After the announcement of variationtheory.com at mathsconf15 I must admit I was very excited to try out some of the resources in my lesson. So when I was faced with product of prime factors I decided to see if there would be a variation theory resource out there and there was! https://variationtheory.com/2018/02/26/product-of-prime-factors/
I took the advice of the creators of variation theory (Craig Barton, Ben Gordon and Jess Prior) and implemented this into my lesson as one activity. My full lesson is here: product of prime factors product of prime factor treasure hunt
I used the two examples from the variation theory worksheet to introduce the class to prime factor decomposition and then introduced the activity to them. I used my two examples as models as how to set out their work and how to create an expectation. In the case of the examples they were 24 and 48 and so the students suggested that their expectation would be that the second question would be double the first number. At this point I was happy that they understood what was required and gave them the well thought out questions.
What became apparent from their work is the misconceptions that the students hold. (see picture below)
The first question was to write 10 as product of prime factors (hence why there is no expectation) and then the second question was to write 20 as the product of prime factors. You can see that the student has correctly identified that this would be double but they have not understood that to double means to multiply by 2, they have instead doubled the powers (double 2 x 5 would be 2 squared multiplied by 5 squared). This in itself is genuinely not something I would have necessarily considered before, This is not the only student to have made this error, it highlighted to me that my students may not be necessarily making links between topics. It dawned on me that perhaps my students are not consciously
aware that the product of prime factors is simply the original number written in a different way. I would be confident that if I asked my students to double 10 that they could tell me this would be the operation of mutiplying by 2 and would write it as 2 x 10, if they were then aware that 2 x 10 can be written as 2 x 2 x 5 then they would achieve the answer of 2 squared multiplied by 5. The misconception of course could lie with the index laws however in our examples we had discussed writing the product of prime factors using indexes so I am reluctant to say that indices would be the main cause.
What I really like about this activity is that it did indeed make my students think. They were not mindlessly carrying out a procedure, they were having to stop and think about the links between the numbers and use this to make predictions. This is the first time that I did this with my students and so naturally I got the comments ‘do I have to make an expectation?’ and ‘I don’t know what to expect’. I stuck to the advice of the creators and insisted that my students make predictions. When students said that they didn’t know what to expect I challenged them o
n that and dug a little deeper. My students did know how to make a predictions but didn’t necessarily know how to write their thoughts down (again, a very important skills that our students need to hone).
I personally think that this was a very worthwhile activity. It made my students think and encouraged them to make links between different skills. It highlighted to me as the teacher misconceptions that my students held and it meant that I could address them in the lesson and help them move forward.
If you are unsure as to whether use a variation theory activity or not I would certainly encourage you to use one in your lesson.