5633
/www/nrich/html/content/id/5633/
1
2011-02-01T00:00:01
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<p>Draw a number line on a piece of paper, marked from 0 to 20, like this:</p>
<div><mdo:image alt="number line from 0 to 20" height="37" src="numberline.gif" width="550"></mdo:image></div>
<p>(You could print off <a href="/content/id/5633/numberlines.doc">this sheet</a> which has several number lines drawn on it.)</p>
<p>This challenge is a game for two players.The first player chooses two numbers in this grid and either $\times$ or $\div$.</p>
<div><mdo:image alt="grid of numbers: 2, 3, 4, 5, 10, 12, 25, 36, 100; multiply and divide symbols" height="216" src="fourgo.gif" width="304"></mdo:image></div>
<p>He or she then marks the answer to the calculation on the number line.The second player then chooses two numbers and either $\times$ or $\div$, and marks that number in a different colour on the number line.</p>
<p>If the answer is too big or too small to be marked on the number line, the player misses a go.</p>
<p>The winner is the player to get four marks in a row with none of their opponent's marks in between.</p>
<p>What good ways do you have of winning the game?</p>
<p>Does it matter if you go first or second?</p>
<p><em>[This game is adapted from a SMILE Centre card.]</em></p>
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<br></br>
<p class="editorial"><span class="editorial">Arti wrote to us to say:</span></p>
<p>There are two things that are not defined which need a definition:<br></br>
- If player A marked a number in the number line, can player B mark it later?<br></br>
- If player A used two numbers from the square, can player B use one or both those numbers?</p>
<p class="editorial"><span class="editorial">These are excellent questions, Arti. I'd be interested to know what you decided. To start with, I think I'd try to play with the rules that once a number has been marked, it can't be marked again later and that it doesn't matter which numbers your partner chooses, you can choose any you like. But perhaps you decided
differently?</span></p>
<p class="editorial"><span class="editorial">Rowena from Christ Church Primary told us:</span></p>
<br></br>
<p>I played this game with my Mum and neither of us won. We played it again and my Mum let me win!<br></br>
We decided to list all the possible whole number answers. They were 2, 3, 4, 5, 6, 8, 9, 10, 12, 15 and 20. Once we knew these, it was easy to choose numbers to block the opponent and not let them get 4 in a row.<br></br>
You can only win if your opponent makes a mistake or lets you win!<br></br>
</p>
<p class="editorial"><span class="editorial">Thank you, Rowena - a good idea to make a list of the whole number answers.</span></p>
<p class="editorial"><span class="editorial">I wonder whether you could change the game so that it was easier to win?</span></p>
<br></br>
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<mdoxml version="1.0"><div class="embed">
<h2>Four Go</h2>
<p>Draw a number line on a piece of paper, marked from 0 to 20, like this:</p>
<div><mdo:image alt="number line from 0 to 20" height="37" src="numberline.gif" width="550"></mdo:image></div>
<p>(You could print off <a href="/content/id/5633/numberlines.doc">this sheet</a> which has several number lines drawn on it.)</p>
<p>This challenge is a game for two players.The first player chooses two numbers in this grid and either $\times$ or $\div$.</p>
<div><mdo:image alt="grid of numbers: 2, 3, 4, 5, 10, 12, 25, 36, 100; multiply and divide symbols" height="216" src="fourgo.gif" width="304"></mdo:image></div>
<p>He or she then marks the answer to the calculation on the number line.The second player then chooses two numbers and either $\times$ or $\div$, and marks that number in a different colour on the number line.</p>
<p>If the answer is too big or too small to be marked on the number line, the player misses a go.</p>
<p>The winner is the player to get four marks in a row with none of their opponent's marks in between.</p>
<p>What good ways do you have of winning the game?</p>
<p>Does it matter if you go first or second?</p>
<p><em>[This game is adapted from a SMILE Centre card.]</em></p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/5633&part=">This game</a> gives children the opportunity to estimate answers to calculations in a motivating context and gives plenty of practice in multiplication and division. Playing strategically involves higher-order thinking and the need to think ahead. </div>
<br></br>
<h3>Possible approach</h3>
<div>You could play the game on the board against another adult (or a child who has been told the rules), but without saying anything about the rules to the whole group. You could write down each calculation as you go along. Invite the class to watch the game in action and after a few moves, ask them to suggest what they think the rules might be. How might the game be won?
Having discussed the possibilities as a whole class, explain the rules as described in the problem itself and set the children off playing in pairs.</div>
<div> </div>
<div>As they play, you should find that if each child wants to win, they will automatically scrutinise the answers of their opponent carefully and this means that the pupils will be required to explain and justify their thinking, and to check their own calculations thoroughly. There is also scope here for the children themselves to make decisions about the rules of the game. For example, will
they allow a player to multiply or divide a number by itself, or must the two numbers chosen be different? Will they notice that the zero can't be used?</div>
<div> </div>
<div>During the plenary, encourage learners to explain any strategies that they developed. You may also wish to draw attention to some particularly good examples of justification and explanation which you heard as the children played the game, or examples of insightful comments in general. </div>
<br></br>
<h3>Key questions</h3>
<div>How are you deciding which number to aim for next?</div>
<div>Can you find a winning strategy?</div>
<br></br>
<h3>Possible extension</h3>
<div>Children can be encouraged to tweak the game and to try out their new version. For example, they might change the number line, the grid of numbers, the operations, the number of numbers needed to win ...</div>
<br></br>
<h3>Possible support</h3>
<div>You might wish to have calculators available and/or adapt the grid/number line to suit the children with whom you are working. </div>
<br></br>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Where on the number line do you want to put a mark? How will you
make that answer from the numbers in the grid?<br></br>
You could use a calculator to check your answers if your opponent
doesn't agree!<br></br>
How will you stop your opponent from getting four in a line before
you?<br></br>
Are there any good places on the number line to start? Are there
some places which might not be so good to start? Why?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">This game gives children the opportunity to estimate answers to
calculations in a motivating context and gives plenty of practice
in multiplication and division. (Of course the numbers and the
number line could be adapted to suit the level of your class.) As
they play, encourage pairs to talk to each other about the numbers
and operations they are choosing, and why. You should find that if
each child wants to win, they will automatically scrutinise the
answers of their opponent carefully and this means that the pupils
will be required to explain and justify their thinking, and to
check their own calculations thoroughly. You might also wish to
have calculators available. There is also scope here for the
children themselves to make decisions about the rules of the game.
For example, will they allow a player to multiply or divide a
number by itself, or must the two numbers chosen be
different?<br></br></mdoxml>
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Four Go
This challenge is a game for two players. Choose two numbers from the grid and multiply or divide, then mark your answer on the number line. Can you get four in a row before your partner?
Calculations and Numerical Methods
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