Simplify algebraic expressions involving algebraic fractions. Grade 6 Algebraic Fractions Simplify algebraic expressions involving algebraic fractions.

Lesson Plan Lesson Overview Progression of Learning Objective(s) Simplify algebraic expressions involving algebraic fractions Grade 6 Prior Knowledge Fraction operations (numerical) Algebraic manipulation (including factorising quadratics) Duration Provided algebraic skills are strong (expanding and factorising quadratics) content can be taught with practice time within 60 minutes. Resources Print slides:4, 6, 8, 10, 12 Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) Recap key algebraic skills needed for solving quadratics (expanding brackets and simple factorising) Give students slide 4 and 6 printed. To work independently to complete (content should be recap of prior knowledge). Teacher circulation to check all are confident in these two key skills. Students can self mark their work using slide 5 and 7. 15 Factorising quadratics Introductory “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 8 for students. Differentiated practice at factorising quadratics. Print slide 10 for students. 20 Simplifying fractions by looking for common factors Give students slide 12 printed. Using slide 13 show how can simplify looking for common factors in each of the term and then simplify the factors to the lowest possible for each term. Students to copy example and write a note in the white box for the method. 5 Simplifying fractions by factorising Using slide 14 show how can simplify by factorising expressions and then looking for common factors to cancel. Students to copy example and write a note in the white box for the method. Practice questions on slide 12 to then be completed independently by students. Review answers using slide 15. Reasoning question on slide 16. There are no OCR exam questions in the SAMS which examine this skill in isolation. Further learning to be completed on adding/subtracting algebraic fractions and solving equations involving algebraic fractions before reviewing related exam questions. Next Steps Adding and subtracting algebraic fractions Solving equations involving algebraic fractions Assessment PLC/Reformed Specification/Target 6/Algebra/Algebraic Fractions

Key Vocabulary Factorise Simplify Quadratic Linear Collect Algebraic Fraction Denominator Product

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 1

Expanding Brackets

Factorising - Simple 3x + 6 x2 + 7x 3e2 + 5e 8y2 – 4xy 8a2 + 12a 3xy2 – 6xy Student Sheet 2

Factorising - Simple 3x + 6 x2 + 7x 3e2 + 5e 8y2 – 4xy 8a2 + 12a 3xy2 – 6xy 4y(2y - x) 4a(2a + 3) 3xy(y – 2)

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 3

Factorising - Quadratics 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 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 4

Factorising - Quadratics 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…

Simplify an algebraic fraction EXAMPLES PRACTICE 3𝑎𝑏+6 𝑏 2 3𝑏𝑐−12𝑏𝑑 12𝑎𝑏 4𝑎 2𝑎𝑏−8𝑎𝑏𝑐 2𝑎 2 𝑏+4𝑎𝑏𝑐 𝑥 2 +7𝑥+6 𝑥 2 +8𝑥+12 2𝑥 2 +7𝑥+6 2𝑥 2 −7𝑥−10 3𝑥 2 +17𝑥+10 3𝑥 2 −𝑥−2 Write the algebraic fractions in their simplest terms. 3𝑏+ 𝑏 2 +2 𝑏 2 −𝑏−2 A rectangle has an area of 4x 2 −7x−15 square units, its width is x-3 units. What is the length of the rectangle? Student Sheet 5

Simplify an algebraic fraction Look for common factors in each of the terms. Simplify the factors to the lowest possible for each term. 3𝑎𝑏+6 𝑏 2 3𝑏𝑐−12𝑏𝑑 Each of the 4 terms in the fraction have a multiple of 3. They also have a ’b’. So each term can be divided by 3 and b. There is nothing else common in all 4 terms so this is complete. 1 2 3𝑎𝑏+6 𝑏 2 3𝑏𝑐−12𝑏𝑑 1 4 𝑎+2𝑏 𝑐−4𝑑

Simplify an algebraic fraction Look for multiples of a number of letter in every term –if there are not any, can you factorise the expressions? Simplify the factors to the lowest possible for each term. 3𝑏+ 𝑏 2 +2 𝑏 2 −𝑏−2 Both of the terms are quadratic in terms of b. They can both be factorised. Both have a complete identical bracket so it can be cancelled out in the numerator and denominator. (𝑏+1)(𝑏+2) (𝑏−2)(𝑏+1) (𝑏+1)(𝑏+2) (𝑏−2)(𝑏+1) = 𝑏+2 𝑏−2

Practice 3b 1−4𝑐 𝑎+2𝑐 𝑥+1 𝑥+2 𝑥+2 𝑥−5 𝑥+5 𝑥−1 Write the algebraic fractions in their simplest terms. 12𝑎𝑏 4𝑎 2𝑎𝑏−8𝑎𝑏𝑐 2𝑎 2 𝑏+4𝑎𝑏𝑐 𝑥 2 +7𝑥+6 𝑥 2 +8𝑥+12 2𝑥 2 +7𝑥+6 2𝑥 2 −2𝑥−15 3𝑥 2 +17𝑥+10 3𝑥 2 −𝑥−2 3b 1−4𝑐 𝑎+2𝑐 𝑥+1 𝑥+2 𝑥+2 𝑥−5 𝑥+5 𝑥−1

Reasoning. A rectangle has an area of 4𝑥 2 −7𝑥−15 square units, its width is x-3 units. What is the length of the rectangle? Area ÷width = length so: 4𝑥 2 −7𝑥−15 𝑥−3 Factorise and simplify: (4𝑥+5)(𝑥−3) 𝑥−3 The length is 4x+5 units.