When I give this question to colleagues, I get a range of answers: One-quarter, one-fourth, one divided by four. And often, 'one over four'. Admittedly, this is something I have said over the years. At a talk given by Cathy Bruce last year, I realised that this might be problematic: for the number 25 we don't say 'two beside 5' so we shouldn't use positional language ('one over four') to describe a fraction. A simple switch in the language we use can help our students from developing misconceptions.
I've been thinking a lot about how to use the Concrete-Diagrammatic-Symbolic approach to multiplying fractions to get help students learn how to multiply fractions. My concern is that if I jump straight to a symbolic explanation (multiply numerators, multiply denominators) students are at risk of developing more misconceptions.
Starting with multiplying a whole number by a fraction, I'll use pattern blocks to model things like 5 x half:
or 7 x third:
8 x two-thirds:
Diagrammatically, I could use a number line to show these, or an interactive fraction tool:
Now I can develop this to symbolic notation:
At this point my aim is to get students to articulate what happens symbolically when we multiply a whole number by a fraction. I'm finding that this shift from Diagrammatic to Symbolic is smoother if I develop the Symbolic alongside the Diagrammatic and not treat them as separate domains. All the time I am doing this I must remember to say the fractions using the correct terminology (e.g. eight x two-thirds as opposed to eight times two over three). Also, I need to make sure that students transfer their knowledge of commutativity so that they understand that 5 x one-half is the same as one-half of 5.
When looking at multiplying a fraction by a fraction, I will introduce this concretely by folding Post-it notes. For example to do 1/3 of 3/4 , fold a post-it note into thirds and shade one-third blue. Now fold into quarters and shade three-quarters yellow. Notice how the post-it note is now divided into 12 parts (why?) and since 3 of those parts are shaded blue AND yellow, then1/3 x 3/4 equals three-twelfths or, more simply, one-quarter. (How might a student use this post-it note to prove that three-twelfths is equivalent to one-quarter?)
This method is not so easy for 3/5 x 5/6 though, so a diagram will be more useful now:
Notice that the overlap here has 15 units shaded (why?) and that the whole comprises of 30 units (again, why?). Hence three-fifths times by five-sixths is fifteen-thirtieths which simplifies to one-half.
As students do more of these diagrammatically, they will be in a better position to generalise by using this diagram:
This then gives reason to our symbolic rule.
Interestingly, having shared these strategies with colleagues and parents, a surprising number of people comment how they now understand when fractions are multiplied. If they understand it better, they will have less misconceptions, and, as a result, they are less likely to make mistakes.