Factorising algebraic expressions with powers

When factorising algebraic expressions with powers students often struggle to identify the highest common factor when it involves an algebraic term. For example, factorising 3h + 12 as 3(h + 4) is attempted correctly much more often than factorising 3h2 + 12h as 3h(h + 4). In this lesson students learn how to identify the highest common factor of expressions that include algebraic terms.

Recognising letter terms as possible factors

The starter activity helps students to understand factors are not restricted to numerical terms but could also include letters (or unknowns).

To help with this I have included some possible factor pairs of 36x2y. When I first taught this one student made an interesting point of including 72x2 and 1/2 y as a factor pair. Algebraically this works but we would not typically include fractions or decimals when finding the factor pairs of a number.

factorising algebraic expressions with powers

I ask students to work on mini-whiteboards to write at least 5 factor pairs of the term 40ab2c. For the purpose of this activity factor pairs involving fractions and decimals are encouraged as this comes up later in the lesson. A couple of the higher ability students include factor pairs such as 40abd and bcd-1 which I applaud.

Factorising algebraic expressions with powers

When factorising terms involving powers students need to understand the basic rules of indices. As we work through the first couple of examples I write out each term as a product of the highest common factor. For example,

r3t + rt2

r3t = rt × r2

rt2 = rt × t

rt(r2 + t)

6w2y – 8wy2

6w2y = 2wy × 3w

– 8wy2 = 2wy × – 4y

2wy(3w – 4y)

8u3c2 – 20u2c

8u3c2 = 4u2c × 2uc

– 20u2c = 4u2c × -5

4u2c (2uc – 5)

I work through the first questions in this way and ask the class to attempt the next two on mini-whiteboards with similar working out. After this students work in their exercise book to match the equivalent expressions.

Complex factorisations for the most able

As students work independently through the questions on the third slide I challenge the most able to factorise expressions similar to those in the extension. I remember a similar question appearing in the final GCSE paper a couple of years back. The examiner’s report noted most students didn’t have a clue.

To factorise 16(f + d)2 + 8f + 8d most students only recognise 8 as a common factor. We discuss the need to factorise 8f + 8d and rewrite the expression as 16(f + d)2 + 8(f + d). It is now easier to see 8 and (f + d) are both common factors. 16(f + d)2 + 8f + 8d factorises to 8(f + d) (2f + 2d + 1).

Linking factorising algebraic expressions with powers to perimeter and area

In the plenary we investigate factorising the algebraic area of a rectangle to find possible perimeters. I ask everyone to include taking out the highest common factor for one of their solutions.

factorising algebraic expressions with powers

The plenary takes about 8 minutes with students working in pairs on mini-whiteboards.

Supporting the less and challenging the most able

For the less able students I emphasise the need to find the highest common denominator as they quite often only partially factorise by taking out the numerical factor.

For more able students I include problems involving negative powers such as 18x-2y + 27xy-2 = 9xy(2x-3 + 3y-3) This requires a greater understanding of the rules of indices.

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes:

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Mr Mathematics Blog

How to Simplify Surds

When learning how to simplify surds students need to understand the difference between a rational and irrational number. Rational numbers include integers and terminating and repeating decimals. They can be written as a fraction with both the numerator and denominator as integers. An irrational number is a number which, in its decimal form does not […]

Sharing an Amount to a Given Ratio

There are three common methods for sharing an amount to a given ratio.  Depending on the age group and ability range I am teaching I would choose one approach over the other two. The three methods are: Using fractions Unitary method Using a table In this blog I will demonstrate each of the three methods […]

Plotting Scatter Graphs and Understanding Correlation

To introduce plotting scatter graphs and understanding correlation I ask students to think about the relationships between different variables and to describe how they might be related. Here’s my starter activity which students discuss in pairs then present to me on mini-whiteboards. When the students have had time to discuss the matching pairs we talk […]