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PostPosted: Sun Sep 14, 2008 12:10 am 
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I was wondering the methodology used for doing some maths questions when practicing from past papers.
..example....
Two numbers add up to 130 and their difference is 26. Find the numbers.

There are several methods of solving this, one would be simultaneous equations which I assume 11plus students would not be taught.
The only other methods I can think of is to reason the problem out which is not easy if the numbers are not simple.

Any Ideas

Thanks


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PostPosted: Sun Sep 14, 2008 5:25 am 
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1. 130/2 = 65 Gives you the half way point
2. 26/2 = 13 The amount above and below the halfway point
3. 65-13 = 52
4. 65+13= 78
5. Check 52+78 = 130

I think you'll find the numbers at this level are generally simple when you have to go through a several step solution.

I'm sure there are more complicated ways of working it out as you point out but not within the skill level they should have.


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PostPosted: Sun Sep 14, 2008 9:05 am 
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Or just add 130 and 26 which is 156.

156 divided by two gives you one number and you can quickly get the other ..


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PostPosted: Sun Sep 14, 2008 9:14 am 
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or simpler!


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PostPosted: Sun Sep 14, 2008 9:27 am 
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It will always work if you are given the sum and difference of two numbers.

If you want to know why then

a + b = 130

a - b = 26

adding gives

2a = 156 [because the b and the -b give zero]

a = 78


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PostPosted: Sun Sep 14, 2008 11:21 am 
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Thanks for your responses

My concern was not how to get the answer but the methods used and how it ties in with maths topics.

Example:-
If I am dealing with mixtures, fractions, percentages, ratio & proportion.......say Sand and cement.....3 parts sand to 1 part cement.....ratio is 3

to 1 or 3:1. Question what fraction is cement?. ..... Students often say 1/3 cement...WRONG.

I can explain this as...... There are 4 parts in total..... 3 of the parts are Sand and 1 of the parts is cement and the 4 parts together make up

the WHOLE mixture. So 3/4 is Sand and 1/4 is cement. There is a logical link between the methods. If a student is familiar with fractions and we

are introducing Ratio & Proportion we can use their previous knowledge of fractions to explain it.

I will answer the above responses turn:-

TonDad..... Do I explain we are using averages or mean values..... The total of the two is 130 so if we divide by 2 we get the average of mean value

which would be the actual value if the difference was Zero. In our case there is a difference so one number is larger than the other. That is one

will be more than average and one will be less than average. We are only dealing with 2 numbers so the amount one is above the average will be the

same as the amount the other is below the average value. We could show this on a Graph or Bar Chart. The amount one is above + the amount the other

is below is the difference and as both amounts are the same each must be half the total difference of 26. so divide 26 by 2.

ADD this to the mean value of 65 to get the larger number ( 65 + 13= 78 )
SUBTRACT this from the mean of 65 to get the smaller number ( 65 - 13 = 52 )

-------------------------------------------

Guest55.... How do I explain to a student that I add the sum of the numbers + their difference and then divide by 2.

In Your second response you are using the methods of simultaneous equations.


My Answer:-

Show the problem in diagram form using a number line 130 units long, showing the larger and smaller number.


x-------Larger----------------x----small----x

0------------------------------------------->130


Now show the small number subtracted from the larger number to leave the difference.


x--difference---x----small----x.............>130

This gives the equation as shown below

x-----26--------x----small----x----small----x this equals 130

If we now write down the equation using S for small

26 + S + S = 130

From the numberline above we can explain that if we remove the 26 from the LHS we have to subtract it from the RHS
This gives S + S = 130 - 26
we can explain that S + S is equal to 2 x S is equal to 2S

So 2S = 104
S = 52

Smaller number is 52 and larger is 130 - 52 = 78


------------------------------------------------------

This will probably provoke some thought, Thanks again


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PostPosted: Sun Sep 14, 2008 12:18 pm 
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Muggins - ratio is level 6 understanding really - you have highlighted the key 'issue' - sometimes we compare parts sometimes we want to express as a fraction of the whole. I usually use sweets to share out between two people 2 to one, three to the other - we can then compare in two ways. Using concrete objects is important at the beginning.

I love number lines, so if your child can use them that's great. I would not use the equations with children until Secondary but logically I think children can 'see' the method in my first post.


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PostPosted: Sun Sep 14, 2008 5:56 pm 
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I asked DD2, working for 11+, how she would do this question. She very quickly gave me the answers and it seems she worked it out by taking 26 from 130 to give 104, halving that to give 1st no. 52 then adding 26 back on to give 2nd no. 78. Seems to be a method she's worked out for herself, rather than taught method but it seems very quick!


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PostPosted: Sun Sep 14, 2008 6:25 pm 
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Yes, that is exactly my point. Some can do it by manipulating the numbers until they get the answer but if you ask them why they did a particular step they would not know why.

When you take 26 from 130 .... Did DD2 know what was left?..... you are left with twice the smaller number. This can be seen from the numberlines above. Then divide by 2 to get the smaller number.

What about a similar type problem:-

50 people go on a day trip. Some go on the bus only, some others also go to a show while they are there. Tickets for the bus are £3.00, Tickets for the show are £5.00. The total money collected is £300.00. How many went to the show?.

Wat will be the methodology used here?
-----------------------


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PostPosted: Sun Sep 14, 2008 7:04 pm 
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Muggins, Yes she did know that 104 was twice the smaller number, which was why she halved it! The method myself and school had shown her was the half 130/half the difference 26/ add and take 13 away from half as detailed above, but I knew at the time she didn't like that method. I'm pleased she's thought about it about it and worked out something that works and she can do v. quickly.

For the other prob, if 50 people go on the bus at £3 each that would cost £150. £150 left, divided by five means that 30 people went to the show as well. Seems like a simple two-step calculation once you work out what you need to find out. Probably simpler than working out the difference prob for some kids, if they are don't have a method and are using trial and error. Maybe I misunderstood your question?


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