Additional Mathematics for the OCR syllabus - Algebra 2 AS Mathematics Algebra – Manipulation of brackets Lesson A2 This presentation revisits expanding brackets & factorising simple & quadratic expressions. A3 will extend this method to solving equations by factorisation. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Objectives Be confident in the use of brackets Be able to factorise linear expressions It is assumed that pupils have these skills, and this is a quick revision! These skills are prerequisite for later work on the factor & remainder theorem. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Review of expanding brackets Additional Mathematics for the OCR syllabus - Algebra 2 Review of expanding brackets Example 1 multiply term by term Expand (x + 3) (x + 5) collect like terms x2 + 5x + 3x + 15 x2 + 8x + 15 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Alternatives for expanding (x + 3)(x + 5) grid method smiley face x +5 +3 x2 +5x (x+3) (x+5) +3x +15 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Example 2 Perfect square! Expand (x + 4)2 (x + 4) x2 + 4x + 16 x2 + 8x + 16 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of learning this standard result! Remember (a + b)2 = a2 + 2ab + b2 Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Example 3 Difference of 2 squares! Expand (x - 8) (x + 8) x2 + 8x - 8x - 64 x2 - 64 Remember (a-b)(a+b) = a2 - b2 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of learning this standard result! Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Example 4 – A harder one! Expand (x2 + 2x + 1) (x - 2) multiply term by term collect like terms + x - 2 x3 - 2x2 + 2x2 – 4x x3 -3x - 2 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of using a systematic method. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Example 5 Expand (x + 4) (x - 3) (2x + 1) multiply any two brackets multiply remaining bracket (x2 + x – 12) (2x + 1) collect like terms 2x3 + x2 + 2x2 + x – 24x - 12 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of using a systematic method. 2x3 + 3x2 – 23x - 12 Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Factorising This involves taking out any common factors. Try to spot the HCF by inspection. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 (i) common factors Example 1 Factorise 12x – 18y The HCF of 12x & 18y is 6 6(2x – 3y) Check your answer by expanding the brackets Example 2 Factorise 6x2 – 21x The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. ALSO What if the answer to example 1 is given as 2(6x +9y)? Why is this not correct? Stress the importance of checking answers by multiplying out the brackets! The HCF of 6x2 & 21x is 3x 3x(2x – 7) Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 (ii) grouping Example 3 Factorise ax + ay + bx + by The first two terms have common factor a, the last two have common factor y a(x + y) + b(x + y) There is now a common factor of (x + y) The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of using a systematic method. The next slide looks at this method more closely. (x + y)(a + b) Check your answer by expanding the brackets. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 To illustrate this:- a(x + y) + b(x + y) let z = x + y az + bz z(a + b) but z = x + y Hopefully this explanation helps! However if it doesn’t, forget about it! (x + y)(a + b) …..as before! Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Example 4 Factorise xy + 2x + 3y + 6 The first two terms have common factor x, the last two have common factor 3 x(y + 2) + 3(y + 2) There is now a common factor of (y + 2) (y + 2)(x + 3) The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of using a systematic method & checking the answer by expanding the brackets. Check your answer by expanding the brackets. Written by HVaughan (North Chadderton) & LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 2 Example 5 Factorise 2x - 2xy - y + y2 2x(1 - y) - y(1 - y) For this method to succeed, both brackets should be the same, i.e both (1 - y) (1 - y)(2x - y) Example 6 Factorise 6a + 3ab - 2b - b2 The examples come up line by line. Get the pupils to predict the next line of the solution, discuss alternative solutions. Stress the importance of using a systematic method & checking the answer by expanding the brackets. Extra practice on this topic would be beneficial. See the Additional mathematics for OCR textbooks, or any A’ level textbook for questions. Check your answer by expanding the brackets 3a(2 + b) - b(2 + b) (2 + b)(3a - b) Written by HVaughan (North Chadderton) & LDobson (Blue Coat)