Getting them to get it

Strategies to teach Mathematics in the Secondary School so it sticks

Phil Murray

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15

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Number Algebra Geometry Statistics

We, the leaders of learning in mathematics, are a family. We share a passion. We strive to inculcate it in the young people we lead. We support each other with ideas, encouragement, understanding and wit. We want each other to be successful out there in the big wide world, the classrooms in which we perform.

But those of us who are the adolescents in the family, the emerging teachers, sometimes feel that our parents (heads of department, established practitioners, mentors) don’t have enough time for us. We need help, and we need it now. But they have their own problems to deal with and can’t always help us when we need it. And being parents, they sometimes forget what it was like to grow into a fully-fledged mathematical pedagogue, or Mathemagogue.

Left to our own devices, we search around the department and we browse the web, desperate for material to make our next performance a success. We find seemingly limitless supplies of text books, worksheets, games, activities, applications, posters, presentations. All about what the learner sees, but very little about how the teacher teaches.

We also may feel a little intimidated by those who advocate a purity of method that doesn’t seem altogether helpful when faced with the task of moving the disaffected Year 10 a step further on. There are those who rail against what they see as tricks and shallow mnemonics that don’t teach the proper underlying mathematics. I have some sympathy with this view, but is not applicable to every classroom.

What we really need is simple guidance in the practical pedagogy. If we are going to stand up in front of them tomorrow and introduce the concept of upper bounds for the first time since they last forgot it, what is the essential idea we have to get across, and how are they prone to misunderstand, and what will resonate and help them not to forget it all again? Even if it doesn’t build the proper understanding, if we have recall, we have something to build on. Tina Cardone’s book, Nix the Tricks, sets out to discourage “tricks” but actually is quite a helpful listing for those happy to use them!

This book is designed to help emerging leaders of secondary mathematical learning at Key Stage 4 by providing comprehensive pedagogical guidance, topic by topic, across the full curriculum to age 16. It provides full coverage for GCSE Foundation and Higher and for IGCSE Core and Supplementary, geared to the UK Grade 9-1 system. It does not aim to cover the additional curriculum for GCSE Additional Maths or for IGCSE Additional Maths.

For many topics, the pedagogical idea is the tried and tested notion that established mathematics teachers everywhere have applied successfully for years. For some topics, the idea is more novel and quirky. Retention of the idea is the challenge for many of our students, and we need to have a wide variety of approaches with effective memory tags. Our subject has so many ideas we expect them to remember, and the more diversity we can introduce, the more we improve their chances.

“Oh, yeh, it’s that river fing, innit?”

(see section on Ratio) can be a heart warming moment in the last days of revision before the exam!

The book is designed to be practical, durable and portable, always within an arm’s reach when the planning moment arrives. Each topic is presented on a single page, with a clear guide to the order in which the idea can be presented. Visual images are offered where appropriate, but the aim is not to create a slide presentation for you to copy. Indeed, I firmly believe that slides are over used. The dynamic Mathemagogueprefers to react to the evolving mood and receptivity of the learners, using a combination of prepared images and free writing. The ideal medium is the smart whiteboard with a pre-prepared file containing only those images it would take too long to draw in real time in front of the class, and allowing the idea to blossom from the hand of the teacher in front of their very eyes.

Guidance is given as to common misconceptions for the teacher to anticipate, intercept, warn against. Connections are offered to other parts of the curriculum, inside and outside Mathematics.

The Mathemagogue’s Toolkit at the back of the book takes its inspiration from Paul Ginnis’ fantastic Teacher’s Toolkit, and offers some additional practical teaching techniques with a particular application to mathematical learning.

Anyone who reads and uses this book will accelerate their progress to be an accomplished leader of mathematical learning. Before long they will have ideas to add to this collection. The family eagerly awaits such ideas, and it will be my privilege to incorporate them in future editions.

Phil MurrayEngland, 2018

Navigation Introduction

OK, so our students have been doing maths for quite some time now. They have seen the basics. So it’s time to consider a different order in which we think about this strand, changing the presented priorities so they have more to do with what matters most for the rest of a teenager’s mathematical syllabus rather than what is a natural evolution of mathematical discovery for a five year old.

And what matters most in Number at this level of Mathematics? Ratio.

We need to combat the natural inclination of students to think first about additive relationships between numbers and get them more comfortable in multiplicative relationships. Starting the KS3 or KS4 scheme of work with a week or more on ratio, and then bouncing into other applications in similarity, mensuration, graphs, speed, proportionality and so on gives a clear message as to what we think matters. See what your Head of Maths thinks.

Here’s a way of looking at the secondary school number map, and the order of topics in this section is guided by the map:

Number

Ratio

4 Operations

BIDMAS Negatives

Ordering

Factors & Multiples

Primes, Squares, Cubes

Decomposition

HCF & LCM

Fraction Combinations

Fractions to Percentages

Percentage Manipulation

Decimal Conversion Place Value

Standard Index Form

Compound Problems

Indices

Surds

Fraction Equivalence

Decimal Recurrence

ApproximationDivisibility

Proportionality

Percentage Change

Scheme of Work Approach 1

Number Ratio - Finding shares given a total and a ratioThe Ratio RiverThe classic problem with teaching ratio to teenagers is that they all too easily forget whether they are supposed to be multiplying or dividing. And this is no surprise really, since those of us more at ease with the process will glibly flit between the two according to what the numbers concerned appear to make easier. To provide comfort and security, welcome to the Ratio River which puts everything in a proper place so we don’t get lost.

Typical Problem: Divide 56 sweets in the ratio 3:5

We start with the conventional starting point for any ratio question, by adding the parts, but we tell our students to write the numbers nicely spread out like this:

Then we draw a river underneath and tell a story about wanting to cross the river to get to the real world.

Then we ask where the 56 total belongs in the real world? Opposite which of our numbers on the screen? Somebody will be brave enough to suggest the 8, and then we can ask what we have to do to cross from the ratio world to the real world. A little bit of patient questioning will lead to a discovery that multiplying by 7 will do the trick.

And the final step is to add a x7 under the 3 and the 5 as well. The correct shares of 21 and 35 are then easy to see.Students need to be reminded of the importance of writing the correct answer to an examination question on the dotted line!The beauty of this method is that exactly the same approach can be taken with 3 part ratios, or with finding the total given a part.There is opportunity for collaboration with Food Technology on this topic. Find some recipes for smoothies and set an exercise to scale the recipe up and down. Maybe even set a homework that involves the student making the smoothies for their family?

Unitary RatioA particular skill involving reduction of the ratio to a form such as 1:1.66 needs also to be covered. This is a straightforward extension of the river method above, though it needs to be considered carefully since a true ratio of for example 3:5 will not be accurately reflected by a decimal rounding such as 1:1.66 and we may need to advise students either to be prepared to adjust answers by judicious rounding, or, much better, to use fractional ratios such as 1:5/3.

3 + 5 = 8

3 + 5 = 8

3 + 5 = 8x7

56

2

Number 4 OperationsMental ImagesWe talk about the order of operations under BIDMAS on the next page, but here let’s take a look at a collection of ideas for quicker mental addition, subtraction, multiplication and division.

Dividing by 2,3,4,5Divisible by 2 we hope all students have mastered as “even” by the time they get to KS4. The extension to divisible by 4 can be shown to correspond to a list of 25 two digit “leap year” alternatives at the end of an integer, eg2552. But did we ever take the time to look at the patterns of those options? Here’s a check for divisibility by 4 on the right. May be easier to teach as an alternating sequence: “ends in a 0,4 or 8, then… ends in a 2 or a 6, then… ”Divisible by 3: add up the digits, and if that sum is divisible by 3 so is the number, eg2568: 2+5+6+8=21 so 2568 divisible by 3. Yes, but Why? (see eponymous book by Southall).

Last digit divisible by 4 and penultimate digit even

eg 24 68 4320

Last digit not divisible by 4 but even, and penultimate digit odd

eg 36 52 5192

The Slavonic abacus is a variation of the traditional abacus with the beads coloured in groups of 5x5. An example is shown on the left, but we want our students just to imagine the array, like the image on the right. When they are asked to attempt the classically difficult to remember 7x8=56 they just have to imagine that it is made up of one lot of 25, plus 5x3=15, plus 2x5=10 and a remaining 2x3. That’s 25 plus 15 plus 10 plus 6. Some students use a more accelerated technique of 8x5=40 plus 8x2=16.

Multiplying with a Mental Image of the Slavonic Abacus

Multiplying by 9What about the patterns of numbers divisible by 9? Up to 90 there is a cute trick of holding up all ten digits and then if you want 9x9 dropping the 8th digit, leaving 8 on the left and 1 on the right, representing the correct answer of 81. Above 90 this clearly won’t work, but the same trick applies as for 3 – if the sum of the digits is divisible by 9, so is the number.

Difference of SquaresMore able students might appreciate an application of the “Difference of Squares” principle for an impressive display of mental arithmetic. If you have two numbers to multiply that are equidistant either side of a round number, try this:

23 x 17 = (20+3) x (20-3) = 20² - 3² = 391

Reinforce mental arithmetic skills through the use of starter games such as FizzBuzz

Divisible by 5: easy enough to see if it is divisible by just checking if it ends in a 5 or a 0, but surprisingly few good mathematicians think of doubling and dividing by 10 as a quick way of actually dividing by 5.

SubtractionTradtional methods of subtraction using tabular formats with lots of borrowing and carrying are never very secure on transition from primary to secondary, and techniques recently more widely adopted based on Singapore Bar principles tend to be more effective. The essential principle behind these modern principles is to represent difference visually as the difference in lengths of a collection of bars and portions of bars. 73 - 59 would be shown as

And then subtracting the red bar leaves three portions to be added: 1+10+3=14.

Eventually through familiarity students get to process this as a round-up from the smaller number, plus however many tens, plus the units in the larger number.

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3 bars

Number Ordering FractionsWriting or Drawing?Putting numbers in order is an early challenge for mathematical learners, but even less able students can gain confidence from successfully addressing ordering of integers and ordering of decimals:

3, 5, -2, -8, 14

Just encourage them to draw a number line with a central zero and place the numbers where they see fit.

0.72, 0.717, 0.7087, 0.7The common misconception is to think of 0.7087 as having something to do with the integer 7087 and so considering it bigger than 0.717. Overcome this by guidance to “make all the decimals the same length first by adding zeroes0.7200, 0.7170. 0.7087, 0.70000.7000, 0.7087, 0.7170, 0.7200 in order.

It gets more difficult with fractions. Conventional wisdom encourages students to convert the fractions to decimals first and then to place them in order as above. However, with some denominators it is not easy to convert, particularly if you have never been comfortable with long division.

Rulers Make Orders!An alternative visual approach requires no calculation, just some neat work with a ruler and pencil on squared paper. For example, let’s look at the relative size of 4/9 and 3/7.

-8 -2 3 50

For 4/9 draw a line through the point with co-ordinates 4,9 to the top (suggest 10 squares high) of the grid (the red line). For 3/7 do the same through the point 3,7 (the green line).Label the lines, and the labels will be your fractions in the right order!Simple!And if we are required to compare fractions with decimals and percentages, mark the latter along the blue scale at height 10. There is of course some concern about exam questions that require the student to show working. The student should find a grid somewhere in the paper to do this trick and then say “I drew triangles to compare the fractions on page X” next to the original answer box. I’d be confident of asking the supervising examiner to condone that.

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Decimal OrderingFor some lower ability groups you may need to allocate some support time to ordering decimal numbers, particularly where numbers of varying length of decimal placing are considered. Here there needs to be an insistence on making all the numbers the same length before attempting to order them, to avoid the common misconception of making 2.3 less than 2.39.

2 . 2 8 5

2 . 3

2 . 2 7

2 . 2 8 5

2 . 3 0 0

2 . 2 7 0

2 . 2 7 0

2 . 2 8 5

2 . 3 0 0

10

1 100%

We have to admit there is a challenge in persuading students of the importance and purpose of algebra. To make this work we have studiously and recurrently to introduce algebraic techniques through the medium of functional skills problems where the point of it all is more evident. This can contribute to a greater motivation for the student to explore what algebra can do for them (though I am a realist enough to know that this is not an easy task. It’s what you came into Maths teaching for, what you aspire to be good at, and what will make mathematicians of your students if you can succeed at it).In this map I am not trying to claim that there is a unique pathway between aspects of algebra, more one possible route that can help secure the acceptance that algebra is there for a reason.The order of topics that follows is guided by this map.Be warned. Traditionalists will be surprised and possibly disturbed…

Linear Graphs

Simultaneous Equations

Inequalities

Expanding Brackets

Construct Equations

Trial & Improvement

Quadratic Graphs

Quadratic Factorising

Quadratic Equations

Advanced Graphs

Rules for Sequences

Rearranging Formulae

Patterns & Sequences

Substitution

Factorising

Linear Equations

Algebraic Manipulation

Real Life Graphs

Algebra

Graphic Inequalities

Linear Programming

Quadratic Sequences

Quadratic Expansion

Functions

Quadratic Formula

Completing the Square

Algebraic Fractions

Parallel & Perpendicular

Circle Equations

Set Notation

OtherSequences

Scheme of Work ApproachAlgebra 31

Differentiation

Trial & ImprovementUsing a Sledgehammer to Crack a NutThere is a tried and tested (pardon the pun) method of doing this, which involves a table with guesses progressively honing in on the best answer. Which is fine, but it doesn’t help students who have no intuitive idea of the progression of powers to see what’s happening. Much more powerful is a number line approach, which we base on the idea of the fairground sledgehammer, as it matches up better with intuition.So the traditional method first for completeness.Imagine we are asked to solve x³ - 5x = 27 to the nearest decimal point.Write out a table like this. For most problems, starting with 3 is a good guess, and then we progressively approach the answer we are looking for, commenting each time to help us get closer. If you ask your students to suggest the next option they will often massively under- or over-state the next guess, but that’s fine. All part of the learning curve.

So, how about the alternative method to help see the progression and understand the process? Draw a fairground scale in the form of an upside down T. Then mark the target somewhere in the middle. Try 3 on the calculator and get 12. Then try 4, get 44, and mark it roughly in the right place. Carry on till you get to the Final Five. Then the right answer is the nearest 1dp on the opposite side to the Final Five.

Use a spreadsheet to generate 4 or 5 examples, each starting with x³ but the progressing through a simple +x, then –ax, and so on. Make sure your explained example uses ax though or you will see some find it hard to progress.

Algebra

x x³ - 5x to match 27 Comment

3 27-15 = 12 Far too small

4 44 Too big

3.5 25.375 1.625 too small

3.6 28.656 1.656 too big

At this point students are likely to want to say 3.5 is the best answer to 1dp because it seems closer, but watch this…

3.55 26.98888 Very close

So the right answer if we found it, must be between the answers for 3.55 and 3.6. That means it rounds to 3.6, so 3.6 is the answer. We have to drive our students to do this Final Five test to decide rather than just a visual inspection.

27

3 12

27

3 12

4 44

3.5 25.375

27

3 12

4 44

3.5 25.375

3.6 28.656

3.55 26.98888

rounding

33

Encourage flowery answers in the comment box or the comment clouds to increase the fun, but also to get a better sense of the performance of guesses and improve the sense of progression of powers.

Linear ProgrammingIs you in or is you out?This topic requires a secure knowledge of both graphic inequalities and simultaneous equations before it can be attempted.Think carefully about the real life examples you wish to choose in order to enthuse the students most about the capability that this topic presents. For more able students, whether they pursue Maths at A-level or not, this is a topic that delivers real advantage in business to those who get it. If students argue that their parents have never used it in business, counter that this shows even more the opportunity to beat your competitors because so few people make use of it!Good real life examples to choose from are as follows:• Running a market stall to sell varieties of chilli sauce• A calculator company deciding how many of different types to make (see Purplemath for a great worked

example).• Hiring coaches of different sizes for a school trip (see Pearson D1 Edexcel textbook)• Required daily intake of vitamins from different food products.The main challenge faced is in the translation from a paragraph of text describing the problem to a set of equations and inequalities. Many students find it difficult to create the line equations, particularly with statements like “we need to use twice as many oranges as apples”. A good entry point for this activity is a “show me what works” exercise.

Encourage easy plots like the origin, (0,1) or (1,0). Is it red or green? Very much helps identify the side of the line that’s in the feasible region.

Algebra

x

Give each pair of students a personal whiteboard with a pre-printed cartesian grid and a red, green and black pen. One student has the green pen to mark vertices that work with a cross and the other a red pen to mark vertices that don’t.Then follow a sequence of commands like this• Make the y-axis oranges and the x-axis tomatoes• Each orange has 50mg of Vitamin C• Each tomato has 20mg of Vitamin C• I need at least 400mg a week. Show me some combinations

that work and don’t work• Draw the boundary line• Identify the equation of this line• OK, rub out your crosses and leave the line. Label it C• Each orange has 20 units of Vitamin A• Each tomato has 80 units of Vitamin A• I need 1000 units a week. Show me some combinations that

work and don’t work• Draw the boundary line• Identify the equation of this line. Label it A• Oh and I like tomatoes best so I’d like more than twice as

many tomatoes as oranges. Show me what works.

x

x

xx

xxxx

You can inject a bit of fun into this by picking a working combination that the closest guess wins a prize. And then optimise and get them to guess the improved combination.

With definition of feasible region sorted, the next challenge the students come across is the accuracy of the objective line method. Some students are not particularly proficient at sliding the ruler parallel. Show them how to use two 15cm rulers to do this.

When it comes to vertex testing re-kindle the “Is you in or is you out” idea to encourage the evaluation of vertices, Perhaps introduce another colour (blue) to write the objective value on a series of vertices. This is particularly useful when integer testing is relevant. The more this process can seem intuitive first and calculated second, the more we will have a deep and effective understanding.This is one of many aspects of mathematics where two quite different methods are available (objective line and vertex testing). As a teacher we have a duty to introduce our students to both methods with no preference, allowing them to select the one that suits their learning style best.Be ready for students to lose the plot with unscaffolded problems that don’t guide them through the necessary steps. Encourage them to formulate the objective optimisation equation first, so that they choose appropriate quantities to define and label for the constraints.

x xx x

48

Circle EquationsNoughts and CrossesThe DfE require that students can recognise and use the equation of a circle with centre at the origin; and that they can find the equation of a tangent to a circle at a given point. This is a new GCSE requirement as part of the 2015 curriculum.The best way to reinforce the equation of the circle is to provide a discovery worksheet as illustrated on the right with a series of circles drawn on squared paper, and have the students discover that the Pythagorean equation extends to create the equation of a circle. In practical terms it is hard to provide enough diversity of example with normal paper and rulers since centimetre square paper is too large a scale for anything other than a 5cm base to be easy on the numbers. One way round this is to use the typically available 0.7cm square paper and to create and artificial paper ruler with units spaced to match the squares, a bit like that shown in the example.

Algebra

Touching the C-BottomThe tangent equation is a little more challenging. The simplest way to develop the equation of a tangent unfortunately involves a lot of distinct steps, which many students will struggle to remember. So we want to give them a method that makes the best compromise between number and simplicity of steps.This is a topic that we will only expect our top students to grasp; students who are already comfortable with trigonometry and can calculate the tan of an angle easily.

So for the tangent to the circle x²+y²=169 at the point P(5, -12) we first find the tan of the angle from the y-axis.

tan 𝜃𝜃 = ⁄−512.

Then we add x times the tangent to the y value to find the y-axis intercept

𝑦𝑦 + x tan 𝜃𝜃 = 𝑐𝑐In this case:

−12 + 5 × ⁄−512 = 𝑐𝑐

𝑐𝑐 = ⁄−16912

And so the equation of the tangent is

𝒚𝒚 = ⁄𝟓𝟓𝟏𝟏𝟏𝟏 𝒙𝒙 − ⁄𝟏𝟏𝟏𝟏𝟏𝟏

𝟏𝟏𝟏𝟏

θy

xtanθ

It is at this point (or perhaps when they have done a second example that the sharpest knives in the block will say “is it a coincidence about the 169?” at which point you can demonstrate that the generalised version of the above mathematics is

𝑦𝑦 − 𝑥𝑥 𝑥𝑥1𝑦𝑦1

= 𝑐𝑐. 𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎 𝑃𝑃 𝑦𝑦12 + 𝑥𝑥1

2 = 𝑐𝑐𝑦𝑦1

Some teachers might choose to provide the generalist formula first, but there is a lost opportunity to share the source first, and sharing the source will in this case help create something more memorable. But for completeness, here is the formula:

𝑦𝑦 = −𝑥𝑥 𝑥𝑥1𝑦𝑦1

+ 𝑟𝑟2

𝑦𝑦1

The concept of the tangent connecting from the value of y to c, the value of the y-intercept, or the c-bottom, is the way we provide linguistic and visual recall.

θ

Latin lesson for the day: “tangent” means “touching”

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DifferentiationSliding Down the SlopeMost specifications do not include differentiation, but the IGCSE extended specification does.To begin with, we should focus on the topic qualitatively, examining different curves and their gradients. We already know from our work on straight lines of the form y=mx+c that they all have constant gradients, indicated by the size and sign of m. Show the class a curve of 𝑦𝑦 = 𝑥𝑥2, perhaps using a graphing package such as Desmos. Set it up so that you also have the derivative curve as the example shows below.

Algebra 62

Now that you have whetted the appetite, let’s go back to the 𝑦𝑦 = 𝑥𝑥2 curve and see if we can address what is happening quantitatively. Create a cartesian grid with the curve on it. Mark the points (1,1) and (2,4). Gain agreement that what we are after is the slope of the green tangent line. Ask how good the orange line is at giving the answer. Ask if moving the top red dot from x=2 to x=1.5 would give a better answer? x=1.25? 1.1?At this point you may want to use a dynamic model like the Geogebra file offered by the Further Maths Support Programme (your school has probably got a login, but if not take a look at their CPD offerings which can give you access):

Connect from the transition from x2 to 2x (and x3 to 3x2 if you decided to go this far) to the formulaic approach for differentiating polynomials (drop the power and multiply by the old power).We aid retention of the approach by illustrating it a• sliding the power down by one, • and sliding the redundant old power down to be

part of the co-efficient. There needs to be a veritable shoal of practice to secure this principle, which in addition to simple natural number exponents should progress to include reciprocals, roots and expressions and fractions requiring rendering to simple index before differentiating.

By varying a, you can show through the blue tangent how the slope changes from steeply negative to gently negative, to horizontal, which we call stationary, on to gently positive and steeply positive. You could follow on with a graph of 𝑥𝑥3. For this the g(x) function required above is 𝑎𝑎3 − 3𝑎𝑎 + 3 𝑎𝑎2 − 1 (

)𝑥𝑥 −

𝑎𝑎 . Try a bit of recreational maths at the weekend to see where this came from!

Once students have learned how to obtain a differentiated result, they are expected to use it to find turning points, places where the result is zero, and to identify whether they are maxima or minima. Whilst the technique of second differentiation could be taught as an advance on A-level work, it is probably simpler at this stage to advocate adjacent testing as the example below shows.

Find the co-ordinates of the turning point of the curve y = x2 + 4x − 3, and identify their nature.

𝑑𝑑𝑦𝑦𝑑𝑑𝑥𝑥 = 2𝑥𝑥 + 4,

stationary at x=-2, y=4+8-3=-7Test at x=-1.9 y=-6.99Test at x=-2.1 y=-6.99Turning point is minimum

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Set Notation & Venn Diagrams∈ or ∉, that is the QuestionLearning about sets is in some respect like learning about the introductory elements of algebra. There doesn’t initially seem to be any point to it, until we start to apply it to other problems. At GCSE level, however, we need to introduce the topic as a standalone concept, capable of being separated from some of the early applications, particularly in probability, to which we apply it. So initially there is a set of definitions for students to get used to, with associated notation that uses characters they have not come across before. It doesn’t help that the characters used to represent set properties are abbreviations derived from language which is itself old-fashioned and less meaningful to students. A Venn diagram is a visual explanation of the concept represented by each example of set algebra as follows:

But all of this is very dry and uninteresting taken on its own. Time for some drama. Students should be used to the idea of tableaux from their Drama classes. Use the following set of tableau captions to create a set of situations that groups of students can set up, explain, and have photographed with a nice big card in front of them with the relevant piece of set notation and an appropriate Venn diagram provided. Clearly the words can be tailored to fit the circumstances and students on the day.

see Probability in Statistics for applications of these concepts.

Algebra

Set Notion Set Algebra Venn RepresentationEverything (or everything being considered by a problem). Classically referred to as the Universal Set.

ξ

Nothing, or the Empty Set ∅

A set of elements (things) obeying a rule.An element is in a setAn element is not in a set

{a, b, c, …}a ∈ Ad ∉ A

The elements that are common to two sets. Often called the Intersection. Means “and” so talk about the ∩ as the “n” in Fish ‘n Chips

A ∩ B

The elements that are part of either or both sets. Often referred to as the Union. Means “or”, The ∪ is easily represented as Union.

A ∪ B

The elements that are not in a set (sometimes referred to as the Complement Set A’

The number of elements in a set. n(A)

a b c

Tableau Instruction Set Notation

Jimmy and Emma are the only ones in our team who have black hair and are wearing a jumper.

A ∩ B

Ant and Dec are sad because the rest of the group are not talking to them A’

Hands up if you’ve got a pencil or a ruler (which includes or both) A ∪ B

Squares and rhombi are Parallelograms but trapeziums aren’t {s,r} ∈ P t ∉ P

The murderer must be not dead and not female D’ ∩ F’

The victim has black hair and is not male B ∩ M’

A

65

def

Something of an inversion in this strand. Weaker students like it more than other parts. Stronger students like it less apart from where the algebra sneaks in (eg in trigonometry).It’s to do with different modes of representation (check out Bruner if you want to find out more) and that the way in which we prefer to access and store knowledge evolves as we grow, from enactive (doing), through iconic (seeing), to symbolic (describing). Those who have developed further mathematically like being in the symbolic and experience some frustration in being pulled back to the iconic or enactive, until they have the confidence and insight to recognise that mathematics is about all three.It is for this reason that I favour starting our geometry work with lots of visual work, and only bringing the symbolic algebra and reasoning in to support and facilitate. Another map for you, again suggesting that we use functional skills to provide the “port of entry”: CGP produced a very good array of functional skills problems for both more able and less able groups to use for such introductory work. Published in 2010 it is no longer in print but can still be found from partner sellers on Amazon by searching the ISBNs: 9781 84762 5151 for less able students and 9781 84762 5137 for more able students.The order of topics that follows is guided by this map.

Linear Measure

Similarity

Ratio Measures

The Seven Angle Facts

Loci

Translation

Vectors

Enlargement Sine Rule

Circle Measure

Tesselation

Area & Perimeter

Construction Triangles

Construction Bisection

Nets: 2D-3D

Functional Skills

Volume

Symmetry

Reflection & Rotation

Congruence

Composite Transformations

Trigonometric Graphs

Vector Proofs

2D & 3D Measure

Cosine Rule

Right Angle Trigonometry

Angles in Polygons

Bearings

RATApplications

Pythagoras

Circle Theorems

Sectors

Rectilinear Area

Triangular Area

Scheme of Work ApproachGeometry

Surface Area

Solids & Shapes

Money & Finance

66

LociSwing the LeadA good way to introduce this, if the weather is nice, is to get outdoors with a couple of nice pieces of rope (30m of 12mm braided rope is best). Stand at a central point on flat ground and ask all the students to stand at a point 4 metres away from you (gives a circumference of around 25m, enough for comfort!). Pick one student to take the end of a 4m section of rope and ask them to walk round in a circle from you, asking those touched by the rope to sit down and those touched by the walking student to raise both arms. (Students who enjoy Scouting are very likely to have played a game where a rope is swung around by a leader to trip those scouts who don’t jump at the right moment – not a game that most schools are prepared to sanction!) Use the result to reinforce that the locus of points equidistant is a circle. Using an enactive-to-visual strategy, tell them you will ask them to record this in their exercise books when you get back to the class.Using a similar process for other commonly considered loci:• The lozenge as points equidistant from a line• The line bisector as points equidistant from two points• The angle bisector between two lines• The set of outside points equidistant from a triangle or rectangle• The set of inside points equidistant from a triangle or rectangle.Once these six loci have been properly acted out, head back to the classroom to a task to record each of them in a revision section of the exercise book, and move to work through an example set. 10 ticks offers a very good set that has a real-life flavour to them (Sheet 7509).Extension examples, for the enactive or visual phases, include:• The points twice as far from A as from B• The points equidistant from a circle and its tangent.• For each point of a polygon, the points the same distance from that point as the furthest-point-inside-

the-polygon. Done for a pre-drawn heptagon this enables students to discover how the 50p piece was designed to have a constant diameter as it rolls. More able students would then be able to appreciate why a 12 sided coin such as the new £1 coin would create headaches for vending machine operators.

For revision and consolidation of understanding of loci there are several concepts to emphasise:• For points equidistant round the outside of a convex

polygon, there are curved corners, each a sector of a circle with the vertex at its centre, and connecting straight lines parallel to the edges of the polygon. By contrast, the locus of points equidistant from the polygon but inside has sharp corners, a faithful fractional enlargement of the original.

• However, if the polygon has a concave vertex, the story is reversed. Have your most able students try to mark the inside and outside equidistance loci for this shape, or have the whole class do it my teamwork with the rope inside.

• So why are the inside and outside equidistance loci for a circle both faithful enlargements of the original? Does this extend to other shapes (ellipse? cardioid?)? Why?

For a different process to examine loci, look at the activity builder for loci on the graphing site Desmos.

Geometry 70

Cosine RuleAdjusting Pythagoras with a “Little Bit”A good way to introduce the cosine rule is to explain that it is a way of adjusting the familiar Pythagoras for cases when we haven’t got a right angled triangle. Provide pairs of students each with an assembly of two laminated card strips held together by a paper fastener as shown. Ask them to arrange the strips in a right angle and to measure the distance between the centre of the gold dots (one of which is the top of the fastener) and satisfy themselves that Pythagoras works in this case. Then ask them to move the vertical strip so the angle is reduced and measure again.It’s a little bit shorter.So wouldn’t a good equation in this case be: 𝑎𝑎2 = 𝑏𝑏2 + 𝑐𝑐2 − 𝒂𝒂 𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍 𝒃𝒃𝒍𝒍𝒍𝒍?The question is what that little bit is. For most groups it is too challenging to show the basis for the calculation at the outset, but a good exploratory discussion can lead students to expect that the little bit needs to have the following features:• To be zero when the angle is 90°• To get larger as the angle drops to zero• To be “plus a little bit” if the angle increases to more than 90• To be bigger when b or c are bigger.When you identify that the little bit is 2bc cosA this should be seen as acceptable. Most boards do not require the formula to be memorised, and so this approach of emphasising the rationale rather than the detail is appropriate. To avoid undue focus on a, b and c, think of the little bit as “twice the other sides by the cosine”.

Putting it into PracticeIt is important to recognise the key relationships in the cosine rule. As we argued for the sine rule, the use of ABCabc is less helpful than a sense of the geometric association, and the concept of the angle sandwich as the start point helps students see the applicability of the rule. In the example opposite the angle of 56 is sandwiched between the two sides, and the opposite side can then be sourced from our adjusted Pythagoras formula:

𝑥𝑥2 = 42 + 82 − 2 × 4 × 8 × cos 56 → 𝑥𝑥 = 6.65

Finding the AngleFinding the angle is often seen as a more difficult challenge, and teachers vary between drilling the alternative rearranged equation for students whose algebra is too weak to be able safely to rearrange it themselves, and suggesting using the substitution in the original form of the equation with progressive simplification until the inverse cosine can be addressed to the calculator. Neither is especially easy for students. Let’s go back to the “little bit” idea. We know that the little bit is twice the other sides and the cosine. Which are the “other sides”? Oh, yes, the sides of the angle sandwich, so calculate as shown in the box. What is this little bit? The adjustment that shows how much smaller the opposite side square is than the sum of the sandwich side squares. Calculate it as in the box. Then form the equation. Easy. (The examiner might be a bit surprised by the terminology but what does that matter if the critical mark scheme elements are evident?)

4

x

8

56

4

x

8

7

Little bit: 2 × 4 × 8 × cos 𝑥𝑥 = 64 cos 𝑥𝑥

Pythagoras adjustment:42 + 82 − 72 = 31

Equation:64 cos 𝑥𝑥 = 31

𝑥𝑥 = cos−1 3164

= 61°

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Geometry 87

This strand can be approached from the viewpoint of fair representation, and can be most effectively kick started through a journalistic investigation project (collaboration with the English Department, now there’s an idea!).Once we have got their emotional commitment to the idea that statistics is about fair representation and reporting, the topic can naturally proceed in substrands relating to data collection, analysis, representation and interpretation. At every stage, the fairness debate should be rekindled. Probability is a separate matter altogether. The probability concepts shown in orange do not command a lesson each, but should be woven into the other lessons – a summary discussion page is provided in this section. The order of topics that follows is guided by this map.

Data Analysis

Questionnaire Design

Data Collection

ProbabilityConcepts Predictions

Experimental Probability

Box Plots

Stem & Leaf Diagrams

Mean Mode & Median

Sampling

Data Representation

Journalistic Investigation

Histograms

Comparing Data Sets

Calculations from tables

Cumulative Frequency

Pie Charts

Probability Scale

Interpretation

Quartiles

Time Series

Correlation

Tree Diagrams

Two Way Tables

Probability Spaces

Venn Diagrams

Combined Probability

Statistics Scheme of Work Approach

Scatter Charts

104

Journalistic InvestigationRead All About ItHave you come across the “Set 2 Challenge?”. In the second ability set in a Year 10 group you can expect to find several examples of this individual:• very intelligent, highly literate and articulate• in the top set wherever ability grouping is applied elsewhere in the school• disaffected with regard to mathematics, possibly aggressively rejecting its relevance• and because of the above attributes, capable of diverting other students away from the beauty of

Maths.Today, we have an opportunity to turn that around, even if only briefly, by making journalism the subject of a Maths lesson.Procure enough copies of two recent quality newspapers for your class to conduct an investigation into the use of data in journalism. Across the two newspapers you should be able to find enough instances of the potential issues identified below for them to be able to build a report or display material giving opinions on potential distortion of issues that concern them through incomplete, inaccurate or misrepresented information.To be fair, many newspaper articles are exemplary in their fair representation of data, so your students should be encouraged to find and celebrate particularly good examples as well.

Catalogue of Issues to Search for:Chart issues• Has the choice of axis scales unfairly exaggerated changes in data? Particularly by not starting from 0?• Has the use of oblique, solid or pictogram-based bars distorted the representation?• Has the right decision been made between line, bar or pie charts?• Have stacked bars been arranged in a way that makes it harder to see data the presenter doesn’t want

you to notice?Ratio issues• Has the right decision been made between reporting actual (absolute) numbers versus percentages

(ratios)?• If changes are reported, are these meaningful without equal prominence being given to the base the

change is from, or would a ratio or percentage have been a fairer representation?• Is a change from a previous period fair? A poor example would be comparing flu patients in January with

flu patients in October. • Is a multi-year percentage change meaninglessly presented for exaggerated effect?Deception issues• Has the paper selected an unrepresentative example, such as an extreme low temperature in a remote

hillside in North Wales rather than the average across the country?• Has another relevant variable been ignored, for example by reporting rising total graduate first year

earnings without acknowledging the rising numbers of graduates, unfairly representing the increase in earnings for individual graduates?

• Has a time series of data been unfairly indexed to the best or worst start year to show an exaggerated outcome?

• Has the size of classes of data been ignored, such as claiming that teachers and nurses are dangerous drivers because they are the profession with the most accidents, ignoring that there are many more teachers than say piano tuners!

Fog IndexWhile we have the newspapers to hand, it may be worth comparing the fog index of, say, an article on economics in the Financial Times with an article on last night’s reality TV contest. For guidance on how to calculate a fog index, and what it is, consult https://en.wikipedia.org/wiki/Gunning_fog_index.

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105Statistics

Reach up to the Top RightA further method of illustrating distribution is the cumulative frequency chart. This has the advantage of facilitating predictions about different slices of the overall distribution.

Statistics Cumulative Frequency

Valuing a positionOn the diagram on the right, we could find out the median speed of the 60 cars by starting at 30 on the y-axis (since this is an estimate we don’t make the “sample-plus-one” correction) drawing a horizontal line to the curve and abseiling down to an answer of 120.

Finding a proportionIn the other direction, we could predict how many of the cars exceeded the 130 km/h speed limit by projecting up from 130 on the x-axis to the curve and thence back to the frequency level of 52. Note that this would deliver an answer that 8 cars exceeded the limit, an example of a subtraction calculation very popular with examiners, and frankly very often necessary in real world applications.

Use it, then build itThis topic is often taught by first showing how to generate the chart, and only somewhat later helping students to use it for interpretation. As a result the interpretation work is often rushed or squeezed. Far better to follow on from box plot work by showing a series of examples of cumulative frequency charts and seeing how useful they can be for making predictions about more than just the medians and quartiles.Then, with the justification established, there is perhaps more motivation to see how to create them.

Build itSo how do we build it, and what do we need to beware of?The normal start point is a given frequency table, perhaps like this one:

The first job is to create a new row for the table, entitled “Cumulative frequency”, calculated as a running total across the page, like this

Students should plot the upper limit of each range on the x-axis (this is the main source of misunderstanding on this topic – overcome it by talking about how the curve “reaches up to the top right” and so you must plot at the top right of every box) and the frequency on the y-axis. (For all statistical diagrams, they should expect to plot frequencies on the y-axis). So the first plot point above would be (20,3), then (30,9) and so on. This will normally yield the S-shaped curve shown above, but if they have forgotten to calculate cumulative frequency and just used frequency, they will just end up with a frequency distribution bell curve.Examination questions normally include reference to the final cumulative total in the first sentence, against which the cumulative frequency row can be checked.

Time 10<t≤20 20<t≤30 30<t≤40 40<t≤50 50<t≤60 60<t≤70

Frequency 3 6 10 17 11 3

CumulativeFrequency

3 9 19 36 47 50

Two SweepsOnce all the points have been plotted, a smooth curve needs to be drawn, and for this the page should be turned to allow two half sweeps. Starting from the middle plot, the right-hander should draw a sweeping arc up to the top, then turn the book upside down and draw a sweep to the bottom. The left-hander does the opposite. This is very satisfying!

1 2

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115

Strategy Proof Concepts – Principle of RigourAlways, Sometimes, NeverProof is something often neglected, or delivered somewhat apologetically. It doesn’t fit the standard pedagogical model, so teachers are less confident as they approach the lesson. As we progressively learn to emphasise problem solving properly (teaching them to fish rather than serving them fishfingers) the necessary teaching style will probably become more comfortable. There is also a tendency to treat proof as a concept only of interest to top sets, because it is often difficult and because the independent thinking is less evident to us in other sets. But there is another aspect of proof, the principle of rigour, that is important at every level of mathematical ability. And for questions our students will in future contemplate not just in the arena of mathematics.An important technique to embed the idea of rigour at a very basic level is to create a set of laminated voting circles as shown in the margin and using them regularly to check student understanding across a range of topics. The emergence of a recognition that there is a difference between a “sometimes” answer and an “always” answer is the start of the acceptance of the importance of rigour. Continuing repetition of the mantra “Always, sometimes, never?” develops a good mathematical mindset.

So should we plan to teach a lesson on the subject of rigorous proof? Probably not. Better to ensure through our medium term planning that the concept is deployed regularly through the rest of the curriculum. Listed below are some key occasions when extension work or plenary activity can benefit both the topic being considered and the principle of rigour.• During teaching on sequences, exploring the behaviour of odd and even numbers, through the concept

of numbers conforming to 2n+1 and 2n formats, and potentially extending this to patterns of divisibility using the format kn. Examining proofs involving consecutive numbers using n, n+1, n+2 etc..

• With more able students in trigonometry, taking the time to show why the sine and cosine rules work, ideally through student discovery exercises, and then engaging students in building proof statements for display.

• When working on circle theorems providing model expositions of proofs and explaining why it is important to present the argument clearly.

EuclidIf time permits, dedicate a couple of lessons to exploring Euclidean geometry. There are five principal theorems of plane Euclidean geometry:

• the sum of the angles in a triangle is 180 degrees,• the Bridge of Asses, • the fundamental theorem of similarity, • the Pythagorean theorem, • and the invariance of angles subtended by a chord in a circle.

The Bridge of Asses theorem is repeated here, so-called because in medieval times it was suggested it was a threshold to indicate whether someone was up to the task of understanding geometry – if not they were one of the asses who could not cross. The theorem states that a triangle with two equal sides has two equal angles. Start with a triangle in which AB and AC are equal Create equal length lines AD and AE as respective extensions of AB and AC The triangles ADC and AEB can then be shown to be congruent on a SAS basis because

˂DAC=˂EAB (same angle) AD=AE (drawn to same length) AB=AC (assumed same originally)

Then the triangles DBC and ECB can be shown to be congruent on a SAS basis because ˂BDC=˂BEC (corresponding angles in congruent triangles ADC and AEB) BD=CE (both equal length extensions of BD and CE) CD=BE (corresponding sides in congruent triangles ADC and AEB

So ˂DBC=˂ECB (corresponding angles in congruent triangles DBC and ECB So ˂ABC =˂ACB (each 180° - ˂DCB and ˂ECB respectively) QED

It is good to demonstrate this theorem fully to show the importance of the rigour in providing reasons for every statement.

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127

Always

A

B C

D E

Navigation Mathemagogue’s Hardware ToolkitAs a trainee teacher, you may be financially challenged. All those course fees, prior student debts, lack of earnings. But you are also challenged for time, so there is a physical toolkit of devices and resources you may want to assemble or purloin as best you can. Some of them may be provided by your Maths department, but others you may choose to acquire for yourself. Or perhaps kind relatives can think of buying you these for your birthday or other celebrations? Think of it as a Pinterest for the Maths teacher:For the performer in you (bear in mind that technology models change quickly, so more up to date options might be available when you go searching…)

• A “presenter” such as the Kensington. This enables you to control your slide presentation from anywhere in the classroom. Spookily for your students (and commanding some easy respect) even from your pocket! Presenting from the back of the class brings a new dynamic of sharing a view with the class, making it feel like you are “with” them rather than “to” them, aside from any behaviour management gains that may result.

• A graphics tablet, such as the Ugee M708 . This is different from an iPad or other tablet computer. It runs much more simply, and thus much more powerfully, and enables you to draw on the presentation plane much more naturally from the surface of your desk than a mouse can. Don’t upgrade to extra capabilities, as all you will do is slow down the machine’s movement response!

• A visualiser, such as the Hue HD camera. When you are coaching students on physical skills such as the use of compasses and protractor, the visualiser really brings the ideas to life.

• An A4 laminator to have at home. Get home to see your loved ones and still be able to create great resources. OK so the school should really pay for this, but isn’t your time at home worth more?

All of these products can be had somewhere in the region of £30 at the time of writing. Well worth it.For you as a teacher, and for the students, some practical low-tech resources as well.

• A set of lolly ice sticks on which to write the names of every student in each class and to keep in a class bundle for random questioning. Effectively 2p each at Wilko!

• Some lengths of rope for outdoor or floor display of graphs and loci. An excellent source is www.cheaprope.co.uk, and make sure you go for braid-on-braid rope as it performs much more to your bidding. I’d have three 10m lengths of 10mm in my toolkit. If you want to be particularly smart, consider some endless loops for illustration of circle and Venn diagram problems. If you want to draw excellent circles on the board, fashion a 20cm string with a loop at both ends to act as a compass.

• A class set of laminated postcards with a large A B on one side and C D on the other. Excellent for multiple choice exercises. Students can hold them up with a hand covering up whichever of the facing pair they are not voting for. Remember that multiple choice is an extremely powerful way of teaching about misconceptions.

• A few packs of cards for probability lessons, along with packs of nice big dice. See if you can get hold of a 15cmx15cmx15cm foam die for class presentation.

• A class set of dry wipe whiteboards with a pre-drawn cartesian grid on one side to speed up questioning in graph-based lessons. And while you’re at it, a set of pipe cleaners to help students model a curve before they draw it.

• And a visit to your local DIY store to pick up supplies of mosaic tiles, tile spacers, pipe sections for model cylinders, and maybe treat yourself to a laser measurer for a bit of fun.

130

Statistics

This book was conceivedwith the trainee secondary Mathematics

teacher in mind, because its author saw how frequently new teachers would struggle to identify what were the critical issues about the lesson they were about to teach for the

first time tomorrow, and their poor beleaguered Heads of Department were too

busy to help.

Dramatic, Kinaesthetic, Mnemonic, VisualStrategies to teach Mathematics in the Secondary School so it sticks

Practical advice for how to teach each topic tomorrow morning. Guidance on what students find hard to follow, how to explain the idea clearly, the misconceptions to manage, and how to provide for future recall. Not a textbook, not a treatise on teaching, but useful immediate support for the mathematical pedagogue.

Phil Murray has a MA in Mathematics from the University of Oxford. After a career in management consulting he switched to become a secondary school mathematics teacher in 2005, and more recently has moved to teacher training as a Visiting Tutor for Initial Teacher Education and as a course writer and presenter at the Further Mathematics Support Programme.

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Number Algebra Geometry