
0! Both of them include -2 in them.</div>
<div style="clear: both;">
<div style="clear: both;">
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<p style="clear: both; font-style: italic;">Test</p>
<div style="clear: both; font-style: italic;"></div>
<div style="clear: both; font-style: normal;">This is a test to see
whether each pattern contains 3 negative numbers one 0, and 4 other
numbers.</div>
<div style="clear: both; font-style: normal;"></div>
<p style="clear: both; font-style: normal;">Numbers 54, 55, 56,
57:</p>
<div style="clear: both; font-style: normal;"></div>
<div style="clear: both; font-style: normal;"></div>
<div style="clear: both; font-style: normal;">1, 54 + 55 + 56 + 57
= 222<br></br>
<div style="clear: both;">2, 54 - 55 - 56 - 57 = -144<br></br>
<div style="clear: both;">3, 54 + 55 - 56 + 57 = 110<br></br>
<div style="clear: both;">4, 54 - 55 + 56 - 57 = -2<br></br>
<div style="clear: both;">5, 54 + 55 + 56 - 57 = 108<br></br>
<div style="clear: both;">6, 54 - 55 - 56 + 57 = 0<br></br>
<div style="clear: both;">7, 54 + 55 - 56 - 57 = -4<br></br>
<div style="clear: both;">8, 54 - 55 + 56 + 57 = 112</div>
<div style="clear: both;"></div>
<div style="clear: both;">Test complete.</div>
<div style="clear: both;"></div>
<p style="clear: both;"><span style="font-style: italic;">New Rule
- Consecutive Numbers backwards</span></p>
<div style="clear: both;"></div>
<p style="clear: both;">Numbers 63, 62, 61, 60:</p>
<div style="clear: both;"></div>
<div style="clear: both;">1, 63 + 62 + 61 + 60 = 246<br></br>
<div style="clear: both;">2, 63 - 62 - 61 - 60 = 120<br></br>
<div style="clear: both;">3, 63 + 62 - 61 + 60 = 124<br></br>
<div style="clear: both;">4, 63 - 62 + 61 - 60 = 2<br></br>
<div style="clear: both;">5, 63 + 62 + 61 - 60 = 126<br></br>
<div style="clear: both;">6, 63 - 62 - 61 + 60 = 0<br></br>
<div style="clear: both;">7, 63 + 62 - 61 - 60 = 4<br></br>
<div style="clear: both;">8, 63 - 62 + 61 + 60 = 122</div>
<div style="clear: both;"></div>
<p style="clear: both;">Here, there are no negative numbers - -4
became 4 and -2 became 2 !</p>
<div style="clear: both;"></div>
<p style="clear: both;">Numbers 04, 03, 02, 01:</p>
<div style="clear: both;"></div>
<div style="clear: both;">1, 4 + 3 + 2 + 1 = 10<br></br>
<div style="clear: both;">2, 4 - 3 - 2 - 1 = -2<br></br>
<div style="clear: both;">3, 4 + 3 - 2 + 1 = 6<br></br>
<div style="clear: both;">4, 4 - 3 + 2 - 1 = 2<br></br>
<div style="clear: both;">5, 4 + 3 + 2 - 1 = 8<br></br>
<div style="clear: both;">6, 4 - 3 - 2 + 1 = 0<br></br>
<div style="clear: both;">7, 4 + 3 - 2 - 1 = 4<br></br>
<div style="clear: both;">8, 4 - 3 + 2 + 1 = 4</div>
<div style="clear: both;"></div>
<p style="clear: both;">There is one negative number again -2, -4
has changed to two +4 [not a sum].</p>
<div style="clear: both;"></div>
<p style="clear: both;">Comparing [with the other rule]: Answer 6
stays as 0. The negative numbers have become positive, [normal]
numbers [i.e. -2 = 2 -4 = 4 ] This is because you take lower
numbers, and add higher numbers.</p>
<p style="clear: both;"> </p>
<p class="editorial" style="clear: both;">Pupils from William
Harding Combined School and also from Gorseland Primary School
investigated these consecutive numbers as well. They tried many
different sets of four consecutive numbers and all agreed with what
Stuart (from William Harding) wrote:</p>
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<div>I found out that there are 8 sums for every set of consecutive
numbers 1 full add,1 full subtract, 3 with 2 adds and 3 with 2
subtracts.</div>
<br></br>
<p class="editorial">The group from Gorseland also found three
rules (if the numbers are in ascending order):</p>
<ul>
<li>When the order of the signs is +, -, -, then you always get
-4</li>
<li>When the order of the signs is -, -, +, then you always get
0</li>
<li>When the order of the signs is -, +, -, then you always get
-2</li>
</ul>
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<h2>Consecutive Numbers</h2>
<br></br>
<span style="font-style: italic;">Well I wonder how often you have noticed that there are numbers around the place that follow one after another</span> $1, 2, 3, \ldots$ <span style="font-style: italic;">etc.? Sometimes they appear in reverse order when a countdown is happening for a launch of a rocket. But usually they happen in an order going up, like when you read through a book and notice the
page numbers. These kinds of numbers are called consecutive numbers, you may have heard of the word before - it simply means that they are whole numbers that follow one after another.</span><br></br>
<br></br>
<br></br>
<span style="font-style: italic;">You can start anywhere [</span> $3, 4, 5, 6, \ldots$ <span style="font-style: italic;">etc. or</span> $165, 166, 167, 168, \ldots$ <span style="font-style: italic;">etc.] and they can be explored in a number of different ways, when they are not counting anything particular. This investigation is about using the idea of consecutive numbers and gives us other
numbers that we can explore much further and find out all kinds of things. You may very well discover things that NO ONE else has discovered or written about before, and that's GREAT!!</span><br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">So this is how it starts.</span> You need to choose any four consecutive numbers and place them in a row with a bit of a space between them, like this:<br></br>
<br></br>
<mdo:image alt="4, 5, 6, 7" height="31" src="b_1.gif" width="271"></mdo:image><br></br>
<br></br>
When you've chosen your consecutive numbers, stick with those same ones for quite a while, exploring ideas before you change them in any way. Now place $+$ and $-$ signs in between them, something like this :<br></br>
<br></br>
<span class="editorial" style="font-style: italic;">4 + 5 - 6 + 7</span><br></br>
<span class="editorial" style="font-style: italic;">4 - 5 + 6 + 7</span><br></br>
<br></br>
and so on until you have found all the possibilities. You should include one using all $+$'s and one that includes all $-$'s.<br></br>
<br></br>
<br></br>
Now work out the answers to all your calculations (e.g. <span class="editorial" style="font-style: italic;">4 - 5 + 6 + 7 = 12</span> and so on). Are you sure you've got them all?<br></br>
<br></br>
If so, try other sets of four consecutive numbers and look carefully at the sets of answers that you get each time. It is probably a good idea to write down what you notice. This can lead you to test some ideas out by starting with new sets of consecutive numbers and seeing if the same things happen in the same way.<br></br>
<br></br>
You might now be doing some predictions that you can test out...<br></br>
<br></br>
<br></br>
FINALLY, it is good to ask the question "I wonder what would happen if I ... ?"<br></br>
You may have thought up your own questions to explore further. Here are some we thought of:<br></br>
<br></br>
"What would happen if I took the consecutive numbers in an order going down, instead of up?"<br></br>
"What would happen if I only used sets of 3 consecutive numbers?"<br></br>
"What would happen if I used more consecutive numbers?"<br></br>
"What would happen if I changed the rule and allowed consecutive numbers to include fractions or decimals?"<br></br>
"What would happen if I allowed a $+$ or $-$ sign before the first number?"<br></br>
<br></br>
<br></br>
<span class="editorial">This problem was chosen as a favourite for the NRICH 10th Anniversary website by Bernard Bagnall. Find out why Bernard selected it in the</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=31&part=note">Notes</a></div>
<br></br>
<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=31&part=">This problem</a> has the capacity to interest young and old alike. I have used it with a wide range of attainment levels, and new things keep on being found out. It offers opportunities to work together when sharing results and making decisions as to which consecutive numbers to look at next.</div>
<br></br>
<h3>Possible approach</h3>
<div>It is sometimes useful to suggest to the students that they are being detectives and seeking out links, relations and reasons.</div>
<br></br>
<div>To introduce the problem, go through what consecutive means, getting suggestions from the pupils for the starting number. It is good to let the pupils select the three operations and to take four or five examples, but not to discuss how many possibilities there are at this stage.</div>
<br></br>
<div>Most children find some connections between the eight answers that they find. The first finding is usually that all the answers are even. The fact that $0$, $-2$, and $-4$ appear with every group of four consecutive numbers suggests the question "why?" leading to interesting discussions about the occurrence of negative numbers.</div>
<br></br>
<h3>Key questions</h3>
<div>Do you think you've found all the possibilities?</div>
<div>Tell me about your answers.</div>
<div>Do you notice anything about your answers?</div>
<div>Can you explain why these things always happen?</div>
<br></br>
<h3>Possible extension</h3>
<div>I have found that all the students who have been involved in this investigation have got very excited as various observations are made, patterns seen and questions asked. The most enjoyable times for me have been hearing ten year olds using their own form of algebra and coming to some powerful [for them] realisations about why every one has a $0$, $-2$ and $-4$.</div>
<div>The problem has also been the starting point for some pupils to be able to ask "I wonder what would happen if ...?" And in this case it's been:</div>
<div>... we used more consecutive numbers each time?</div>
<div>... we had a starting point in the negative numbers?</div>
<div>... we took consecutive to mean going up in $2$s?</div>
<div>... we were allowed to use fractions or decimals in between the whole numbers?<br></br>
and for negatives look at <a href="http://www.nrich.maths.org.uk/5868">Consecutive Negative Numbers.</a></div>
<br></br>
<h3>For the exceptionally mathematically able</h3>
<h4 style="font-weight: 400;">These pupils would be encouraged to work on proofs. They could also begin to make comparisons - say between using four consecutuve numbers and six consecutive numbers. Some learners may want to examine other properties of the answers for any set of four consectuive numbers and this could lead on to generalisations.</h4>
<h3>Possible support</h3>
<div>On the odd occasions that pupils needed support I have found that putting a number of pupils together to work as a sharing group is all that has been necessary.</div>
<br></br>
<div class="editorial">These notes are taken from writings by Bernard Bagnall who has used this activity more than sixty times and chose it as his favourite problem on the NRICH site.</div>
<br></br>
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Having chosen your $4$ consecutive numbers you place $+$ and $-$
signs in between them. <br></br>
<br></br>
You have $3$ empty spaces, so how many possible combinations are
there for arranging the $+$ and $-$ signs?<br></br>
<br></br>
Are you sure you have found them all? How will you know? <br></br></mdoxml>
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Consecutive Numbers
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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