7471
/www/nrich/html/content/id/7471/
1
2011-06-01T00:00:00
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<p>In this problem it is not the squares that jump, you do the jumping!<br></br>
It is not a race, but a game of skill. You have to be able to look ahead.<br></br>
The idea is to go round the track in as few jumps as possible, keeping to the rules.</p>
<p>The first line of the track looks like this:</p>
<div class="imagedisplay" style="width:100%;overflow:auto;"><mdo:image alt="bit of track" src="JSqs.png" style="height: 39px; width: 300px;"></mdo:image></div>
<p>You start on the green square which tells you that you can jump forward either one or three squares. Here is the whole track:</p>
<div style="text-align: center;"><mdo:image alt="whole track" src="JSqs2.png" style="width: 300px; height: 371px;"></mdo:image></div>
<p>You make your way round the track and finish on the red square with 'end' on it.</p>
<p>If you land on a square which has 2 and 3 on it, you can jump forward - or back - either 2 or 3 squares.</p>
<p>If the square has 2 and 0 on it, you can jump forward - or back - only 2 squares.</p>
<p>If the square has 0 and 0 on it, you cannot jump at all. You have to go right back to the beginning and start again!</p>
<p>You can download a larger version <a href="/content/id/7471/JumpingSqs.pdf">here</a>.</p>
<p>You can do this on your own or with a friend.</p>
<p>You can count your jumps by making a note on paper whenever you jump or by counting out twenty counters and taking one from the pile every move.</p>
<p>What is the least number of jumps you can make to get round the whole track?</p>
<p>Which squares do you need to land on?</p>
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<mdoxml version="1.0"><span class="editorial">Well done to everyone who had a go at this.
Here are some of your solutions.</span> <br></br>
<br></br>
<span class="editorial">Kimberley managed to get round in 12
moves:</span> <br></br>
<br></br>
(All moves are forward) You start on 1\3 so if you move 1 that
takes you to 4\1 (1 move). Then, move 4 places which takes you to
0\1 (2). After that, move 1 space to another 0\1 (3 moves). Then
move the following numbers, 1 (4 moves), 3 (5 moves), 2 (6 moves),
2 (7 moves), 4 (8 moves), 4 (9 moves), 2 (10 moves), 1 (11 moves),
then finally move the final 4 spaces (12 moves) and you come to the
end. <br></br>
<br></br>
<span class="editorial">Nicholas from Purleigh did the same route
but made a mistake on his first go so he counted on before deciding
which number to move to on his second go.</span> <br></br>
<br></br>
<span class="editorial">Jonah, Class 9a (Angmering), Erin (Wingate
Primary) and Indi (Broomfields Junior School) all did it in 11
moves.</span> <br></br>
<br></br>
<span class="editorial">Here, Abby and Claire (Springfield Primary
School) explain how they did it:</span><br></br>
<br></br>
First we went forward 3 and that led us to a square that said 1/3.
Then we went forward 3 again which took us to 1/0. So we went
forward 1 and that led us to 4/3. Then we just went forward 3 not 4
which took us to 3/2. Then we went forward 2 which led us to 2/1.
So then we went forward 2 so that we didnt land on 0/0. we ended up
on 4/0. Now we go forward 4 that led us to 1/4. Then we went
forward 4 which led us to2/2. So we went forward 1 which led us to
to 4/1. Then we went forward 4 which led us to the end!!! So in the
end we went round the track in 11 moves. <br></br>
<br></br>
<span class="editorial">James, Emerald and Chris from Myland
Primary School told us about their methods:</span> <br></br>
<br></br>
We tried only going forwards. These are some of the routes that get
from the start to the end. <br></br>
Start 3, 3, 1, 3, 2, 2, 4, 4, 2, 1, 4, end <br></br>
Start 3, 3, 1, 3, 2, 2, 4, 4, 2, 1, 1, 3 end <br></br>
Start 1, 4, 1, 1, 3, 2, 2, 4, 4, 2, 1, 4, end <br></br>
Start 1, 4, 1, 1, 3, 2, 2, 4, 4, 2, 1, 1, 3 end <br></br>
Start 1, 1, 2, 2, 1, 3, 2, 2, 4, 4, 2, 1, 4, end <br></br>
Start 1, 1, 2, 2, 1, 3, 2, 2, 4, 4, 2, 1, 1, 3 end <br></br>
We tried all the starting combinations with 1 and 3. We found that
no matter what numbers you chose you always land on the square 4/3.
Then if you choose 4 then you get stuck at 0/0, so you have to move
3 onto 3/2. If you choose 3 you land on 0/0, so you have to move 2.
The middle part of the route is always the same 3, 2, 2, 4, 4, 2,
1. The quickest way to get to 4/3 is 3, 3, 1 or 3, 1, 3. The
quickest way to end is to choose 4 on square 4/1 near the end. We
had different strategies and tried out different ideas.<br></br>
<br></br>
James: "With only forwards moves it is impossible to land on the
squares 3/1, 2/1, 2/3, 2/4, 1/0, 3/3, 3/0, 0/1."<br></br>
Chris: "I tried getting back to the start from the 2/4 square near
the end. It wasn't possible, I always ended up on the 0/0s. 2/3,
2/2, 2/1 and 3/1 to the left of the end square were dead ends and
there is no way of jumping over them." <br></br>
Emerald: "I didn't like to land on a square with a zero, because
then I didn't have a choice." "Going backwards you will always end
up on 0/0, no matter which of the inside squares you start on."
<br></br>
<br></br>
<span class="editorial">Indigo class from Unicorn tried something a
bit different: </span> <br></br>
<br></br>
Jo, Ollie, Rudi, Freya, Oliver and Joseph tackled the problem. They
worked in pairs moving a counter around the board. To begin with
they just said how many moves they could do it in but hadn't
recorded it. They then came up with the method of moving the
counter and recording the jumps on a piece of paper. The least
number of jumps they all did it in was 11 with no jumps backwards.
<br></br>
<br></br>
+3 +3 +1 +3 +2 +2 +4 +4 +2 +1 +4 (everyone got this solution)
<br></br>
<br></br>
Joseph and Oliver investigated starting with +1 followed by +4 but
this resulted in 12 jumps. The last 8 jumps are the same as the
first sequence of jumps but they have 4 moves before this rather
than 3. <br></br>
<br></br>
+1 +4 +1 +1 +3 +2 +2 +4 +4 +2 +1 +4 <br></br>
<br></br>
Jo was quick to spot that you didn't want to land on the box with
2/0 in it in the bottom row as going forward 2 put you on a 0/0
box. Going back 2 put you on 2/4 box so you then had to go back 4
and so on therefore adding in lots of extra jumps! <br></br>
They discussed the possibility of doing it in less than 11 moves
but realised that in each case bar one they had moved the maximum
number of spaces. They had to choose 3 moves instead of 4 moves at
the bottom of the first column to avoid the situation described by
Jo above. <br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Jumping Squares</h2>
<p>In this problem it is not the squares that jump, you do the jumping!<br></br>
It is not a race, but a game of skill. You have to be able to look ahead.<br></br>
The idea is to go round the track in as few jumps as possible, keeping to the rules.</p>
<p>The first line of the track looks like this:</p>
<div class="imagedisplay" style="width: 100%; overflow: auto;"><mdo:image alt="bit of track" src="JSqs.png" style="width: 300px; height: 39px;"></mdo:image></div>
<p>You start on the green square which tells you that you can jump forward either one or three squares. Here is the whole track:</p>
<div style="text-align: center;"><mdo:image alt="whole track" src="JSqs2.png" style="width: 300px; height: 371px;"></mdo:image></div>
<p>You make your way round the track and finish on the red square with 'end' on it.</p>
<p>If you land on a square which has 2 and 3 on it, you can jump forward - or back - either 2 or 3 squares.</p>
<p>If the square has 2 and 0 on it, you can jump forward - or back - only 2 squares.</p>
<p>If the square has 0 and 0 on it, you cannot jump at all. You have to go right back to the beginning and start again!</p>
<p>You can download a larger version <a href="/content/id/7471/JumpingSqs.pdf">here</a>.</p>
<p>You can do this on your own or with a friend.</p>
<p>You can count your jumps by making a note on paper whenever you jump or by counting out twenty counters and taking one from the pile every move.</p>
<p>What is the least number of jumps you can make to get round the whole track?</p>
<p>Which squares do you need to land on?</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/7471&part=">This problem</a> will encourage children to work systematically, and think and plan ahead. The activity can be done by one child working alone, but might be better if two work together.<br></br>
<h3>Possible approach</h3>
<div>You could start by looking at the first line of the track with the whole group. Explain the rules of the challenge and invite children to talk to a partner about what their first move might be. Ask for some suggestions and encourage good explanations of their choices. This initial discussion will allow you to reinforce the rules and make sure that learners do not count the square
they are on when jumping forward (or back).</div>
<div> </div>
<div>Ask the group for suggestions as to how they will keep track of the number of moves they've made. Allow them to choose a way that suits them and have available all sorts of equipment that might help, for example whiteboards, paper, counters, number lines, number squares, digit cards ...</div>
<div> </div>
<div>Once learners have got the idea of the task, suggest that they work either on their own or in pairs to try and find the smallest number of jumps that are needed to get to the centre. <a href="/content/id/7471/JumpingSqs.pdf">This sheet</a> gives the full track which can be printed and copied (perhaps even laminated).</div>
<div> </div>
<div>After some time, you may wish to draw the group together for a brief discussion about progress so far. It might be helpful to ask some pairs to share what they've done, for example you may notice that they have recorded which squares they have jumped on in a good way. Keeping a record not just of the number of jumps but where the jumps are to and from might help children tweak and
improve their total number of jumps.</div>
<div> </div>
<div>At the end of the lesson bring the group together again. What is the least number of jumps that were made to get round the whole track? Is that the very best way to jump round or does the group think that there might be a better way still to find?</div>
<br></br>
<h3>Key questions</h3>
<div>Where could you land next?</div>
<div>Which move might be better? Why?</div>
<div>Have you thought of jumping backwards?</div>
<div>How are you keeping count of your moves?</div>
<div>What is the least number of jumps that you made to get round the whole track?</div>
<div>Do you think that is the very best way to jump round? How do you know?</div>
<br></br>
<h3>Possible extension</h3>
Learners could make their own 'jumping squares' track for others to try or perhaps they could introduce a different rule using the same track.<br></br>
<h3>Possible support</h3>
It might be that an adult could keep count of the number of moves made so this takes out one level of detail for the children to attend to.<br></br>
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<mdoxml version="1.0">Where could you land next? <br></br>
Which move might be better? Why? <br></br>
Have you thought of jumping backwards? <br></br>
How are you keeping count of your moves? <br></br></mdoxml>
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Jumping squares
In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.
Numbers and the Number System
Counting
Using, Applying and Reasoning about Mathematics
Trial and improvement
Using, Applying and Reasoning about Mathematics
Working systematically
Admin
Lower primary mapping document