1. AS Maths Ms Parr: C1 -Algebraic expressions -Inequalities -Differentiation -Applications of differentiation -Transforming graphs Mr Corbridge: C1 -Surds and indices -Equations and quadratic functions -Simultaneous equations -Coordinate geometry and the straight line -Coordinate geometry of the circle S1C2

2. Organising your files C1 File - Topics -Algebraic expressions -.... -Coordinate geometry of the circle -Assessment sheets and progress tests -Completed past papers -Course overview/spec? For each topic.... -Lesson notes, worked examples, handouts etc. -Exercises (classwork and homework)* -Further independent study/practice * on separate sheets of paper

3. What you need to bring to my lessons Textbook Formula book? A4 paper, pen, pencil, rubber, ruler, calculator C1 folder or as a minimum – current topic(s) section(s) – current tracker/assessment sheet Complete C1 folder when requested Homework when required

4. C1 ALGEBRAIC EXPRESSIONS Wed 10/9/14 LO: (i) review test (ii) Recap components of algebraic expressions (iii) understand function notation

5. Starting A level test Mark my paper Review your paper

6. Algebraic expressions Write down an example of: (i) An expression (ii) An equation (iii) An identity (iv) A formula (v) An inequality

7. Some important terminology Variable Constant Term Expression Polynomial Coefficient

8. C1 ALGEBRAIC EXPRESSIONS Fri 12/9/14 LO: (i)Understand function notation (ii)Add, subtract, multiply polynomials

9. Mappings Sketch the following curves: 1)y = 2x + 1 2)y = x 3 3)y = 1/x 4)x 2 + y 2 = 9 These all represent relationships between x and y. The first 3 are formulae for calculating y given x.

10. Function notation The formulae can also be written using function notation. 1)y = 2x + 1 2)y = x 3 3)y = 1/x f(x) = 2x + 1 g(x) = x 3 h(x) = 1/x or f:x →2x + 1 or g:x → x 3 or h:x → 1/x

11. Examples f(x) = 3x 2 + 4 f(y) = f(2) = f(-3) = f(x) - 2 = f(a – 2) = f(x) = x(x+3) f(y) = f(2) = f(-3) = f(x) - 2 = f(a – 2) =

12. Polynomials A polynomial in x is an expression of the form where a, b, c, … are constant coefficients and n is a positive integer. Examples of polynomials include: Polynomials are usually written in descending powers of x. 3 x 7 + 4 x 3 – x + 8 x 11 – 2 x 8 + 9 x 5 + 3 x 2 – 2 x 3.and The value of a is called the leading coefficient. They can also be written in ascending powers of x, especially when the leading coefficient is negative, as in the last example.

13. Polynomials A polynomial of degree 1 is called linear and has the general form ax + b. A polynomial of degree 2 is called quadratic and has the general form ax 2 + bx + c. A polynomial of degree 3 is called cubic and has the general form ax 3 + bx 2 + cx + d. A polynomial of degree 4 is called quartic and has the general form ax 4 + bx 3 + cx 2 + dx + e. The degree, or order, of a polynomial is given by the highest power of the variable.

14. Using function notation Polynomials are often expressed using function notation. For example, f ( x ) = 2 x 2 – 7 However, for polynomials, we often use the letter p instead of f, hence p ( x ) = 2 x 2 – 7

15. Adding and subtracting polynomials When two or more polynomials are added, subtracted or multiplied, the result is another polynomial. Find a) f ( x ) + g ( x ) b) f ( x ) – g ( x ) a) f ( x ) + g ( x ) = 2 x 3 – 5 x + 4 + 2 x – 4 Polynomials are added and subtracted by collecting like terms. = 2 x 3 – 3 x For example: f ( x ) = 2 x 3 – 5 x + 4 and g ( x ) = 2 x – 4 b) f ( x ) – g ( x ) = 2 x 3 – 5 x + 4 – (2 x – 4) = 2 x 3 – 5 x + 4 – 2 x + 4 = 2 x 3 – 7 x + 8

16. Suppose p(x) has degree 4 and q(x) has degree 5. What is the degree of f(x) + g(x)? What about f(x) – g(x)? Suppose f(x) and g(x) both have order 5. What is the order of f(x) + g(x)? Adding and subtracting polynomials

17. PRACTICE Ex 2B Q 1-3

18. Multiplying polynomials Don’t forget DOTS! Remember how to multiply two linear expressions together to form a quadratic - for example, (3 x – 2)(2 x – 1) =

19. Multiplying polynomials When two polynomials are multiplied together every term in the first polynomial must by multiplied by every term in the second polynomial. ( 3 x 3 – 2)( x 3 + 5 x – 1) = (Check: do you have the right number of terms?) Try this:

20. Multiplying polynomials Suppose p ( x ) = ( 3 x 3 – 2). Find -2 xp ( x ) Let p(x) = 3 x 3 – 2 and q ( x ) = x 3 + 5 x – 1. Find p ( x ) q ( x ). Example 1 Example 2 Find (3x + 2)(x - 5)(x 2 – 1) Example 3 (What is the degree of p ( x ) q ( x )?)

21. Multiplying polynomials Find the coefficient of x 2 in the expansion of (2x + 3)(3x 2 – 2x + 4). Example 4

22. Multiplying polynomials The product (Ax + B)(2x – 9) = 6x 2 – 19x – 36, where A and B are constants. Find A and B. Example 5

23. Function Notation and Multiplying Polynomials Homework (from sheet) - Ex 3A (excl Q 5 & 7) - Miscellaneous Exercise 9

24. Homework I will... set on Friday collect on Wednesday mark Wed/Thu return on Friday You will... complete it mark it (usually) re-attempt if necessary present it properly hand it in on time Individual study Go over material from lessons. Follow up problem areas. Practise by completing further questions.

25. C1 ALGEBRAIC EXPRESSIONS Wed 17/9/14 LO: Factorise expressions (including quadratics) Warm-up: Mini-test

26. Marking/Collecting... Function Notation and Multiplying Polynomials - Ex 3A - Misc. Ex. 9

27. Factorising expressions (mini-test)

28. Factorising quadratics a x 2 + b x + c c = 0 Example 1:10 x 2 – 2 x

29. Factorising quadratics a x 2 + b x + c b = 0 Example 2: x 2 – 9

30. Factorising quadratics a x 2 + b x + c b = 0 Example 3:25 x 2 – 81

31. Factorising quadratics a x 2 + b x + c b = 0 Example 4:5 x 2 – 80

32. Factorising quadratics a x 2 + b x + c b = 0 Example 5: x 2 + 9

33. Factorising quadratics a x 2 + b x + c b = 0 Example 6: x 2 + 9

34. Factorising quadratics a x 2 + b x + c a = 1 Example 7a: x 2 + 5 x + 6 Example 7b: x 2 - 5 x + 6 Example 7c: x 2 - x - 6 a = -1 Example 7d: 12 + 4 x - x 2

35. Factorising quadratics a x 2 + b x + c a ≠ ± 1 but a or c is a prime number Example 8a:5 x 2 – 8 x - 4 Example 8b:6 x 2 + 13 x - 5

36. Factorising quadratics a x 2 + b x + c Worst case scenario: neither a nor c is prime Example 9:4 x 2 - 5 x - 6 Either try all combinations: Or.....

37. Example: 4x 2 - 5x - 6 i) Find 2 numbers that multiply to give ac and add to give b. Factorising ax 2 + bx + c – Guaranteed method ii) Split the “x” term. iii) Factorise in pairs. iv) Complete the factorisation.

38. Factorising Practice/Homework Ex 2C Q 2-5 Ex 2D all Qs First and last part of each Q only

39. C1 ALGEBRAIC EXPRESSIONS Fri 19/9/14 LO: Multiply and divide algebraic fractions

40. Reviewing... Factorising - Ex 2C Q 2-5 - Ex 2D all Qs Any issues?

41. Returning... Function Notation and Multiplying Polynomials - Ex 3A - Misc. Ex. 9

42. Homework review Missing and mystery homework C3/X see me at end of lesson Questions not attempted Careless errors – “-” signs – Expanding brackets – Collecting like terms – Simple arithmetic Incorrect marking -4 ≠ 4 4.1346... ≠ 4 2x 2 – 4x + 4 ≠ x 2 – 2z + 2

43. Presentation Your name Title/Exercise number Margin and question numbers (and a, b etc.) Questions in the right order Sequence within a question (↓) Legible writing Not cramped Presentable. Rewrite if necessary!

44. Given p(x) = x 2 - 4x + 3, q(x) = x 2 – x – 4, find p(x) – q(x) p(x) – q(x)= x 2 - 4x + 3 - x 2 – x – 4 p(x) – q(x)= x 2 - 4x + 3 – (x 2 – x – 4)

45. Given p(x) = x 2 - 4x + 3, q(x) = x 2 – x – 4, find p(x)q(x) x 4 – x 3 - 4x 2 - 4x 3 + 4x 2 + 16x + 3x 2 – 3x - 12 x 4 – 5x 3 + 3x 2 + 13x - 12 p(x)q(x)= x 4 – x 3 - 4x 2 - 4x 3 + 4x 2 + 16x + 3x 2 – 3x - 12 = x 4 – 5x 3 + 3x 2 + 13x - 12

46. 3 x -1 = -3 2 = 9 + 2 = 11 Given f(x) = 3x 2 + 2, find f(-1). 3 x -1 = -3 Should have squared the -1 first (BIDMAS) 2 ( ) (-3) 2 not -3 2 = 9 -3 2 = -9 not 9 - + 2 ≠ (-3) 2 ≠ 9 + 2 f(-1) = f(-1) = is missing = 11 ≠ 3 x -1 ≠ -3 2 f(-1) = 3 x (-1) 2 + 2 = 3 x 1 + 2 = 5

47. Homework review Ex 3A Q 4c Misc 9 q 3b, 4 (We did one just like it!!!)

48. Simplify fractions See teaching notes 19/9

49. Multiply fractions See teaching notes 19/9

50. Divide by a fraction See teaching notes 19/9

51. Algebraic fractions Read P 35-37 Examples 22-25 Practice Ex 2E Q 4 all, Q5 a-e

52. Factorising/Fractions Homework Ex 2C Q 2-5 Ex 2D all Qs (from Wed) Ex 2E Q 4 all, Q5 a-e

53. C1 ALGEBRAIC EXPRESSIONS Wed 24/9/14 LO: Add and subtract algebraic fractions

54. Returning... Function Notation and Multiplying Polynomials - Ex 3A - Misc. Ex. 9 (Late entries) SEE ME: Peravin, Zendell, John, Ehimen, Shaquille, Cameron

55. Collecting... Factorising Ex 2C Q 2-5 Ex 2D all Qs x and ÷ fractions Ex 2E Q 4 all, Q5 a-e

56. Adding and subtracting fractions Consider: 3434 3535 =+ 20 + = 27 20 Write as a single fraction in its lowest terms. The LCM of 3 x and y is ___.3 xy 1512

57. Adding and subtracting fractions Write as a single fraction in its lowest terms. The LCM of x 2 and x is ___. x2x2 Consider: 3535 4 15 =+ + = 13 15 94

58. Write as a single fraction in its lowest terms. The LCM of x + 3 and 2( x + 3) is ______.2( x + 3) Start by factorizing where possible: Adding and subtracting fractions

59. Write as a single fraction in its lowest terms. Notice that this becomes – 8. Adding and subtracting fractions

60. Read pp 34-35 Examples 20&21 Practice Ex 2E Q2

61. Alegebraic expressions - review Revision homework for weekend P 39 Ex 2F middle column

62. Returning... Function Notation and Multiplying Polynomials - Ex 3A - Misc. Ex. 9 (Late entries) SEE ME: Peravin, John, Ehimen, Cameron, Tim, Shanil

63. C1 ALGEBRAIC EXPRESSIONS Wed 1/10/14

64. Write the following as single fractions in their lowest terms. Adding and subtracting fractions – trickier examples 1 a + bcbc a) 1 - z xyxy b) c) 1 - 1a1a x + 1 1 - 1x31x3 d) e) 1b1b + 1a1a abab - baba