Tag: teaching fractions

Posted in Elementary School Math, Number Sense

Why do we ‘invert’ the divisor in division of fractions

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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

fraction divided by wholeHere 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

fraction divided by a fraction

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:

dividing by fraction

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’:

dividing by fraction

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.

Posted in Elementary School Math, Number Sense

What are fractions and what does it mean to understand them?

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Negative numbers, the irrationals, and imaginary numbers are not that easy to make sense of for many students. But this is something understandable. One only needs to check-out the histories of these numbers. The mathematicians themselves took a long time to accept and make sense of them. But fractions? How can something so natural, useful, and so much a part of our everyday life be so difficult? Didn’t we learn what’s half  before we even learn to count to 10? I’m sure this was true even with our brother cavemen. So how come the sight of a fraction enough to scare the wits out of many of our pupils and yes, adults, too?

Fractions are used to represent seemingly unrelated mathematical concepts and this is what makes these numbers not easy to make sense of and work with. In mathematics, fractions are used to represent a:

  1. Part-whole relationship – the fraction 2/3 represents a part of a whole, two parts of three equal parts;
  2. Quotient – 2/3 means 2 divided by 3;
  3. Ratio – as in two parts to three parts; and
  4. Measure – as in measure of position, e.g, 2/3 represents the position of a point on a number line.

Of these four, it is the part-whole relationship that dominates textbooks. For many this conception is what they all know about fractions. While it is also the easiest of the four to make sense of, students requires series of learning activities to fully understand part-whole relationship . Crucial to this notion is the ability to partition a continuous quantity or a set of discrete objects into equal sized parts. Below are sample tasks to teach/assess this understanding. They call for visualizing skills.

Of course understanding fractions involve more than just being able to use them in representing quantities in different contexts. There’s the notion of fraction equivalence, which is one of the most important mathematical ideas in the primary school mathematics and a major difficulty. This difficulty is ascribed to the multiplicative nature of this concept. There’s the notion of comparison of fraction which includes finding the order relation between two fractions. And if your students are having a hard time on comparing fractions you can check their understanding of equivalence of two fractions. It could be the culprit. And let’s not forget the operations on fractions. An understanding of the procedure for adding, subtracting, multiplying, and dividing of fractions depends on students’ depth of understanding of the different ways fractions are conceived, on the way fractions are used to represent quantities, on the idea of equivalent fractions, and on order relation between fractions, and  many others such as the meaning of the operations themselves.

A study has been conducted categorizing students levels of conception of fractions, at least up to addition operation. Just click on the link to read the summary.