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Understanding Multiplication

Discussion in 'Mathematics' started by larrylemur, Jan 2, 2012.

  1. Can any one clarify <u>exactly </u>what the multiplication symbol signifies:
    Does 2x3 mean
    1) 2, 3 times i.e. 2 + 2+ 2 OR
    2) 2 groups of 3 i.e. 3 + 3
    I am well aware the numierical <u>result</u> will be the same but that does not actually define what "X" means. I am also aware one can show the net result of either gives the same numerical result by using an array - but that still does not answer the question - what does "X" actually mathematically mean?
    Multiplication is now being taught in Primary schools using both these methods : once one has covered fact that the numerical result of 2x3 = the numerical result of 3X 2 , it dosen't really pose any problem. However before getting to this point one has to be able to explain clearly what for example 2x3 means: this is not helpful when some schools (after the introduction of the Primary Strategy) teach one meaning & other schools the other.
    I would also be interested how secondary school maths teachers would explain what the multiplication sign meant and whether teaching the meaning of the multiplication sign to mean either of the above meanings causes problems further on when teaching maths in secondary school.
     
  2. DM

    DM New commenter

    Does it have to be either?
    Does 0.2 x 0.3 mean
    1) 0.2, 0.3 times OR
    2) 0.2 groups of 0.3
    If I had to provide a definition I would say it is the scaling of one number by the other.
     
  3. So how do you explain the "x" symbol to seven year olds using language and diagrams they would understand?
    Would it be accurate/correct to say 2x3 CAN mean 2 groups of 3 <u>OR</u> 2, 3 times?
     
  4. bombaysapphire

    bombaysapphire Star commenter

    Yes. This can be demonstrated using a rectangular array. 2 rows of 3, turn it round to get 3 rows of 2.
    I like to use the expression "lots of". I find it extends well into other areas of Maths and is easy for all students to understand.
    It is useful if children understand that 5x7 gives the same answer as 7x5, even just because it is much easier to count up in 5s than to count up in 7s.
     
  5. I would say it means both of those things. A 2 by 3 rectangle and a 3 by 2 rectangle not only have the same area, they are the same rectangle (albeit rotated by 90 degrees).
     
  6. PaulDG

    PaulDG Occasional commenter

    As a matter of mathematics, it can be either.
    In the real world, the entities that that particular element of maths is being used to model might well work with only one of the interpretations - obvious examples being breeding pairs of some animals.
     
  7. Can you define multiplication of animals, mathematically?
     
  8. Paul might be thinking of using the 'x' as a shorthand for crossing, such as crossing a donkey with a mule. (That's my best guess).
     
  9. PaulDG

    PaulDG Occasional commenter

    That's not my intention.
    My point is that maths has "pure" and "applied" aspects.
    Pure maths makes no distinction between 2 X 3 and 3 X 2.
    In any particular application, such as breeding animals which pair-bond for a significant period of time, "3 pairs" has a very different applied effect when compared with "2 triads".
    Or perhaps consider 3 pieces of wood 2m long and 2 pieces 3m long - both give 6 linear metres, but neither sets may be useful for bridging a gap and perhaps only one of the sets would give a pleasing esthetic without visible joints when used for skirting board.
     
  10. Does this mean when creating an array to show 7x5 that it does not matter which way the children draw the array as it makes no difference to the answer if they have drawn 7x5 or 5x7?
    I will be teaching year 3 as an NQT in September and would personally be of the opinion that it does not matter as the children can see the answer is the same either way but have seen marking telling children that the answer is correct but the array shows the incorrect multiplication. Can someone clear this up for me?
    Thank you
     
  11. carriecat10

    carriecat10 Established commenter Community helper

    technically 7x5 means 7 ... 5 times. In any number sentence the first number is the size of the group followed by the operation. Once there is a context attached to the numbers it makes more sense; so if the 7 represents pence, cows, metres etc you can see this is then 5 lots of those 7 things.
    It is important for children to understand that the outcome of 5x7 will give the same answer as they will know some multiplication facts better than others. Being able to turn arrays around is useful to show this.
    Once this links with other operations, this makes more sense to children. So supporting understanding that you CAN subtract a bigger number from a smaller number.
    Carrie [​IMG]
     
  12. afterdark

    afterdark Lead commenter

    is there a convention you mean?
    My intuition says the first one. But it is easier to teach the commutative property rather than attribute an order rule. I explain in my next response.

     
  13. afterdark

    afterdark Lead commenter

    draw a large grid on a single sheet. show the class. establish an operation and how many elements [dots].
    Now rotate the sheet 90 degrees.
    Ask the children has the number of dots changed?
    Now ask which calculation?
    You can now establish commutative rule rather than choosing between 7 x 5 and 5 x 7.
    The marking may well be have been done with the best of intentions. However do you sometimes tell the children to turn their book [90 degrees] sideways?
    Teaching children that order is not important operations x and + is more use than having
    2 X 3 = 2 +2 +2 instead of 3 + 3
    Time would be better spent explaining that subtraction and division are not commutative.
    BTW loustar86, you are correct IMHO.

     
  14. Thank you for your help
     
  15. yes - but for ks 1 and 2 learning, the problem is we write tables as 1x5=5, 2x5=10, 3x5=15 - and that is so much easier to grasp as '1 lot of 5' 2 lots of 5'
    if i ruled the world, tables would be turned round and read '5x1=5, 5x2=10' etc to match
    i wish i knew the correct way to show/display/explain this consistently at ks2 - i just think there isn't one - we have had discussions on this in school wrt displays
    (i should get out more, shouldn't I? [​IMG] )
     
  16. but that's way more complicated than 3x2 and 2x3 , as gender matters - not all pairs or triads are equal!
    wrt to the wood - isn't that about units - neither 2x (3 metres of wood) nor 3 x (2 metres of wood) is the same as (2x3) metres of wood [or (3x2) metres of wood]
    there is a resource out there by the late, unlamented weebecka where she objects to using numerical multiplication rather than scaling to describe a tree 3 times higher than a 7m tree because she appears to hold that
    3 x (7m tree) - which she shows as three 7m trees on top of each other - is the same as (3x7)m tree - i would say those are quite different

     
  17. As a secondary maths teacher, introducing multiplication of fractions, it was useful to indicate that "of" means the samer as "x", so that "half of 16 is 8" means the same as "half x 16 = 8". From that viewpoint, 2 x 3 could be taken as 2 of 3, hence the second item would be more useful (i.e. 2 x 3 = 3 + 3).

     
  18. now - that would work with times tables
     

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