1. Straight line in real-life Equation given any two points www.mathsrevision.com Gradient Revision Nat 5 The General Equation of a straight line. Best – fitting straight line Straight Line Simple Gradient Equation of a line gradient and y-intercept Exam Type Questions Higher Form of straight Line y – b = m(x – a)

2. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Nat 5 In pairs “Explain the rules for fractions”

3. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 2.Calculate simple gradients. 1.We are learning to calculate simple gradient using a right angle triangle 1.Gradient is : change in vertical height divided by change in horizontal distance The Gradient of a Line S4

4. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com The Gradient Change in vertical height Change in horizontal distance The gradient is the measure of steepness of a line The steeper a line the bigger the gradient Difference in x -coordinates Difference in y -coordinates S4

5. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com The Gradient 3 4 3 2 5 2 6 3 S4

6. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.1 Ch6 (page 50) www.mathsrevision.com Straight Line Nat 5

7. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Nat 5 9 8 5 Q1.Is this triangle right angled ? Explain

8. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.We are learning to find the equation of a straight line. 1.Understand how to calculate the gradient and identify y-intercept. Straight line Nat 5 2.Be able to write down equation of a straight line in the format y = mx + c.

9. 21-May-15Created by Mr.Lafferty Maths Dept Almost all straight lines have the equation of the form y = mx + c Gradient y - intercept www.mathsrevision.com Straight Line Equation y intercept is were line cuts y axis Slope left to right upwards positive gradient Slope left to right downwards negative gradient Nat 5 lines are parallel if same gradient

10. m = gradient x1x1 21-May-15www.mathsrevision.com 10 y-axis O x-axis www.mathsrevision.com The gradient using coordinates x2x2 y2y2 y1y1 Nat 5

11. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com The gradient formula is : It is a measure of how steep a line is A line sloping up from left to right is a positive gradient A line sloping down from left to right is a negative gradient The gradient using coordinates Nat 5 Remember parallel means same gradient

12. 21-May-15www.mathsrevision.com 12 y-axis O x-axis The gradient using coordinates Find the gradient of the two lines. Nat 5

13. 21-May-15www.mathsrevision.com 13 The gradient using coordinates Write down the gradient and y intercept for each line. Nat 5 (a) y = -3x - 5 (b) 4y - 8x = 24 m = - 3c = - 5 m = 2c = 6 Write the equation given the gradient and y intercept. (a)m = 1.5 c = 1 (b)m = -2 c = - 4 y = 1.5x + 1 y = -2x - 4 Challenge

17. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.2 Ch6 (page 52) www.mathsrevision.com Straight Line Nat 5

18. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Q1.A house 4 years ago is valued at £50 000. Calculate it’s value if it has increased by 5%. Q2.Calculate 3.36 x 70 to 2 significant figures. Q3.Write down the 3 ways of factorising. Nat 5

19. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 2.Know how to find y-intercept using substitution. Be able to write down straight line equation in y = mx + c format. 1.We are learning to find the equation of a straight line given any two points. 1.Know how to calculate the gradient using formula. Straight Line Nat 5

20. 21-May-15www.mathsrevision.com Equation of a Straight Line To find the equation of a straight line we need to know Two points that lie on the line ( x 1, y 1 ) and ( x 2, y 2 ) OR The gradient and a point on the line m and (a,b) y = mx + c www.mathsrevision.com Nat 5 We will look at this later

21. 21-May-15 -10 y Find the equations of the straight lines below given the coordinates. (4,1) and (8,5) using Sub (4,1) into y = mx + c x y 1 = 1 x 4 + c c = 1 - 4 c = -3 y = x -3 (-6,7) and (-2,3) Sub (-2,3) into y = mx + c x y c = 3 - 2 c = 1 y = -x + 1 Demo

22. Find the equation of the straight line passing through the points (4, 4) and (8,24). Solution y = mx + c www.mathsrevision.com Nat 5 Equation of a Straight Line Using the point (4,4) and the gradient m = 5 sub into straight line equation 4 = 5 x 4 + cc = 4 - 20 = -16 Equation : y = 5x - 16 y = mx + c 

23. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.3 Ch6 (page 54) www.mathsrevision.com Straight Line Nat 5

24. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Q1.The points ( 1, 4) and (3, 11) lie on a line. Find the gradient of the line. Q2.Are the two lines parallel. Explain your answer y = x + 2 and y = 2x + 2 Nat 5

25. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.We are learning to model real life situations using straight line theory 1.Be able to work out gradient and y intercept using a graph. Modelling Using Straight Line Equation 2.Form an equation for any straight line graph. Nat 5

26. A LINEAR equation can and is usually written in the form Note : Other letters can be used y = mx + c www.mathsrevision.com Nat 5 Real – Life Straight Line The following equations are NOT linear y = mx 2 + c v = ut + a v = ut 2 + a Why ? Not allowed powers greater than 1

27. www.mathsrevision.com Real-Life Straight Line 3102 3 1 4 2 6 5 h0123P0246 (0,0) (1,2) (3,6) (2,4) P h P = 2h Nat 5 John earns £2 an hour Using the table draw a graph of P against h Write down a formula connecting P and h. How much did he get paid for working 48 hours. Find the gradient and P- intercept P = 2 x 48 P = £96 m = 2 c = 0

28. Modelling Real – life The cost for hiring a plumber per hour is shown below (a)Calculate the gradient. (b)What is the value of V when T = 0 (c)Write down an equation connecting V and T. Pick any 2 points (7,100) (0,30) 30 V = 30 T+ 10 (d)Find the cost for a plumber for 10 hours? T = 10C = 10 x 10 + 30 = £130

29. 80 -5 Modelling Real – life The graph shows how a the volume of water tank drains over time. (a)Calculate the gradient. (b)What is the value of V when T = 0 (c)Write down an equation connecting V and T. Pick any 2 points (8,40) (0,80) 80 V =T+ (d)How long before the tank is empty? V = 00 = -5 x T + 80T = (-80) ÷(–5) = 16mins

30. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.4 Ch6 (page 55) www.mathsrevision.com Real – Life Straight Line Nat 5

31. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Q1.Prove that the coordinates ( 1,1) (2, 2) and (3,3) lie on the same line. Q2.Are the two lines parallel. Explain answer y = -4x + 1 and 6y - 24x = 12 Nat 5

32. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.We are revising gradient. 1.Use gradient formula to calculate gradient. Gradient - Revision Nat 5

33. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com The gradient formula is : It is a measure of how steep a line is A line sloping up from left to right is a positive gradient A line sloping down from left to right is a negative gradient The gradient using coordinates Nat 5 Remember parallel means same gradient

34. www.mathsrevision.com Higher Outcome 1 21-May-15www.mathsrevision.com 34 Gradient Facts m < 0 m > 0 m = 0 x = a y = c Sloping left to right up has +ve gradient Sloping left to right down has -ve gradient Horizontal line has zero gradient. Vertical line has undefined gradient.

35. www.mathsrevision.com Nat 5 Gradient - Revision Find gradients PQ, QR and RP

36. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.5 Ch6 (page 58) www.mathsrevision.com Gradient Revision Nat 5

37. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Q1.Prove that the coordinates ( 1,1) (2, 2) and (3,3) lie on the same line. Q2.Are the two lines parallel. Explain answer y = -4x + 1 and 6y - 24x = 12 Nat 5

38. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.We are learning alternative way to find straight line equation. 1.Know Nat 5 version of straight line equation y – b = m(x – a) Higher Form of Straight Line Equation 2.Form an straight equation from information given. Nat 5

39. 21-May-15www.mathsrevision.com Equation of a Straight Line To find the equation of a straight line we need to know Two points that lie on the line ( x 1, y 1 ) and ( x 2, y 2 ) OR The gradient and a point on the line m and (a,b) y = mx + c www.mathsrevision.com Nat 5

40. www.mathsrevision.com Nat 5 = The Equation of the Straight Line The Equation of the Straight Line y – b = m (x - a) The equation of any line can be found if we know the gradient and one point on the line. O y x x - a P (x, y) m A (a, b) y - b x – a m = y - b (x – a) m Gradient, m Point (a, b) y – b = m ( x – a ) Point on the line ( a, b ) ax y b Demo

41. Find the equation of the straight line passing through the points (3, -5) and (6,4). Solution y - b = m(x – a) www.mathsrevision.com Nat 5 Equation of a Straight Line Using the point (6,4) and the gradient m = 3 sub into straight line equation Equation : y = 3x - 14 y – b = m(x - a) y – 4 = 3(x - 6) OR Equation : y = 3x - 14 y – b = m(x - a) y – (-5) = 3(x - 3)

42. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.6 Ch6 (page 59) www.mathsrevision.com Real – Life Straight Line Nat 5

43. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Q1.Find the equation of the straight line passing through and (4, 5) and parallel to the line 2x + y = 1 Q2.Expand (x - 2)(x 2 + 3x + 1) Nat 5

44. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.We are learning the general form for a straight line. 1.Know the format for the general form of a straight line. General Equation of a Straight Line 2.Be able to rearrange general form. Nat 5

45. www.mathsrevision.com Nat 5 Straight Line Facts Another version of the straight line general formula is: ax + by + c = 0 You need to be able to arrange this form to identify Key information i.e. Gradient and y–intercept. Demo

46. www.mathsrevision.com Simultaneous Equations Nat 5 Straight Lines 21-May-15Created by Mr. Lafferty Maths Department www.mathsrevision.com Find the gradient and y-intercept for equations below x - y – 4 = 0and4x + 2y + 12 = 0 Rearrange ( BALANCING METHOD) x – y – 4 = 0 y = x - 4 m = 1c = - 4 Rearrange ( BALANCING METHOD) 4x + 2y + 12 = 0 y = -2x - 3 m = -2c = - 6

47. 21-May-15Created by Mr.Lafferty Maths Dept Now try N5 TJ Ex 6.7 Ch6 (page 61) www.mathsrevision.com Real – Life Straight Line Nat 5

48. 21-May-15Created by Mr.Lafferty Maths Dept Starter Questions www.mathsrevision.com Q1.The points ( 5, 7) and (7, 21) lie on the same line. Find the gradient of the line. Q2.Are the two lines parallel. Explain answer y = -2x + 1 and y = 2x + 1 Nat 5

49. 21-May-15Created by Mr.Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.How to model real life situations using straight line theory 1.Be able to work out gradient and y intercept using a graph. Best Fit Straight Line Equation 2.Form an equation for any straight line graph. Nat 5

50. x c = 0 (b)Draw in the best-fit line and find an equation relating height and weight. 2.57 0 Best-Fitting Straight Line Data was collected on pupils weight and height. Data is plotted below. (a)Is there correlation between height and weight. Pick any 2 points (70,180) h =w+ (d)What height is a pupils who weights 40 kgs? w = 40h = 2.57 x 40 = 102.8 cms x (0,0) Yes, a positive correlation

51. c = 1818 -1.25 (b)Draw in the best-fit line and find an equation relating value and year. Best-Fitting Straight Line A survey was carried out on the value of cars depending on their age The data is plotted below. (a)Is there correlation between value and age. Pick any 2 points (8,8) v =y+ (d)What is the cost of a car after 4years? y = 4v = -1.25 x 4 + 18 = 13 x (0,18) Yes, a negative correlation x Value = £13 000

55. 21-May-15Created by Mr.Lafferty Maths Dept Now try Sheet provided www.mathsrevision.com Nat 5 Best Fit Straight Line Equation