Multiplication can be spoken about and expressed multiple ways.... multiplied by, times, groups, sets, rows etc, etc. And more importantly, the way we verbalise these operations can unwittingly confuse children. If we say ‘two multiplied by four’, that implies 2 + 2 + 2 + 2, e.g. four plates with two cookies on each, four sets of two. However, if we say ‘two times/groups/rows of four’, that implies 4 + 4, e.g. two plates with four cookies on each. While we understand that the order for multiplication doesn’t matter and the answer for both of these is the same since there is still 8 cookies (i.e. the commutative property of multiplication), children don’t inherently understand this, especially at the introductory stage. And when represented concretely/pictorially, both images would also look different:   Therefore, for consistency, initially we should write and say ‘sets/groups/rows of’ and when we use the multiplication symbol (×) we will say ‘times’. And when we meet a multiplication sentence eg 4 x 5 it should be interpreted as 4 sets of/groups of/rows of 5. Then, when the children are comfortable translating a concrete/pictorial representation into a number sentence, and vice versa, they can then be introduced to the commutative property or "turnaround facts" similar to this: 5 x 4 could be seen as 5 plates of 4 cookies which is 20 cookies 4 x 5 could be seen as 4 plates of 5 cookies which is still 20 cookies So the answer is the same, but it looks different i.e. "same value, different appearance" I also get the students to use a 100 dots grid (array) when exploring this:  3 x 6 = 3 rows of 6 6 x 3 = 6 rows of 3 There are still 18 dots each time but each arrangement/array looks different. However, if you rotate/turn the 100 dots around, the arrangements/arrays begin to look very similar, hence the phrase "turn-around facts" ie 6 x 3 = 3 x 6 The beauty of this type of 100 dots grid is that it can also be used to explore the distributive property of multiplication i.e. that in multiplication we can break apart groups/arrays into smaller/easier/more familiar groups/arrays and then re-combine them to get an answer, ie 3 rows of 6 is clearly the same as 3 rows of 5 plus 3 rows of 1 3 x 6 = (3 x 5) + (3 x 1) = 15 + 3
Many thanks also to Kristin Gray for her inspiring blog that reminded me that it might be worth looking at this here. |
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