Beelines

737 /www/nrich/html/content/00/10/six6/ 1 2012-11-12T15:47:57 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Take a look at the video below: <video controls="controls" height="300" src="beelines.mp4" tabindex="0" width="400"> </video> If you can't see the video, click below for a description. <div class="toggle"> If I choose the point (5,5) and draw a line segment joining the point to the origin, my line passes through 5 grid squares. If I choose the point (4,3) and draw a line segment joining the point to the origin, my line passes through 6 grid squares. If I choose the point (6,4) and draw a line segment joining the point to the origin, my line passes through 8 grid squares.</div> If you know the coordinates of the point at the end of the line, are you able to predict how many squares the line will pass through? If I drew the line joining the origin to the point (50,37) how many grid squares will it pass through? How can you be sure without drawing it? If I drew the line joining the origin to the point (96,72) how many grid squares will it pass through? How can you be sure without drawing it? Can you work out how many grid squares a line passes through, if you are given the coordinates of the two endpoints? You could also investigate the number of grid lines crossed... <div class="framework">Notes and Background Working out which grid squares a straight line crosses allows you to create algorithms for drawing straight lines on a computer, where each pixel is a grid square. Read more about line drawing algorithms <a href="http://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm">here</a>.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> No one sent us a complete solution to this problem, but here is the start of a solution which shows some good mathematical thinking and a systematic way of working. We decided to investigate what happens if you keep the $x$ coordinate the same and change the $y$ coordinate. Here is our table to show the number of squares and gridlines crossed when the $x$ coordinate is $5$. <div style="text-align: center;"><mdo:image height="193" width="385" alt="" src="beelines.png"></mdo:image></div> We noticed a pattern for the number of squares crossed, which works for every $y$ coordinate we tried except $5$. If you add together the $x$ and $y$ coordinates and take away 1, you get the number of squares crossed. This is because for example to get from (0,0) to (5,3) you go along 5 squares and up 3 squares, meaning that you travel through 8 squares altogether, but that counts the corner square twice so you need to take away 1. It doesn't work for (5,5) because you go diagonally through the corners of the squares instead of cutting through the edges. Can anybody build on this thinking and explain the patterns found? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <div> Working on this problem will give students an opportunity to make and justify conjectures that link numerical and geometrical ideas. It offers students valuable experience of working on coordinate geometry and will develop their understanding of factors and multiples. </div> <h3>Possible approach</h3> Show the video below, or introduce the problem in the same way. You could use this <a href="https://nrich.maths.org/737?part=clue">Geogebra applet</a> or download this <a href="/content/00/10/six6/beelines.ggb">Geogebra file</a>. <video controls="controls" height="300" src="beelines.mp4" tabindex="0" width="400"> </video> "In a while I'm going to give you the coordinates of the point at the end of a line, and challenge you to tell me how many squares the line will pass through." Students could work in pairs, choosing coordinates and keeping a record of their results. Once students have a selection of results, invite them to look for patterns. If they find it hard to spot any relationships, here are some prompts: "Group together all your results where the x and y coordinate are the same. What do you notice? Will it always happen?" "What about results where the x coordinate is twice the y coordinate? Or three times?" "What about results when the two coordinates have no factors in common?" To finish off, choose some pairs of coordinates and challenge students to work out quickly how many squares the line will pass through. This is a very good activity for students to display their answers on mini-whiteboards. Finally, take some time to discuss the geometrical reasons why a straight line joining the origin to a point with coprime coordinates will not go through any grid squares, and the reason why we are able to calculate the number of squares the lines will go through without needing to draw and count. <h3>Possible extension</h3> This <a href="http://en.wikipedia.org/wiki/Bresenham%27s_line_algorithm">Wikipedia article</a> about line-drawing algorithms may be of interest to students who are keen on computer programming. You could challenge students to find a way to work out whether a given line passes through a given square on the grid (without drawing!). <h3>Possible support</h3> <a href="http://nrich.maths.org/2292">Coordinate Patterns</a> might provide a suitable preliminary activity. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Start by drawing some lines and recording how many squares they cross. What do you notice about the coordinates of lines that pass through the corners of grid squares? You may wish to experiment with the GeoGebra applet below. Be patient as it may take some time to load. Alternatively, you can install <a href="http://www.geogebra.org">GeoGebra</a> on your computer and download and run the <a href="/content/00/10/six6/beelines.ggb">GeoGebra file</a> yourself. <mdo:applet archive="http://jars.geogebra.org/webstart/4.0/geogebra.jar" code="geogebra.GeoGebraApplet" codebase="/content/00/10/six6/" datafile="" height="500" title="Java(TM)" width="550"><param name="filename" value="http://nrich.maths.org/content/00/10/six6/beelines.ggb"></param> <param name="framePossible" value="true"></param> <param name="showResetIcon" value="true"></param> <param name="enableRightClick" value="true"></param> <param name="showMenuBar" value="false"></param> <param name="showToolBar" value="true"></param> <param name="showToolBarHelp" value="true"></param> <param name="showAlgebraInput" value="false"></param></mdo:applet></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Original Problem This problem is about straight lines joining the origin to different points (with integer coordinates) in the first quadrant. Here are some examples: <div style="text-align: center;"><mdo:image alt="Grid" height="311" src="Grid1.png" width="529"></mdo:image></div> For some of these points one coordinate is a factor of the other, for some the coordinates have a common factor, but for others the coordinates are coprime (that is they have no common factor except 1). The lines cross the grid in different ways: <ul> <li>some lines form neat diagonals across the squares of the grid (for example the line from the origin to (5,5)),</li> <li>others create diagonals of rectangles (for example the line from the origin to (2,6) creates a diagonal which crosses two identical rectangles,</li> <li>some (for example the line from the origin to (11,1)) forms a single diagonal.</li> </ul> Can you find any relationships between the number of squares that lines cross and the coordinates of their end points and explain why they work? Can you find any relationships between the number of grid lines that lines cross and the coordinates of their end points and explain why they work? Can you describe any relationships between the coordinates of the end points of lines, the lengths of the lines and the lengths of the diagonals of any rectangles they cross? Here is the solution which arrived only just in time to be included! Dear Cambridge, We have found a solution to the October Six puzzle Beeline. First we drew up a table showing the number of squares crossed in certain examples <table> <tbody> <tr> <td>P</td> <td>Q</td> <td>squares crossed</td> </tr> <tr> <td>1</td> <td>2</td> <td>2</td> </tr> <tr> <td>1</td> <td>3</td> <td>3</td> </tr> <tr> <td>1</td> <td>4</td> <td>4 etc</td> </tr> <tr> <td>2</td> <td>3</td> <td>4</td> </tr> <tr> <td>2</td> <td>5</td> <td>6</td> </tr> <tr> <td>2</td> <td>7</td> <td>8</td> </tr> <tr> <td>3</td> <td>4</td> <td>6</td> </tr> <tr> <td>3</td> <td>5</td> <td>7</td> </tr> <tr> <td>3</td> <td>7</td> <td>9</td> </tr> </tbody> </table> From this we can see that the number of squares crossed is likely to be P+Q - 1 We can prove this by showing that when you cross from (0,0) to (3,4) you are going along 3 squares to the right and up 4 squares. However you cannot count the corner square twice therefore it is 3 squares (P) to the right plus 4 squares (Q) up take away one for the corner square, or P+Q -1. Prav and Sheli from the North London Collegiate School Maths Puzzle Club. </mdoxml> 2 3 0 0 0 1 0 Beelines Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses? Using, Applying and Reasoning about Mathematics Generalising Coordinates and Coordinate Geometry Coordinates Sequences, Functions and Graphs Linear functions Information and Communications Technology smartphone Secondary Mapping Document Coordinate geometry Secondary Mapping Document DisplayCabinet