1. Algebraic Fractions
Mathematics Grade 10
Teacher Guide
Algebraic Fractions
? overview
Series
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InRthis series, we establish the basic concepts of factorisation and then look at some methods of
simplifying algebraic fractions. We move on to multiply, divide, add and subtract algebraic fractions.
E
2+2=4
Curriculum links
The lessons in this series link to the following Learning Outcomes and Assessment Standards of the
National Curriculum Statement:
Learning Outcome 2: Functions and Algebra
10.2.4 Manipulate algebraic expressions by:
(b) factorising trinomials;
(c) factorising by grouping in pairs;
(d) simplifying algebraic fractions with monomial denominators.
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R
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2+2=4
E Educational
approach
Algebraic fractions are used in many areas of mathematics. If learners can’t simplify algebraic
expressions, this can slow their progress in all other areas of mathematics. Algebraic fractions use
several mathematical processes, including basic arithmetic, factorising, multiplying, dividing and
simplifying.
We build the skills needed to work with algebraic fractions, starting with easy examples and
progressing to examples that require factorising, multiplying or dividing and finding a lowest common
denominator. Through the series, make sure that your learners know the meaning of the instruction
‘simplify’. At times, we need to factorise in order to simplify and the first steps of an example don’t
always look simple! We explain the difference between trying to cancel terms and cancelling factors.
We also remind learners to simplify as far as possible – to leave the answer in the simplest form.
We explain and demonstrate only one or two examples of each concept or method that we use. We
leave it to you to reinforce the learning with many examples for learners to work on in class and for
homework.
To get the full benefit of the lessons, your learners need to engage actively with the concepts
presented. So, when you preview the videos, think about how to introduce each lesson and what
follow up activities will be useful. Also watch out for places in the video where you can pause to have
a class discussion, or ask learners to complete an activity or solve a problem posed in the video. We
have used this pause icon
to suggest some of these places to you.
Mathematics Grade 10
Teacher Guide
Algebraic Fractions
Series at a Glance
Lesson title
Lesson Outcomes
By the end of this lesson, the learner should
be able to:
1. Revision of Factorisation
• factorise algebraic expressions
2. Simplification of Fractions
• factorise using a common factor
3. Simplification of Fractions
• simplify an algebraic fraction by finding a
common factor that has two terms
4. Simplification of More Advanced
Fractions
• simplify expressions that include the
difference of squares
5. Simplification of More Advanced
Fractions
• simplify expressions that include trinomials
6. Fractions including Trinomials
• factorise trinomials
7. Multiplication of Fractions
• multiply algebraic fractions
8. Multiplication of Fractions
• multiply algebraic fractions that require the
changing of a sign
9. Division of Fractions
• divide algebraic fractions
10. Finding the Lowest Common Multiple
• find the lowest common multiple (LCM)
11. Addition and Subtraction of Fractions
• add and subtract algebraic fractions
12. Algebraic Fractions
• recognise each type of algebraic fraction
• perform all operations on algebraic
fractions.
Mathematics Grade 10
Teacher Guide
Algebraic Fractions
G Teaching Guidelines
Lesson 1: Revision of Factorisation
We present products and factors in terms of the area of a rectangle. This gives learners a visual
understanding of a product as length times breadth to get area. The factors of this area are the length
and the breadth of the rectangle. We then move on to identify and take out the common factor in
simple algebraic expressions.
Task
We provide two simple examples for learners to factorise. Supplement this task with more examples
from a textbook.
Lesson 2: Simplification of Fractions
We link an understanding of factors and ‘cancelling’ common factors with dividing in a fraction. Check
that your learners can identify the separate terms in an expression. Learners often cancel terms in a
fraction without checking that this can be done. You could use a numeric example to convince them
that we cannot cancel terms when we divide. For example, 2 + 2 . If they cancel one of the 2s with
4
the 4 in the denominator, they will not get the correct answer. This is because they need to consider
the whole of the numerator as one term. The dividing line can be thought of as a bracket that holds
the numbers at the top together.
Task
We provide two simple examples for learners to factorise. Supplement this task with more examples
from a textbook.
Lesson 3: Simplification of Fractions
We show how the common factor in an algebraic expression can have two terms within it. For
example the term a – b could be a common factor in an expression. Then we move onto a method
of grouping terms in an algebraic expression in order to identify the common factor. Learners should
develop the habit of using brackets to group the terms such as (a – b) to remind them that the whole
expression in brackets is a factor.
Task
Supplement this task with more examples from a textbook. Check that learners realise that the
expression in question 1 cannot be simplified further.
Lesson 4: Simplification of More Advanced Fractions
We revise how to factorise and simplify an expression that is a difference of two squares. Then we
use this to simplify more complicated algebraic fractions. Learners may need help to recognise the
difference of squares within an algebraic expression.
Task
We give a more complex algebraic fraction to simplify. Learners need to combine the methods they
have learnt so far. They need to take out the common factors in the numerator and the denominator,
factorise the difference of squares in the numerator, and cancel the common factors in the numerator
and the denominator. Supplement this task with more examples from a textbook.
Mathematics Grade 10
Teacher Guide
Algebraic Fractions
Lesson 5: Simplification of More Advanced Fractions
We explain how to factorise a trinomial using the area of a rectangle. We use an example with two
positive factors, but it is possible to use this method to find a factor with a negative number in it as
well. This lesson is used to ‘make sense’ of factorising trinomials.
Task
This task is quite challenging, as you need to show x – 1 in terms of length.
Lesson 6: Fractions including Trinomials
In this lesson we revise how to factorise trinomials by focusing on the sum and the product of
numbers, rather than using a diagram. Then we use this knowledge to simplify algebraic fractions.
Task
Note that the task requires learners to change the factor x – 3 into 3 – x in order to simplify. We
explain how to do this in lesson 8.
Lesson 7: Multiplication of Fractions
In this lesson we learn how to multiply algebraic fractions by working with factors. We need to
remember the difference between factors and terms to decide whether we can simplify a fraction or not.
We establish the important concepts using numbers without any variables. Remember that when
we multiply fractions, we multiply the numerators by the numerators and the denominators by the
denominators. Then we move onto algebraic examples and show how to cancel factors with each
other. Remind learners that what they are really doing is dividing through by the same factor in
the numerator and in the denominator, which means that they are not changing the value of the
expression. They are merely multiplying by 1 over 1, so the value of the expression stays the same.
Task
Learners need to multiply two complex fractions. Remind them that their final answer must be in the
simplest form. Supplement this task with more examples from a textbook.
Lesson 8: Multiplication of Fractions
We show learners how to manipulate an expression for the purpose of simplifying it. One way to do
this is to take a common factor of –1 out of a bracket. For example, b – a can be written as –1(a – b).
This is easy to understand in a numeric example such as 7 – 2 = –(2 – 7).
Task
Learners need to use the methods they have learnt so far to do this task. Supplement this task with
more examples from a textbook.
Mathematics Grade 10
Teacher Guide
Algebraic Fractions
Lesson 9: Division of Fractions
We establish a solid understanding of division. It is not enough to give learners a rule that says
you can flip over a number or invert it. Learners are then likely to use this rule in the wrong places,
because they don’t really understand it. We discover that a number multiplied by its reciprocal always
gives an answer of 1. Division is the inverse operation of multiplication. Instead of dividing, you may
multiply by the reciprocal. Once we have established this principle, we use it to simplify more complex
algebraic fractions that involve division.
Task
Supplement this task with more examples from a textbook.
Lesson 10: Finding the Lowest Common Multiple
We establish a visual and concrete understanding of equivalent fractions. Then we use this to explain
the concept of the lowest common multiple of both numbers and algebraic expressions. We calculate
the LCM of several examples.
Task
This should be revision work that prepares the learners for finding the lowest common denominator
of algebraic fractions in the next lesson.
Lesson 11: Addition and Subtraction of Fractions
We use the lowest common multiple (LCM), or lowest common denominator (LCD), to simplify
algebraic expressions that are a sum or difference of two algebraic fractions. Learners are given a
four-step process to follow. Learners need to change the denominators to the LCD in each fraction.
Then they need to multiply each term of the numerator by the same thing as they multiplied the
denominator by, to ensure that they are multiplying the whole fraction by 1. Remind learners to be
systematic in their work, using each step of the process and checking their accuracy as they go.
Task
Supplement this task with more examples from a textbook.
Lesson 12: Algebraic Fractions
We use all the tools established in this video series. Learners should now be able to identify what
steps are needed for different types of questions that are asked. Allow learners to analyse problems
for themselves and make their own decisions. We work through several examples that need a
combination of the steps explained in different lessons. Some of our examples challenge learners to
work with binomial denominators, so this prepares them for work in the Grade 11 curriculum.
Task
This task requires learners to summarise the terms they have learnt and the steps to follow when
simplifying algebraic fractions. We suggest you use their summary and an appropriate activity from a
textbook to assess whether learners are able to manipulate algebraic expressions successfully.