Tricks Too Tips 1: Hcf

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 postal service summarises the content of that workshop for those who were unable to attend. I receive got quite a lot to comprehend therefore I human face I'll demand to write 3 or 4 weblog posts. In this i I'm going to explicate the rationale for the workshop as well as depict option methods for finding a Highest Common Factor. In subsequent posts I'll comprehend 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 create you uncovering the Highest Common Factor of 2 numbers? Do yous piece of occupation a Venn method? List the factors? Use the Euclidean Algorithm? Are in that location 'better' methods that yous don't know about? (how create nosotros define 'better'?). These are the variety of questions I desire to explore. The aim of my workshop was to encourage people to reverberate on their dependent area noesis 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 conduct the explicate things?

Most teachers flora pedagogy habits during their grooming as well as NQT year. The method they piece of occupation the start fourth dimension they learn a topic volition in all likelihood remain with them throughout their career - unless they brand a concious effort to assay a novel method. 

One of the few things I recall nearly GCSE maths was that I solved equations yesteryear 'moving price over the equals sign' (the 'magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore 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 comprehend much inwards the agency of mathematical methods, therefore 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 receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for matrix multiplication that I'd happened to stumble across online.

During my PGCE I was asked to learn a lesson on Highest Common Factor. I recall my mentor showing me the 'Venn Method'. As a final result of that conversation I used the Venn Method for years, until yesteryear jeopardy a colleague mentioned an alternative. Let's human face at that option method now, as well as a release of others. Will yous stick with the method yous know as well as love, or volition yous assay something new?

Highest Common Factor as well as Lowest Common Multiple

I've identified 6 methods for finding the HCF as well as LCM of 2 numbers. I'll explicate each method hither as well as position whatever 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 view precisely what they're trying to arrive at here. List all the factors of the 2 numbers as well as uncovering the biggest release that's inwards both lists - that's your HCF. List all the multiples of 2 numbers as well as uncovering the smallest release that's inwards both lists - that's your LCM. Simple! Shame it's therefore time-consuming. And, inwards my experience, students sometimes immature lady factors from their list. One agency to avoid this is yesteryear listing factors using a pairing method similar this:

Factor rainbows are a pretty option (see this article from the NCTM).

2. The Venn Method

This is a pop method inwards the UK. First, nosotros demand to create a prime number gene breakdown. By the way, if you're pedagogy prime number factorisation therefore yous mightiness similar these lovely magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore 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 comprehend much inwards the agency of mathematical methods, therefore 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 receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for factor tree activities from Don Steward.

Once yous receive got the prime number factors of each number, push clitoris a Venn diagram as well as house the mutual factors inwards the intersection of 2 sets, as shown inwards the illustration below.

tutorvista.com

The HCF is the production of the elements inwards H5N1 intersection B (ie 2 x 2 x 2 x 2 inwards the illustration above) as well as the LCM is the the production of all the elements inwards H5N1 matrimony B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that United Kingdom of Great Britain as well as Northern Ireland of Britain as well as Northern Republic of Ireland GCSE students are non even therefore familiar with this terminology (ie intersection as well as union), but they volition hold out 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 therefore they oft can't recall which 'bit' is the HCF. If they create recall the method therefore they in all likelihood don't receive got a clue why it works.

Confusingly, it seems that some people piece of occupation a dissimilar Venn method which involves putting all factors (not prime number factors) into a Venn diagram as well as identifying the highest gene inwards the intersection (see illustration below). This is some other cast of the listing method described higher upwards - it's just a dissimilar agency of organising the list. Let's telephone band this Lenn Method - it's a hybrid of Venn as well as Listing.

http://edtech2.boisestate.edu/brianroska/506/finalproject/gcf.html

3. Prime Factor Pairing

An alternative to the Venn Method is to create the prime factorisation but therefore skip the Venn. Write the prime number factors of each release out as shown inwards the illustration below therefore it's slow to view which factors appear inwards both release - the production of these is the HCF. This method is featured inwards magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore 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 comprehend much inwards the agency of mathematical methods, therefore 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 receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for this post yesteryear Don Steward.

4. Euclidean Algorithm

I dear the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to view 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 occupation it with students... Would you?

Note that this method doesn't plough over yous the Lowest Common Multiple, but it's easily found i time you've got the Highest Common Factor. 

This looks similar a pain, but cancelling helps - inwards the illustration higher upwards I found the HCF of sixty as well as 84, therefore to uncovering the LCM I multiply sixty yesteryear 84 as well as dissever yesteryear the HCF.

5. The Indian Method

I've stopped using the Venn Method at schoolhouse as well as straightaway piece of occupation this instead. I'm non certain it's actually called the Indian Method, I'm solely calling it that because of this video. At my schoolhouse nosotros telephone band 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 slow to explore why it industrial plant too.

Say nosotros desire to uncovering the Highest Common Factor as well as Lowest Common Multiple of 315 as well as 420. Write downward the 2 numbers, therefore (to the left, as inwards my illustration above) write downward any mutual factor. I've chosen 5. Now dissever 315 as well as 420 yesteryear v as well as write the answers underneath (63 as well as 84 inwards this case). Keep repeating this procedure until the 2 numbers receive got 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, uncovering the production of the numbers on the left as well as the numbers inwards the bottom row (to uncovering the LCM, human face for the L shape).

6. The 'Upside Down Birthday Cake Method'

I should scream this method because I proceed spotting it online (video here). The solely departure betwixt this as well as the Indian Method is that hither nosotros tin solely take prime factors. This seems unnecessary - the Indian Method is quicker. In the illustration below, why dissever yesteryear 2 if yous location larger mutual factors? Why non start yesteryear dividing yesteryear 4, 6 or 12?

Source: https://uk.pinterest.com/pin/13229392628819982/

So that's it - 6 methods for finding a HCF as well as LCM.

Integers
I actually similar this magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore 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 comprehend much inwards the agency of mathematical methods, therefore 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 receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for set of questions from Don Steward. The next query confused i of my brightest Year 10s:

"A individual has a rectangular plot of dry reason measuring 8.4m yesteryear 5.6m. To survey the release of dandelions they desire to dissever it every bit into the minimum release of foursquare plots. What is the size of each foursquare plot as well as how many such squares volition in that location be?"

My student's approach to this query was to effort to uncovering the HCF of 8.4 as well as 5.6 using the Indian Method. This is what she did:

But she realised that her reply made no sense. 28 can't hold out the HCF of 8.4 as well as 5.6. Can yous view where she went wrong? Although yous tin dissever 8.4 yesteryear seven (indeed, yous tin dissever 8.4 yesteryear anything), it doesn't hateful than seven is a gene of 8.4. The numbers on the left must hold out factors of the numbers at the top. Non-integers don't receive got factors. H5N1 improve approach would receive got been to convert the measurements to centimetres as shown below. 

Factors vs Multiples

If your students fighting to recall the departure betwixt factors as well as multiples therefore they mightiness uncovering this helpful: for factors, mean value of a factory (where separate parts are set together). For multiples, mean value of multi-packs eg if cokes are sold inwards multi-packs of 6 therefore I tin purchase 6, 12, eighteen etc. These ideas are taken from this resource from the thechalkface.net.

Conclusion

Did yous know all of these methods? Will yous assay something new? Please allow me know of whatever skillful methods that I've missed.

Even if you create upwards one's hear to stick with the method you're currently using, at to the lowest degree you've now reflected on how yous learn this topic. Teachers rarely receive got the chance to interruption as well as reflect.

This postal service should hold out read amongst 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 postal service (give me a few days to write it!) I'll comprehend methods for pedagogy sequences, linear graphs as well as surds.

5 Maths Gems #27

Hello too welcome to my 27^th gems post. This is where I percentage v of the best didactics ideas I've seen on Twitter recently. I'll survive brief today because I experience similar I'm drowning inward marking, reports too lesson planning at the moment! Thankfully it's alone ane calendar week until the Easter holidays, hence there's low-cal at the cease of the tunnel.

1. Which One Doesn't Belong?
This website is pure genius from @MaryBourassa. 'Which One Doesn't Belong?' is 'a website dedicated to providing thought-provoking puzzles for math teachers too students alike. There are no answers provided equally at that spot are many different, right ways of choosing which ane doesn't belong'.  Here's a duet of examples:

Ask students to honor a argue why each ane doesn't belong. What a fantastic tool for generating interesting shape discussion. Mary welcomes contributions to the website - details too templates are available Pythagoras Lesson. In this lesson Pythagoras problems were seat around the room too students worked inward pairs to solve the problems - they had to introduce their instructor amongst equally many right answers equally they could inward an hour. They had 3 ‘hint’ tokens that they could merchandise inward for assist at whatsoever point. I similar this sentiment - I guide maintain a habit of offering likewise much help, which tin survive counterproductive. The express provide of hint tokens volition encourage perseverance inward students - they'll alone enquire for assist when they really demand it.

Spiderbox from illustrativemathematics.org

4. Surds
Two lovely surd resources were shared on Twitter this week. First upward was @aap03102's excellent surds magic square.

And then @MathsPadJames (of awesome website mathspad.co.uk) shared this bright activity:

Followed past times some other fix of examples:

5. Sticky Maths

Tricks In Addition To Tips 3: Quadratics

highest mutual factor and methods for sequences, linear graphs too surds.

The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:

Pause for a infinitesimal too expect at this question, because it's a non bad instance of the modify inwards difficulty inwards the novel GCSE. Note the utilisation of business office annotation too the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning betoken aren't integers. This is a notable footstep upward from the type of questions asked inwards electrical current GCSE exams.

For the purpose of this post, let's consider the business office y = x² - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?

1. Vertex Form
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:

y = (x - 3)² + 1

Now nosotros tin sack position the turning betoken at in 1 trial away. I've e'er explained it a chip similar this:

"The (x - 3)²  is squared thence it tin sack never move negative. The lowest it tin sack move is zero. It's null when x is 3. The lowest possible y value is 0 + 1. So nosotros know that the minimum is at (3,1)..."

This explanation is inwards draw alongside my idea procedure - it's the agency I position the turning betoken - but my students actually create out alongside it. Of all the things I teach, this is the explanation that gets the most blank looks! So this yr I tried a dissimilar approach when revising this theme alongside my Year 11s. This fourth dimension I relied on my students' noesis of graph transformations. I told them to intend of the graph y = (x - 3)² + 1 equally a transformation of the graph y = x². It's been translated iii units correct too 1 unit of measurement up. The vertex moves from (0,0) to (3,1). They constitute this approach actually slowly - it made a lot to a greater extent than feel to them. Suddenly all my students were able to notice the vertex of a quadratic function.

As long equally students own got studied graph transformations thence this approach seems to work. This instruction gild is worth bearing inwards hear when designing a Scheme of Work. 

From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...

2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.

By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.


3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.

4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office  y = x² + 4x + 5  - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.

Preparing for the novel GCSE
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.

CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.

The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.