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Basic Skills in Higher Mathematics

Basic Skills in Higher Mathematics. Mathematics 1(H) Outcome 1. Robert Glen Adviser in Mathematics. Mathematics 1(Higher). Outcome 1 Use the properties of the straight line. Straight lines. Mathematics 1(Higher). Outcome 1 Use the properties of the straight line. PC Index.

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Basic Skills in Higher Mathematics

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  1. Basic Skills in Higher Mathematics Mathematics 1(H)Outcome 1 Robert GlenAdviser in Mathematics

  2. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line Straight lines

  3. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC Index Click on the one you want PC(a) Gradients and straight lines PC(b) Gradients and angles PC(c) Parallel and perpendicular

  4. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Gradients and straight lines

  5. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Index Click on the section you want 1 What is gradient? 2 The gradient of a line 3 The equation of a line given its gradient and the intercept on the y - axis 4 The equation of a line given one point on the line and the gradient 5 The equation of a line given two points on the line

  6. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 1 1 What is gradient?

  7. Mathematics 1(Higher) 1.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? The gradient (slope) of this roof is 2m 2 2m = 3m 3 3m

  8. Mathematics 1(Higher) 1.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? 3m The gradient (slope) of this roof is 3m 3m = 1 3m

  9. Mathematics 1(Higher) 1.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? The gradient (slope) of this roof is 3m 3m 3 7m = 7m 7

  10. Mathematics 1(Higher) 1.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? 3m 3 Gradient = 7 7m Check this:The steeperthe slope, the greater the gradient. 2m 2 Gradient = 3 3m 3m Gradient = 1 3m

  11. Mathematics 1(Higher) 1.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D 4 5

  12. Mathematics 1(Higher) 1.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D 4 5 Click on the letter of the correct answer

  13. Mathematics 1(Higher) 1.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? Sorry, wrong answer What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 Gradient = vertical horizontal C D 4 5 Have another go!

  14. Mathematics 1(Higher) 1.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D 4 5 Click on the letter of the correct answer

  15. Mathematics 1(Higher) 1.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 1 What is gradient? What is the gradient of this roof ? 5m 5 4 3m A B 4 5 4m 3 3 C D Correct! 4 5 End of Section 1

  16. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 2 2 The gradient of a line

  17. Mathematics 1(Higher) 2.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B  Read all lines from left to right A  Line AB is uphill from left to right x  Line AB has a positive gradient mAB 0

  18. Mathematics 1(Higher) 2.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line P uphill B  Read all lines from left to right A downhill  Line PQ is downhill from left to right Q x  Line PQ has a negative gradient mPQ  0

  19. Mathematics 1(Higher) 2.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line P uphill B  Read all lines from left to right A downhill Q x  Line PQ has a negative gradient mPQ  0  Line AB has a positive gradient mAB 0

  20. Mathematics 1(Higher) 2.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B (9, 6) change in y Gradient = change in x 3 A 39 (0, 3) mAB = 9 13 = x

  21. Mathematics 1(Higher) 2.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B (9, 6) change in y Gradient = 1 change in x 3 1 A 3 1 39 (0, 3) mAB = 3 13 = x Note: we could have measured the gradient like this

  22. Mathematics 1(Higher) 2.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line (0, 7) P change in y Gradient = change in x downhill -6 -6 9 mPQ = Q (9, 1) 23 = - 9 x

  23. Mathematics 1(Higher) 2.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line (0, 7) P change in y -2 Gradient = change in x 3 downhill -2 -6 9 mPQ = 3 -2 Q (9, 1) 23 3 = - x Note: we could have measured the gradient like this

  24. Mathematics 1(Higher) 2.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line uphill B (9, 6) change in y Gradient = change in x 6 - 3 A 6 - 3 9 - 0 (0, 3) mAB = 9 - 0 39 = x 13 =

  25. Mathematics 1(Higher) 2.10 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient y 2 The gradient of a line (0, 7) P change in y Gradient = change in x downhill 1 - 7 1 - 7 9 - 0 mPQ = Q (9, 1) -6 9 = 9 - 0 x 23 - =

  26. Mathematics 1(Higher) 2.11 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y B (x2 , y2) A formula to memorise A (x1 , y1) y2 - y1x2 - x1 x mAB =

  27. Mathematics 1(Higher) 2.12 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y B (x2 , y2) A formula to memorise A (x1 , y1) y2 - y1x2 - x1 x mAB =

  28. Mathematics 1(Higher) 2.13 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y B (6 ,5) 1Calculate the gradient of line AB y2 - y1x2- x1 A (2 , 3) mAB = x 5 - 36 - 2 = 24 = Did you getthis answer? 12 =

  29. Mathematics 1(Higher) 2.14 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y 2Calculate the gradient of line CD. D (6 ,2) y2 - y1x2- x1 mCD = x C (2 , -1) 2 - (-1)6 - 2 = 34 Did you getthis answer? =

  30. Mathematics 1(Higher) 2.15 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 2 The gradient of a line y E (-3 , 3) 3Calculate the gradient of line EF. y2-y1x2- x1 mEF = x F (5,-1) -1 - 35 - (-3) = -48 Did you getthis answer? = 12 = - End of Section 2

  31. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 3 3 The equation of a linegiven its gradient and theintercept on the y - axis

  32. Mathematics 1(Higher) 3.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y Find the equation of line KL which has a gradient of ½ and passes through the point (0, 3). (x, y) L  (0,3)  m = ½ K y - 3x - 0 12 mKL= = O x  y - 3 = ½ x The equation of KL isy = ½ x + 3 y = ½ x + 3

  33. Mathematics 1(Higher) 3.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y Find the equation of line KL which has a gradient of ½ and passes through the point (0, 3). (x, y) L  (0,3)  m = ½ K Formula:y = m x + c O x The equation of KL isy = ½ x + 3

  34. Mathematics 1(Higher) 3.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y (x, y) L  The equation of line with gradient m and intercept c is: (0, c) m  K y = m x + c O x Memorise this

  35. Mathematics 1(Higher) 3.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept y P 1 Find the equation of line PQ which has a gradient of -2 and passes through the point (0, 5).  (0,5) m = -2  (x, y) Use the formula O x Q The equation of PQ isy = -2 x + 5

  36. Mathematics 1(Higher) 3.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept F y  (x, y) 2 Find the equation of line EF which has a gradient of ¾ and passes through the point (0, -3). O x m = ¾  (0,-3) Use the formula E The equation of EF is y = ¾ x - 3

  37. Mathematics 1(Higher) 3.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 3 The equation of a line given gradient and intercept End of Section 3 You should now do Section A1 questions 1 - 10 on page 3 of the Basic Skills booklet.

  38. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 4 4 The equation of a linegiven one point on the line and the gradient

  39. Mathematics 1(Higher) 4.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y L Find the equation of the linethrough the point (4, 3) with gradient 3. (x, y)  m = 3  (4,3) y - 3x - 4 mKL= = 3 O x  y - 3 = 3(x - 4) K  y - 3 = 3x - 12 The equation of KL is y = 3x - 9  y = 3x - 9

  40. Mathematics 1(Higher) 4.2 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y L Find the equation of the linethrough the point (4, 3) with gradient 3. (x, y)  m = 3  (4,3) O x Formula:y - b = m (x - a) K The equation of KL is y = 3x - 9

  41. Mathematics 1(Higher) 4.3 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y L (x, y)  The equation of the linethrough the point (a,b) with gradient m is : m  (a, b) y - b = m (x - a) O x K Memorise this

  42. Mathematics 1(Higher) 4.4 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y Q 1Find the equation of the linethrough the point (-1, 2) with gradient 2. (x, y)  m = 2 (-1,2)  Use the formula x P O The equation of PQ is y = 2 x + 4

  43. Mathematics 1(Higher) 4.5 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y Q 1Find the equation of the linethrough the point (-1, 2) with gradient 2. (x, y)  (a, b) m = 2 y - b = m (x - a) (-1,2)  y - 2 = 2 (x - (-1)) x P y - 2 = 2 (x + 1) O y -2 = 2 x + 2 The equation of PQ is y = 2 x + 4 y = 2 x + 4

  44. Mathematics 1(Higher) 4.6 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y 2Find the equation of the linethrough the point (6, -2) with gradient ½. N m = ½  O x  (6, -2) M (x, y) Use the formula The equation of MN is 2y = x - 10

  45. Mathematics 1(Higher) 4.7 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y 2Find the equation of the linethrough the point (6, -2) with gradient ½. N m = ½  O (x, y) x  (6, -2) y - b = m (x - a) M y - (-2) = ½ (x - 6) (a, b) y +2 = ½(x - 6) Multiply both sides by 2to clear the fraction. 2y + 4 = x - 6 The equation of MN is 2y = x - 10 2y = x - 10 orx - 2y -10=0

  46. Mathematics 1(Higher) 4.8 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y R  3 Find the equation of the linethrough the point (-1, 4) with gradient 2/3 . m = -2/3 (-1,4) (x, y)  S O x Use the formula The equation of RS is 3y = -2x + 10

  47. Mathematics 1(Higher) 4.9 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient y R  3 Find the equation of the linethrough the point (-1, 4) with gradient 2/3 . m = -2/3 (-1,4) (x, y) (a, b)  y - b = m (x - a) O x S y -4 = (x - (-1)) -2/3 y- 4 = -2/3 (x + 1) Multiply both sides by 3to clear the fraction. 3y - 12 = -2(x + 1) The equation of RS is 3y = -2 x + 10 3y = -2 x + 10 or 2 x + 2y -10=0

  48. Mathematics 1(Higher) 4.9 Mathematics 1(Higher) 4.10 Outcome 1 Use the properties of the straight line Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 4 The equation of a line given one point and the gradient End of Section 4 You should now do Section A1 questions 11 - 20 on page 3 of the Basic Skills booklet.

  49. Mathematics 1(Higher) Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient Section 5 5 The equation of a linegiven two points on the line

  50. Mathematics 1(Higher) 5.1 Outcome 1 Use the properties of the straight line PC(a) Determine the equation of a straight line given two points on the line or one point and the gradient 5 The equation of a line given two points on the line y  B (6, 4) Find the equation of the linejoining the points A (3, 1) and B (6, 4) . (a, b) (3,1) m = 1  A x O Step 1 Calculate the gradient Step 2 Calculate the equation y2 - y1x2- x1 4 - 16 - 3 y - b = m (x - a) mAB = = Choose A (3, 1) as thepoint on the line.i.e. a = 3, b = 1 y - 1 = (x - 3) 1 33 = y - 1 = x - 3 (You get exactly thesame answer if youchoose B.) = 1 y = x - 2

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