MUL 04

This week's tasks involve structural arithmetic. Students are given the value of an arithmetic expression and asked to find the value of a closely related expression. We focus on expressions that involve division and that contain fractions.

Monday: Some students may not realise that it is possible, and potentially quicker, to find the value of the second expression by comparing it to the first. So they might need to be initiated, eg through a whole-class discussion, into the 'game' being played here (including the fact that the 'game' only works because we have contrived to modify our expression in a particular way!).

 

This is a relatively simple task but far from trivial for some students. Once students spot that the dividend in the second expression is 2 more than in the first, they still need to make use of two important ideas. 

First, that we can find the value of the second expression by adding the value of 2÷6 to the value of the first expression. This involves an implicit understanding that division is distributive over addition, albeit only in one direction: (28592 + 2)÷6 is equal to 28592÷6 + 2÷6; however 28594÷(2+4), say, is not equal to 28594÷2 + 28594÷4.

Second, students need to appreciate that a division like 2÷6, where the dividend is smaller than the divisor, has a solution in the realm of rational numbers, in this case the rational number ⅓.

Tuesday: Here students need to home in on the observation that the dividend has (simply) been increased by 1. They then need to evaluate 1÷⅙.

It is interesting to see how students evaluate 1 ÷ ⅙, and whether they can make use of a range of methods. Also, how readily do they come to terms with the fact that the result (the quotient, 6) is bigger than the dividend, 1?

 

Wednesday: The dividend has increased by +10, so the task can be solved by finding the value of 10 ÷ 2½.

How do students evaluate 10 ÷ 2½ ? Can they use both formal and informal methods?
A rather neat method is to double the two numbers:
10 ÷ 2½ = 20 ÷ 5 = 4.
A more grounded approach is to use repeated addition:
2½ + 2½ + 2½ + 2½ = 10;
What other methods do students use?
When students have worked through the task, you might want to suggest that they check by using a calculator, as a way of consolidating the link between decimals and fractions.

Thursday: The dividend has increased by 1, so we need to find 1÷2½.

How do students evaluate 1÷2½ ? Do they use a formal method (1 ÷ 2½ = 1 ÷ 5/2 = 1×⅖ = ⅖)? Or do they change 1÷2½ into 2÷5? Or do they make use of the knowledge from the previous task that 10÷2½ = 4? When students have worked through the task, you might want to suggest they check by using a calculator.

Friday: Here the divisor is increased by a factor 10, so the quotient is reduced by a factor 10.

If students do try to solve the task by dividing the value of the first expression by 10, how do they set about this? And how do they write the result - using fractions, or decimals, or perhaps an ambiguous mixture of both, such as 691.3⅖?