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Knowing how to find the factors of a number enables students to connect with topics such as prime, square and perfect numbers. You can also link factors to area, perimeter and algebra. I am sure there are many more. The highest common factor for a pair of numbers has obvious applications in fractions and solving real life problems.

In this blog I want to show you how I connect to and introduce a range of topics through factors both in class and as homework activities.

At the start of a lesson on primes I challenge students to think of at least 10 numbers that have exactly two factors. I avoid phrasing it as having a factor of 1 and itself as this includes the number 1 which is not prime. A common misconception when listing primes is to include 9, 21 and 27. By working out the primes this way students are less likely to fall into this trap.

A nice homework activity is to have students list the first 12 numbers that have an odd number of factors. This way they consolidate their learning of factors and prepare for the next lesson on square numbers. An extension activity could be to have students explain why these numbers have an odd number of factors.

Wikipedia defines a Perfect Number as a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. Finding perfect numbers is my favourite plenary when teaching factor pairs as students use a trial and error to find the perfect number thus embedding factor pairs. This activity works well for all abilities because it is really easy to differentiate. I ask less able students to find a perfect number between 1 and 10, answer is 6. More able students have to find one between 1 and 30, answer 28.

To consolidate factor pairs at the start of the following lesson I present a rectangle with a fixed area. Students can earn a House Point for finding the dimensions of a rectangle with the i) the minimum and ii) the maximum perimeter for a rectangle with a fixed area of 24cm^{2}. More able students will consider decimal dimensions, such as 0.1 cm by 240 cm to create a perimeter of 480.2cm. Very few students have ever found the smallest perimeter using the square root of 24cm^{2}.

A nice plenary when teaching algebraic products is to find factor pairs for an algebraic expression, e.g., the factor pairs of 30x^{2}y. Students will typically list 30 and x^{2}y or 30x^{2 }and y but forget factor pairs such as 5x^{2} and 6y. A further extension to this could be to have a rectangle with area 30x^{2}y and students investigate different perimeters.

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