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# Which ones are you using?

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1. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Real life cross/curricular links? Where are we in our journey? Which ones are you using? PLT Skills LESSON OBJECTIVES Always aim high! We are learning to: • Finding connections between different words. (Which PLT skills?) • Accurately solve equations, one linear and one non-linear, with graphs. (Grades A/A*) AUTHOR www.mistrymaths.co.uk

2. Which ones are you using? BRAIN IN GEAR Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills EXAMPLE DITDIONA can be rearranged to make ADDITION TASK Work out the following Mathematical anagrams: Quadratic Radius Intersection EXTENSION Develop your own Mathematical anagrams as above as a creative thinker.

3. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager STARTER PLT Skills TASK Work out x: Find the area of: Write 0.00023 in standard form 1) 3) 2) 3x + 9 4x - 6 7cm x + 23 2x + 4 5cm -4 2.3 x 10 8cm = 12 x 8 7 4x - 6 x + 23 2x + 4 = 360 3x + 9 + + + = 12cm 10 2 = 360 10x + 30 - 30 10x - 30 = 330° 3 9 2 ÷ 10 ÷ 10 EXTENSION x = 33° Work out x: Value of: 4) 5) x Power - Reciprocal 130° 7m Root 2m 140° Pentagon adds to 540° 1 115° 6m = 86° 9m 3 ) ( 9 11 11 1 1 x = x = 3 2 6 2 27 3 Work out x: x = 540° - 86° - 130° - 140° - 115° x = 6 x = x = 69° x = 33m Area of a triangle = 11m 2m 2 Area of a triangle = 48cm x 6m

4. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING Quadratic Linear EXAMPLE 1 1) 2 Find the approximate solutions to the pair of equations y = x + x – 2 and y = 2x + 3 by graphical means. x Set up a table for the quadratic and linear equations (get the x values from the x-axis) Table for quadratic y = x + x – 2 2 x 4 0 10 10 -2 4 -2 18 0 x x Substitute in the x values and work out y x x - 2 - 2 - 2 16 1 16 0 4 9 - 2 = = - 2 = = + 1 + 4 - 4 - 2 - 2 = 9 + 0 = - 2 + 3 - 2 = 4 1 - 3 = - 1 = + 2 - 1 x Table for linear y = 2x + 3 x x x 1 -3 -1 -5 11 9 3 5 7 x x x Substitute in the x values and work out y x 2(-3) + 3 2(2) + 3 2(-1) + 3 2(-4) + 3 2(3) + 3 2(4) + 3 2(-2) + 3 2(0) + 3 2(1) + 3 x x -6 4 -2 -8 + 3 = = + 3 + 3 = = + 3 + 3 6 0 8 -4 2 = + 3 = + 3 = = = + 3 + 3 x Approximate solutions → (-1.8,-0.6) (2.8,8.6) 2 2 2 2 2 2 2 2 2 (4) + (4) - 2 (-1) + (-1) - 1 (-2) + (-2) - 2 (3) + (3) - 2 (-4) + (-4) - 2 (2) + (2) - 2 (0) + (0) - 2 (-3) + (-3) - 2 (1) + (1) - 2

5. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 1) 2 Find the approximate solutions to the pair of equations y = x + 3x – 2 and y = x by graphical means. Table for quadratic y = x + 3x – 2 2 x 2 -4 8 -2 2 -2 -4 16 26 x x x x Table for linear y = x x x -1 -3 -2 -4 4 3 0 1 2 x x x x x x x x Approximate solutions → (0.7,0.7) (-2.7,-2.7)

6. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 2) 2 Find the approximate solutions to the pair of equations y = x - 3x – 6 and y = 2x by graphical means. Table for quadratic y = x - 3x – 6 2 x 34 22 4 -6 -8 -2 12 x x x Table for linear y = 2x x x 16 -8 -4 0 4 12 8 x x x x Approximate solutions → (-1,-2) (6,12) x x

7. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 3) 2 Find the approximate solutions to the pair of equations y = x - 3x + 1 and y = 2x - 1 by graphical means. x x Table for quadratic y = x - 3x + 1 2 x 11 -1 -1 1 5 19 1 x x x Table for linear y = 2x - 1 x 11 1 3 9 -1 5 7 x x x x x x Approximate solutions → (0.5,0) (4.6,8.2)

8. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 4) 2 Find the approximate solutions to the pair of equations y = x - 3 and y = x + 3 by graphical means. x Table for quadratic y = x - 3 2 x x x x 13 -2 -2 1 1 -3 22 22 13 6 6 x x Table for linear y = x + 3 x x x x x 4 7 8 -1 6 3 5 2 -2 0 1 x x x x x Approximate solutions → (-2,1) (3,6) x

9. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 5) 2 Find the approximate solutions to the pair of equations y = x - 3x - 2 and y = 2x - 3 by graphical means. Table for quadratic y = x - 3x - 2 2 x x x -2 -4 -4 8 6 2 -2 2 38 26 16 x Table for linear y = 2x - 3 x x x x -11 3 -13 -9 -7 -1 1 5 -5 -3 7 x x x x Approximate solutions → (4.8,6.6) (0.2,-2.6) x x x

10. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING EQUATION OF A CIRCLE x + y = r 2 2 2 with centre (0,0) Equation of a circle → Radius of a circle Draw the circles of the equations given below: 2 2 x + y = 5 2 (b) (a) 2 2 x + y = 3 2 Radius of circle Radius of circle

11. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING EQUATION OF A CIRCLE x + y = r 2 2 2 with centre (0,0) Equation of a circle → Radius of a circle Draw the circles of the equations given below: 2 x + y = 49 2 (d) Radius = 7 (c) 2 2 2 x + y = 16 2 4 7 Radius = 4

12. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 2 (GRADE A) 1) 2 2 Find the approximate solutions to the pair of equations x + y = 25 and y = 1 - x by graphical means. Table for linear y = 1 - x x 2 5 x 2 4 3 1 -1 6 5 0 -2 -3 -4 -5 7 x x Radius for circle 2 x + y = 25 2 x Radius = 5 x x x x x x x Approximate solutions → (-3,4) (4,-3)

13. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 2 (GRADE A) 2) 2 2 Find the approximate solutions to the pair of equations x + y = 4 and y = x + 1 by graphical means. Table for linear y = x + 1 x 2 2 x 0 1 3 2 4 5 6 7 -5 -4 -3 -2 -1 x x Radius for circle 2 x + y = 4 2 x x Radius = 2 x x x x x x Approximate solutions → (-1.8,-0.8) (0.8,1.8)

14. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 2 (GRADE A) 3) 2 2 Find the approximate solutions to the pair of equations x + y = 9 and y = x - 1 by graphical means. Table for linear y = x - 1 2 3 x -2 -1 1 0 2 3 4 5 -7 -6 -5 -4 -3 x x Radius for circle 2 x + y = 9 2 x x Radius = 3 x x x x x x x Approximate solutions → (-1.6,-2.6) (2.6,1.6)

15. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING EXTENSION (GRADE A*) 1) 2 (a) Find the approximate solutions to the pair of equations y = x + 3x - 4 and y = 5x - 5 by graphical means. (b) What is special about the intersection of the two graphs? 2 2 (c) Show that 5x – 5 = x + 3x – 4 can be re-arranged to x - 2x + 1 = 0 2 (d) Factorise and solve x – 2x + 1 = 0 (e) Explain how the solution in part (d) relates to the intersection of the graphs. 2) 2 (a) Find the approximate solutions to the pair of equations y = x + 2x + 3 and y = x - 1 by graphical means. (b) What is special about the intersection of the two graphs? 2 2 (c) Rearrange x – 1 = x + 2x + 3 into the form ax + bx + c = 0 2 (d) Work out the discriminant b – 4ac for the quadratic on part (c) (e) Explain how the discriminant relates to the intersection of the graphs.

16. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING EXTENSION ANSWERS(GRADE A*) 1) 2 (a) Find the approximate solutions to the pair of equations y = x + 3x - 4 and y = 5x - 5 by graphical means. (1,0) There is only one point of intersection. 2 2 x + 3x – 5x - 4 + 5 = 0 x – 2x + 1 = 0 ( )( ) = 0 x x - 1 - 1 x – 1 = 0 + 1 + 1 x = 1 Only one solution as the line is a tangent to the curve.

17. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING EXTENSION ANSWERS(GRADE A*) 2) 2 (a) Find the approximate solutions to the pair of equations y = x + 2x + 3 and y = x - 1 by graphical means. There is no solution. The graphs do not intersect. 2 2 2 2 x + 2x – x + 3 + 1 = 0 x + x + 4 = 0 x + x + 4 = 0 b - 4ac a = 1 c = 4 b = 1 2 = 1 - 4 x 1 x 4 = -15 The discriminant is negative and you cannot find the square root of a negative number. Therefore, there are no solutions.

18. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills TRIGONOMETRIC ANGLES MINI-PLENARY 1 – SPOT THE MISTAKE 2 Find the approximate solutions to the pair of equations y = x - 2x - 1 and y = 2x - 2 by graphical means. Table for quadratic y = x - 2x - 1 2 x x x -1 -2 0 8 2 2 7 34 23 14 16 x 7 -1 14 Table for linear x y = 2x - 2 x x x -6 5 -12 -8 -6 0 2 6 -4 0 8 x -10 -2 4 x x x x Approximate solutions → (0.3,-1.5) (3.7,5.5) x

19. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING MINI-PLENARY 2 – SPOT THE MISTAKE 2 2 Find the approximate solutions to the pair of equations x + y = 4 and y = x + 1 by graphical means. Table for linear y = x + 1 x x 2 2 x 0 -1 3 2 4 5 6 7 -5 -4 -3 -2 -1 x x Radius for circle 2 x + y = 4 2 x x Radius = 2 x x x x x x Approximate solutions → (-3.2,-2.2) (2.2,3.2) (-1.8,-0.8) (0.8,1.8)

20. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager • LINK BACK TO OBJECTIVES • Accurately solve equations, one linear and one non-linear, with graphs. What grade are we working at? DISCOVERY PLT Skills

21. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING PLENARY ACTIVITY– EXAM QUESTION (GRADE A*) 2 The quadratic graph has equation y = ax + bx where a and b are integers. Points P(-1,10) and Q(4,0) lie on this graph. The straight line is y = x. Find the co-ordinate of the point marked R. You must show your working. To find the co-ordinates of R, we need to find where the equations of the graphs intersect 16a + 4b = 0 y a – b = 10 x4 . Solve the simultaneous equations 16a + 4b = 0 y = x . P y = x 4a – 4b = 40 + (-1,10) x y By substitution of x for y, R 40 20a = . a 2 = 2 0 = 2x - 9x By substitution, x Q a – b = 10 By factorisation, (4,0) x 2 – b = 10 y 0 = x(2x – 9) x(2x – 9) = 0 Equation for (-1,10) Equation for (4,0) – b = 8 x = 0 or 2x - 9 = 0 b = -8 2 2 2 2 2 2 2 2 y = ax + bx y = ax + bx 0 = a(4) + b(4) 10 = a(-1) + b(-1) x = 2x - 8x y = 2x - 8x y = 2x - 8x y = ax + bx 2x = 9 Sub in (-1,10) for x and y Sub in (4,0) for x and y Sub in for a and b x = 4.5 10 = a - b 0 = 16a + 4b y = 4.5 a - b = 10 16a + 4b = 0 R(4.5,4.5) (6 marks)

22. What grade are we working at? Where are we in our journey? What have you learnt? Draw your brain In your brain, write or draw everything you can remember about graphing inequalities and shading the region defined by it. It can be a skill or a reflection, or something else that might be prominent in your brain.

23. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity How well do you understand the task? . I fully understand I don’t understand I nearly understand www.mistrymaths.co.uk

24. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity WWW (What Went Well) EBI (Even Better If) On your post it notes… Think about how you can improve your work. www.mistrymaths.co.uk

25. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 1) 2 Find the approximate solutions to the pair of equations y = x + 3x – 2 and y = x by graphical means. Table for quadratic y = x + 3x – 2 2 Table for linear y = x

26. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 2) 2 Find the approximate solutions to the pair of equations y = x - 3x – 6 and y = 2x by graphical means.

27. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 3) 2 Find the approximate solutions to the pair of equations y = x - 3x + 1 and y = 2x - 1 by graphical means.

28. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 4) 2 Find the approximate solutions to the pair of equations y = x - 3 and y = x + 3 by graphical means.

29. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 1 (GRADE A) 5) 2 Find the approximate solutions to the pair of equations y = x - 3x - 2 and y = 2x - 3 by graphical means.

30. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 2 (GRADE A) 1) 2 2 Find the approximate solutions to the pair of equations x + y = 25 and y = 1 - x by graphical means.

31. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 2 (GRADE A) 2) 2 2 Find the approximate solutions to the pair of equations x + y = 4 and y = x + 1 by graphical means.

32. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING TASK 2 (GRADE A) 3) 2 2 Find the approximate solutions to the pair of equations x + y = 9 and y = x - 1 by graphical means.

33. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING EXTENSION (GRADE A*) 1) 2 (a) Find the approximate solutions to the pair of equations y = x + 3x - 4 and y = 5x - 5 by graphical means. (b) What is special about the intersection of the two graphs? 2 2 (c) Show that 5x – 5 = x + 3x – 4 can be re-arranged to x - 2x + 1 = 0 2 (d) Factorise and solve x – 2x + 1 = 0 (e) Explain how the solution in part (d) relates to the intersection of the graphs. 2) 2 (a) Find the approximate solutions to the pair of equations y = x + 2x + 3 and y = x - 1 by graphical means. (b) What is special about the intersection of the two graphs? 2 2 (c) Rearrange x – 1 = x + 2x + 3 into the form ax + bx + c = 0 2 (d) Work out the discriminant b – 4ac for the quadratic on part (c) (e) Explain how the discriminant relates to the intersection of the graphs.

34. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills SOLVING BY GRAPHING PLENARY ACTIVITY– EXAM QUESTION (GRADE A*) 2 The quadratic graph has equation y = ax + bx where a and b are integers. Points P(-1,0) and Q(4,0) lie on this graph. The straight line is y = x. Find the co-ordinate of the point marked R. You must show your working. y . y = x . P R . x Q (6 marks)