Pedagogy for the Transition of Mathematics Continuity  ALGEBRA  GEOMETRY  CALCULUS 1 PRESENTER: GLENROY PINNOCK MSc. Applied Mathematics; Marine Engineering Diploma; Mathemusician; PhD. (EDUC)pending JANUARY 14TH, 2011

RADICALS What is a radical quantity in mathematics? E.g.,√a Factor for algebraic expression:x 2 - y 2 2 Rationalize √a Factors for algebraic expressions:x 2 + y 2, x 3 - y 3, x 3 + y 3, x 4 - y 4, x 4 + y 4 LAWS OF INDICES (RECALL SIX (6) OF THEM)

PASCAL TRIANGLE/BINOMIAL EXPRESSIONS (a + b) 0 (a + b) 1 (a + b) 2 (a + b) 3 (a + b) 4 (a + b) 5 TOOLS FOR COEFFICIENTS IN ALGEBRAIC EXPANSION

QUADRATIC THEORY 4 What is the discriminant to the formula on the left? Clearly, it’s b 2 - 4ac.  What happens if this term is negative? So, we can now discuss the significance of the discriminant. a.The roots of a quadratic equation are imaginary/complex, if the discriminant is negative. b.If the discriminant is greater than 0, then the roots are real. c.If b 2 = 4ac, then the roots are real and equal. d.By the way, notice that these roots are basically the solutions of a quadratic equation. PROVE THIS FORMULA! y = ax 2 + bx + c The solution x = -b+√ (b 2 - 4ac) 2a Recall that if the roots of the above quadratic equation is and β. What is the sum of the roots and the product of the roots? Write down the equation, whose roots are α 3 and β 3 ? α

TRIGONOMETRIC EQUATIONS 5 What does that equation mean for the positive 0.5? What happens if 0.5 is negative? Solve:sine 2θ = 0.5, sine 3θ = 0.5 WOW! What is the difference for the number of solutions, for the equations above? Solve: sine 2 2θ = 0.25 We are ultimately breaking down 2 nd degree trig equations into a 1 st degree equation or factor. sine θ = 0.5 Clearly sine θ = ½ Do you remember trig-ratios? Do you recall the quadrant system?

THE POLYNOMIAL [P(x)] This can be written as 9 = 2 (4) + 1. Conceptually, we can say 9 represents a polynomial, 2 is the quotient, 4 is the divisor, and 1 is the remainder. Let us now divide x 2 – 1 by x + 1. Clearly, x 2 – 1≡ Q (x + 1) + 0. How did I get a zero? So, generally speaking P (x) ≡ Q (x + a) + R. NB: It is advisable to represent a remainder theorem/factor theorem problem in this format. Remember to use the synthetic rule to reduce tedious long division. Also, you need to know how to solve a pair of simultaneous equations.

LOGARITHMS 7 Always recall the laws of indices and indicial equations whenever you are doing log problems. What is the logarithm of a number? Now consider log = ? 10 1 = 10 Similarly, log e 100 = y. Clearly, e y = 100. Now we can say it’s the definition of a logarithmic quantity.

8 GEOMETRY What is the difference between a horizontal line & a slanted line? Let’s consider the slanted line: B (x 1, y 1 ) A (x 2, y 2 )  Two parallel lines will have equalgradients.  When one line is perpendicular to another, the product of their gradients is -1. Consider the points A(2, 3) and B(-1, -2). What is the equation of the line? Recall y = mx + c *c is the y intercept and m is the gradient. Also, the gradient of a straight line is also equal to the tangent of inclination (tanθ) of the slanted line. y 1 - y 2 x 1 - x 2 Clearly, grad AB = What is the gradient of AB?

9 GEOMETRY (CONT’D) Construct a parallelogram from the points given above, and also find the area of the figure. Prove that the parallel sides of the parallelogram have equal gradients. Construct a rhombus with points of your choice, and find the area of the rhombus. Prove that the diagonal of the rhombus intersects at NB: The coordinates for the point of intersection of two lines is basically the solution of two simultaneous equations. A = ½ bh b h area of a right-angled triangle Area of a non-right angled triangle A= ½ ab sin C This equation also can be written in two other forms. What are these forms?

10 STATISTICS  the multiplication law and the addition law  cumulativefrequency curve (OGIVE)  upper quartile, lower quartile, median, inter-quartile range, semi- inter quartile range  percentiles  frequency polygon  permutation & combination  probability distribution  Poisson distribution  binomial distribution  orientation of the sample space diagram  normal distribution

11 CALCULUS Remember that this represents a straight line. Concept of a limiting value. AB = Recall gradient (grad) of a straight line: In the case of a curve, the gradient is found by considering the differential expression, namely, dy, f ׳, f x dx δ y δ x As a novice, we can say dy ≡ δ y dx δ x δ y δ x 0 So, what is your interpretation of the limiting value? As δ x 0 that is the time dy = δ y dx δ x y 1 -y 2 x 1 -x 2

CALCULUS (CONT’D.) 12 Clearly dy represents the gradient dx which is the tangent of the angle of inclination. Differentiation by formula y = x n ∴ dy dx Finally, to differentiate means to find dy from y. dx = nx n-1 Differentiate this equation: y = x 3 – x -2 + x 4  differentiation by first principles  differentiation of a product  differentiation of a quotient  differentiation of trigonometrical expressions  implicit differentiation  differentiation by approximation  differentiation express by rate of change DISCUSSION

13 INTEGRATION B A The area under the line AB is ? B A However, the area under the curved AB is found by integration. Let us consider this curve to be y = x 2 for x ≥ 0. The area under the curve is ydx. On the other hand, the volume under the curve is y 2 dx, if the revolution is done about the x axis through What would be the formula for the volume of revolution about the y axis?

14 INTEGRATION Integration is considered to be anti-differentiation (anti-derivative) Given that y = x 2 dy= 2x dx So, the integral of 2x can be written as 2x By considering the formula dy= x n dx ∴ y = x n+1 n + 1 Clearly, 2x.dx = x 2 Finally, we can now say integration is to find y whenever dy dx

15 INTEGRATION (CONT’D.)  Integration of a binomial expression to the n power  Integration of a trigonometric function  Integration by parts IN A NUTSHELL THE INTEGRAL SYMBOL REPRESENTS SIGMA NOTATION IN INTEGRATION THEORY. DISCUSSION Glenroy ‘Ajaniah’ Pinnock