5647
/www/nrich/html/content/id/5647/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<br></br>
We're going to look at opening safes!<br></br>
Many have dials on them, and you turn the dials - as you see in
these two pictures - to open them.<br></br>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="297" width="209" alt="safe2" src="PictSafe2.jpg"></mdo:image> <mdo:image height="139" width="144" alt="safe1" src="PictSafe1.jpg"></mdo:image></div>
<div style="text-align: center;"></div>
<div style="text-align: left;">So I'll show you a simple dial first
with just the numbers $0 - 5$ on it.</div>
<br></br>
<mdo:image height="284" width="263" alt="60" src="60.jpg"></mdo:image><br></br>
To open the safe, the dial has to have $2$ next to the arrow, like
this;<br></br>
<mdo:image height="268" width="259" alt="60t" src="60t.jpg"></mdo:image><br></br>
<br></br>
The question for you is, how much turning of the dial did we have
to do to get the $2$ at the top?<br></br>
The next safe we get to has a different dial, it has the numbers $0
-7$. <br></br>
How much turning to get the number $5$ at the top? [Shown in the
second picture.]<br></br>
<mdo:image height="280" width="259" alt="45" src="45.jpg"></mdo:image><mdo:image height="275" width="259" alt="45t" src="45t.jpg"></mdo:image> <br></br>
The number that you have to get at the top is often called the
"combination" of the safe. These three different safes all start
with $0$ [zero] at the top. You have to find the amount of turning
to get to the combination, shown in each second picture:<br></br>
<mdo:image height="285" width="504" src="Pic30%2Bt.jpg" alt="30+t"></mdo:image><br></br>
<br></br>
<mdo:image height="313" width="502" alt="40+t" src="Pic40%2Bt.jpg"></mdo:image> <br></br>
<br></br>
<mdo:image height="312" width="501" src="Pic15%2Bt.jpg" alt="15+t"></mdo:image><br></br>
Now have a go at these different safes. Remember they would all
start with $0$ [zero] at the top, so, how much turning to get the
required combinations shown below?<br></br>
<br></br>
<mdo:image height="425" width="606" alt="all" src="picAllTurn2jpg.jpg"></mdo:image><br></br>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">George and Dominic from St Nicolas C of E
Junior School sent us very clearly explained solutions to this
problem. Thank you! Here is what they wrote for the first part:</p>
<div>1. For the first question I worked it out by seeing that
number 3 was at $180^\circ$ and that meant I needed to divide 180
by 3, giving the angle in between each number. The answer was
$60^\circ$. By knowing this I could work out that number 2 was
$120^\circ$ away from zero anticlockwise and was $240^\circ$
clockwise from zero.</div>
<br></br>
<div>2. To work this out I did almost the same as I did with
question 1 except that I divided 180 by 4 instead. This gave me
$45^\circ$ between each number. By multiplying $45^\circ$ by 3
(which is how many numbers 5 is away from zero ) I got $135^\circ$.
This meant that I would have to either turn the dial $135^\circ$
clockwise or $225^\circ$ anticlockwise.</div>
<br></br>
<div>3. This was a little easier to work out because number 3 was a
quarter of the way round the dial meaning to get to it I either had
to turn it $90^\circ$ anticlockwise or $270^\circ$ clockwise.</div>
<br></br>
<div>4. This was harder than the previous questions because there
wasn't a number at the $180^\circ$ point. To work it out I divided
$360^\circ$ by 9. This gave me how many degrees I would have to
turn the dial to get to the next number. The answer was $40^\circ$.
This showed me that to get to number 3 I had to turn it either
$120^\circ$ anticlockwise or $240^\circ$ clockwise.</div>
<br></br>
<div>5. At first sight this looked really tricky but after thinking
about it wasn't. Number 12 is $180^\circ$ away from zero. Half of
12 is 6 and half of 180 is 90, meaning that to get to the number 6
I have to either turn the dial $90^\circ$ anticlockwise or
$270^\circ$ clockwise.</div>
<br></br>
<p class="editorial">George and Dominic continued to explain how
they had gone about the second part of the solution:</p>
<div>The next five were the hardest questions because we couldn't
see where the zero was at the start.</div>
<br></br>
<div>A. For this question I worked out that 12 was $180^\circ$ from
zero and 180 divided by twelve is 15. This means that each section
is $15^\circ$ wide. There are 9 spaces between zero and nine which
means that to work out the answer we have to times 9 by 15 which
equals 135. So to get to 9 I have to turn the dial either
$135^\circ$anticlockwise or $225^\circ$ clockwise.</div>
<br></br>
<div>B. We found this one quite easy because again 6 was a quarter
of the way round, so this meant that we had to turn the dial
$90^\circ$ clockwise or $270^\circ$ anticlockwise.</div>
<br></br>
<div>C. First I worked out that 360 divided by 6 is 60 meaning that
each section is $60^\circ$ around the dial. Using this information
I worked out that I would have to turn the dial either $120^\circ$
clockwise or $240^\circ$ anticlockwise.</div>
<br></br>
<div>D. On this one I knew that 6 was at $180^\circ$ but you had to
work out what 5 was. So I did 180 divided by 6 which equals 30.
Then I did 180-30 which equals $150^\circ$ so it was $150^\circ$
anticlockwise and $210^\circ$ clockwise.</div>
<br></br>
<div>E. Firstly for this I divided 360 by 9 which gave me 40. I
then multiplied 40 by 5 which gave me 200. Knowing this I could
work out that to get to 5 I either have to turn the dial
$200^\circ$ anticlockwise or $160^\circ$ clockwise.</div>
<br></br>
<div>We ended the lesson by writing down 3 instructions for someone
else to follow which would give our combination. Sophie, another
girl in our group, wrote down: Using dial D, turn the dial
$240^\circ$ anticlockwise. Then from that point, turn the dial
$210^\circ$ clockwise. Then from this point, turn the dial
$90^\circ$ anticlockwise. What's my combination?</div>
<br></br>
<div>We worked out that it was 8 1 4.</div>
<br></br>
<p><span class="editorial">Very well done to you both. You
obviously worked hard on this activity. Thank you too, to Eve and
Rachel from Castle Carrock Primary who also sent in well-explained
solutions.</span></p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>How Safe Are You?</h2>
<br></br>
<br></br>
We're going to look at opening safes!<br></br>
Many have dials on them, and you turn the dials - as you see in these two pictures - to open them.<br></br>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="safe2" height="297" src="PictSafe2.jpg" width="209"></mdo:image> <mdo:image alt="safe1" height="139" src="PictSafe1.jpg" width="144"></mdo:image></div>
<div style="text-align: center;"> </div>
<div style="text-align: left;">So I'll show you a simple dial first with just the numbers $0 - 5$ on it.</div>
<br></br>
<mdo:image alt="60" height="284" src="60.jpg" width="263"></mdo:image><br></br>
To open the safe, the dial has to have $2$ next to the arrow, like this;<br></br>
<mdo:image alt="60t" height="268" src="60t.jpg" width="259"></mdo:image><br></br>
<br></br>
The question for you is, how much turning of the dial did we have to do to get the $2$ at the top?<br></br>
The next safe we get to has a different dial, it has the numbers $0 -7$.<br></br>
How much turning to get the number $5$ at the top? [Shown in the second picture.]<br></br>
<mdo:image alt="45" height="280" src="45.jpg" width="259"></mdo:image><mdo:image alt="45t" height="275" src="45t.jpg" width="259"></mdo:image><br></br>
The number that you have to get at the top is often called the "combination" of the safe. These three different safes all start with $0$ [zero] at the top. You have to find the amount of turning to get to the combination, shown in each second picture:<br></br>
<mdo:image alt="30+t" height="285" src="Pic30%2Bt.jpg" width="504"></mdo:image><br></br>
<br></br>
<mdo:image alt="40+t" height="313" src="Pic40%2Bt.jpg" width="502"></mdo:image><br></br>
<br></br>
<mdo:image alt="15+t" height="312" src="Pic15%2Bt.jpg" width="501"></mdo:image><br></br>
Now have a go at these different safes. Remember they would all start with $0$ [zero] at the top, so, how much turning to get the required combinations shown below?<br></br>
<br></br>
<mdo:image alt="all" height="425" src="picAllTurn2jpg.jpg" width="606"></mdo:image><br></br>
<br></br>
<br></br>
<br></br>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=5647&part=">activity</a> is a rather different way of giving pupils some experience of turning and measuring angles in degrees. It may particularly appeal to those pupils who like visualising something real.</div>
<br></br>
<h3>Possible approach</h3>
<div>You might want to make a version of one of the dials out of two pieces of card, fixed in the middle with a paper fastener. In this way, you could ask the class to visualise the turning and offer their solutions with explanations, before checking their thoughts using the card model.</div>
<br></br>
<div>How this is approached and pupils' thoughts will vary a lot according to their age and experience. It might be alright for the youngest learners to give an answer that is an anticlockwise direction but they perceive it as clockwise, just because it's the numbers that are rotating. For older pupils it would be a good discussion point for them to consider in which direction the turning occurs.
Are their answers for anticlockwise or clockwise turning?</div>
<br></br>
<div>You could print out <a href="/content/id/5647/HowSafeAreYouImages.pdf">this sheet</a> of the dials for children to work on in pairs.</div>
<br></br>
<h3>Key questions</h3>
<div>Which way are you rotating/turning your hand?</div>
<div>Is it more than half a full turn each time? Can you be more exact?</div>
<div>How many degrees there are in a circle?</div>
<br></br>
<h3>Possible extension</h3>
<div style="text-align: left;">Pupils with some knowledge of the $360^\circ$ complete turn will probably be able to have a go at the later questions. More advanced pupils would be able to create their own problems for other pupils to answer. They might like to use <a href="/content/id/5647/HowSafeAreYou.doc">these images</a> as printable dials for those harder questions they may want to
set.</div>
<div style="text-align: center;"> </div>
<div style="text-align: center;"> </div>
<br></br>
<h3>Possible support</h3>
<div>A circular protractor would be useful for some pupils. Some preliminary discussion could be had about the turning involved in an analogue clock.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Does it matter in which direction we turn the dial?<br></br>
Is it more than half a full turn each time? Can you be
more exact?<br></br>
Perhaps you know how many degrees there are in a circle?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
0 to 2 on first dial is; <br></br>
<div style="margin-left: 40px;">1/3 of a turn anticlockwise or
120°</div>
<div style="margin-left: 40px;">2/3 of a turn clockwise or
240°</div>
<div style="margin-left: 40px;"></div>
<div>0 to 5 on the second dial is;</div>
<div style="margin-left: 40px;">5/8 of a turn anticlockwise or
225°</div>
<div style="margin-left: 40px;">3/8 of a turn clockwise or
135°</div>
<br></br>
<div>The next 3 are;</div>
<br></br>
<div>1/4 turn, a right angle, 90° anticlockwise OR 3/4
turn, 3 right angles, 270° clockwise</div>
<br></br>
<div>1/3 turn, 120° anticlockwise OR 2/3 turn,
240° clockwise</div>
<br></br>
<div>1/4 turn, a right angle, 90° anticlockwise OR 3/4
turn, 3 right angles, 270° clockwise</div>
<br></br>
<br></br>
<div>Last 5;</div>
<br></br>
<div>A/ 9/24 [or equivalent] , 135° anticlockwise OR
15/24 [or equivalent] , 215° clockwise</div>
<br></br>
<div>B/ 3/4, 270° anticlockwise OR 1/4, 90°
clockwise</div>
<br></br>
<div>C/ 2/3, 240° anticlockwise OR 1/3, 120°
clockwise</div>
<br></br>
<div>D/ 5/12, 150° anticlockwise OR 7/12,
210° clockwise</div>
<br></br>
<div>E/ 5/9, 200° anticlockwise OR 4/9, 160°
clockwise.</div>
<br></br>
<br></br>
<br></br></mdoxml>
2
3
0
1
0
0
0
How Safe Are You?
How much do you have to turn these dials by in order to unlock the
safes?
2D Geometry, Shape and Space
Angles
Admin
Upper primary mapping document