6878
/www/nrich/html/content/id/6878/
1
2011-02-01T00:00:01
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<mdoxml version="1.0"><div style="text-align: center;">Here's something a little different ...</div>
<div style="text-align: center;">It is an opportunity to work with many aspects of mathematics.</div>
<br></br>
It would be good to work with other people for this challenge.<br></br>
You'll need to split up into five groups.<br></br>
Each group will need to have copies of one of the pictures below. (<a href="/content/id/6878/FingersHands.pdf">This sheet</a> contains one of each picture.)<br></br>
<br></br>
In our group, find out how many fingers (including thumbs) there are in the picture you have been given.<br></br>
Talk to others in your group about how you came to that total and compare your way with other people's ways. Make sure you agree on the number as a whole group.<br></br>
You may like to record what you have done in some way.<br></br>
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<div style="text-align: center;"><mdo:image alt="6H" height="282" src="6HandsiStock_000009944326XSmall.jpg" width="425"></mdo:image></div>
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<mdo:image alt="8H" height="185" src="8HandsiStock_000003964660XSmall.jpg" width="650"></mdo:image><br></br>
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<div style="text-align: center;"><mdo:image alt="10H" height="282" src="10HandsiStock_000008679078XSmall.jpg" width="425"></mdo:image></div>
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<div style="text-align: center;"><mdo:image alt="12H" height="282" src="HANDSiStock_000002904280XSmall.jpg" width="426"></mdo:image></div>
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<div style="text-align: center;"><mdo:image alt="14H" height="283" src="14HandsiStock_000003535401XSmall.jpg" width="424"></mdo:image></div>
<br></br>
Swap pictures with another group.<br></br>
How many fingers, including thumbs, this time? Did you change the way you worked it out?<br></br>
Once again, talk to others in your group about how you each came to that total.<br></br>
<br></br>
You could swap your pictures again and keep going until you have had a chance to see all five pictures.<br></br>
<br></br>
As a class share the different ways that you counted.<br></br>
Are there some ways that were more popular for particular pictures than others? Why was this, do you think?<br></br>
<br></br>
<br></br>
There is another activity on NRICH, <a href="http://nrich.maths.org/5722&part=" linkindex="8">Alien Counting,</a> which starts like this:-<br></br>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="" height="129" src="aliens.gif" width="525"></mdo:image></div>
<br></br>
Well here are five Aliens, all from different planets.<br></br>
They all count using one "hand" except the last one who uses his six tentacles.<br></br>
So let me introduce them:<br></br>
<br></br>
1st - Serious "Dubo" who counts in twos;<br></br>
2nd - The two headed "Threebee" who counts in threes;<br></br>
3rd - The laughing "Quort", who counts in fours;<br></br>
4th - The green-headed "Chindi" who (you guessed it) counts in fives;<br></br>
Finally - The four-eyed "Senda" counting in sixes.<br></br>
<br></br>
What if Dubo, Threebee, Quort, Chindi or Senda had been there in the hand pictures instead of humans?<br></br>
Which Aliens could be in which pictures if you wanted the total to be the same as with humans?<br></br>
What numbers of fingers would there be if each human in each picture were to be replaced by an Alien of your choice?<br></br>
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<p xmlns="http://www.w3.org/1999/xhtml"><span class="editorial">We
had many solutions to the numerical questions we posed. From the
United Kingdom: Louis, Lucy and Gabriel from Ickford, Taylor from
Hillocks, Kieran from Gosforth, Shane and Sameer from Haberdashers,
Rohan from Canon Lane.</span></p>
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<p xmlns="http://www.w3.org/1999/xhtml"><span class="editorial">Charlotte from Laurence Jackson school sent in
the following straight-forward ideas that resemble the other
solutions sent in.</span></p>
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Solution $1$: count the number of hands and times that number by
$5$ which is the number of fingers and thumb on a person's hand.<br xmlns="http://www.w3.org/1999/xhtml"></br>
Solution $2$: simply count the fingers in the picture.<br xmlns="http://www.w3.org/1999/xhtml"></br>
<br xmlns="http://www.w3.org/1999/xhtml"></br>
<p xmlns="http://www.w3.org/1999/xhtml"><span class="editorial">Luke from High Wycombe Maths Master Class sent
in,</span></p>
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There are $250$ hands altogether if you count in $5$s it makes it
much easier. Hint: the first picture has $30$.<br xmlns="http://www.w3.org/1999/xhtml"></br>
<br xmlns="http://www.w3.org/1999/xhtml"></br>
<p xmlns="http://www.w3.org/1999/xhtml" class="editorial">Katrina,
Abigail and Aiden from St. Andrew's sent in the solutions clearly
as:</p>
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Picture $1 - 30$ fingers<br xmlns="http://www.w3.org/1999/xhtml"></br>
Picture $2 - 40$ fingers<br xmlns="http://www.w3.org/1999/xhtml"></br>
Picture $3 - 50$ fingers<br xmlns="http://www.w3.org/1999/xhtml"></br>
Picture $4 - 60$ fingers<br xmlns="http://www.w3.org/1999/xhtml"></br>
Picture $5 - 70$ fingers<br xmlns="http://www.w3.org/1999/xhtml"></br>
We thought the best way to count the fingers was to count the hands
in $5$s. You could count pairs of hands in $10$s or count the
number of hands and multiply them by $5$ to get the number of
fingers.<br xmlns="http://www.w3.org/1999/xhtml"></br>
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<p xmlns="http://www.w3.org/1999/xhtml" class="editorial">From the
United States of America, solutions were sent in from Conor at Burr
Elementary School and Caroline from Estes Hills.</p>
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<p xmlns="http://www.w3.org/1999/xhtml" class="editorial">Thank you
all for your answers and we welcome any later ones as well.</p>
<p xmlns="http://www.w3.org/1999/xhtml" class="editorial">Did any
of you try the last part of the problem, where the challenge was to
count using aliens' hands? Do let us know if you've had a go.</p>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
This <a href="http://nrich.maths.org/6878&part=" linkindex="8">activity</a> has been designed for pupils to work on
in a group so that they share their thoughts and ideas with
others.<br></br>
<h3>Possible approach</h3>
<div>Split the class into five groups and give each group one of
the pictures from <a href="/content/id/6878/FingersHands.pdf">this
sheet</a>. (It may be useful to have multiple copies of each
picture, depending on the number of children in a group.)</div>
<br></br>
<div>To begin with, allow time for everyone individually to work
out the total number of fingers, including thumbs, on their
picture. Then encourage them to discuss their methods amongst the
whole group. You could suggest that each group makes a poster which
explains all the different ways of counting that they have
used.</div>
<br></br>
<div>Rotate the pictures so that each group now has a new picture.
Once again, ask them to work out the total number of fingers, then
to share methods again. If new ways of working out the answer have
been used, these could be added to the poster.</div>
<br></br>
<div>Once all the pictures have been seen by every group, you can
bring the whole class together. They could display their posters
and everyone could have time to look at the other groups' posters.
Then, ask them to comment on what they have done and seen. Were
there some methods that lots of people used? Why do they think that
is? Were there some methods that they wouldn't have thought of? Did
any of them change the way they worked out the total number of
fingers for different pictures? Why?</div>
<br></br>
<div>It may be that questions arise that can be followed through to
a greater depth or you could use the alien context at the foot of
the problem to challenge the children to find the totals with some
of the aliens' hands.</div>
<br></br>
<br></br>
<h3>Key questions</h3>
<div>Tell me what you think.</div>
<div>How did you work out the total?</div>
<div>Do you agree with what s/he said?</div>
<div>Why?</div>
<div>Can you tell others about that?</div>
<br></br>
<h3>Possible extension</h3>
<div>In addition to considering Alien hands, this activity could be
extended further into the aliens counting in different bases
according to the number of fingers they have on a hand.</div>
<br></br>
<h3>Possible support</h3>
<div>Some pupils may benefit from having some paper cut-outs of
hands.</div>
<br></br>
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<mdoxml version="1.0">You really need to be able to talk with someone else to do this
activity.<br></br>
Are there other ways that you could count the fingers?<br></br></mdoxml>
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[We're counting thumbs as fingers!]<br></br>
Ordering the pictures by the number of people we would have $30,
40, 50, 60, 70$ fingers.<br></br>
Pupils may have counted/calculated these in many different
ways.<br></br>
The first picture many have resulted in things like;<br></br>
pupils saying "$5,10,15,20,25,30$",<br></br>
<br></br>
counting hands $1,2,3,4,5,6$ and $6$ lots of $5$ make $30$,<br></br>
<br></br>
counting pairs of hands as representing each person $1,2,3$ ten
$3$'s are $30$,<br></br>
<br></br>
counting each finger
$1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30$.<br></br>
<br></br>
saying there are $4,8,12,16,20,24$ fingers and $1,2,3,4,5,6$
thumbs, so $24+6=30$.<br></br>
<br></br>
Some pupils will nearly alway count in twos so we'd have
$2,4,6,8,10,12,14,16,18,20,22,24,26,28,30$<br></br>
<div style="text-align: center;">----------</div>
<br></br>
<br></br>
Moving onto those pupils who have a go at Alien Counting and then
returning to this challenge we could have things like this
occuring;<br></br>
<br></br>
"To keep the same number of 'fingers' in each picture we could
have<br></br>
$5$ Threebees or $3$ Chindis to make up picture $1$<br></br>
$10$ Dobus, $10$ Quorts or $4$ Chidis to make up picture $2$<br></br>
$5$ Chindis to make up picture $3$<br></br>
$15$ Dobus, $10$ Threebees, $6$ Chindis or $5$ Sendas to make up
picture $4$<br></br>
and just $7$ Chindis to make picture $5$."<br></br>
<br></br>
"If we keep the number of human/aliens the same for each picture we
could have the following totals for pictures;<br></br>
picture $1$ --- $12$, $18$, $24$, $30$, $36$<br></br>
picture $2$ --- $16$, $24$, $32$, $40$, $48$<br></br>
picture $3$ --- $20$, $30$, $40$, $50$, $60$<br></br>
picture $4$ --- $24$, $36$, $48$, $60$, $72$<br></br>
picture $5$ --- $28$, $42$, $56$, $70$, $84$"<br></br>
<br></br>
----------------------------------------------<br></br>
[Also - What numbers would the aliens say for all of the above,
since they do not count in tens.<br></br>
So for example $28$ in human counting, it would be for;<br></br>
Dubo $11100$ [one $2x2x2x2$, one $2x2x2$, one $2x2$, no $2$'s and
no units]<br></br>
Threebee $1001$ [one $3x3x3$, no $3x3$, no $3$'s and one
unit]<br></br>
Quort $130$ [one $4x4$, three $4$'s and no units]<br></br>
Chindi $103$ [one $5x5$, no $5$'s and three units]<br></br>
Senda $44$ [ four $6$'s and $4$ units]<br></br>
<br></br>
What if different aliens were having to make up some of the
pictures? Using the same number of aliens as humans did in the
beginning.<br></br>
For example we need $3$ different Aliens for the first picture.
]<br></br>
<br></br>
"If those numbers were given in the ways that the aliens count we
would have;<br></br>
Picture $1$ -- $1100, 200, 120, 110, 100$<br></br>
Picture $2$ -- $10000, 220, 200, 130, 120$<br></br>
Picture $3$ -- $10100, 1010, 220, 200, 140$<br></br>
Picture $4$ -- $11000, 1100, 300, 220, 200$<br></br>
Picture $5$ -- $11100, 1120, 320, 240, 220$<br></br>
<br></br>
<div style="text-align: center;">----------</div>
<div style="text-align: left;">For different Aliens making up
pictures $1, 2, 3$</div>
<br></br>
<div style="text-align: left;">We could have for picture $1$;</div>
<div style="text-align: left;">Dubo, Threebee, Quort</div>
<div style="text-align: left;">Dubo, Threebee, Chindi</div>
<div style="text-align: left;">Dubo, Threebee, Senda</div>
<div style="text-align: left;">Threebee, Quort, Chindi</div>
<div style="text-align: left;">Threebee, Quort, Senda</div>
<div style="text-align: left;">Quort, Chindi, Senda</div>
<br></br>
<div style="text-align: left;">We could have for picture $2$</div>
<div style="text-align: left;">Dubo, Threebee, Quort, Chindi</div>
<div style="text-align: left;">Dubo, Threebee, Quort, Senda</div>
<div style="text-align: left;">Dubo, Threebee, Chindi, Senda</div>
<div style="text-align: left;">Dubo, Quort, Chindi, Senda,</div>
<div style="text-align: left;">Threebee, Quort, Chidi, Senda</div>
<br></br>
<div style="text-align: left;">And for picture $3$</div>
<div style="text-align: left;">All $5$ of them.</div>
<br></br>
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Fingers and Hands
How would you count the number of fingers in these pictures?
Calculations and Numerical Methods
Addition & subtraction
Numbers and the Number System
Counting
Calculations and Numerical Methods
Multiplication & division
Numbers and the Number System
Number bases