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A great opportunity to introduce multiplications of fractions! 1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fractio

A great opportunity to introduce multiplications of fractions! 1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too. When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents. The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important! To do this, we need to multiply the fractions together like so: (Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls). So in this case we would have: (1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3). This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing: (1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).

quietbritishjim 6y ago
I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still feels like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation. Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it). Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.
danbruc 6y ago
I would suggest to replace 1/2 with 3/6 since the number of students at each table is what matters, not the number of tables, which only yields the correct result if each table has the same number of students.

quietbritishjim 6y ago

I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still feels like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation. Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it). Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.

danbruc 6y ago

I would suggest to replace 1/2 with 3/6 since the number of students at each table is what matters, not the number of tables, which only yields the correct result if each table has the same number of students.