There are many methods for finding the Highest Common Factor as well as Lowest Common Multiple of ii or to a greater extent than numbers. For years I've been didactics pupils to pose the prime number factors into a Venn Diagram, every bit described here.
I recently discovered an option method that is impressively quick as well as simple. It is described inwards this video every bit the 'Indian Method'. It's like to the 'upside downwards birthday cake method' but it's much quicker because at that spot is no requirement to operate primes.
Say nosotros desire to notice the Highest Common Factor as well as Lowest Common Multiple of 24 as well as 36. Write down the ii numbers, then (to the left, every bit inwards my representative below) write downwards whatsoever mutual component (ie 2, 3, 4, half dozen or 12). I've chosen 6. Now separate 24 as well as 36 past times half dozen as well as write the answers underneath (4 as well as half dozen inwards this case). Keep repeating this procedure until the ii numbers have no mutual factors (ie 2 as well as iii below). Now, your Highest Common Factor is merely the production of numbers on the left. And for the Lowest Common Multiple, notice the production of the numbers on the left as well as the numbers inwards the bottom row. It's slowly to retrieve which is which - to notice the LCM, await for the L shape.
It's hence quick! And simple! Try it.
Don Steward features an option method inwards this weblog post. He mentions that you lot tin notice a LCM past times dividing the production of the ii numbers past times their HCF ie inwards this example, (24 x 36)/12 = 72.
See my Resources Library for resources for didactics HCF as well as LCM.
I lately presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks as well as Tips: Clever Methods for Explaining Mathematical Concepts'. This shipping service summarises the content of that workshop for those who were unable to attend. I direct maintain quite a lot to embrace thus I await I'll demand to write iii or 4 spider web log posts. In this 1 I'm going to explicate the rationale for the workshop as well as depict choice methods for finding a Highest Common Factor. In subsequent posts I'll embrace sequences, linear graphs, surds, quadratics, chemical compound measures as well as a few to a greater extent than bits as well as pieces.
Workshop aim How produce you detect the Highest Common Factor of 2 numbers? Do you lot piece of employment a Venn method? List the factors? Use the Euclidean Algorithm? Are at that topographic point 'better' methods that you lot don't know about? (how produce nosotros define 'better'?). These are the form of questions I desire to explore. The aim of my workshop was to encourage people to reverberate on their acre of written report cognition as well as the effectiveness of their explanations. I also hoped that everyone would larn novel methods that they mightiness consider using at school.
What determines the agency nosotros pick out the explicate things?
Most teachers constitute instruction habits during their grooming as well as NQT year. The method they piece of employment the outset fourth dimension they instruct a theme volition belike remain with them throughout their career - unless they brand a concious elbow grease to essay a novel method.
One of the few things I recollect virtually GCSE maths was that I solved equations past times 'moving price over the equals sign' (the 'magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then I was convinced that it was 'the' agency to solve equations.
Source: https://mrjasonto.wordpress.com
When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't embrace much inwards the agency of mathematical methods, thus nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years direct maintain non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through risk encounters. My outset always spider web log shipping service was virtually an choice method for matrix multiplication that I'd happened to stumble across online.
During my PGCE I was asked to instruct a lesson on Highest Common Factor. I recollect my mentor showing me the 'Venn Method'. As a final result of that conversation I used the Venn Method for years, until past times risk a colleague mentioned an alternative. Let's aspect at that choice method now, as well as a position out of others. Will you lot stick with the method you lot know as well as love, or volition you lot essay something new?
Highest Common Factor as well as Lowest Common Multiple
I've identified vi methods for finding the HCF as well as LCM of 2 numbers. I'll explicate each method hither as well as position whatsoever pros as well as cons.
1. Listing
Source: mathx.net
There's no impairment inwards the listing method. It's brilliant inwards price of underlying conceptual agreement - students tin reckon precisely what they're trying to hit here. List all the factors of the 2 numbers as well as detect the biggest position out that's inwards both lists - that's your HCF. List all the multiples of 2 numbers as well as detect the smallest position out that's inwards both lists - that's your LCM. Simple! Shame it's thus time-consuming. And, inwards my experience, students sometimes fille factors from their list. One agency to avoid this is past times listing factors using a pairing method similar this:
Factor rainbows are a pretty choice (see this article from the NCTM).
2. The Venn Method
This is a pop method inwards the UK. First, nosotros demand to produce a prime number cistron breakdown. By the way, if you're instruction prime number factorisation as well as then you lot mightiness similar these lovely magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then I was convinced that it was 'the' agency to solve equations.
Source: https://mrjasonto.wordpress.com
When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't embrace much inwards the agency of mathematical methods, thus nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years direct maintain non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through risk encounters. My outset always spider web log shipping service was virtually an choice method for factor tree activities from Don Steward.
Once you lot direct maintain the prime number factors of each number, clitoris a Venn diagram as well as house the mutual factors inwards the intersection of 2 sets, as shown inwards the instance below.
tutorvista.com
The HCF is the production of the elements inwards Influenza A virus subtype H5N1 intersection B (ie 2 x 2 x 2 x 2 inwards the instance above) as well as the LCM is the the production of all the elements inwards Influenza A virus subtype H5N1 spousal human relationship B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that U.K. GCSE students are non withal familiar with this terminology (ie intersection as well as union), but they volition last nether the novel GCSE syllabus.
Even though I taught the Venn method for years, I'm non a huge fan of it. In my experience, students are ok with filling inwards the Venn diagram but as well as then they oftentimes can't recollect which 'bit' is the HCF. If they produce recollect the method as well as then they belike don't direct maintain a clue why it works.
Confusingly, it seems that some people piece of employment a dissimilar Venn method which involves putting all factors (not prime number factors) into a Venn diagram as well as identifying the highest cistron inwards the intersection (see instance below). This is some other shape of the listing method described inwards a higher house - it's just a dissimilar agency of organising the list. Let's telephone telephone this Lenn Method - it's a hybrid of Venn as well as Listing.
An alternative to the Venn Method is to produce the prime factorisation but as well as then skip the Venn. Write the prime number factors of each position out out as shown inwards the instance below thus it's slowly to reckon which factors appear inwards both position out - the production of these is the HCF. This method is featured inwards magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then I was convinced that it was 'the' agency to solve equations.
Source: https://mrjasonto.wordpress.com
When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't embrace much inwards the agency of mathematical methods, thus nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years direct maintain non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through risk encounters. My outset always spider web log shipping service was virtually an choice method for this post past times Don Steward.
4. Euclidean Algorithm
I honey the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to reckon how straightforward it is.
The method (including why it works) is explained inwards James Tanton's video below. I actually similar this method but for some argue I'm hesitant to piece of employment it with students... Would you?
Note that this method doesn't hand you lot the Lowest Common Multiple, but it's easily found 1 time you've got the Highest Common Factor.
This looks similar a pain, but cancelling helps - inwards the instance inwards a higher house I found the HCF of lx as well as 84, thus to detect the LCM I multiply lx past times 84 as well as separate past times the HCF.
5. The Indian Method
I've stopped using the Venn Method at schoolhouse as well as instantly piece of employment this instead. I'm non certain it's actually called the Indian Method, I'm entirely calling it that because of this video. At my schoolhouse nosotros telephone telephone it the Korean Method because a Korean pupil introduced it to us! I've found that my students actually similar it. It's difficult to acquire wrong. It's slowly to explore why it plant too.
Say nosotros desire to detect the Highest Common Factor as well as Lowest Common Multiple of 315 as well as 420. Write downward the 2 numbers, as well as then (to the left, as inwards my instance above) write downward any mutual factor. I've chosen 5. Now separate 315 as well as 420 past times five as well as write the answers underneath (63 as well as 84 inwards this case). Keep repeating this procedure until the 2 numbers direct maintain no mutual factors (ie 3 as well as 4 above). Now, your Highest Common Factor is only the production of numbers on the left. And for the Lowest Common Multiple, detect the production of the numbers on the left as well as the numbers inwards the bottom row (to detect the LCM, aspect for the L shape).
6. The 'Upside Down Birthday Cake Method'
I should holler this method because I proceed spotting it online (video here). The entirely divergence betwixt this as well as the Indian Method is that hither nosotros tin entirely withdraw prime factors. This seems unnecessary - the Indian Method is quicker. In the instance below, why separate past times 2 if you lot spot larger mutual factors? Why non start past times dividing past times 4, 6 or 12?
So that's it - vi methods for finding a HCF as well as LCM.
Integers I actually similar this magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then I was convinced that it was 'the' agency to solve equations.
Source: https://mrjasonto.wordpress.com
When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't embrace much inwards the agency of mathematical methods, thus nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years direct maintain non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through risk encounters. My outset always spider web log shipping service was virtually an choice method for set of questions from Don Steward. The next query confused 1 of my brightest Year 10s:
"A soul has a rectangular plot of dry soil measuring 8.4m past times 5.6m. To survey the position out of dandelions they desire to separate it as into the minimum position out of foursquare plots. What is the size of each foursquare plot as well as how many such squares volition at that topographic point be?"
My student's approach to this query was to endeavour to detect the HCF of 8.4 as well as 5.6 using the Indian Method. This is what she did:
But she realised that her respond made no sense. 28 can't last the HCF of 8.4 as well as 5.6. Can you lot reckon where she went wrong? Although you lot tin separate 8.4 past times vii (indeed, you lot tin separate 8.4 past times anything), it doesn't hateful than vii is a cistron of 8.4. The numbers on the left must last factors of the numbers at the top. Non-integers don't direct maintain factors. Influenza A virus subtype H5N1 improve approach would direct maintain been to convert the measurements to centimetres as shown below.
Factors vs Multiples
If your students fighting to recollect the divergence betwixt factors as well as multiples as well as then they mightiness detect this helpful: for factors, intend of a factory (where separate parts are position together). For multiples, intend of multi-packs eg if cokes are sold inwards multi-packs of 6 as well as then I tin purchase 6, 12, eighteen etc. These ideas are taken from this resource from the thechalkface.net.
Conclusion
Did you lot know all of these methods? Will you lot essay something new? Please permit me know of whatsoever proficient methods that I've missed.
Even if you create upwardly one's heed to stick with the method you're currently using, at to the lowest degree you've now reflected on how you lot instruct this topic. Teachers rarely direct maintain the chance to interruption as well as reflect.
This shipping service should last read with Ed Southall's post Complements #9 LCM as well as HCF which explains the underlying concepts.
The whole presentation from my workshop is here. In my adjacent shipping service (give me a few days to write it!) I'll embrace methods for instruction sequences, linear graphs as well as surds.
Hello as well as welcome to my 31st gems post. This is where I percentage roughly of the best instruction ideas I've seen on Twitter.
1. Ratio tables The novel GCSE is going to accept an increased focus on proportional reasoning. I enjoyed @MissNorledge's post virtually using ratio tables for non-calculator conversions. It's a logical agency to construction thinking.
This approach tin hold upward used for whatsoever sort of conversion. I despair when I overhear my Year 12s desperately trying to retrieve the 'formula' for converting betwixt radians as well as degrees. "Do I multiply yesteryear 180 therefore split yesteryear pi...? Or is it the other agency round?". It's far easier to memorise π radians = 180o therefore brand proportional adjustments. Simple.
2. Trigonometric Graph Transformations Jon Orr (@MrOrr_geek) has written a serial of Trigonometric Function Transformation Challenges inwards Desmos. These await peachy for instruction trig graph transformation - inwards each challenge, students accept to operate out the equation of the transformed business office - an instance is shown below.
3. Diagnostic Assessment
I actually similar the await of Alexander Cameron's (@AlexandeCameron) diagnostic assessments. Here's an example:
For to a greater extent than data virtually Alexander's assessments come across his TeachMeet presentation.
4. Number Properties I too similar this fantabulous activity shared by @DJUdall.
5. Lowest Common Multiple Hot Dogs I've never take in to role this video from Father of the Bride inwards a lesson on Lowest Common Multiple.
Update If you lot haven't seen my Pret homework website therefore arrive at accept a look. Lots of teachers tell me that they role Pret homeworks real successfully. I dear seeing examples of students' operate - these impressive examples of Pret homeworks were shared by @missradders.
Did you lot come across the Venn Diagram Subject Knowledge Check I produced ? This is designed for teachers who haven't taught Venn Diagrams earlier therefore they tin create for the novel GCSE. I computer program to brand these for every novel GCSE topic.
Speaking of GCSE, presumably you lot saw the Ofqual announcement this calendar week virtually the difficulty of the novel maths GCSE. Expect to a greater extent than Sample Assessment Materials yesteryear the halt of June. If you lot missed the announcement, there's a brusk video summary below (what's alongside the weird modify inwards photographic television set camera angle?!). This extract from the Ofqual study is actually interesting - it ranks the GCSE questions inwards fellowship of difficulty.
If you lot missed this post from mathwithbaddrawings.com virtually U.K. vs USA mathematical terminology therefore arrive at accept a read, it's real entertaining. I'm pleased to tell I've never called an index an indice!
Finally, if you lot haven't seen the updated speaker listing for the upcoming National Mathematics Teacher Conference therefore arrive at accept a look. Less than i calendar month to learn straightaway - promise to come across you lot there.
I'll move out you lot alongside this mathematical limerick, shared yesteryear @MrBenWard.
