Tag Archives: factor

350. Quadratic factor puzzle

Back in posts 95. Quadratic puzzles and 322. Quadratic puzzles I’ve looked at how to approach factorising and solving quadratic equations/expressions in a ‘gentle’ way.

Time to take off the kid gloves!

I have an awesome class of 13 year olds who are starting out on quadratic manipulation. They are great, but there are a significant number who rush their work and skip steps of working out because they ‘know what they are doing’. Really? Let’s see …

I gave the class twelve quadratic expressions and asked them to factorise them, then to spot any common themes. What I didn’t tell them was that all of the factors used were combinations of x, 2x, +/-1 and +/-5. If they were sloppy with their attention to detail, their solution would look like the solution to a different expression. Essentially a difficult easy task.

It soon sorted out those who had at true understanding of factorising a quadratic from those who’d lucked their way through easier questions.

I’ve shared the presentation and pdf version below. I’ve added in two slides where you can cut out the expressions to use as more of a card sort. You’ll notice that there are no 4x^2 expressions – I was focussing on solutions with only one x co-efficient greater than one. Although I used this as a starter, you may wish to use it as a longer activity, depending on your class.

Solving quadratic expressions (PPT)

Solving quadratic expressions (PDF)

99. Factor Races

I rather like teaching prime factor decomposition as you can assess lots of numerical skills within the topic. I can easily cover:
*Tests of divisibility
*Quick recall of multiplication facts
*Prime numbers
*Factor/Multiple misconceptions
*Powers & Index notation
*Venn diagrams*
*Vocabulary related to all the above

Many people already use prime factor trees to teach this topic, but if you are unfamilar with them here is a quick summary:

Find two numbers that multiply to give the top number.


Repeat for each branch, circling the prime numbers. These are like the fruit on the end of the branch.


Repeat until every branch has ‘fruit’ at the end.


Write out the factors, in numerical order, as a multiplication.


Collect like factors into index notation.


And that’s how to make a prime factor decomposition tree.

The Race
You will need as many pupils as you can standing at your board, all equipped with a whiteboard pen. Depending on which room I am in, I get about 10 pupils out.

Their team mates sit near them – it is up to you as to whether calculators are allowed. Only the person at the board can write.

You call out a number and every team must work out the prime factor decomposition on the board. The winning team is the first to write the number as a product of prime factors.

Teaching Point
Once everyone has completed the task, leave the calculations on the board. You can now ask for comments and corrections. The class should notice that even though the number was split up differently, they all got the same answer. If they didn’t, the class can check for errors.

I like to use this as a plenary or a recap starter. It effectively demonstrates that even though your brain chose to breakdown the calculation differently, you are still correct. This can be a confidence boost to those pupils who think there is only one possible method and don’t ‘get’ that method. Maths is about the strategies and skills to solve problems, not just one approved technique.

¤ To be covered in the next blog post

88. Factor (Nasty) Game

Today, it’s a classic maths game which has been passed around for at least 12 years, if not longer. It develops basic number skills such as multiplication, division, factors, primes, squares and place value.


A whiteboard (or wall or floor)
Ten digit cards (0-9): you could also just write the digits, but cards are more fun/tactile.
Mini-whiteboards (or paper)
Calculators are optional

Before you start
You need between 2 and 5 teams. Five gives you more tension/excitement.

Each team needs a whiteboard.
Indicate on the board where each team puts their cards. They need tens and units columns.

Blu tak the number cards to the board.

Basic game
Each team takes it in turn to pick a card and place it in one of their columns. This is repeated so that every team has a two digit number. If you have five teams, there will be no leftover digits.

The winning team is the one whose number had the most factors. It is up to the teams to prove this by giving all the calculations of factors. Other teams may challenge their accuracy. You can also discuss why some numbers have an odd number of factors (eg square numbers) or exactly two factors (prime numbers).

Time to get Nasty
I said this game was called Factor Nasty … and it is.

There is only one change to the game rules: you can pick a digit and put it in any team’s answer box. They can use their number skills to really stitch each other up.

Imagine Team A puts a 0 in Team C’s tens column. Then Team B puts a 3 in Team C’s units column. Team C can’t do anything to improve their number, but they can make life tricky for Team A and B. Meanwhile, Teams D and E ignore them and create really good answers.

Have fun trying this as a starter or plenary!