factorising – Chelekmaths

Spot the difference! Which wording are you used to and why?

Asking “What is Factorising?”

The majority of the time, when I hear this question asked in class I have heard the word “Brackets” come up in the response. As a teacher when I hear this answer I generally take it as correct – The student has clearly remembered part of the method behind factorising and is stating a key symbol they use during that process. But there would usually be silence when I asked how factorising related to its root – Factors. What are the factors of 3x+3 – why does it feel strange to describe (x+1) as a factor?

Using a Factor Tree to Practise Factorising

When students are taught to write numbers as the product of their prime factors we tend to use a “tree” to split the numbers (As shown below)

Example of Prime Factor Tree with Prime Numbers circled

I decided to see if I could use this same method to practise students ability to factor expressions as well and then create a link between what you are doing in both case – Splitting something into a product of their factors.

So I made a worksheet and gave it to my students to see what they would make of it. As the sheet contains a lot of structure I let them initially work through it in pairs and without giving them too much introduction so that they could do all the thinking themselves. Before this lesson we had previously practised how to factorise different simple linear expressions (Using the worksheet here) so we had already seen and discussed some of the maths behind expanding and factorising expressions.

The Worksheet

(The worksheet is attached here – I encourage you to try it yourself before continuing!)

The purpose of the worksheet is for students to recognise that when they are factorising the parts they have inside and outside the bracket are the factors of the expression. Intuition that will help with many similar problems later in their mathematical careers.

Worksheet starts with an example and builds from there – fill in the blanks!

The sheet uses the idea of a “Slow Build” to get students to start from something they are comfortable with and slowly build to deeper questions

Moving onto using the same method for splitting expressions into factors

After quickly discussing questions 4 – 6 with the class they were then left to attempt the rest of the worksheet while I circulated to check on progress. The sheet contains various questions and hints that students are required to write an answer for which lead to some fantastic discussion throughout.

Example of hints and questions for encouraging deeper thought. In particular looking at the different ways you can start to split equivalent expressions.

The rest of the worksheet contains more practise including looking at harder examples eventually moving to expressions with multiple variables that they hadn’t seen before (Although in the context of this worksheet were answered quite successfully!)

This Question on the left gets the students to ask if there is a more efficient way of answer the same question leading to a factor of 6 on the outside of the bracket later on. As a class we discussed the differences between prime factorisation and splitting things into their factors and also using brackets when writing something in its factorised form. This question on the left encapsulates lots of these thoughts at once.

Factor(is)ing

I am undecided as whether I like using the word Factorising when looking at products of factors or whether I prefer the more american Factoring. The (is) in Factorising feel slightly unnecessary and Factoring seems to convey just as simply that we are going to be looking into factors. Let me know which one you use!

Post Credits

Overall this worksheet created some very interesting discussion with my students and also led to increased fluency when dealing with factoring questions in later lessons. The idea that factors can also be algebraic expressions and not just integers was also a great source of conversation in the classroom during the lesson.

I have included an image of the whole worksheet below in case you are unable to view the download. Answers are available through the Resources page.

The whole worksheet – click it to download

Thanks for reading!

NJK

The Problem

Question taken from Brilliant.org – The most glorious of websites (Ans: a=4 b=3 can you show why)

As with most of the Maths that I think about I got pretty obsessed with Factorising by Grouping after doing a question on Brilliant.org. (I will not spend too much time here talking about how much I love Brilliant.org but I absolutely absolutely love Brilliant.org and think that anyone that is interested in Maths and Problem Solving would love scrolling through its hallowed courses)

I was also in the process of discussing the factor theorem with my year 11 students and was repeatedly asked “Do I just use trial and error to find a factor”. As well as this technique works it doesn’t feel all that satisfying and certainly there is more fun maths to be found here if we look around.

The Fun Maths

The start of the Slow Build worksheet – start with something they have seen before

When putting together exercises to try and practise this skill I started with using factorising by grouping to factorise quadratics – something that the students were confident in and had seen before.

As the worksheet progressed the questions slowly increase in difficulty, with the scaffolding being taken away in steps.

Cubics joining the party

Eventually moving on to some all together more tricky cubics that require a splitting of the middle terms. This is a technique that is often used to factorise quadratics where the coefficient of the x squared term >1 but I had never seen it used to factor cubics.

Its these questions that I think were particularly fun to play around with as it is not immediately obvious how to split our terms nicely. I would encourage you to try to complete these questions as well as the rest of the questions on the sheet.

Try these questions yourself!

Post Credits

As we continued to try this out with different cubics it felt like we were once again using trial and improvement to work out how to split these trickier cubics up. It didn’t feel like a fruitless exercise as working out why your choices weren’t working and trying to choose better options feels like it has great benefit in become more fluent in your algebraic manipulation and also was just a fun way to practise lots of smaller expanding and factoring skills. It is always fun to practise these skills using deeper problems!

I spent a long time trying to come up with a clear method for factoring cubics by grouping when then grouping isn’t immediately obvious and I am still struggling! If you come up with anything please let me know at chelekmaths@gmail.com – I am excited to keep learning!

This is a link to the worksheet – click to download. You can also find it in the Slow Build section of the website as well as on our resources page.

NJK