1. Turning points and completing the square
Grade 7
Turning points and completing the square
Deduce turning points by completing the square
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2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Deduce turning points by completing the square
Grade
7/8
Prior Knowledge
Completing the square
Quadratic graphs
Duration
50 minutes
Resources
Print slides: 16 to 20
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.
Give students slide 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 how the marks are allocated. 25
Next Steps
Assessment
PLC/Reformed Specification/Target7/Algebra/Turning points and completing the square

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

4. Completing the square Write an equivalent expression in the form
(x a)2 – b
Half of 6 = 3
32 = 9
x2 + 6x =
(x + 3)2 - 9
Check by multiplying out the brackets and simplifying
(x + 3)2 – 9 =
x2 + 6x
= x2 + 6x

5. Completing the square – now you try:
Write equivalent expressions in the form (x a)2 – b
x2 + 8x =
(x + 4)2 - 16
x2 - 3x =
(x - 1.5)2 – 2.25

6. 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

7. Turning points and completing the square
The equation of a curve is y = f(x) where f(x) = x2 + 4x + 6
The diagram shows a sketch of the graph .
A is the minimum point of the curve.
Write down the coordinates of A. y
Complete the square
x2 + 4x + 6 = (x + 2)2 – 4 + 6
= (x + 2)2 + 2
The minimum value of y is 2 and occurs when x = -2 A x
A (-2,2)

8. Turning points and completing the square
Use completing the square to find the minimum point of the curve y = x2 + 12x + 7 .
Complete the square
x2 + 12x - 7 = (x + 6)2 –
= (x + 6)
The minimum value of y is -29 and occurs when x = -6
Minimum point is (-6,-29)

9. 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 x2 + px + q?
(x + 3) = x2 + 6x
= x2 + 6x + 17

10. 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.

11. Exam Question – Specimen Papers

12. Exam Question – Specimen Papers

13. Exam Question – Specimen Papers

14. Exam Question – Specimen Papers

15. Exam Question – Specimen Papers

16. Completing the square Write an equivalent expression in the form
(x a)2 – b
Half of 6 = 3
32 = 9
x2 + 6x =
(x + 3)2 - 9
x2 + 8x =
x2 - 3x =
Student Sheet 1

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

18. Turning points and completing the square
The equation of a curve is y = f(x) where f(x) = x2 + 4x + 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 = x2 + 12x + 7 .
If a quadratic graph has a minimum turning point (-3,8), what is the equation of the graph in the form x2 + px + q?
Student Sheet 3

19. Exam Question – Specimen Papers
Student Sheet 4

20. Exam Question – Specimen Papers
Student Sheet 5