1. Derive an equation Grade 5
Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution in context
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2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution in context
Grade 5
Prior Knowledge
Algebraic skills – simplifying by collecting like terms, expanding single and double brackets
Basic shape knowledge – equal sides, equal angles, perimeters, areas
Solving equations (including simple simultaneous equations)
Duration
Provided prior knowledge is secure (algebra skills in particular simplifying and solving equations) content can be taught with practice time within 90 minutes. May decide to split to focus on lengths in one learning episode and then angles in the next.
Resources
Print slides: 4, 6, 8, 14, 18, 24
Equipment
Progression of Learning
What are the students learning?
How are the students learning? (Activities & Differentiation)
Recapping key algebraic skills – simplifying by collecting like terms
Give students slide 4 printed. To work independently to complete (content should be recap of prior knowledge). Teacher circulation to check all are confident with simplifying. Discuss any key misconceptions. Show students slide 5 so that they can self mark their work. 10
Recapping key algebraic skills – expanding brackets
Give students slide 6 printed. To work independently to complete (content should be recap of prior knowledge). Teacher circulation to check all are confident with expanding brackets. Remind students that when expanding double brackets they will need to simplify as well. Discuss any key misconceptions. Show students slide 7 so that they can self mark their work.
Derive an equation where two algebraic expressions are equivalent, and use algebra to support and construct arguments
Give students slide 8. Students have 4 different types of questions related with making two algebraic expressions equal. Allow students to attempt al questions independently before working through each question with them using slides 9 – 13. 20
Derive equations from worded situations and solve the equations where necessary.
Give students slide 14 printed. This is extending understanding to be able to form equations from worded problems.. Show students slides 15 – 17 to explain. 15
Derive equations using angle knowledge and solve the equations where necessary.
Give students slide 18 printed. This extending the concept to angles expressed algebraically. Show students slides 19 – 23 to explain how they can form algebraic expressions related to angle knowledge.
Arguing mathematically that two algebraic expressions are equivalent in exam questions (from specimen papers)
Give students slide 18. This includes 2 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
More complex algebraic arguments involving quadratics
Assessment
PLC/Reformed Specification/Target 5/Algebra/Algebraic Argument

3. Key Vocabulary
Construct an argument Equivalent Expression Algebraic

4. Simplifying 3x + 7y + 2x – y 3a + 2h + a + 3h
6g – 5h – 4g + 2h
a + a + a p x p x p a x c x 3 3 x c x a 2e x 3f
Student Sheet 1

5. Simplifying

6. Expanding Brackets 2(x - 4) + 3(x + 5) 5(y – 2) -2(y – 3) 2m(m + 3)
5(y + 4t – 2)
x(x2 + 2)
(t + 2) (t + 4)
(x – 5) (x + 3)
(2x + 1) (x – 4)
(2t – 3) (t + 5)
Multiply Out – AQA foundation
Student Sheet 2

7. Expanding Brackets

8. Constructing algebraic arguments
Show that for all values of x
(x + 3)(x - 4) = x2 - x -12
(x + 3)(x - 4)(x + 1) = x3 – 13x – 12
Show 2(x + 3) can be written in the form 2x2 + 12x + 31
Use the diagram to show algebraically that: (3x – 2)2 = 9x2 - 12x + 4
Show algebraically that the perimeters of these shapes are identical
Student Sheet 3

9. Constructing algebraic arguments
Show that (x + 3)(x - 4) = x2 - x -12
Show that (x + 3)(x - 4)(x + 1) = x3 – 13x - 12
For all values of x

10. Constructing algebraic arguments
Show that (x + 3)(x - 4) = x2 - x -12
x2 – 4x + 3x -12 = x2 – x -12
Show that (x + 3)(x - 4)(x + 1) = x3 – 13x - 12
x3 + x2 – x2 - x - 12x -12 = x3 – 13x - 12
For all values of x

11. Constructing algebraic arguments
Show 2(x + 3) can be written in the form 2x2 + 12x + 31
2(x + 3) = 2(x2 +6x + 9) + 13
= 2x2 + 12x
= 2x2 + 12x + 31

12. Constructing algebraic arguments
Rectangle
Triangle
12+ 20x
2(5 + 2x)
4(2x - 1)
5(x + 3) 2x
Show algebraically that the perimeters of these shapes are identical
5(x + 3) x 2 + 4(2x - 1) x 2 =
10 x x – 8
= 26x + 22
2(5 + 2x) x + 2x =
10 + 4x x
=26x + 22

13. Problem solving and reasoning
Use the diagram to show algebraically that: (3x – 2)2 = 9x2 - 12x + 4
6x x – = 12x – 4
Area of square 3x x 3x = 9x2
9x2 - (12x – 4)
= 9x2 - 12x + 4 3x
(3x – 2)2 3x
2(3x – 2) = 6x - 4 4 2
2(3x – 2) = 6x - 4
2 x 2 = 4 2

14. How to derive an equation – worded problems
Five sandwiches and four drinks cost £20. Two sandwiches and five drinks cost £8. Find the cost for one sandwich.
A rectangle has a perimeter of 32cm. Its length is 4cm longer than its width. What is its width?
A rectangle has an area of 50cm2. Its length is double its width. Find the perimeter.
A shirt and two trousers cost £20. Four shirts and two trousers cost £50. Find the price difference between the two items.
The perimeter of the garden pictured is 48 metres. Jim wants to cover the garden with grass seed. Each pack of grass seed can cover 8m2 and cost £5.20. How much will it cost to cover the garden?
Student Sheet 4

15. How to derive an equation – worded problems
Five sandwiches and four drinks cost £20. Two sandwiches and five drinks cost £8. Find the cost for one sandwich.
We can call sandwiches ‘s’ and drinks ‘d’.
5s + 4d = 20
x 5
2s + 5d = 8
x 4
Solve!
25s + 20d = 100 _
8s + 20d = 32
17s = 68
s = 4
Can we calculate the cost for one drink?

16. How to derive an equation – worded problems
A rectangle has a perimeter of 32cm. Its length is 4cm longer than its width. What is its width?
A rectangle has an area of 50cm2. Its length is double its width. Find the perimeter.
A shirt and two trousers cost £20. Four shirts and two trousers cost £50. Find the price difference between the two items.
w = 6
w = 5, l = 10, P = 30cm £5

17. Problem Solving and Reasoning5
The perimeter of the garden pictured is 48 metres. Jim wants to cover the garden with grass seed. Each pack of grass seed can cover 8m2 and cost £5.20. How much will it cost to cover the garden?
a + 1 B A
5 + a
(5 + a) + a (a + 1) + a + (2a + 1) = 48
6a + 12 = 48
a = 6
Area of the shape = A = 6 x 11 = 66m2
Area of the shape = B = 6 x 7 = 42m2
Total area = = 108m2
108 ÷ 8 =13.5, so 14 packets needed
14 x 5.20 = £72.80

18. ABC is an isosceles triangle. Find the sizes of each angle.
Algebraic Arguments - Angles
ABC is an isosceles triangle. Find the sizes of each angle.
Student Sheet 5

19. Algebraic Arguments - Angles
Triangle
180°
Quadrilateral
360°
4x + 6x + 2x = 180°
Guide
x x + 7 = 360°

20. Triangle 180° Find the value of y 2y + 3y + y = 180° 6y = 180°
Guide
y = 180° 6
y = 30°

21. Isosceles Parallelogram y = 52 Opp angles = 52 + 52 = 104
3x – 15 = 2x + 24
180 – 104 = 76
Guide
Then can solve for x
14x + 6 = 76
Then can solve for x

22. How to derive an equation
2x + 1
3x - 4 A B C 1)
ABC is an isosceles triangle. Find the sizes of each angle.
Since ABC is an isosceles triangle, angle C is equal to angle B, 2x + 1.
2x + 1
As we know angles in a triangle add up to 180°, we can write the equation as follows:
Angle A + Angle B + Angle C = 180
(3x - 4) + (2x + 1) + (2x + 1) = 180
Simplify!
7x – 2 = 180
Solve!
7x = 182
x = 26

23. How to derive an equation
2x + 1
3x - 4 A B C 1)
ABC is an isosceles triangle. Find the sizes of each angle.
Now that we have found the value of x, we can substitute it into each expression.
Is there a way of checking our answer?
Angle A = 3x – 4
Angle B = 2x + 1
Angle C = 2x + 1
3(26) – 4 = 74°
2(26) + 1 = 53°
2(26) + 1 = 53°

24. Exam Questions – Specimen Papers
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Student Sheet 6

25. Exam Questions – Specimen Papers

26. Exam Questions – Specimen Papers

27. Exam Questions – Specimen Papers

28. Exam Questions – Specimen Papers