Teacher Insight Pages (TIPS)
TIPS pages are quick tips for mathematics teachers. They have been compiled by Ed Staples and Paul Turner, the authors of Mathematical Whetstones Sharpening Understandings and Primary Concepts. The tips are focused on material from the senior secondary or A Level curriculum. You are welcome to download them however we request that you please acknowledge the authors if you are distributing the material.
| tip_1_form_and_symmetry_published.pdf |
TIP 2
Carlyle and the Quadratic
How do you find the roots of a quadratic equation using a pencil, ruler and compass?
Carlyle and the Quadratic
How do you find the roots of a quadratic equation using a pencil, ruler and compass?
| tip_2_carlyle_and_the_quadratic_final_.pdf |
TIP 3
Fermat Area
Fermat Area
Pierre de Fermat, well over a decade before Isaac Newton was born, had found a way of determining areas under certain polynomial curves even though the connection between area and gradient hadn’t been made. This tip shows the technique for finding the area under a simple parabola.
It’s ingenious thinking.
In school settings, sequences and series (including APs and GPs) are usually discussed well before the calculus starts, and the topic of finding areas under curves is typically introduced with finding area approximations using upper and lower rectangles.
Why not use Fermat’s technique as well?
Can you use a geometric series to find the area under the curve? 🤔
It’s ingenious thinking.
In school settings, sequences and series (including APs and GPs) are usually discussed well before the calculus starts, and the topic of finding areas under curves is typically introduced with finding area approximations using upper and lower rectangles.
Why not use Fermat’s technique as well?
Can you use a geometric series to find the area under the curve? 🤔
| tip_3_fermat_area.pdf |
TIP 4
Lill's Quadratic Solution
Lill's Quadratic Solution
Eduard Lill (1830-1900) was an Austrian engineer who studied maths in Prague. He chose a military career and later worked as an engineer for the Austrian Northwestern railway. Besides his work in traffic and transport research, Lill is remembered for his technique of graphically locating roots of polynomial functions.
Lill invented a technique of solving a polynomial equation using a set-square! This TIP raises far more questions than it answers. Sliding a set-square over a mysterious construction graphically ‘solves’ a quadratic equation.
Evaluating the polynomial at any given point directly involves up to calculations. The headmaster William Horner in 1819 observed that, when polynomials are re-expressed as interlinked linear expressions, the number of calculations reduces to . Lill’s brilliance was to utilise Horner’s scheme in a construction containing inconspicuous similar triangles that correspond to these linear expressions.
If you haven’t seen it before, the technique will look mystifying. Your curiosity will be aroused, and you’ll need to do more reading (there’s a blog listed at the end of the tip which explains more and shows a variety of examples). But persist! The collaboration believes that it’s worth spending the time looking into it.
It would make a great investigation for any senior school classroom.
Lill invented a technique of solving a polynomial equation using a set-square! This TIP raises far more questions than it answers. Sliding a set-square over a mysterious construction graphically ‘solves’ a quadratic equation.
Evaluating the polynomial at any given point directly involves up to calculations. The headmaster William Horner in 1819 observed that, when polynomials are re-expressed as interlinked linear expressions, the number of calculations reduces to . Lill’s brilliance was to utilise Horner’s scheme in a construction containing inconspicuous similar triangles that correspond to these linear expressions.
If you haven’t seen it before, the technique will look mystifying. Your curiosity will be aroused, and you’ll need to do more reading (there’s a blog listed at the end of the tip which explains more and shows a variety of examples). But persist! The collaboration believes that it’s worth spending the time looking into it.
It would make a great investigation for any senior school classroom.
| tips_4_lills_quadratic_solution_final.pdf |
TIP 5
Horner's Scheme
Horner's Scheme
Not so long ago, teachers taught ‘tricks’ of calculation. I can remember them. For example, it’s easier to find 50% of $21 than it is to find 21% of $50, even though it gives the same result. Tests of divisibility were emphasised and learning things like the times tables up to 12, square numbers to 20 and powers of 2 up to 10 were encouraged. Then electronic technology arrived and our reliance on the new tools followed quickly thereafter.
As a mathematician in the 19th century, the work involved producing a decent sketch of a degree 5 polynomial function was prolific. There can be up to 20 calculations (additions and subtractions) to perform to derive each y value and you would need at least 10 of them, perhaps more.
But a teacher, a chalkie from England, by the name of William George Horner, found an insightfully efficient way to do it. Don’t think this is just history though! Algorithm efficiency is becoming a key concept in the modern world of mathematics. Check it out!
As a mathematician in the 19th century, the work involved producing a decent sketch of a degree 5 polynomial function was prolific. There can be up to 20 calculations (additions and subtractions) to perform to derive each y value and you would need at least 10 of them, perhaps more.
But a teacher, a chalkie from England, by the name of William George Horner, found an insightfully efficient way to do it. Don’t think this is just history though! Algorithm efficiency is becoming a key concept in the modern world of mathematics. Check it out!
| tip_5_william_horner_final.pdf |
TIP 6
Dirichlet's Rational Approximation
Dirichlet's Rational Approximation
How many socks do we need to pull from a drawer containing many socks of just three colours to be sure of obtaining two of the same colour? And if you can’t answer that, perhaps try writing down a 27-word sentence where each word starts with a different letter.
Dirichlet’s pigeonhole principle is powerful and versatile. In TIP 6 it’s applied to a proof that shows that a rational approximation, to any desired level of accuracy, can always be found for the fractional part of an irrational number. Moreover, the proof alerts us to a way to determine that rational number.
For example, restricting the approximation to a two-digit numerator and denominator, 41/56 pops out as a close approximation to the fractional part of the square root of 3. The author hunted that one down with a spreadsheet because, armed with the proof, he knew he would find it.
Takes a while to really get this one, but when you do, it’s a great “ah ha!” moment.
Dirichlet’s pigeonhole principle is powerful and versatile. In TIP 6 it’s applied to a proof that shows that a rational approximation, to any desired level of accuracy, can always be found for the fractional part of an irrational number. Moreover, the proof alerts us to a way to determine that rational number.
For example, restricting the approximation to a two-digit numerator and denominator, 41/56 pops out as a close approximation to the fractional part of the square root of 3. The author hunted that one down with a spreadsheet because, armed with the proof, he knew he would find it.
Takes a while to really get this one, but when you do, it’s a great “ah ha!” moment.
| tip_6_dirichlets_rational_approximation.pdf |
TIP 7
Quarter-Square Tables
Quarter-Square Tables
What is the difference between a quarter of the square of the sum of 12 and 7 and a quarter of the square of their difference? Compare that to their product.
One of the most ingenious 19th Century financial tools ever developed was a QS table. Bankers, clerks, auctioneers and others were continually faced with painstaking manual multiplications. QS tables rescued them. They became extremely popular because they were compact, easy to use and minimised error because the need to multiply was completely eliminated.
An algebraic idea out of ancient Mesopotamia was given new life in the early 1800’s by, among others, a French auctioneer by the name of A. Voision. If you ever come across a little book entitled (in French) Tables of Quarter Squares, with Logarithms of whole numbers from 1 to 20,000, grab it! They’re as rare as hen’s teeth, and they’ve got nothing to do with logarithms!
A great investigation for a high school class. Open it up!
One of the most ingenious 19th Century financial tools ever developed was a QS table. Bankers, clerks, auctioneers and others were continually faced with painstaking manual multiplications. QS tables rescued them. They became extremely popular because they were compact, easy to use and minimised error because the need to multiply was completely eliminated.
An algebraic idea out of ancient Mesopotamia was given new life in the early 1800’s by, among others, a French auctioneer by the name of A. Voision. If you ever come across a little book entitled (in French) Tables of Quarter Squares, with Logarithms of whole numbers from 1 to 20,000, grab it! They’re as rare as hen’s teeth, and they’ve got nothing to do with logarithms!
A great investigation for a high school class. Open it up!
| tip_7_qs_tables_final_version.pdf |
TIP 8
Integration Technique
Integration Technique
How do we want our students to believe in mathematical arguments and procedures? Should our claims in the classroom rest on our authority as local ‘experts’? Perhaps we prefer to appeal to the word of a respected textbook.
Much better, we suggest, is that students become convinced because they understand the logic of the concepts being discussed.
The study of calculus can be prone to a procedural kind of learning in which a student memorises the steps to be taken in resolving questions of a certain type but may not really understand what is being done. In this TIP we look at a tiny fragment in the topic of integration as an illustration.
Does it look familiar?
Much better, we suggest, is that students become convinced because they understand the logic of the concepts being discussed.
The study of calculus can be prone to a procedural kind of learning in which a student memorises the steps to be taken in resolving questions of a certain type but may not really understand what is being done. In this TIP we look at a tiny fragment in the topic of integration as an illustration.
Does it look familiar?
| tip8_integration_technique_.pdf |
TIP 9
Average of Roots
Average of Roots
The roots of a polynomial function and all of its derivatives (first, second, third, etc.) are related in a very interesting way. You may never have considered this relationship before. All is explained in TIP 9 along with a proof of this remarkable connection. Why not share it with your students? It’s a great classroom exercise!
| tip_9_average_of_roots.pdf |
TIP 10
Power of a Point Theorem
Power of a Point Theorem
Imagine a ray, drawn from an external point P, that is tangent to the circle at the point of contact T. Let h equal the square of the distance PT. Then h is known as the power of the point P.
TIP 10 is a straightforward introduction to a beautiful geometric theorem called the Power of a Point Theorem, formulated by the great geometer Jacob Steiner (1796 – 1863). There is much more to learn, but if you are looking for a proof utilising similar triangles, this could be it!
TIP 10 is a straightforward introduction to a beautiful geometric theorem called the Power of a Point Theorem, formulated by the great geometer Jacob Steiner (1796 – 1863). There is much more to learn, but if you are looking for a proof utilising similar triangles, this could be it!
| tip_10_power_of_a_point_theorem_2.pdf |
TIP 11
The Sine Rule
The Sine Rule
The Sine Rule (or the Law of Sines) expresses that remarkable symmetric relationship between the three sides of a triangle and the angles that are opposite them. Specifically, any side divided by the sine of its opposite angle is a constant, say d. How big or how small can d get? Can there be triangles with different sides and angles yet with the same d? Does d have any significance to anything? Is it even pedagogically worthwhile mentioning d in maths classes? Before you look at TIP 11, ask yourself, have you ever considered d in your classroom? Something to think about.
| tip_11_the_sine_rule.2.pdf |
TIP 12
Dodgson's determinants
Dodgson's determinants
Charles Dodgson (1832-1898) invented an ingenious method for finding determinants. He called it the ‘method of condensation’. Through a series of order reductions on a matrix, the evaluation of its determinant is gradually condensed out like a water droplet squeezed out of a mass of moist air. There is an exquisite charm about the way it works.
It takes a few minutes to see just how beautiful it is, and it’s certainly worth showing in a classroom for that reason alone! But if you want to know more, read the 2006 article Lewis Carroll’s condensation Method for evaluating determinants by A Rice and E Torrence. We have the link if you want it.
https://www.maa.org/sites/default/files/pdf/Mathhorizons/pdfs/nov_2006_pp.12-15.pdf
It takes a few minutes to see just how beautiful it is, and it’s certainly worth showing in a classroom for that reason alone! But if you want to know more, read the 2006 article Lewis Carroll’s condensation Method for evaluating determinants by A Rice and E Torrence. We have the link if you want it.
https://www.maa.org/sites/default/files/pdf/Mathhorizons/pdfs/nov_2006_pp.12-15.pdf
| tip_12_dodgsons_determinant_.pdf |
TIP 13
By Infinite Descent
By Infinite Descent
The concept of irrational numbers, when understood properly, must at first be quite confronting to many high school students; numbers that really exist, but which cannot be expressed as the ratio of two integers. There is an infinite number of integers to choose from but no two of them will ever do the job!
Every proof is based on exclusion from the set of rationals, and one of these is known as proof by infinite descent. It’s not too difficult to understand and is well suited to the classroom. Nevertheless, students can be left with an uneasy feeling about numbers that are not rational. They might be a little unsure about what or even where they are.
Every proof is based on exclusion from the set of rationals, and one of these is known as proof by infinite descent. It’s not too difficult to understand and is well suited to the classroom. Nevertheless, students can be left with an uneasy feeling about numbers that are not rational. They might be a little unsure about what or even where they are.
| tip_13_by_infinite_descent_.pdf |
TIP 14 & 15
Gardner Triples, What, How and Why
Gardner Triples, What, How and Why
Think of the reciprocals of the numbers 1, 2 and 3 as the tangents of three angles. Two of the angles add to the third. This relationship lies at the heart of the renowned three-square problem. One night I began thinking about the possibility of other sets of integer triples that might exhibit the same summative property.
I managed to find a few others and then a light came on. The set 1, 2, 3 is not just the first three positive integers, but is also three consecutive Fibonacci numbers. It turned out that triples like 3, 5, 8 and 8, 13, 21 and 21, 34, 55 etc. all exhibited the same property!
But there were other more general types of triple lurking, and only some of those were Fibonacci related. One year on and three articles later, we had properly defined the ‘Gardner Triple’ and the ‘Gardner triangle’. TIP 13 illustrates how to create them and, in Tip 14, Paul Turner establishes why they exist. Fascinating stuff!
I managed to find a few others and then a light came on. The set 1, 2, 3 is not just the first three positive integers, but is also three consecutive Fibonacci numbers. It turned out that triples like 3, 5, 8 and 8, 13, 21 and 21, 34, 55 etc. all exhibited the same property!
But there were other more general types of triple lurking, and only some of those were Fibonacci related. One year on and three articles later, we had properly defined the ‘Gardner Triple’ and the ‘Gardner triangle’. TIP 13 illustrates how to create them and, in Tip 14, Paul Turner establishes why they exist. Fascinating stuff!
| tip_14_gardner_triples_how.pdf |
| tip_15_gardner_triples_why.pdf |
TIP 14 & 15 Supplement
The following supplement provides a little more insight into Gardner Triples, particularly in terms of their application to certain mathematical series. It’s fascinating stuff! The supplement also shows how Machin’s formula for Pi can be developed using the concept.
You must read it!
You must read it!
| ss_to_tip_1415_gardner_triples.pdf |
TIP 16 The Rule of 72
Knowing the “Rule of 72” was standard fare for any bank teller in the early to mid 1900s. In those days, without computers or electronic calculators to refer to, cumbersome books of tables were used to determine things like future values of investments, mortgage calculations, annuity instalments etc.
Reference to the rule was made by Luca Pacioli in Venice in 1494. It estimates the time an investment will take to double, given some interest rate. Getting an estimate has its advantages. A client would always appreciate an estimate, and the rule made a great check on the table look up procedure. With copious tables and books to search through, mistakes could easily be made. A great tip to pass on to students.
Reference to the rule was made by Luca Pacioli in Venice in 1494. It estimates the time an investment will take to double, given some interest rate. Getting an estimate has its advantages. A client would always appreciate an estimate, and the rule made a great check on the table look up procedure. With copious tables and books to search through, mistakes could easily be made. A great tip to pass on to students.
| tip_16_the_rule_of_72.pdf |
TIP 17 The Lambert W Function
It was in1983 that I first noticed three real solutions to the equation 2^x=x^2. Two of them were obvious, but the third was a weird negative number. I had no idea how to get to it analytically, but after a few minutes of trial and error I reckoned it to be about –0.76666. I thought at first it might be the rational number -23/30…but, alas, it wasn’t.
Some years later, I went to an Australian Mathematics Conference workshop and the presenter talked about a thing called the Lambert W function. It was a difficult workshop for me, but I remember him saying that maths was about form and symmetry, and that, one day, the Lambert W function would be a button on a scientific calculator. If you’re up for it, here is a peak into a fascinating contemporary mathematical tool.
Some years later, I went to an Australian Mathematics Conference workshop and the presenter talked about a thing called the Lambert W function. It was a difficult workshop for me, but I remember him saying that maths was about form and symmetry, and that, one day, the Lambert W function would be a button on a scientific calculator. If you’re up for it, here is a peak into a fascinating contemporary mathematical tool.
| tip_17_the_lambert_w_function.pdf |
TIP 18 Simpsons Paradox
The Bletchley Park code breaker and statistician Edward Simpson (1922-2019) was believed to be the first to describe the statistical paradox. He discovered that a trend could appear in several different groups of data but then disappear or reverse when those groups are combined.
The paradox is instructive and is highly recommended to all high school students studying statistics. Don’t miss this one!
The paradox is instructive and is highly recommended to all high school students studying statistics. Don’t miss this one!
| tip_18__simpsons_paradox_.pdf |
TIP 19 Games of Lotto
In the casino game of roulette, the player easily visualises the chance of winning. The fraction associated with the probability of correctly forecasting a winning number becomes immediately evident when the chip is placed onto a grid square of a standard grid of 37 or 38 numbers.
In the more deceptive commercial and state-run games of Lotto, that visualisation is not at all apparent. The size of the grid required in a game of 45-ball lotto for a player to fully comprehend the probability of winning the major prize with a single ticket would need to be over 200,000 times larger than the standard roulette grid. A few simple calculations show that about four football fields placed side by side are required to lay out just one of these ‘lotto grids’.
Getting students to visualise these very small probabilities in this sort of way is critically important. Writing fractions with large denominators doesn’t really cut it! Here is an idea worth considering.
In the more deceptive commercial and state-run games of Lotto, that visualisation is not at all apparent. The size of the grid required in a game of 45-ball lotto for a player to fully comprehend the probability of winning the major prize with a single ticket would need to be over 200,000 times larger than the standard roulette grid. A few simple calculations show that about four football fields placed side by side are required to lay out just one of these ‘lotto grids’.
Getting students to visualise these very small probabilities in this sort of way is critically important. Writing fractions with large denominators doesn’t really cut it! Here is an idea worth considering.
| tip_19_games_of_lotto.pdf |
TIP 20 The Babylonian Quadratic Formula
There is an interesting way to multiply two numbers. Suppose we think about the product 12 and 18. Half their sum is 15 and half their difference is 3. The difference in the squares of 15 and 3 is 216, which is the product we want. When halving sums and differences you might get fractional bits, but these will disappear when you take the difference in their squares. Try one yourself to check.
The Babylonians were the first to document the technique, and we can use their ingenuity to derive the general quadratic formula. A great classroom activity! A precious insight for students! Open it up and follow the logic. Can you see an assignment in this?
The Babylonians were the first to document the technique, and we can use their ingenuity to derive the general quadratic formula. A great classroom activity! A precious insight for students! Open it up and follow the logic. Can you see an assignment in this?
| tip_20_babylonian_quadratic_formula.pdf |
The String Cut Problem
| string_cut_problem_pt_2_circle_and_square.pdf |
| string_cut_problem_pt_3_circle_and_n-gon.pdf |