This is a neat way I found to teach fraction multiplication. It seems to stick well with students, and gives a very good mathematical idea behind what's going on.

2 different colored pencils/crayons/something that draws.

A piece of blank printer paper

Let's say we want to multiply 3/7 and 5/8. Take a piece of paper and draw six straight lines, from left to right, on the paper (attempt for regular spacing, but nobody's perfect). This leaves seven spaces. Have the student fill in any three of the lines that they want in one color. Now, draw seven lines from top to bottom on the piece of paper (again, regular spacing helps), partitioning the paper into 8 parts. Ask the student how many boxes have been formed as a result of this. Many of the students have done some sort of exercise similar to this when attempting to learn integer multiplication, so generally in my experience they have been able to answer this easily, revealing that the fraction must be somehow divisible by the product of the numbers in the denominator. Now, let the student fill in any five columns with the other colored pencil, and count the number of boxes that have two colors in them. It's easy for them to see there are fifteen boxes colored in, and that this is the product of the terms in the numerator. The proportion of boxes filled in with two colors to all of the boxes is 15/56.

Let me know how it goes!

Note: The idea for this came from W.W. Sawyer's The Mathematician's Delight. It's a great book on the nature of math that I cannot recommend more highly to (late high school/college) students, teachers, or anyone fascinated by solving puzzles or finding patterns.

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