Grade 7 Turning points and completing the square

Grade 7 Turning points and completing the square Deduce turning points by completing the square If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl. org. uk

Lesson Plan Lesson Overview Objective(s) Deduce turning points by completing the square Prior Knowledge Completing the square Quadratic graphs Duration 50 minutes Resources Print slides: 16 to 20 Grade 7/8 Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) Completing the square method Give students slide 16 – recap on completing the square. Demonstrate method using slide 4. Students to complete 2 further questions. Provided prior knowledge of this is secure continue. 5 What are turning points Give students slide 17 to complete notes. Using slide 6 explain the nature of turning points and how their coordinates can be easily found using completing the square. 5 Deduce turning points by completing the square Give students slide 18. 3 questions related to turning points to be completed independently – review the answers using slide 7, 8 and 9. Extension questions on slide 10. 15 Deduce turning points by completing the square in exam questions (from specimen papers) Give students slide 19 and 20. This includes 5 exam questions related to objective. Students to use notes from lesson. Ensure that all steps are shown. Relate to mark scheme to show the marks are allocated. 25 Next Steps Assessment PLC/Reformed Specification/Target 7/Algebra/Turning points and completing the square

Key Vocabulary Turning point Minimum turning point Maximum turning point Gradient Complete the square Expression Roots Real roots

Completing the square Write an equivalent expression in the form (x ± a)2 – b Half of 6 = 3 x 2 + 6 x = (x + 3)2 -9 32 = 9 Check by multiplying out the brackets and simplifying (x + 3)2 – 9 = x 2 + 6 x + 9 - 9 = x 2 + 6 x

Completing the square – now you try: Write equivalent expressions in the form (x ± a)2 – b x 2 + 8 x = (x + 4)2 - 16 x 2 - 3 x = (x - 1. 5)2 – 2. 25

Turning points Here are the graphs of y = f(x) and y = g(x) y y x x Here are the turning points. Here the gradients of the graphs are zero (x ± a)2 – b If + a then x coordinate is –a If – a then x coordinate is +a b is the y coordinate

Turning points and completing the square The equation of a curve is y = f(x) where f(x) = x 2 + 4 x + 6 The diagram shows a sketch of the graph. y A is the minimum point of the curve. Write down the coordinates of A. Complete the square x 2 + 4 x + 6 = (x + 2)2 – 4 + 6 = (x + 2)2 + 2 A A (-2, 2) x The minimum value of y is 2 and occurs when x = -2

Turning points and completing the square Use completing the square to find the minimum point of the curve y = x 2 + 12 x + 7. Complete the square x 2 + 12 x - 7 = (x + 6)2 – 36 + 7 = (x + 6)2 - 29 The minimum value of y is -29 and occurs when x = -6 Minimum point is (-6, -29)

Problem Solving and Reasoning If a quadratic graph has a minimum turning point (-3, 8), what is the equation of the graph in the form x 2 + px + q? (x + 3)2 + 8 = x 2 + 6 x + 9 + 8 = x 2 + 6 x + 17

Problem Solving and Reasoning What do you understand by the terms roots and real roots of a quadratic equation? Can you show the difference using graphs? Explain the difference between a minimum turning point and a maximum turning point.

Exam Question – Specimen Papers

Exam Question – Specimen Papers

Exam Question – Specimen Papers

Exam Question – Specimen Papers

Exam Question – Specimen Papers

Completing the square Write an equivalent expression in the form (x ± a)2 – b Half of 6 = 3 32 = 9 x 2 + 6 x = (x + 3)2 - 9 x 2 + 8 x = Student Sheet 1 x 2 - 3 x =

Turning points Here are the graphs of y = f(x) and y = g(x) y y x Student Sheet 2 x

Turning points and completing the square The equation of a curve is y = f(x) where f(x) = x 2 + 4 x + 6. The diagram shows a sketch of the graph. A is the minimum point of the curve. Write down the coordinates of A. Use completing the square to find the minimum point of the curve y = x 2 + 12 x + 7. Student Sheet 3 If a quadratic graph has a minimum turning point (-3, 8), what is the equation of the graph in the form x 2 + px + q?

Exam Question – Specimen Papers Student Sheet 4

Exam Question – Specimen Papers Student Sheet 5