5469
/www/nrich/html/content/id/5469/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<div style="text-align: center;"><mdo:image alt="Diagonal pattern" height="406" src="diagonals.gif" width="403"></mdo:image></div>
<br></br>
<br></br>
<p>Take some time to look at the route followed by the arrows in this diagram.<br></br>
Then look away and describe the path to a friend.<br></br>
<br></br>
If the pattern of arrows continues for ever, which point will the route visit immediately after (18,17)? Explain how you know.<br></br>
<br></br>
How many points will be visited before the route reaches the point (9,4)?<br></br>
Explain how you worked it out.<br></br>
<br></br>
<br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6171&part=">Click here for a poster of this problem</a>.</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p><span class="editorial">After (18, 17), the route will visit (19, 16). Lots of people got this right - let's have a look at just a few of their solutions:</span></p>
<p><span class="editorial">Tiberiu noticed:</span></p>
<p>I see that the route goes upwards through the points with coordinates that are equal (eg. 18,18) and downwards through the points with coordinates that are adjacent (eg. 18,17). Therefore the next coordinate is (19,16).</p>
<p><span class="editorial">Lawrence, from Chircher's College, worked it out like this:</span></p>
<p>Looking at the diagram you can tell that if the x coordinate of the bottom right point on a line is even, the arrows in the line go down and right, otherwise, they go up and left. The point (18,17) is on the line with bottom right point (18+17-1,1) = (34,1). Because 34 is even, the arrows go down and right, so the next point is (19,16).</p>
<p><span class="editorial">Alex, from Dixons City Academy, spotted the same thing, in a different way:</span></p>
<p>The numbers all lie on lines with equations x+y=n, where n is some integer. If n is odd, then the line begins at (n-1, 1). These arrows point towards the bottom right. If n is even, the line begins at coordinate (1, n-1) and the direction of the line is towards the top left. Notice that 18+17=35, so the coordinate (18, 17) is on the line x+y=35.</p>
<p><span class="editorial">Segev, Adam, Yoav, Daniel and Jonathan from JFS answered the second part of our question:</span></p>
<p>There are 74 points visited before the route reaches the point (9,4). The number of points visited follows the triangle numbers.<br></br>
Number of points visited up to (1,1): 1<br></br>
Number of points visited up to (2,1): 3<br></br>
Number of points visited up to (3,1): 6<br></br>
Number of points visited up to (4,1): 10<br></br>
etc.</p>
<p><span class="editorial">There were far too many great solutions to list here - thanks to everyone who submitted one!</span></p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
<p><a href="http://nrich.maths.org/public/viewer.php?obj_id=5469&part=">This problem</a> offers a good opportunity for students to discuss images and find convincing arguments for their solutions.</p>
<p>Reuben Hersh has written that:</p>
<p>"In the classroom, convincing is no problem. Students are too easily convinced. Two special cases will do it."</p>
<p>This problem offers an opportunity to ensure that students are justified in generalising from the particular cases that they have selected.</p>
<p> </p>
<h3>Possible approach</h3>
<div>Show the diagram. It's available as a <a href="/content/id/5469/Route%20to%20Infinity.ppt">PowerPoint</a>, or you could print out the <a href="/6171">poster</a>.</div>
<div> </div>
<div>"Have a look at this image. In a moment I'm going to remove it, and I want you to be able to describe the route that the arrows take to your partner."</div>
<div>Give students a short while to look at the image, then remove it.</div>
<div> </div>
<div>"Without using paper or pencil, can you describe the route to each other?"</div>
<div>Once they have done this, show them the image again to check that what they have described is indeed what they saw.</div>
<div> </div>
<div>"I'd like one person in each pair to turn their back to the screen and list the coordinates in the order in which they're visited, and your partner to look at the screen and check. When you make a mistake, swap over. See how far you can get."</div>
<div> </div>
<div>Once students have spent some time listing the coordinates, bring the class together.</div>
<div>"I wonder if you can work out where the route will take you after visiting the point (18,17)? Spend a short while thinking about it on your own, then discuss it with your partner, and together develop a convincing explanation for your answer to share with the class."</div>
<div> </div>
<div>As students are working, if they get stuck you could offer the following hint:</div>
<div>"What do you notice about the coordinates of the points visited when the arrows are sloping upwards/downwards?"</div>
<div> </div>
<div>Students' explanations are likely to refer to specific examples on the visible grid. It is important to insist on clearly justified arguments that refer to the generality - a key question to ask is "How do you know it will <strong>always</strong> happen?".</div>
<div> </div>
<div>Finally, introduce the last question: "I wonder if you can work out how many points the route will pass through before reaching (9,4)? Again, you may want to start by working on your own before discussing it with your partner, and then developing a convincing explanation to share with the class."</div>
<div> </div>
<div>While pairs are talking, circulate and eavesdrop on discussions, correcting any misconceptions and making a mental note of any students with clear explanations.</div>
<div>Bring the class together and invite those students with interesting or elegant strategies to present their ideas to the rest of the class.</div>
<div> </div>
<h3>Possible extension</h3>
<div> </div>
<div>Challenge students to design a route that will cover every grid point on an infinite coordinate grid in all 4 quadrants, and to create some questions (and answers), like those above, to go with their design. They could then swap with a partner.</div>
<div> </div>
<div>The thinking involved in this problem could lead onto some investigation into countable infinity. <a href="/2756">This article</a> by Katherine Korner would make a good starting point.</div>
<br></br>
<h3>Possible support</h3>
<p>Before working on this problem, spend time on <a href="/6288">Cops and Robbers</a> and <a href="/2292">Coordinate Patterns</a> to develop students' fluency with coordinates.</p>
<p><br></br>
</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Take a look at the co-ordinates of the points along the route.
<br></br>
<br></br>
What do you notice about the order in which the route visits the
points?<br></br>
<br></br>
When do the arrows point up, and when do they point down?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Yes, the route will visit (18,17) - it goes through all points
(x,y) where x and y are positive integers. At some stage, the route
will pass along the diagonal corresponding to the sum of the
co-ordinates being x+y, and it hits all points on that diagonal
that have both co-ordinates positive.<br></br>
<br></br>
At (18,17) the route will be going down and to the right, because
this is what it does on diagonals where the sum of the co-ordinates
is odd. This means that the next point visited will be
(19,16).<br></br>
<br></br>
The route visits 74 points before reaching (9,4). <br></br>
It passes through 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 =
(1/2) x 11 x 12 = 66 points that lie on diagonals completed by the
route.<br></br>
The diagonal passing through (9,4) will be going from top left to
bottom right (as 9 + 4 = 13 is odd). Before it reaches (9,4), the
route will have passed through the points with sum of co-ordinates
equal to 13 where the x co-ordinate is one of 1, 2 3, 4, 5, 6, 7
and 8, so there are 8 such points.<br></br>
So the route will have visited 66 + 8 = 74 points before it reaches
(9,4).<br></br></mdoxml>
2
4
0
0
1
1
0
Route to infinity
Can you describe this route to infinity? Where will the arrows take you next?
Coordinates and Coordinate Geometry
Coordinates - first quadrant
Numbers and the Number System
Triangle numbers
Admin
Workshop
Information and Communications Technology
smartphone
Using, Applying and Reasoning about Mathematics
Visualising
Using, Applying and Reasoning about Mathematics
Generalising
Secondary Mapping Document
Coordinate geometry