Fractions for algebra and arithmetic: 4.5 - Part 2

Teaching Strategies

In preparation for algebra, students need to strengthen their knowledge of fraction operations (so practice is important here and textbooks can supply exercises). They also need to work increasingly with fractions where the numerator and denominator are expressions rather than single numbers. The activities below cover familiar concepts such as equivalent fractions, but in an increasingly complex setting.

Activity 1: Write the fraction focuses on students translating verbal descriptions into written expressions and attending to order of operations.
Activity 2: Making friends with quirky quotients is an awareness-raising activity where students are provided with examples of fractions that might be unexpected or appear difficult.
Activity 3: Cancelling links work on equivalent fractions with an increasingly efficient process of cancelling common factors.

Activity 1: Write the fraction

In this short activity, teachers give common verbal descriptions and students have to write the matching mathematical expression, using correct mathematical notation. Students can also use a calculator to compare correct and incorrect interpretations of fractions.

First students write expressions from verbal descriptions. Start with simple descriptions and gradually become more complex (as appropriate for current requirements of other work). For example,

one half; two quarters; three over six
53 plus 12 all over 13; 53 plus 12 over 13
2 times 4 minus 3 all divided by 17; 4 minus 3 all times 2 divided by 17
4 squared over the sum of 15 and 48
1 + 4 squared over 15 plus 48 (this is included to emphasise that verbal expressions can be ambiguous – there are several reasonable meanings if we say this)

Discuss the expressions that students have written, making points such as:

the order in which operations are done affects the answers (calculate values to demonstrate this, if necessary)
verbal expressions are clumsy and often ambiguous – we need mathematical notation
when we talk about expressions in class, we need to be careful to indicate the structure.

Students may need assistance to calculate verbal or written expressions on a calculator.

Is this expression calculated as 21 ÷ (35 + 64) or 21 ÷ 35 + 64?

When students are familiar with pronumerals, the same exercise can be repeated using examples such as:

x over 2; three x over 2; four over y; four divided by y; x multiplied by 0.05
x plus 2 all over 3
2 x minus 3 all divided by 2 equals 17
y plus 2 is equal to 2y minus five
3 by x squared over 2 is equal to 48
y plus 3 is equal to y over 2 plus 6.

Finally, reverse the direction of the activity by asking students to create their own complex fractions and corresponding verbal descriptions. Students should hide the mathematical notation, read out the verbal fraction description to a partner who writes translates it back into mathematical notation, which they both compare with the original.

Note: About the word ‘over’. When children start talking about fractions in primary school they use terms such as ‘two fifths’, but when they get to secondary school they use terms such as ‘x over y’ to describe both fractions and division x ÷ y.

Activity 2: Making friends with quirky quotients

The key teaching strategy here is to provide students with quirky examples of fractions to focus on developing meaning and to decrease the extent to which students are perturbed when complex fractions arise in calculation.

It is important that students consider, rather than avoid, these examples as they are likely to appear when substituting into simple algebraic expressions or solving algebraic equations.

Present students with a number of fraction-like expressions, written in ways that might be unexpected and that might look difficult. For each fraction get students to quickly estimate the value, write it as a division or a product, and calculate the exact value. For example, the Fraction 5a (below) may look unusual to students, but recognition that the fraction can be rewritten as calculation 5b helps students give an approximate (or exact) value for the number. This expression can be further clarified through Calculation 5c (how many tenths in four and a half – answer 45).

Provide a number of other examples where classroom discussion of simplification can help develop students’ confidence in dealing with unusual fractions. Some examples are:

Activity 3: Cancelling

Prior to this level, students will have found equivalent fractions (ie equal fractions), but they may not have used the shorthand ‘cancelling’ process that is very useful for algebra. In addition, it is likely that they will mostly have dealt with familiar fractions and they often can simplify these almost intuitively without explicitly thinking about the factors involved. This 5 step activity begins by showing the role of common factors in making fractions equal (equivalent), and then moves students towards the shorthand cancelling process.

Step 1: See a new pattern in sequences of equivalent fractions

Consider the following sequence of equivalent fractions, which students will know well:

Some students see the pattern as ‘numerators going up by 4, denominators going up by 5’. This is correct, but it is not helpful for algebra. Instead we need them to see the multiplicative pattern:

Step 2: Illustrate the role of common factors with diagrams

Illustrate why cancelling common factors gives an equivalent (equal) fraction, as in the diagram.

Diagram showing that:

The top bar is divided into twelfths. There is a common factor of 3, so we can group both the 15 and the 12 small regions into a whole number of groups of 3. This produces 4 larger regions shaded out of 5, which is 4 fifths of the bar.

We can write this process symbolically as shown in the image below. As preparation for algebra, it is important that students become familiar with fractions in the factorised, ‘unclosed’ form.

Step 3: An alternative justification using fraction operations

Students should see the link between fraction algorithms illustrated in the image below, and equivalent fractions. These justify cancelling.

Equivalent fractions and factors

Fraction operations justifying cancelling

For example:

Step 4: Cancelling by explicitly writing out the common factors
The most efficient way of simplifying fractions involves finding any common factor of the numerator and denominator, cancelling that, and then looking for common factors of the newly obtained simpler equivalent fraction. The process is repeated until the fraction is reduced to its simplest form (which means there are no more common factors to be found).

and with many flexible variations such as

Get students to do a few examples such as 20/50, 15/25 and 16/64, writing them out step by step in the way shown above, and then build up to examples with more factors and factors that are harder to find. Writing out the factors is suggested as a bridge to algebra, where the factors often appear explicitly in the fraction (e.g. xy/xyz)

Step 5: Learning the shorthand process
Finally, demonstrate the shorthand procedure (see below). Stress again that the aim is to find a fraction equivalent to (this means equal to!) the first fraction (and to the second one!) Practise this by simplifying other fractions of similar complexity.

Think

Write

What is a common factor of 128 and 240?
Any factor will do, so choose 4.

Divide both the numerator and denominator by 4 to give a simplified, but equivalent fraction (32/6).

What is a common factor of 32 and 6?
Any factor will do, so choose 2.

Now divide the numerator and denominator by 2. The simplified fraction is 16/3.

Any more common factors?
No. So the fraction is in its simplest form.

(This working is shown as a series of steps, but students would normally show this on the one fraction)

Equation worked out: 128/24

The working that a student using this method would show is

Equation short answer: 128/24

Indicator of Progress

For Fractions for Algebra and Arithmetic indicators of progress, see: Part 1