Firstly here's the Excel file: Fraction Multiplication (simple). Right-click, "Save Target As", and select where you want the file to be saved.
There's another version of this file called Fraction Multiplication (extra) which shows more of the stages in the justification. However, I prefer the simpler version, it feels less cluttered. I just make sure I'm giving a good commentary, with lots of references to the underlying concept structure, when I'm using it in the classroom.
Question: What's 2/5 of . . . 1/3, 2/3, 3/4, 3/5, or 4/5 ?
Here's what I say and do if I'm using Fraction Multiplication (extra); with the simple version, what I say will be much the same.
Set the spinners to get 2/5 of 1/3
Show 1/3 at the top of the board or screen
Next look at 1/5 of 1/3 and observe that this is the same as 1/15
Show 2/5 of 1/3 - obviously 2/15
Change the spinner to get 2/5 of 2/3 , and discuss how this answer can be worked out almost instantly from the first one.
2/5 of 2/3 would be . . . 2/15 from the first third, then another 2/15 from the second . . . in other words: 4/15
Move on to think about 2/5 of 3/4
Using fifths of quarters means we will be counting up twentieths
1/5 of 1/4 is 1/20
1/5 of each of the three quarters (1/5 of 3/4 ) is 1/20 three times: 3/20
So another 1/5 (that's two fifths of 3/4 in all) makes another 3/20 , or 6/20 in all.
Sometimes, as in this case, the result can be expressed more simply: 3/10 instead of 6/20
Many of the Excel files offered, month by month, in this section could be used simply to demonstrate and explain key procedures, however, practice and reinforcement can frequently be accomplished successfully, and abler pupils better motivated, if an approach of "differentiation by challenge" is adopted. This means that the same resource is used with all children but the challenge is varied for children at different stages of understanding.
Easier challenges than the main presentation could be as follows :
Find : 1/n of 1/n , where the n in each term is either a 2, 3, 4, or 5,
Likewise : 2/n of 1/n, followed by 3/n, 4/n, 5/n each time of 1/n
And then : 1/n with 2/n, then with 3/n, 4/n, 5/n
More demanding challenges :
What multiplies with 1/4 to get 1/12, 1/16, or 1/20 ?
Or with 2/3 to get 4/9, 8/15, or 1/6 ?
With children in pairs, each pupil can then invent similar puzzles for their partner.
Can you make a general rule (method) of what to do to solve this kind of puzzle?
It seems that multipication of fractions is communtative : that, for example, 2/3 of 4/5 is the same as 4/5 of 2/3. Check this, and maybe a few more, then try to explain why this should be so.
How can you tell in advance which multiplication answers will reduce to a simpler form at the end?