7281
/www/nrich/html/content/id/7281/
1
2011-02-01T00:00:01
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Once there was a garden with lots of paths and little patches of
grass between. It looked a bit like this.<br></br>
<div style="text-align: center;"><mdo:image height="182" width="181" src="7281A.png" alt="4 x 4"></mdo:image></div>
<br></br>
Two children, William and Thea, made up a game.<br></br>
<br></br>
For every patch of grass they passed as they walked to the right on
the picture, they added one.<br></br>
For every patch of grass they passed as they walked to the left on
the picture, they took away one.<br></br>
If they went North (up on the picture), they added two for every
patch of grass, and if they went South (down on the picture), they
took away two for each patch of grass.<br></br>
<br></br>
They started where it says S, and ended where it says E on the
picture below.<br></br>
They began with ten points at S, and added and took away until they
reached E.<br></br>
<div style="text-align: center;"><mdo:image height="270" width="278" src="7281B.jpg" alt=""></mdo:image></div>
These are the paths they took:<br></br>
<div style="text-align: center;"><mdo:image height="270" width="278" src="7281C.jpg" alt="paths they took"></mdo:image></div>
<br></br>
Thea took the yellow path. Remember she started with $10$
points.<br></br>
How many points did she have when she arrived at E?<br></br>
<br></br>
<br></br>
William took the purple path.<br></br>
How many points did he have when he arrived at E?<br></br>
<br></br>
In the picture below, we've taken out the grass and just drawn a
grid. You can imagine that the squares are patches of grass and the
lines are the paths in the garden.<br></br>
<br></br>
We have marked different starting and end points.<br></br>
What would your score be if you walked along the blue path?
<br></br>
And the red path?<br></br>
<div style="text-align: center;"><mdo:image height="369" width="372" src="7281D.jpg" alt="6 x 6"></mdo:image></div>
<br></br>
Make up some more paths between these start and end points.<br></br>
What is your score each time?<br></br>
What do you notice?<br></br>
Can you explain why?
<div> </div>
<div>You can download a copy of the grid <a href="/content/id/7281/7281.pdf">here</a>.</div>
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<p class="editorial">Jack from Heacham Infant and Nursery
School looked at Thea's and William's routes and told us:</p>
The score each time was $16$.<br></br>
It did not matter which route you take, the score was the
same.<br></br>
<br></br>
<p class="editorial">Ellie from Short Wood Primary School looked at
the red and blue routes. She said:</p>
<br></br>
the red path takes the route of:<br></br>
minus $1$, plus $2$, plus $2$ , plus $1$, plus $1$, plus $1$, plus
$1$, plus $2$, minus $1$ which will give you a total score of $18$
points<br></br>
<br></br>
the blue path takes the route of:<br></br>
plus $2$, plus $1$, plus $2$, plus $2$, plus $1$ which will give
you the score of $18$ points<br></br>
<br></br>
So the solution is that if you take the longer red path you will
score exactly the same as the shorter blue path.<br></br>
<br></br>
<p class="editorial">David from Belmont Primary School began to
explain why these two routes gave the same total:</p>
<br></br>
The reason that it is impossible to get any other number than $18$
is because the shortest path from start to finish is $2$ right and
$3$ upwards, leaving you with $18$ ($10+1+1+2+2+2$). Therefore, any
other move to attempt getting less points by subtracting (moving
left/down) would be met by an equal addition of the same number
(moving right/upward). So ultimately any path (no matter how
complex) would lead to $18$. <br></br>
<br></br>
<p class="editorial">Someome from St Matthew's gave a good
explanation too:</p>
I conjecture that the line you draw can be of any length.<br></br>
The quickest way to get from start to finish is in $5$ moves and
this adds up to $18$ (starting with $10$).<br></br>
If you imagine that your line is a brick wall and you make an
extension in the middle of it, then as you move backwards and
forwards, the operations always cancel each other out. E.g. if you
move up $1$, you add $2$ and then if you move down $1$, you take
away $2$. Whatever shapes you make it doesn't matter because the
moves always cancel each other out. <br></br>
<br></br>
<p class="editorial">Well done!</p>
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<h2>The Add and Take-away Path</h2>
<br></br>
Once there was a garden with lots of paths and little patches of grass between. It looked a bit like this.<br></br>
<div style="text-align: center;"><mdo:image alt="4 x 4" height="182" src="7281A.png" width="181"></mdo:image></div>
<br></br>
Two children, William and Thea, made up a game.<br></br>
<br></br>
For every patch of grass they passed as they walked to the right on the picture, they added one.<br></br>
For every patch of grass they passed as they walked to the left on the picture, they took away one.<br></br>
If they went North (up on the picture), they added two for every patch of grass, and if they went South (down on the picture), they took away two for each patch of grass.<br></br>
<br></br>
They started where it says S, and ended where it says E on the picture below.<br></br>
They began with ten points at S, and added and took away until they reached E.<br></br>
<div style="text-align: center;"><mdo:image alt="" height="270" src="7281B.jpg" width="278"></mdo:image></div>
These are the paths they took:<br></br>
<div style="text-align: center;"><mdo:image alt="paths they took" height="270" src="7281C.jpg" width="278"></mdo:image></div>
<br></br>
Thea took the yellow path. Remember she started with $10$ points.<br></br>
How many points did she have when she arrived at E?<br></br>
<br></br>
<br></br>
William took the purple path.<br></br>
How many points did he have when he arrived at E?<br></br>
<br></br>
In the picture below, we've taken out the grass and just drawn a grid. You can imagine that the squares are patches of grass and the lines are the paths in the garden.<br></br>
<br></br>
We have marked different starting and end points.<br></br>
What would your score be if you walked along the blue path? <br></br>
And the red path?<br></br>
<div style="text-align: center;"><mdo:image alt="6 x 6" height="369" src="7281D.jpg" width="372"></mdo:image></div>
<br></br>
Make up some more paths between these start and end points.<br></br>
What is your score each time?<br></br>
What do you notice?<br></br>
Can you explain why?
<div> </div>
<div>You can download a copy of the grid <a href="/content/id/7281/7281.pdf">here</a>.</div>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/7281&part=">This problem</a> will give children a chance to generalise and make predictions. It also provides practice in simple addition and subtraction. It draws out the inverse relationship between these two operations as well as encouraging children to think about the order of operations. </div>
<div> </div>
<div>In relation to the March 2011 theme, 'actions on objects', we can think of 'going right' as an action performed on the starting number of points (and likewise going in the other directions). We can encourage children to find out what properties these actions have when combined one after the other.</div>
<div> </div>
<div>You will need copies of <a href="/content/id/7281/7281.pdf">this sheet</a>, which could be enlarged if more than two children are going to work using the copy.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could start by showing the pictures in the first part of the problem to the whole group, making sure that William and Thea's game is understood. A small scale version could be drawn out on the playground or on the hall floor so that the game can be played practically.</div>
<div> </div>
<div>After this introduction, the group could work in pairs so that they are able to discuss their ideas with a partner, using copies of the sheet. Encourage them to find interesting routes that use subtraction as well as addition. Routes can be drawn using different colours but pairs may well need more than one copy of the sheet.</div>
<div> </div>
<div>At the end of the lesson the group can come together again and discuss their findings. They could show their most interesting and longest routes. What did they notice about the total each time? Were they surprised by this result? Why do they think this happened? </div>
<br></br>
<h3>Key questions </h3>
<div>How will you remember which routes you have tried out?</div>
<div>How will you remember the total of each route?</div>
<div>Can you find a more interesting way to go?</div>
<div>Can you check your answers another way? </div>
<div>Do you notice anything about the totals?</div>
<br></br>
<h3>Possible extension</h3>
<div>Learners could try the Stage 2 version of this problem, <a href="http://nrich.maths.org/7285&part=">Journeys in Numberland</a>.</div>
<br></br>
<h3>Possible support</h3>
Learners might find it useful to have counters to help keep track of the score. Starting with $10$ counters, they could physically add and take away counters, then count them up when they arrive at the end point. <br></br></mdoxml>
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Try it out!<br></br>
You could start with $10$ counters and add more or take them away
as you move along your path. Then, when you reach the end of
the path, you can count the number of counters you have left
to find out your score. <br></br>
How will you make sure you remember the paths you've tried?<br></br>
How will you remember the score each time?<br></br></mdoxml>
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The add and take-away path
Two children made up a game as they walked along the garden paths.
Can you find out their scores? Can you find some paths of your own?
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Generalising
Calculations and Numerical Methods
Addition & subtraction
Calculations and Numerical Methods
Inverses
Admin
Lower primary mapping document