Derive an equation Grade 5 Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution in context 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) 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

Key Vocabulary Construct an argument Equivalent Expression Algebraic

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

Simplifying

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

Expanding Brackets

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)2 + 13 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

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

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

Constructing algebraic arguments Show 2(x + 3)2 + 13 can be written in the form 2x2 + 12x + 31 2(x + 3)2 + 13 = 2(x2 +6x + 9) + 13 = 2x2 + 12x + 18 + 13 = 2x2 + 12x + 31

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 +30 + 16x – 8 = 26x + 22 2(5 + 2x) + 12 + 20x + 2x = 10 + 4x + 12 + 22x =26x + 22

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

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

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?

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

Problem Solving and Reasoning 5 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 + 5 + (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 = 66 + 42 = 108m2 108 ÷ 8 =13.5, so 14 packets needed 14 x 5.20 = £72.80

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

Algebraic Arguments - Angles Triangle 180° Quadrilateral 360° 4x + 6x + 2x = 180° Guide 127 + 5x + 3 + 88 + 10x + 7 = 360°

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

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

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

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°

Exam Questions – Specimen Papers [4] Student Sheet 6

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers