The following sequence of tasks shows how we can teach a procedure for dividing fractions, conceptually. The technique involves the same idea used for finding equivalent fraction or proportions – multiplying the upper and lower number of a proportion by the same number preserves the proportion. This is something that they learned before. The task of division of fractions becomes a simple extension of finding an equivalent fraction.
Teaching Sequence on Division of Fractions
Task 1 – Find fractions equivalent to the fraction 5/8.
This should be easy for learners as all they needed to do is to multiply the same number in the numerator and denominator. This is more of a revision for them.
Task 2 – Find fractions equivalent to
Here they will apply the same idea used in task 1: If you multiply, the same number (except 0) to numerator and denominator, you produce equivalent fractions. You can ask the students to classify the fractions they made. One group I’m sure will have a fraction for numerator and whole number for denominator; another group will have fractions for both numerator and denominator; and, another will have whole numbers for numerator and denominator. The last group is what you want. This fraction is in simplest form. They should be ready for Task 3 after this.
Task 3 – Find the fraction in simplest form equivalent to
From here you can ask the students to express the fraction as a division (this is one of the meaning of fraction – an indicated division) and rework their solution. It should be something like this:
You can challenge your students to find the shortest possible solution of getting the correct answer. It will involve the same idea of multiplying the dividend and the divisor by the same number. I’m sure that after doing the tasks above, they will be able to figure out the following solution which now leads to the the procedure ‘when dividing fractions, just multiply it by the reciprocal of the divisor’:
Procedural fluency does not mean doing calculation with speed and accuracy even without understanding. Remember that procedure is only powerful and useful in problem solving when students understand what it means and why the procedure is such. I suggest you also read my post on what it means to understand fractions and math knowledge needed by teachers to teach fractions and decimals.
Note:
The above lesson is not just about division of fractions. I made it in such a way that weaved in the lesson are the ideas of equivalent fractions, proportion, the property that when you multiply same number to the numerator (or dividend) and to the denominator (divisor) it does not change the value of the quotient, division by 1, etc. Working with the tasks engages students to the same process/technique they will be applying when they work with rational algebraic expressions. The main point is to use the lesson on division of fractions as context to make connections and to teach important ideas in mathematics. I think this is how we should teach mathematics.
