6940
/www/nrich/html/content/id/6940/
8
2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
<p>Before starting to explore Steve's mapping, you might want to watch a short video explaining how to use the NRICH Number Plumber.<br></br>
<mdo:flash height="390" width="670"><param name="allowfullscreen" value="true" ></param><param name="movie" value="/content/id/6940/VideoPlayer.swf" ></param><param name="allowFullScreen" value="true" ></param><param name="allowscriptaccess" value="always" ></param><param name="flashVars" value="flv=Stage3Final.flv" ></param></mdo:flash>Steve has created two mappings which you can access by clicking on the picture below. The initial challenge is to figure out what Steve's mappings do. You can drop some numbers into the mappings, and see what comes out.<br></br>
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Test your ideas by predicting the outputs for some different inputs.<br></br>
Only reveal the hidden parts of the mapping once you are certain you know what is going on.<br></br>
Can you find an input which gives the same output for both mappings?<br></br>
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<span style="font-weight: bold;">Click on the picture to explore Steve's mappings.</span><br></br>
<a href="http://nrich.maths.org/DataFlow/DataFlow.html?config=/content/id/6940/SteveNumberPlumber.xml" style="font-weight: bold;" target="_blank"><mdo:image alt="" height="100" src="icon.png" width="100"></mdo:image></a><br></br>
As you explore Steve's mappings, you will notice points appearing on the graph. The input number is the x coordinate, and the output number is the y coordinate.<br></br>
Create some other function machines using the <a href="/DataFlow/DataFlow.html?config=/content/id/6929/Stage4NumberPlumber.xml">NRICH Number Plumber</a>, and observe the outcomes on the graph.<br></br>
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Mathematicians like to ask themselves questions about what they notice.<br></br>
<span style="font-weight: bold;">What possible questions could you ask?</span><br></br>
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These questions may lead you to make conjectures - something which you believe to be true but need to investigate further in order to convince yourself.</p></mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why work on this question?</h3>
<div>This month's NRICH site has been inspired by the way teachers at Kingsfield School in Bristol work with their students. Following an introduction to a potentially rich starting point, a considerable proportion of the lesson time at Kingsfield is dedicated to working on questions, ideas and conjectures generated by students.</div>
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<div>Working on this question will encourage students to work together, discuss ideas, develop conjectures, suggest new lines of enquiry and solve problems.</div>
<h3><br></br>
Here are the sort of questions that might emerge:</h3>
<div style="margin-left: 40px;">What affects the direction and steepness of a graph?</div>
<div style="margin-left: 40px;">Can I tell from a function where its graph will cross the axes?</div>
<div style="margin-left: 40px;">Which functions give straight lines, and which give curves?</div>
<div style="margin-left: 40px;">When will two quadratic functions intersect?</div>
<div style="margin-left: 40px;">Can I tell from a quadratic function where its graph has a turning point?</div>
<div> </div>
<div>The interactivity could be used as a starting point to encourage students to make conjectures about functions and graphs, in a similar way to how students at Kingsfield School are introduced to the topic.</div>
<div><br></br>
Read the article <a href="http://nrich.maths.org/6808&part=">Kingsfield School - Building on Rich Starting Points</a>, which has links to a description of a Kingsfield teacher's first lesson on functions and graphs, and a video showing how these ideas are put into practice in the classroom.</div>
<div> </div>
<div>For teachers who want to create their own alternatives to Steve's Mapping for use in the classroom, here is an introductory video explaining how to build, load and save your own examples.</div>
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<p><mdo:flash height="390" width="670"><param name="allowfullscreen" value="true" ></param><param name="movie" value="/content/id/6940/VideoPlayer.swf" ></param><param name="allowFullScreen" value="true" ></param><param name="allowscriptaccess" value="always" ></param><param name="flashVars" value="flv=TeacherIntro3.flv" ></param></mdo:flash><br></br>
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To read more about the Number Plumber, visit <a href="http://grumplet.wordpress.com/2010/02/23/the-nrich-number-plumber/" target="_blank">Grumplet's blog</a> where you can comment on how you have used the number plumber and share links to files you have created. We are continuing to develop this resource so your feedback and ideas will be very useful.</p>
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<h3>Key questions</h3>
<div>What do you think these function machines do?</div>
<div>What will happen if we input 5? 13? 100? 0.7? ...</div>
<p>Is it possible to get the same output from both machines using the same input number? Is there more than one way?</p>
<div>What other questions could we ask?</div>
<div>Can you make any predictions about what might happen when we change the function machines?</div>
<div>What's the same? What is different?</div>
<div>Can you explain?</div>
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<h3>Possible extension</h3>
<p>Students can create and investigate their own Rational Functions using the <a href="/DataFlow/DataFlow.html?config=/content/id/6929/Stage4NumberPlumber.xml">NRICH Number Plumber</a></p>
<h3>Possible support</h3>
<p><a href="http://nrich.maths.org/6929">Alison's Mapping</a> provides a starting point using quadratic functions.<br></br>
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</p></mdoxml>
<mdoxml version="1.0">The two functions are<br></br>
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$$y=\frac{2x+9}{x+2}\quad\quad x=\frac{9-2y}{y-2}$$<br></br>
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Note that these have been chosen to be inverse function and also to
have whole number fixed points and dual values on the given
ranges<br></br>
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So, points of interest are $(3, 3)$ and $(-7, 1)$.<br></br>
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The symmetry of the graphs should emerge from plotting, although we
have chosen to restrict to integer values for the initial detective
work part of the problem<br></br>
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Good luck and happy hunting<br></br></mdoxml>
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Steve's Mapping
Steve has created two mappings. Can you figure out what they do?
What questions do they prompt you to ask?
Sequences, Functions and Graphs
Graphs
Sequences, Functions and Graphs
Rational functions
Using, Applying and Reasoning about Mathematics
Making and proving conjectures