Tricks and Tips 3: Quadratics

Last month I presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This is the third in a series of posts summarising the content of that workshop for those who were unable to attend. 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 delegates would learn new methods that they might consider using at school. In today's post I'm covering quadratics, specifically methods for finding a vertex. My previous posts were on methods for finding a highest common factor and methods for sequences, linear graphs and surds.

The vertex of a quadratic graph
This comes up in the new maths GCSE, in questions like this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:

Pause for a minute and look at this question, because it's a great example of the change in difficulty in the new GCSE. Note the use of function notation and the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning point aren't integers. This is a notable step up from the type of questions asked in current GCSE exams.

For the purpose of this post, let's consider the function y = x² - 6x + 10. There are a number of ways to find the coordinates of the turning point - how would you do it?

1. Vertex Form
The term 'vertex form' is not commonly used in the UK. Vertex form is what you get when you complete the square. So we'd write the function as follows:

y = (x - 3)² + 1

Now we can identify the turning point straight away. I've always explained it a bit like this:

"The (x - 3)²  is squared so it can never be negative. The lowest it can be is zero. It's zero when x is 3. The lowest possible y value is 0 + 1. So we know that the minimum is at (3,1)..."

This explanation is in line with my thought process - it's the way I identify the turning point - but my students really struggle with it. Of all the things I teach, this is the explanation that gets the most blank looks! So this year I tried a different approach when revising this topic with my Year 11s. This time I relied on my students' knowledge of graph transformations. I told them to think of the graph y = (x - 3)² + 1 as a transformation of the graph y = x². It's been translated 3 units right and 1 unit up. The vertex moves from (0,0) to (3,1). They found this approach really easy - it made a lot more sense to them. Suddenly all my students were able to find the vertex of a quadratic function.

As long as students have studied graph transformations then this approach seems to work. This teaching order is worth bearing in mind when designing a Scheme of Work. 

From now on, I'm going to use the transformation method. But there are alternatives...

2. A formula
In some countries, students simply memorise a formula. They learn that the x coordinate of the vertex is -b/2a. They then find the y coordinate by substituting that value into the equation.

By memorising this formula, you can find the coordinates of the turning point of any quadratic function without completing the square. At my conference session I showed the video below - watch it to see how the method is explained. I'm not a fan of this approach. I don't want my students to simply memorise a formula - there's no conceptual understanding here.


3. Differentiation
Differentiation is always a pleasure. We don't do calculus at GCSE, but I thought it worth mentioning here that another method to find a turning point of a function is to set the derivative equal to zero. As you can see below, for a quadratic that will always give us x = -b/2a.

4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the two roots. This works because parabolas are symmetrical. Up until recently I thought this approach wasn't possible for quadratics that don't intercept the x axis, but then I discovered James Tanton's method. It's described below for the function  y = x² + 4x + 5  - for more detail and examples, see this curriculum essay or this video.

If we apply this method to our original example, we rewrite y = x² - 6x + 10 as y = x(x - 6) + 10. We can see that two points on this curve are (0,10) and (6,10), so the vertex has x coordinate 3 by symmetry. Simple!

James Tanton has produced a brilliant pamphlet 'Guide to Everything Quadratic' which is helpful for any maths teacher preparing to teach quadratics for the first time. My Algebra and Core AS resource libraries are packed full of recommended resources for teaching this fantastic topic, such as this activity from Susan Wall.

Preparing for the new GCSE
As I was writing this post it occurred to me that there's a lot of really important things that maths departments need to do this term to prepare for the new GCSE. Writing new Schemes of Work is a huge job, as is finding suitable resources for teaching the new GCSE topics.

CPD for maths teachers is also really important. All maths teachers need to be familiar with the new GCSE content - they need to know what's been added and what's been removed. They need to look at lots of example questions.

The other thing that all teachers need to do now is a subject knowledge check - are there any topics on the new GCSE syllabus that you're not familiar with? This is particularly relevant for teachers who've never taught A level maths. Has everyone in your department thought about how to teach the new GCSE topics? It's time for some vital maths department CPD.

Tricks and Tips 2: Sequences, Linear Graphs and Surds

I recently presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This is the second in a series of posts summarising the content of that workshop for those who were unable to attend. 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 delegates would learn new methods that they might consider using at school. In today's post I'm covering sequences, linear graphs and surds. My previous post was on methods for finding a Highest Common Factor.

Linear Sequences

What's the n^th term of the sequence below?

5, 8, 11, 14, 17, ...

It might only take you a second to work it out - how did you do it? This is another one of those topics where there's lots of different approaches. Ask the same question to a random sample of teenagers from across the UK and you'll see a wide range of methods used.

Here I'm going to describe four different methods (there may be more!). As you read them, consider whether you're going to stick with the method you currently use, or try something new.

1. Zeroth term
Inspired by this post on Don Steward's blog, I taught the 0^th term method for the first time this year. To work out the value of q in the n^th term U_n = pn + q, we simply step back from the first term in the sequence (ie q = U₀).

In the example 5, 8, 11, 14, 17..., subtract 3 from the first term to get U₀ = 2. So the n^th term is 3n + 2. This method is so quick that it's now my preferred method for working out n^th terms.

When I taught this method it went well, although a misconception did surface later on. When students were asked to 'write down the first 3 terms of the sequence 4n + 3', some of them gave the answer 3, 7, 11 (ie they started their sequences from n = 0 instead of n = 1). This is something to be aware of next time I teach this method.

Jumping along a line - Median Don Steward

2. Shifting times tables
'Shifting times tables' is a popular method. The idea is that we compare our sequence to a times table.  In the example below, compare the sequence to the 3 times table. Ask your students how to shift the three times table to get the sequence and they'll spot that they need to add two, so the  n^th term is 3n + 2.

If you plan to use this method for the first time then I recommend this NRICH article Shifting Times Tables, which comes with an interactive tool. If you subscribe to MathsPad, they also have an interactive tool for identifying shifts.

This method works well for quadratic sequences too (a topic on the new GCSE syllabus). For example to find the n^th term of the sequence 4, 7, 12, 19, 28 we can compare it to n² and notice that it is shifted up by 3 (ie the n^th term is n² + 3).

3. Formula
It's fairly straightforward to derive the following formula for the n^th term:

Why do we restrict this formula to A level? I once took on a GCSE class in Year 11 and asked them to find some n^th terms in a revision lesson. I was surprised to see them all using 'the A level formula' - their previous teacher had taught them it, and why not? They seemed quite happy with it. I wouldn't use it with my Year 7s though - their algebra skills are very basic when they first meet sequences. In our example, this is what they'd have to do to find the n^th term:

Perhaps this is more effort than necessary. The 0^th term method is considerably quicker.

4. Substitution
This is the way I taught sequences for years. To work out the value of q in the n^th term U_n = pn + q, we substitute a value for U_n, p and n, then solve for q. For example in the sequence below, we know that U_n = 3n + q. We substitute the first term to get 5 = 3 + q. Therefore q = 2.

Thinking about it, this is the same method I'd use to find the value of c in y = mx + c if I know the gradient and a point on the line. In fact given that linear graphs are simply graphical representations of linear sequences, any methods for finding the equation of a straight line work for finding the n^th term of a sequence. I don't do enough to make this connection with my students.

Linear Graphs

A straight line through the point (5, 7) has gradient 4. How would you find the equation of the line?

As I said in my post about linear graphs, I teach this differently at GCSE and A level. Wouldn't it be better to pick my preferred approach and stick with it? 

At GCSE my students write down y = mx + c and substitute values for y, m and x, then solve for c (let's call this Method 1). At A level my pupils use the formula y - y₁ = m(x - x₁) (let's call this Method 2). Look at the steps involved - for this question, the methods are equally efficient.

My current Year 10s had an American teacher last term - she covered linear graphs with them. When we came to revise this topic, they took a while to figure out where to start. Evetually I heard one say, "is this the point-slope thing? What was that formula again?". 

OK, so they'd been taught to call gradient 'slope', that's easily fixed. But they'd also been taught Method 2 and they couldn't remember the formula. This is a problem. Given that there's less to memorise, I think they'd be better off with Method 1. 

It's worth reading @srcav's post The Straight Lines Debate for more analysis and opinion on the two methods.

I find that all my students really struggle with linear graphs. It's one of those topics that frustrates maths teachers because it's hard to see exactly what it is that students find so difficult. I teach it over and over again from Year 8 to Year 12 and my students never seem to remember it from the previous year - I'm clearly doing something wrong!

One suggestion (thanks to @letsgetmathing for this) is to approach this topic from a less algebraic perspective - instead, focus on a table of coordinates. This is demonstrated in the example below. Students have to recognise that the y-intercept is at x = 0 and the gradient is the 'step' in the y values.

If your students struggle with the algebra-heavy Method 1 and Method 2, perhaps this table method might work.

Finally, I recommend that all teachers watch this video from James Tanton. He makes the concept of the equation of a line really clear.

3. Surds
Let's finish with three methods for simplifying surds.

In the book Nix the Tricks (essential reading for all maths teachers), we're shown a teaching trick called 'Jailbreak Radicals'. Thankfully I've never seen this 'zero conceptual understanding method' used in the UK.

Instead, most teachers tell students to identify a square factor and then split the surd accordingly eg √45 = √9√5 = 3√5.

Sometimes students struggle to spot square factors. In this case they might prefer to simplify using prime factors.

A geometric method is explained in the extract from Nix the Tricks below. Try to simplify a few surds like this yourself and see what you think. Notice that you still need to identify a square factor.

Source: Nix the Tricks

The feedback from my conference workshop suggests that some people find this unnecessarily complicated. But if you're going to try it, I recommend this post by @ChrisHunter36 and this resource (pages 5 - 6) by @314Piman.

That's it for today's post. I hope you've found it helpful. Please comment below or tweet me if you know of any alternative methods for teaching these topics. My next post will be all about quadratics - sketching, expanding and factorising.

Tricks and Tips 1: HCF

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.

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

3. Prime Factor Pairing

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?

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

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.

#mathsconf2015

I had a great time at La Salle Education's National Maths Teacher Conference in Birmingham yesterday. It was well worth going up there on Friday night - I had a fantastic evening, enjoying a delicious dinner and the company of fellow tweeters @solvemymaths, @tessmaths, @7puzzle, @MrPatelMaths and @DocendoTim. I was in the bar at midnight to see the start of Pi Day but up bright and early the next morning to make the final preparations for my workshop (with a slightly sore head due to too much wine the night before...). I was a bag of nerves over breakfast pastries but thankfully the lovely Craig Barton (an experienced and popular conference presenter) reassured me with words of wisdom.

@MissBsResources, me and @MrPatelMaths at 12.01am on Pi Day -
thanks to @El_Timbre for the posters and @jennypeek for the photo

At the conference, I really enjoyed David Thomas' session '3 Techniques You Should Know (but probably don't)'. It was helpful to see good examples of bar modelling, algebra tiles and double number lines. I already use some aspects of these methods in my teaching but the presentation gave me plenty of food for thought and there were moments where I could really see the advantages of these approaches. In the second session I saw Robert Wilne speak about developing reasoning. He was an entertaining speaker with lots of good ideas including his 'olympic podium' of operations.

The TweetUp at lunch was fun - Bruno Reddy did a great job organising it. It was amazing to see so many teachers trying to solve @solvemymaths' puzzles on 'the back of an envelope' so to speak. His puzzles are awesome but are all pretty challenging. The few that I've tried to solve have involved a long period of concentration and frustration (followed by satisfaction, so well worth it!). I was really impressed to see teachers getting stuck into these tricky problems.

TweetUp: @MistryMan6's selfie with me and the lovely @MissBsResources 

I presented in the final session of the day to an audience of around 100. The aim of the session was to discuss different approaches to teaching topics such as sequences, linear graphs, surds and quadratics. I had lots of ideas to share and I was pleased that I learnt even more methods from delegates after the session. It was a wonderful opportunity to swap teaching ideas (this is the sort of collaboration we should all do at school but never get time). I was careful not to label any methods as right or wrong, but just to suggest alternatives and - hopefully - encourage people to think about their subject knowledge and delivery. I was pleased to see @danicquinn and @BodilUK enthusiastically discussing methods at the end of the session.

Me presenting at MathsConf III - thanks @DrBennison for the photo

If you were at my session then you'll already have a copy of this workbook which is full of links, and here's the presentation as promised. If you weren't at my session then the presentation won't make much sense, but I'll write a blog post soon to explain all. Give me a week or two to get it written though!

It was nice to end the day with a quick drink at the bar with @letsgetmathing, @MrPatelMaths and @MrKMorrison. One of the best things about these conferences is meeting up with like-minded maths teachers who enthusiastically give up a Saturday for professional development. Over the weekend it was great to catch up with people I met at the last conference and I enjoyed meeting lots of new people too.

Many thanks to the La Salle team for the tremendous effort they put into organising the conference. I'm sad that I probably won't be at the next one (it's on 20^th June - the weekend of my 10^th wedding anniversary) but I will definitely be at MathsConf V in September and I'm already looking forward to it.

Me in the conference programme :)

Best mug ever? Thanks @solvemymaths

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Blog posts that review/mention my workshop:
@BodilUK
@DrBennison
@WorkEdgeChaos 
@mrprcollins