I've blogged and presented about methods for finding a Highest Common Factor and Lowest Common Multiple numerous times, but I still think that The Factor Method* deserves a bit more love.
[*also known as The Ladder Method, and various other names]
When I trained to be a teacher I was told that the Venn Method was THE way to find HCFs and LCMs. A few years later a colleague showed me a very different method that she'd learnt from a student, and this is what started my fascination with methods. I went on to write the book A Compendium of Mathematical Methods.
Never trust anyone who says that a particular method is the best way of doing something because there is absolutely no research to back up their claim (it's a great shame that no one in maths education academia does large scale studies comparing methods - it seems like a big gap in our profession's pedagogical subject knowledge). However, I think it's reasonable to describe something as a favourite method. The Factor Method is definitely my favourite.
I've a made a 28 minute video explaining in detail how to find HCFs and LCMs using the Factor Method. This is a video for teachers, not for students. In it I model a number of examples, including those that might spark discussion. I also talk about how we can use this method to find an HCF and LCM of three numbers, and how we can easily use the Factor Method to work backwards.
I'm not a YouTuber so please forgive the ropey handwriting...
Welcome to my 178th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.
1. Inverse Proportion I really like this tweet from @catrionateaches. When teaching inverse proportion I always talk about constant products but then I head straight into the formula y = k/x. Catriona suggests a subtle change in method.
2. Graph Transformations
MathsPad has published another brilliant interactive tool for the Higher GCSE topic graph transformations. It shows how points on the original curves map on to those on the transformed curves. There is a separate section for quadratic graphs, which is particularly useful for relating completed square form to graph transformations.
There is also a section for trigonometric graphs, where transformations of the graphs of y=sin(x) and y=cos(x) can be explored.
It's great to see a new resource published for teaching graph transformations - I always find that this is the GCSE topic for which it's trickiest to find suitable resources, because many tasks still include stretches (these were removed from the syllabus back in 2017 - GCSE now just covers reflections and translations).
Do check out MathsPad's full range of interactive tools - some of them are free to use even if you don't subscribe (you really should subscribe to MathsPad though!).
3. GeoGebra
It's worth following @geogebra to see ideas for interactive GeoGebra maths resources that can be used for demonstrations in lessons or student activities. Here are a few recent examples:
4. Pythagoras @nathanday314 shared a clever set of questions on Pythagoras' Theorem. There's a lot of challenge here (check out Question 12). Question 3 is deliberately impossible, and note the subtle differences between Questions 3, 4 and 5.
@1stclassmaths has shared GCSE Predicted Paper 1 for both Edexcel and AQA. After Paper 1 takes place in May he will publish Predicted Paper 2s, and after Paper 2 he will publish Predicted Paper 3s. These papers will be very high quality so I recommend following @1stclassmaths for updates.
My school is still seeking an A level teacher to start in September. I can offer 100% A level teaching, or alternatively I can offer a timetable with Year 11 - 13 (or similar) if that's more desirable. Our students are a delight to teach and my team is awesome. If you live in Surrey or South London, this is an amazing opportunity. Please get in touch (email resourceaholic@gmail.com) if you're interested - I'm very happy to arrange a tour or call.
I'm presenting at two maths conferences in the next four weeks. The first is #mathsconf34 which is near Bristol:
And in the Easter holidays I'm presenting at the Shape Up conference in Stratford-Upon-Avon:
Finally, it's been a while since I've hosted a social event so I'm excited about this... more information coming soon.
Hello and welcome to my 31st gems post. This is where I share some of the best teaching ideas I've seen on Twitter.
1. Ratio tables
The new GCSE is going to have an increased focus on proportional reasoning. I enjoyed @MissNorledge's post about using ratio tables for non-calculator conversions. It's a logical way to structure thinking.
This approach can be used for any kind of conversion. I despair when I overhear my Year 12s desperately trying to remember the 'formula' for converting between radians and degrees. "Do I multiply by 180 then divide by pi...? Or is it the other way round?". It's far easier to memorise π radians = 180o then make proportional adjustments. Simple.
2. Trigonometric Graph Transformations
Jon Orr (@MrOrr_geek) has written a series of Trigonometric Function Transformation Challenges in Desmos. These look great for teaching trig graph transformation - in each challenge, students have to work out the equation of the transformed function - an example is shown below.
3. Diagnostic Assessment
I really like the look of Alexander Cameron's (@AlexandeCameron) diagnostic assessments. Here's an example:
Update
If you haven't seen my Pret homework website then do have a look. Lots of teachers tell me that they use Pret homeworks very successfully. I love seeing examples of students' work - these impressive examples of Pret homeworks were shared by @missradders.
Did you see the Venn Diagram Subject Knowledge Check I produced ? This is designed for teachers who haven't taught Venn Diagrams before so they can prepare for the new GCSE. I plan to make these for every new GCSE topic.
Speaking of GCSE, presumably you saw the Ofqual announcement this week about the difficulty of the new maths GCSE. Expect more Sample Assessment Materials by the end of June. If you missed the announcement, there's a short video summary below (what's with the weird change in camera angle?!). This extract from the Ofqual study is really interesting - it ranks the GCSE questions in order of difficulty.
If you missed this post from mathwithbaddrawings.com about UK vs US mathematical terminology then do have a read, it's very entertaining. I'm pleased to say I've never called an index an indice!
Finally, if you haven't seen the updated speaker list for the upcoming National Mathematics Teacher Conference then do have a look. Less than one month to go now - hope to see you there.
I'll leave you with this mathematical limerick, shared by @MrBenWard.
I recently presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This post summarises the content of that workshop for those who were unable to attend. I have quite a lot to cover so I expect I'll need to write three or four blog posts. In this one I'm going to explain the rationale for the workshop and describe alternative methods for finding a Highest Common Factor. In subsequent posts I'll cover sequences, linear graphs, surds, quadratics, compound measures and a few more bits and pieces.
Workshop aim
How do you find the Highest Common Factor of two numbers? Do you use a Venn method? List the factors? Use the Euclidean Algorithm? Are there 'better' methods that you don't know about? (how do we define 'better'?). These are the sort of questions I want to explore. The aim of my workshop was to encourage people to reflect on their subject knowledge and the effectiveness of their explanations. I also hoped that everyone would learn new methods that they might consider using at school.
What determines the way we choose the explain things?
Most teachers establish teaching habits during their training and NQT year. The method they use the first time they teach a topic will probably stay with them throughout their career - unless they make a concious effort to try a new method.
One of the few things I remember about GCSE maths was that I solved equations by 'moving terms over the equals sign' (the 'magic portal method'). This method is now considered to be a shortcut which stands in the way of conceptual understanding. These days it is more acceptable to teach students to solve equations using inverse operations (ie balance the equation by 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me not to use the magic portal method because before then I was convinced that it was 'the' way to solve equations.
Source: https://mrjasonto.wordpress.com
When we were student teachers, we drew our methods from a variety of sources. ITT courses don't cover much in the way of mathematical methods, so we were left to gather ideas from examples in textbooks, our memories from school, observations of teachers and conversations with colleagues. It's these conversations with colleagues that are vital, but we simply don't get enough time for them. Most of the new methods I've encountered over the years have not been through organised CPD (ie 'collaboration sessions') within my department, but instead through chance encounters. My first ever blog post was about an alternative method for matrix multiplication that I'd happened to stumble across online.
During my PGCE I was asked to teach a lesson on Highest Common Factor. I remember my mentor showing me the 'Venn Method'. As a result of that conversation I used the Venn Method for years, until by chance a colleague mentioned an alternative. Let's look at that alternative method now, and a number of others. Will you stick with the method you know and love, or will you try something new?
Highest Common Factor and Lowest Common Multiple
I've identified six methods for finding the HCF and LCM of two numbers. I'll explain each method here and identify any pros and cons.
1. Listing
Source: mathx.net
There's no harm in the listing method. It's brilliant in terms of underlying conceptual understanding - students can see exactly what they're trying to achieve here. List all the factors of the two numbers and find the biggest number that's in both lists - that's your HCF. List all the multiples of two numbers and find the smallest number that's in both lists - that's your LCM. Simple! Shame it's so time-consuming. And, in my experience, students sometimes miss factors from their list. One way to avoid this is by listing factors using a pairing method like this:
Factor rainbows are a pretty alternative (see this article from the NCTM).
2. The Venn Method
This is a popular method in the UK. First, we need to do a prime factor breakdown. By the way, if you're teaching prime factorisation then you might like these lovely factor tree activities from Don Steward.
Once you have the prime factors of each number, draw a Venn diagram and place the common factors in the intersection of two sets, as shown in the example below.
tutorvista.com
The HCF is the product of the elements in A intersection B (ie 2 x 2 x 2 x 2 in the example above) and the LCM is the the product of all the elements in A union B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that UK GCSE students are not yet familiar with this terminology (ie intersection and union), but they will be under the new GCSE syllabus.
Even though I taught the Venn method for years, I'm not a huge fan of it. In my experience, students are ok with filling in the Venn diagram but then they often can't remember which 'bit' is the HCF. If they do remember the method then they probably don't have a clue why it works.
Confusingly, it seems that some people use a different Venn method which involves putting all factors (not prime factors) into a Venn diagram and identifying the highest factor in the intersection (see example below). This is another form of the listing method described above - it's just a different way of organising the list. Let's call this Lenn Method - it's a hybrid of Venn and Listing.
An alternative to the Venn Method is to do the prime factorisation but then skip the Venn. Write the prime factors of each number out as shown in the example below so it's easy to see which factors appear in both number - the product of these is the HCF. This method is featured in this post by Don Steward.
4. Euclidean Algorithm
I love the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to see how straightforward it is.
The method (including why it works) is explained in James Tanton's video below. I really like this method but for some reason I'm hesitant to use it with students... Would you?
Note that this method doesn't give you the Lowest Common Multiple, but it's easily found once you've got the Highest Common Factor.
This looks like a pain, but cancelling helps - in the example above I found the HCF of 60 and 84, so to find the LCM I multiply 60 by 84 and divide by the HCF.
5. The Indian Method
I've stopped using the Venn Method at school and now use this instead. I'm not sure it's really called the Indian Method, I'm only calling it that because of this video. At my school we call it the Korean Method because a Korean student introduced it to us! I've found that my students really like it. It's hard to go wrong. It's easy to explore why it works too.
Say we want to find the Highest Common Factor and Lowest Common Multiple of 315 and 420. Write down the two numbers, then (to the left, as in my example above) write down any common factor. I've chosen 5. Now divide 315 and 420 by 5 and write the answers underneath (63 and 84 in this case). Keep repeating this process until the two numbers have no common factors (ie 3 and 4 above). Now, your Highest Common Factor is simply the product of numbers on the left. And for the Lowest Common Multiple, find the product of the numbers on the left and the numbers in the bottom row (to find the LCM, look for the L shape).
6. The 'Upside Down Birthday Cake Method'
I should mention this method because I keep spotting it online (video here). The only difference between this and the Indian Method is that here we can only remove prime factors. This seems unnecessary - the Indian Method is quicker. In the example below, why divide by 2 if you spot larger common factors? Why not start by dividing by 4, 6 or 12?
So that's it - six methods for finding a HCF and LCM.
Integers
I really like this set of questions from Don Steward. The following question confused one of my brightest Year 10s:
"A person has a rectangular plot of land measuring 8.4m by 5.6m. To survey the number of dandelions they want to divide it equally into the minimum number of square plots. What is the size of each square plot and how many such squares will there be?"
My student's approach to this question was to attempt to find the HCF of 8.4 and 5.6 using the Indian Method. This is what she did:
But she realised that her answer made no sense. 28 can't be the HCF of 8.4 and 5.6. Can you see where she went wrong? Although you can divide 8.4 by 7 (indeed, you can divide 8.4 by anything), it doesn't mean than 7 is a factor of 8.4. The numbers on the left must be factors of the numbers at the top. Non-integers don't have factors. A better approach would have been to convert the measurements to centimetres as shown below.
Factors vs Multiples
If your students struggle to remember the difference between factors and multiples then they might find this helpful: for factors, think of a factory (where separate parts are put together). For multiples, think of multi-packs eg if cokes are sold in multi-packs of 6 then I can buy 6, 12, 18 etc. These ideas are taken from this resource from the thechalkface.net.
Conclusion
Did you know all of these methods? Will you try something new? Please let me know of any good methods that I've missed.
Even if you decide to stick with the method you're currently using, at least you've now reflected on how you teach this topic. Teachers rarely have the opportunity to pause and reflect.
This post should be read alongside Ed Southall's post Complements #9 LCM and HCF which explains the underlying concepts.
The whole presentation from my workshop is here. In my next post (give me a few days to write it!) I'll cover methods for teaching sequences, linear graphs and surds.
There are many methods for finding the Highest Common Factor and Lowest Common Multiple of two or more numbers. For years I've been teaching pupils to put the prime factors into a Venn Diagram, as described here.
I recently discovered an alternative method that is impressively quick and simple. It is described in this video as the 'Indian Method'. It's similar to the 'upside down birthday cake method' but it's much quicker because there is no requirement to use primes.
Say we want to find the Highest Common Factor and Lowest Common Multiple of 24 and 36.
Write down the two numbers, then (to the left, as in my example below) write down any common factor (ie 2, 3, 4, 6 or 12). I've chosen 6. Now divide 24 and 36 by 6 and write the answers underneath (4 and 6 in this case). Keep repeating this process until the two numbers have no common factors (ie 2 and 3 below). Now, your Highest Common Factor is simply the product of numbers on the left. And for the Lowest Common Multiple, find the product of the numbers on the left and the numbers in the bottom row. It's easy to remember which is which - to find the LCM, look for the L shape.
It's so quick! And simple! Try it.
Don Steward features an alternative method in this blog post. He mentions that you can find a LCM by dividing the product of the two numbers by their HCF ie in this example, (24 x 36)/12 = 72.
