7190
/www/nrich/html/content/id/7190/
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2011-02-01T00:00:01
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How many odd numbers are there between $3$ and $11$?<br></br>
<br></br>
How many odd numbers are there between $4$ and $11$?<br></br>
<br></br>
What do you notice? Can you explain your observation?<br></br>
<br></br>
Can you find any other pairs of numbers which have this same number
of odds between them?<br></br>
<br></br>
Can you find a pair of numbers which have four odd numbers between
them? <br></br>
<br></br>
Can you find another pair of numbers which have four odds between
them? And another pair?<br></br>
<br></br>
How would you find a pair of numbers that have five odds between
them? Six odds?<br></br>
<br></br>
How would you explain to someone else how to find a pair of
numbers that have a certain number of odds between them?<br></br></mdoxml>
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<p class="editorial">Thank you for your solutions to this problem.
Many of you were careful to say that you only counted numbers
BETWEEN the two numbers given, so you didn't include the numbers
themselves. That's important to get clear, I think.</p>
<p class="editorial">Matthew and Luca from Dunchurch Boughton
Junior School described what they noticed:</p>
If the two numbers are consecutive <span class="editorial">(I
think here they mean the starting numbers of each set of
numbers)</span> and the last numbers are the same, the answer will
always be exactly the same.<br></br>
<br></br>
Here are some examples:<br></br>
$1$-$9$: $3$, $5$, $7$. = three odds<br></br>
$2$-$9$: $3$, $5$, $7$. = three odds<br></br>
<br></br>
$3$-$15$: $5$, $7$, $9$, $11$, $13$. = five odds<br></br>
$4$-$15$: $5$, $7$, $9$, $11$, $13$. = five odds<br></br>
<br></br>
$123$-$131$: $125$, $127$, $129$. = three odds<br></br>
$124$-$131$: $125$, $127$, $129$. = three odds<br></br>
<br></br>
<p class="editorial">Moana from the Canadian Academy explained what
Matthew and Luca noticed by saying:</p>
We figured out that after an odd number, it's an even number and
since we don't count even numbers it doesn't change. <span class="editorial">(I think by 'it' in the second sentence Moana
means the number of odd numbers.)</span><br></br>
<br></br>
<p class="editorial">James from Edenlode wrote:</p>
3 odd numbers between $3$-$11$!<br></br>
3 odd numbers between $4$-$11$!!!<br></br>
The same number of odd numbers in between!!!!!<br></br>
Because the next odd number after $3$ is $5$!!!!!!!<br></br>
<br></br>
Some numbers with 3 odds in between are $0$ and $6$ or $0$ and
$7$!<br></br>
Some numbers with 4 odds in between are $0$ and $8$ or $0$ and
$9$!!!<br></br>
Some numbers with 5 odds in betwen are $0$ and $10$ or $0$ and
$11$!!!!!<br></br>
Some numbers with 6 odds in between are $0$ and $12$ or $0$ and
$13$!!!!!!!<br></br>
<br></br>
If I was explaining to someone else how to work out how many odds
between two numbers I would say 'If you start on an even number and
land on an even number then the number of odds is half the number
you count on. If you start on an even number and land on an odd
number, then the number of odds is half the number you count on,
plus one more'.<br></br>
<br></br>
I love maths. <br></br>
<br></br>
<p class="editorial">I wonder what happens when you start on an odd
number and land on an even number? Or if you start on an odd
number and land on an odd number? James also asked "Do you
notice anything about my exclamation marks?".</p>
<p><span class="editorial">Perhaps some of you thought about it in
a slightly different way? Let us know if you
did! </span></p>
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<h2>How Odd</h2>
<br></br>
How many odd numbers are there between $3$ and $11$?<br></br>
<br></br>
How many odd numbers are there between $4$ and $11$?<br></br>
<br></br>
What do you notice? Can you explain your observation?<br></br>
<br></br>
Can you find any other pairs of numbers which have this same number of odds between them?<br></br>
<br></br>
Can you find a pair of numbers which have four odd numbers between them? <br></br>
<br></br>
Can you find another pair of numbers which have four odds between them? And another pair?<br></br>
<br></br>
How would you find a pair of numbers that have five odds between them? Six odds?<br></br>
<br></br>
How would you explain to someone else how to find a pair of numbers that have a certain number of odds between them?</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/7190&part=">This problem</a> will help consolidate children's understanding of, and familiarity with, odd and even numbers. It also provides an opportunity for learners to explain and generalise.</div>
<br></br>
<h3>Possible approach</h3>
<div>Depending on your pupils' past experience, you may want to begin with some models of some numbers made from multilink cubes. You could put the cubes together so that they are paired, for example this would be the model for $9$:</div>
<div> </div>
<div style="text-align: center;"> <mdo:image alt="" height="88" src="nine.gif" width="183"></mdo:image></div>
<div style="text-align: left;">Put your multilink models out for the children to see and invite them to talk about the models. You could lead on to ordering the models numerically and pupils could make their own models to fill in some of the missing numbers.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">Having made, for example, models of the numbers from $1$ to $12$ and placed them in numerical order, you could pose a few questions to focus on odds and evens, if this has not already come up in conversation. For instance, you could ask whether there is anything the same about any of the models. Alternatively, you could group the models into two sets,
one of odds and one of evens and invite the group to talk about what you've done.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">You could then pose the first challenge in the problem and step back for learners to have a go in pairs. You may need to have a conversation about what 'between' means, so that, for example, the whole numbers between $3$ and $11$ would not include $3$ or $11$. Allow them to choose how they tackle it - this will be a great assessment opportunity for
you. Once they have had sufficient time, talk together about their methods and conclusions. You can then go on to the other parts of the problem, again leaving it up to the children to decide how they approach it and how they record their solutions. If possible, give the group plenty of time to find lots of examples at each stage.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">You could leave the final part of the problem to the plenary. Talk together about ways of finding two numbers which have, for example, ten odds between them. You may be able to encourage some children to articulate how they would find a pair with any number of odds between them. </div>
<br></br>
<h3>Key questions</h3>
<div>How will you find out how many odd numbers there are between $3$ and $11$, $4$ and $11$ etc.?</div>
<div>Tell me about what you're doing.</div>
<div>How will you remember what you've found so far?</div>
<br></br>
<h3>Possible extension</h3>
<div>You could challenge some children to find good ways of predicting the number of odd numbers between any two numbers. How about investigating the number of even numbers between a pair of numbers? </div>
<br></br>
<h3>Possible support</h3>
<div>It would be good to have multilink and number lines available along with plenty of blank paper and pencils and small whiteboards. Of course some children may want to use particular equipment which you hadn't thought of, so do allow them to choose whatever suits them best.</div>
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<mdoxml version="1.0">You might find it useful to have some cubes or a number line, or
you could draw your own number line.<br></br>
How will you make sure you don't forget what you've found
out?<br></br></mdoxml>
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How Odd
This problem challenges you to find out how many odd numbers there
are between pairs of numbers. Can you find a pair of numbers that
has four odds between them?
Numbers and the Number System
Odd and even numbers
Using, Applying and Reasoning about Mathematics
Generalising
Numbers and the Number System
Counting
Admin
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