Simultaneous Equations (non-linear) Grade 7/8 Simultaneous Equations (non-linear) Solve two simultaneous equations (one linear, one quadratic) algebraically and approximately graphically) If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl.org.uk

Lesson Plan Lesson Overview Progression of Learning Objective(s) Solve two simultaneous equations (one linear, one quadratic) algebraically and approximately graphically) Grade 7/8 Prior Knowledge Rearranging equations Substitution Solving equations Factorising quadratics Duration Provided prior knowledge of basic algebraic skills are secure this content can be taught with practice time within 70 minutes. Resources Print slides: 18 - 25 Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) Starter; Recap of linear equations needed later for factors in quadratics. Teacher led activity on slide 4. Eight equations of the general form ax + b = s. If a student is correct, click on any equation to show the answer and turn sector green, if an answer is incorrect, click on a blank area of colour in the sector to turn it amber. The red circle is timing down one minute from the time at which the start button is pressed. A printable sheet (slide 18) is provided if more practice is needed. 10 Factorising Recap “investigation”; pair the eight factors with the expressions. This can be used to emphasise the role of negative signs (note the x2 ± 3x − 18 expressions). Print slide 19. Differentiated practice at factorising. Later questions use common factors and difference of two squares. Print slide 20. 15 Recap simultaneous linear equations Give students slide 21. Recap of solving linear simultaneous equations. Simultaneous equations where one is not linear can still be solved graphically. Give students slide 22. Using slide 8 demonstrate that the points of intersection are the solutions. Note the difference between linear simultaneous equations where there is only one point of intersection. Discuss why now two points of intersection. 5 Solving simultaneous equations using method of substitution Give students slide 23. Demonstrate using slide 9 the steps to solve by substitution. Students to complete a further question – solution on slide 10. Give students slide 24. Solutions on slide 12. Solving two linear simultaneous equations in two variables algebraically in contextualised problems Give students slide 24. Solving two linear simultaneous equations in two variables algebraically in exam questions (from specimen papers) Give students slide 25. This includes 3 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated. Next Steps Assessment PLC/Reformed Specification/Target 7/Algebra/Simultaneous Equations (Quadratic)

Key Vocabulary Linear Quadratic Intersection Rearranging Substitution Approximate Graphical

You have one minute from the moment you press start 2x − 1 = 0 2x + 1 = 3 x = 1 2 x = 1 3x + 1 = 1 3x + 2 = 0 START x = 0 x = − 2 3 2x + 1 = 9 5x − 1 = 8 x = −2 1 2 x = 1 4 5 You have one minute from the moment you press start 4x = 6 3x = 1 x = 1 1 2 x = 1 3 Click the start button to begin the timer and reveal eight equations. Click the equation to reveal the correct answer; click the colour in the sector to try again

Quadratic Equations; factorising Pair the factors with the expression x2 + 6x + 8 x2 + x − 20 ( x + 6 ) ( x + 3 ) ( x + 4 ) ( x − 4 ) ( x − 6 ) ( x + 2 ) ( x − 3 ) ( x + 5 ) x2 + 3x − 18 x2 − 3x − 18

Quadratic Equations; factorising Factorising All but two of these can be factorised; express them using brackets – and how about explaining why the other two don’t work? SILVER x2 + 7x + 10 x2 + 4x + 6 x2 + 3x − 18 x2 + x − 42 x2 + 10x + 25 x2 + 3x x2 + 16 x2 − 16 x2 − x Fancy a challenge? All but one of these can be factorised… GOLD 2x2 + 7x + 3 3x2 + x − 2 3x2 + x 4x2 − 25 9x2 + 16 4x2 − 8x − 5 Click on a box to see if it will factorise. If it turns red, it won’t. If it turns green, you know what to do…

Simultaneous Equations Linear Recap x-y=4 x + 2y=2 Elimination Substitution 5x + y = 22 x + y = 9 2x - y = 6 3x+1 = y x - y = 2 4x + 3y = 29 x = y + 6 x + y = 14

Simultaneous equations graphically We can solve simultaneous equations where one is linear and one is non linear graphically Step 1: Plot the curves of both graphs x= y -2 y=x² Step 2: Read off the values where the curves intersect There are two intersections so two pairs of solutions: x=2, y=4 and x=-1, y=1

Solve by Substitution x2 = y x=y-2 (y-2)2 = y y2-4y+4= y y2-5y+4= 0 Step 1: Sub equation 2 into equation 1 for x. (1) x=y-2 (2) (y-2)2 = y y2-4y+4= y Step 2: Expand brackets and make equation equal to 0. y2-5y+4= 0 (y-4)(y-1) =0 Step 3: Factorise to solve. Therefore y=4 and y=1 Step 4: Substitute both y values to find both x values When y=4 x=y-2 x=4-2 x=2 When y=1 x=y-2 x=1-2 x=-1 Step 5: Always substitute into the linear equation! Much easier! The solutions are (2,4) and (-1,1) the same as the graph!

Solve by Substitution - 2 x+4 = y (1) x²+2 = y (2) If you plot can see the solutions graphically x2 +2 = x+4 x2-x-2 = 0 (x-2)(x+1) = 0 Therefore x=2 and x=-1 When x=2 x+4=y 2+4=y 6=y When x=-1 x+4=y -1+4=y 3=y The solutions are (2,6) and (-1,3)

Practice y = x2 + x - 21 y = 2x – 1 y=x2+1 y-x=1 x2+y2=26 x-y=6 d) Can you draw the solution graphically?

Practice - Solutions y = x2 + x - 21 y = 2x – 1 y=x2+1 y-x=1 x2+y2=26 d) Can you draw the solution to graphically? a) x = -4 and y = -9 x = 5 and y = 9 b) x = 0 and y = 1 x = 1 and y = 2 c) x = 5 and y = -1 x = 1 and y = -5

Problem Solving and Reasoning The sum of two square numbers is 8, and the difference between the two numbers is 4. Can you identify the two numbers? x2+y2 = 8 (1) x-y=4 (2) Both y values are the same, what does this show? Think about the graph? Step 1: Construct two equations x=4+y Step 2: Rearrange equation 2 and substitute into equation 1 (4+y)2+y2=8 y2+8y + 16+ y2=8 2y2+8y + 8=0 (2y+ 4 )(y+2) =0 Step 3: Solve for y and substitute to find x. y=-2 y =-2 x=2 Solution (2,-2)

Reason and explain If there are no solutions, what does this look like graphically? Is there an instance when there is only one solution? When would this happen? Could I get extra solutions, which are not shown when drawn graphically? Can you justify.

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Student Sheet 1 2x − 1 = 0 2x + 1 = 3 4x + 3 = 0 2x + 7 = 10 If these were the answers, what were the questions? 2x − 9 = 0 2x + 1 = 3 This is the set of questions from the starter slide x = 2 x = − 3 4 4x + 3 = 7 4x + 3 = 2 x = 0 x = 4 2 3 3x + 1 = 0 6x − 5 = 7 x = 1 2 x = −3 5x = −2 2x = −5 x = −3 1 4 x = −1 1 2 Student Sheet 1

Quadratic Equations x2 + 6x + 8 x2 + x − 20 ( x + 6 ) ( x + 3 ) Factorising Pair the factors with the expression x2 + 6x + 8 x2 + x − 20 ( x + 6 ) ( x + 3 ) ( x + 4 ) ( x − 4 ) ( x − 6 ) ( x + 2 ) ( x − 3 ) ( x + 5 ) x2 + 3x − 18 x2 − 3x − 18 Student Sheet 2

Quadratic Equations x2 + 7x + 10 x2 + 4x + 6 x2 + 3x − 18 x2 + x − 42 Factorising All but two of these can be factorised; express them using brackets – and how about explaining why the other two don’t work? SILVER x2 + 7x + 10 x2 + 4x + 6 x2 + 3x − 18 x2 + x − 42 x2 + 10x + 25 x2 + 3x x2 + 16 x2 − 16 x2 − x Fancy a challenge? All but one of these can be factorised… GOLD 2x2 + 7x + 3 3x2 + x − 2 3x2 + x 4x2 − 25 9x2 + 16 4x2 − 8x − 5 On the screen, the ones that factorise change colour and become green; the ones that don’t become red. If you have a coloured crayon… Student Sheet 3

Simultaneous Equations Linear Recap x-y=4 x + 2y=2 Elimination Substitution 5x + y = 22 x + y = 9 2x - y = 6 3x+1 = y x - y = 2 4x + 3y = 29 x = y + 6 x + y = 14 Student Sheet 4

Simultaneous equations graphically Step 1: Plot the curves of both graphs x= y -2 y=x² Step 2: Read off the values where the curves intersect Student Sheet 5

Solve by Substitution x2 = y x+4 = y x = y - 2 x²+2 = y Step 1: Sub equation 2 into equation 1 for x. x+4 = y x = y - 2 x²+2 = y Step 2: Expand brackets and make equation equal to 0. Step 3: Factorise to solve. Step 4: Substitute both y values to find both x values Step 5: Always substitute into the linear equation! Much easier! Student Sheet 6

Problem Solving and Reasoning Practice Problem Solving and Reasoning The sum of two square numbers is 8, and the difference between the two numbers is 4. Can you identify the two numbers? y = x2 + x - 21 y = 2x – 1 y=x2+1 y-x=1 x2+y2=26 x-y=6 d) Can you draw the solution graphically? Reason and explain If there are no solutions, what does this look like graphically? Is there an instance when there is only one solution? When would this happen? Could I get extra solutions, which are not shown when drawn graphically? Can you justify. Student Sheet 7

Exam Questions – Specimen Papers Student Sheet 8