Microsoft PowerPoint - Saturday 9am Six New Gems for C3 and C4

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1 The six gems Six new gems that teachers of 3 and 4 should know MEI onference 200 roof: My favourite proof Scalar roduct and Vector: ool applications, vector spaces with dimension greater than three Differential Equations: Introducing e and giving DEs meaning lgebra: The inomial Expansion Functions: Why they save time Trigonometry: different approach? Some great results Each of these sections contains a gem! My favourite proof Theorem: There are arbitrarily long sequences of consecutive numbers that are non-prime. roof roof: For any natural number n larger than, the sequence n! + 2, n! + 3,, n! + n is a sequence of n consecutive composite integers. onjectures about primes There infinitely many twin primes, pairs of primes with difference 2. Or stronger - olignac's conjecture:for every positive integer n, there are infinitely many pairs of consecutive primes which differ by 2n. There are infinitely many primes of the form n 2 + rocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n 2 and (n + ) 2 for every positive integer n. Goldbach's conjecture asserts that every even integer greater than 2 can be written as a sum of two primes The scalar product

2 b 3-4 a 2 cos (42) (3) (2) 2 2 2 2 2 2 4 2 2 3 ( ) This uses a technique called the

2 Lighting a scene - world without lighting effects World with lighting What is a vector anyway? 5 a 2 b 3 - we can calculate the angle between two vectors. 2 b 3-4 a 2 cos (42) (3) (2) ( ) This uses a technique called the scalar product ngle between normal vector and light source determines how light the surface should appear 2

3 Lighting a Flat Surface How do we work out when this happens? 3

4 4

5 = 360 Demonstration 2D < 360 5

Geogebra version Demonstration 3D ompatibility Social

inema - Every person has a vector associated with

application form Joe and Jenny Joe 4 Jenny?

vector is 2S + M + 0T 5 2 3 2 0 Joe Jenny 5 2 3 2 2 2

6 Geogebra version Demonstration 3D ompatibility Social networking websites Sport - S Music - M Theatre - T inema - Every person has a vector associated with themselves representing their taste from their application form Joe and Jenny Joe 4 Jenny? For example: Joe s vector is 5S + 2M + T 3 Jenny s vector is 2S + M + 0T Joe Jenny ( 3) ( ) Joe.Jenny (52) (2) (0) ( 3)

7 match! Differential Equations Joe loves Jenny 84.98% Recommendations to customers according to taste on e-commerce sites. Why do we need differential equations? Differential Equations are at the core of calculus. Traditionally the key to understanding the physical sciences lso one of the most essential practical tools engineers, economists and others have for dealing with rates of change. E.g. is is the cost of producing X widgets then d/dx is the marginal cost of producing the Xth widget Differential Equations The idea in solving an algebraic equations is to determine from conditions on a number what that number might be. The idea in solving a differential equation is to determine from conditions on the derivative (and higher derivatives) of a changing quantity what that quantity might be at any given time. Differential Equations Newton s laws of motion Laplace s heat and wave equations Maxwell s electromagnetic theory Navier Stokes Equations for fluid dynamics Volterra s predator prey systems Useful classroom demonstration Students don t seem to necessarily understand what they are doing when they solve a differential equation. The emphasis now is more on numerical approximation and computing and less on traditional methods involving limits and infinite processes. 7

Ways to introduce e Ways to introduce e Invest at a rate of 00% interest per annum. If the interest is given once at the end of the year then you get 2 back at the end of the year.

8 Ways to introduce e Ways to introduce e Invest at a rate of 00% interest per annum. If the interest is given once at the end of the year then you get 2 back at the end of the year. If you get 50% interest after sixth months and then another 50% at the end of the year you get back x.5x.5 = 2.25 Ways to introduce e e If you get 25% every three months you get back x.25 x.25 x.25 x.25 = 2.44 With 2.5% every.5 months you get.25 8 = 2.57 Does this sequence keep getting bigger and bigger? Is it bounded? If not, prove it. If it is, does it tend to a limit? This shows us e s key role in banking and compound interest calculations. e makes lots of implausible appearances in problems. e is ubiquitous in mathematical formula, theorems and their proofs. It is intimately involved with trigonometric functions, geometrical figures, differential equations, infinite series. ascal s triangle and the binomial theorem 4. lgebra and the binomial theorem The additive property of ascal s triangle becomes clear when multiplication is set out in a formal way 8

$Extending ascal s Triangle Extending ascal s Triangle Newton had the remarkable insight that the binomial theorem is not only true for positive integers but also for negative and fractional values of$

9 Extending ascal s Triangle Extending ascal s Triangle Newton had the remarkable insight that the binomial theorem is not only true for positive integers but also for negative and fractional values of n. Lines corresponding to negative integers can added to ascal s triangle Functions 5. Functions One important aspect of functions that is often overlooked is that the enable us to save time. For example the chain rule, the product rule and the quotient rule mean that we can differentiate a huge class of functions from just knowing how to differentiate simple ones. great example of this is Lighbot pproaches to trigonometry 6. Trigonometry Usually trigonometry is introduced as being concerned with the ratio of side lengths in right angled triangles Ratio is a difficult concept algebraically and an approach based on similar triangles and enlargement might be better Sometimes over-use of algebra is not the best approach 9

Sin and os.2 Introduce them by getting students to see the output for various values of. 0, 0.7, 0.98 0.34, 0.94 0.50, 0.87 0.8 0.64, 0.77 y 0.

10 Sin and os.2 Introduce them by getting students to see the output for various values of. 0, 0.7, , , , 0.77 y , , , , 0.7 0, x Doing problems using an enlargement technique This then follows very naturally. sin 3? cos r rsin rcos 0.34 ontrast the approaches Tan and Gradient Tan often gets overlooked in -level mathematics and is frequently dealt with as sin/cos. Giving tan attention in its own right and looking at the connection to gradient can pay dividends. 0

11 Defining tan as a length using enlargement xtan 60 tan 60 tan x Defining tan as a length using enlargement Geometric proofs sin sin cos sin sin cos sec tan sin ( + ) sin cos cos 2 + sin 2 = + tan 2 = sec 2 cos cos ( + ) cos cos cos sin Trigonometry Draw the graph of y = sin 2 x Is this a translation of sinx?

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37 92 19 40 19 20 21 1 7 22 1/5 6/30 5/3030 23 24 25 26 1 2 27 1 2 28 29 30 5 8 8 3 31 32 33 34 35 36 37 38 39 A Study Investigating Elementary School Students Concept of the Unit in Fraction in Northern