1. 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)

2. 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)

3. 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)

4. 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)

5. 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)

6. 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)

7. 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)

8. 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)

9. 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)

10. 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)

11. 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)

12. 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)

13. 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)

14. Additional Mathematics for the OCR syllabus - Algebra 2
Example 5
Factorise x - 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 a + 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)