5652
/www/nrich/html/content/id/5652/
1
2011-02-01T00:00:01
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Leah and Tom both have number lines and a counter. They always
start with their counters at zero.<br></br>
<br></br>
Leah's number line goes along from left to right like this:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="38" width="313" alt="0-10 number line" src="NoLine1.gif"></mdo:image></div>
<br></br>
<br></br>
First Leah made a jump of three along her number line and then a
jump of four. Where did she land? <br></br>
<br></br>
Next Leah made a secret jump along her number line. Then she made a
jump of five and landed on $9$. <br></br>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="60" width="275" alt="0-10 number line with 9 marked" src="NoLine3.gif"></mdo:image></div>
<br></br>
<br></br>
How long was her secret jump?<br></br>
<br></br>
Then Leah made a jump of six and another secret jump. She landed on
$10$. <br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="54" width="275" alt="0-10 number line with 10 marked" src="NoLine4.gif"></mdo:image></div>
<br></br>
<br></br>
How long was her second secret jump? <br></br>
<br></br>
Tom's number line goes up and down like this: <br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="313" width="38" alt="0-10 vertical number line " src="NoLine2.gif"></mdo:image></div>
<br></br>
<br></br>
First Tom made a jump of three up his number line and then a jump
of two. Where did he land?<br></br>
<br></br>
Next Tom made a secret jump up his number line. Then he made a jump
of six and landed on $8$. <br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="273" width="61" alt="0-10 vertical number line with 8 marked" src="NoLine5.gif"></mdo:image></div>
<br></br>
<br></br>
How long was his secret jump? <br></br>
<br></br>
Then Tom made a jump of four and another secret jump. He landed on
$7$. <br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="273" width="61" alt="0-10 vertical number line with 7 marked" src="NoLine6.gif"></mdo:image></div>
<br></br>
<br></br>
How long was his second secret jump? <br></br></mdoxml>
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<p class="editorial">Thank you to Daniel from Englefield Green
Infant School and Daina and Emma from Kelso State School for their
solutions to the Number Lines problem. Daniel said:</p>
Leah's first jump landed at $7$, $3+4=7$<br></br>
Leah's first secret jump was $4$, $4+5=9$<br></br>
Leah's second secret jump was $4$, $6+4=10$<br></br>
Tom's first jump landed at $5$, $3+2=5$<br></br>
Tom's first secret jump was $2$, $2+6=8$<br></br>
Tom's second secret jump was $3$, $4+3=7$<br></br>
<br></br>
<p class="editorial">Thank you, Daniel!</p>
<br></br></mdoxml>
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<h2>Number Lines</h2>
<br></br>
Leah and Tom both have number lines and a counter. They always start with their counters at zero.<br></br>
<br></br>
Leah's number line goes along from left to right like this:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="0-10 number line" height="38" src="NoLine1.gif" width="313"></mdo:image></div>
<br></br>
<br></br>
First Leah made a jump of three along her number line and then a jump of four. Where did she land?<br></br>
<br></br>
Next Leah made a secret jump along her number line. Then she made a jump of five and landed on $9$.<br></br>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="0-10 number line with 9 marked" height="60" src="NoLine3.gif" width="275"></mdo:image></div>
<br></br>
<br></br>
How long was her secret jump?<br></br>
<br></br>
Then Leah made a jump of six and another secret jump. She landed on $10$.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="0-10 number line with 10 marked" height="54" src="NoLine4.gif" width="275"></mdo:image></div>
<br></br>
<br></br>
How long was her second secret jump?<br></br>
<br></br>
Tom's number line goes up and down like this:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="0-10 vertical number line " height="313" src="NoLine2.gif" width="38"></mdo:image></div>
<br></br>
<br></br>
First Tom made a jump of three up his number line and then a jump of two. Where did he land?<br></br>
<br></br>
Next Tom made a secret jump up his number line. Then he made a jump of six and landed on $8$.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="0-10 vertical number line with 8 marked" height="273" src="NoLine5.gif" width="61"></mdo:image></div>
<br></br>
<br></br>
How long was his secret jump?<br></br>
<br></br>
Then Tom made a jump of four and another secret jump. He landed on $7$.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="0-10 vertical number line with 7 marked" height="273" src="NoLine6.gif" width="61"></mdo:image></div>
<br></br>
<br></br>
How long was his second secret jump?</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=5652&part=index">This activity</a> is a good one to try once children are familiar with number lines and they have begun to use them for addition. It could be a good way to talk about subtraction.</div>
<br></br>
<div>The problem requires higher-order thinking compared with simply adding or subtracting. It encourages children to be confident at "undoing" operations and therefore introduces them to the idea of subtraction being the inverse of addition and vice versa, although you don't need to use this vocabulary at first.</div>
<br></br>
<div>The idea of having an unknown number (the secret jump) in a calculation (further jumps along the number line) is the essence of algebra and this problem introduces that important concept but in an appropriate context.</div>
<br></br>
<h3>Possible approach</h3>
<div>Start with a number line with which the children are familiar and then introduce a vertical number line if they have not met them before.</div>
<br></br>
<div>Involve the children in suggesting numbers to add using both number lines before introducing the idea of the 'secret jump'. Next, it might be appropriate to try some of your own examples with the group, asking them how they are going to find the secret number of jumps. Encourage pupils to explain how they will do it in their own words and discuss whether or not this will work.</div>
<br></br>
<div>Give children a counter and a number line to work on, either individually or in pairs. <a href="/content/id/5652/5652.pdf">This sheet</a> provides two number lines from $0$ - $10$, one horizontal and one vertical. It is intended to be cut into two and could be enlarged from A4 to A3. The children could then deal with the questions given in the problem and make up some examples to try on each
other.</div>
<br></br>
<h3>Key questions</h3>
<div>Are you starting or finishing with the secret jump?</div>
<div>Can you show me the jump on your number line?</div>
<br></br>
<h3>Possible extension</h3>
Use a number line which goes to up to $20$ or $100$.<br></br>
<h3>Possible support</h3>
<div>Make sure that children are counting the jumps they make, and not the starting point. If there are difficulties with a vertical number line you could try <a href="http://nrich.maths.org/public/viewer.php?obj_id=5896&part=index">this problem</a> first.</div>
<br></br>
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It might help to have a number line of your own to work with.
<br></br>
<br></br>
You could print off <a href="/content/id/5652/5652.pdf">this
sheet</a> of number lines.<br></br>
<br></br>
If Leah did a jump of five to get to $9$, where did she start?
<br></br>
How could you find out?<br></br>
So how big was the jump she had to do to get there from zero?
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Leah: 7, 4, 4<br></br>
<br></br>
Tom: 5, 2, 3<br></br></mdoxml>
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Number Lines
Leah and Tom each have a number line. Can you work out where their
counters will land? What are the secret jumps they make with their
counters?
Algebra
Introducing algebra
Mathematics Tools
Number line/counting stick
Calculations and Numerical Methods
Addition & subtraction
Admin
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