Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3.

Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable.

It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5. 

However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign.

Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only works because they are using childish language. A secondary school teacher is more like to expand 'times' as 'multiplied by'. And so we get '5 multiplied by 3' - if you think about it, this clearly means 'take 5 and reproduce it three times.' So it means 5x5x5.

I think an excellent last word can be given to a Dr Petersen on the Math [sic] Forum: [Multiplication] is a commutative operation that can be modeled in two symmetrical ways as repeated addition (when applied to whole numbers).

Conclusion? The child was as correct to use this formulation as the one being taught, and the teacher was wrong to mark him/her down.


  1. Assumptions being made here. 5x3 can mean 'matrix 5 by 3' which is not the same as 'Matrix 3x5' when the order is agreed along and down?

    1. But the question had nothing to do with matrices. Matrix multiplication is a whole different kettle of bananas.

  2. Pedagogically, I love the approach used in the learning system published by OUP. It introduces and develops concepts along a careful progression, helping prevent children from developing common misconceptions.

    The very first introduction to the concept of multiplication is made through the active carrying of objects. So, reading from left to right, 5 x 3 becomes,
    5... speak ‘5’ and pick up 5 and start to carry them to a location, the ‘maths table’ (paper cups are used initially)
    x speak ‘times’ and say ' this means do the same thing lots of times'.
    Freeze at the start of the process of carrying the cups and ask ‘How many times?’ Read the next symbol
    3 speak ‘3’ and then count as you carry groups of five cups each time... ‘time number one, time number 2, time number 3’
    Then children are then told to look at the maths table and count… 15
    Here is one example of misconceptions it helps to prevent. Children commonly make mistakes with zero in multiplication such as 5 x 0 = 5 and 0 x 5 = 5.
    The approach helps prevent this. In the first case children would pick up 5 cups as though to carry them to a maths table, look at the next instruction – do this how many times? Zero. And promptly put the cups back down without moving them. Then look at the maths table and count. Zero
    In the second example, children would role-play picking up nothing and then carry that nothing to the maths table 5 times. Then look at the maths table and count. Zero.
    It also helps them gain a good first grounding in the cardinal / ordinal distinction – number as a muchness (the holding of 5 cups) and number as a manyness, (counting 3 journeys to the maths table).

    Have you come across this approach? I’d love to know what you think.

    1. I haven’t specifically come across this - in effect it is an embodiment of the ‘multiplied by’ (as opposed to ‘lots of’) interpretation, where the act of multiplying is represented by carrying to the table. It looks rather complicated as written down, but I suspect it feels far more natural when doing it.


  3. Yes, sorry, I was a bit wordy!!

    Here's how it looks when it's done more naturally with a group of 5 and 6 year olds.

  4. About the math test: the story doesn't tell if the difference had been made clear to the student, so could it be held against him? We may never know.

    The weakness in the main article above, is that it refers to the use of English. Which, to speak in mathematical terms, should not be part of the equation.

    Though 3x5 has the same outcome as 5x3, the two are definitely not the same, and we know this from when we first learn the tables of multiplication:
    They (and this is their essence) teach us that 5x3 is 3 more than 4x3 but 3 less than 6x3 (meaning 3+3+3+3+3), whereas 3x5 is 5 more than 2x5 but 5 less than 4x5. (meaning 5+5+5)
    I cannot see why a teacher would explain 5x3 as taking 5 cups 3 times, since this contradicts these very tables (s)he is also teaching. Similarly 5x0 explained cf. these tables, would mean they have to run 5 times with empty hands, not take 5 cups, and then not walk.

    Commutativity is not an argument here. It only says the multiplication can be commutated without disturbing the outcome, but not that the commutated process is equal.
    Going South 2 blocks, then East 3 blocks lands me on the same position as going East 3 blocks, then South 2, but it is a different scene along the way.
    A 3-story building with 5 rooms on each floor has the same combined total of rooms as a 5-story building with 3 rooms on each floor, but the buildings are hardly similar.

  5. My daughter I just got marked wrong for this exact issue.

    Telling a child 5 x 3 is 5 + 5 +5 and not 3 + 3 + 3 + 3 + 3 is not only lying to them, but completely undercuts their education and progression in mathematics.

    1. Just because they have the same outcome it doesn't follow that they're both the same in their process to reach that outcome. 5+2 and 10-3 both have the same outcome but they are not the same process which leads to the outcome.


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