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

Angles in Polygons

There are two key learning points when solving problems with angles in polygons.  The first is to understand why all the exterior angles of a polygon have a sum of 360°.  The second is to understand the interior and exterior angles appear on the same straight line. Students can be told these two facts and […]

Getting Ready for a New School Year

When getting ready for a new school year I have a list of priorities to work through. Knowing my team have all the information and resources they need to teach their students gives me confidence we will start the term in the best possible way.  Mathematics Teaching and Learning Folder All teachers receive a folder […]

Mathematics OFSTED Inspection – The Deep Dive

Earlier this week, my school took part in a trial OFSTED inspection as part of getting ready for the new inspection framework in September 2019. This involved three Lead Inspectors visiting our school over the course of two days. The first day involved a ‘deep dive’ by each of the Lead Inspectors into Mathematics, English […]