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.
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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
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| Source: mathx.net |
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.
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| 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.
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| 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.
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| http://edtech2.boisestate.edu/brianroska/506/finalproject/gcf.html |
3. Prime Factor Pairing
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.
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| 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.
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.
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.
6. The 'Upside Down Birthday Cake Method'
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| Source: https://uk.pinterest.com/pin/13229392628819982/ |
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.
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.
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| 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:
"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?"
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.
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