10 Apr 2023

Algebra

Evaluating Expressions

Factorising – The common factor

10 Apr 2023

Learning Intention Success Criteria

1. To show how to evaluate an expression given values for the letters.

1. Be able to substitute numbers for letters in an expression.

Algebra

2. Use previous knowledge to evaluate expression.

Evaluating Expressions – number for letter

10 Apr 2023

AlgebraEvaluating Expressions – number for letter

Given the following information find the values of :-

a = 3 ; b = 4 and c = -1

BODMAS

5 a 5 3 15 2a c 2 3 ( 1)

22b

2 2b a (4 4) (3 3)

2 4 4

516 9 7

2 b b 32

10 Apr 2023

AlgebraEvaluating Expressions – number for letter

Given the following information find the values of :-

a = 3 ; b = 4 and c = -1

25c 5 ( 1) ( 1) 5 c c 5

22 3 30a b c

(2 3 3) (3 4) (30 ( 1)) (2 ) (3 ) (30 )a a b c

18 12 30 0

10 Apr 2023

Now try Exercise 4Ch2 (page 51)

AlgebraEvaluating Expressions – number for letter

10 Apr 2023

Starter Questions

1. My shape has 1 line of symmetry, 1 pair of equal

angles and adjcent lengths are equal.

What is my shape?

2. Find the highest common f actor f or

(a) 12 and 24 (b) 2x and 10x

3. Calculate 6a+ 5ab when a=(-1) b=(-2)

10 Apr 2023

Learning Intention Success Criteria

1. To show how to reverse the process of removing bracket ‘factorising’.

1. Be able to recognise the HCF for set of values.

Algebra

2. Understand the term factorising.

Factorising – The Common Factor

3. Factorise simple expressions.

F8 = 1 and 8

2 and 4

10 Apr 2023

Example : Find the HCF of 8 and 12.

HCF = 4

F12 = 1 and 12

2 and 6

3 and 4

Highest Common Factor

FactorsInt 2

F ab = 1 and ab

a and b

10 Apr 2023 Created by Mr. Lafferty@mathsrevision.comw

ww

.math

srevis

ion.c

om Example : Find the HCF of ab and 2b.

HCF = b

Fx2 = 1 and 2b

2 and b

Highest Common Factor

F 2h2 = 1 and 2h2

2 and h2 , h and 2h

Example : Find the HCF of 2h2 and 4h.

HCF = 2h

F4h = 1 and 4h

2 and 2h

4 and h

FactorsInt 2

10 Apr 2023 Created by Mr. Lafferty@mathsrevision.com

Factors

Find the HCF for these terms

(a) 16w and 24w

(b) 9y2 and 6y

(c) 4h and 12h2

(d) ab2 and a2b

8w

3y

4h

ab

10 Apr 2023

Factorising

Example Factorise 3x + 15

1. Find the HCF for 3x and 15 3

2. HCF goes outside the bracket 3( )

3. To see what goes inside the bracketdivide each term by HCF

3x ÷ 3 = x 15 ÷ 3 = 5 3( x + 5 )

Check by multiplying out the bracket to get back to where

you started

10 Apr 2023

Factorising

Example

1. Find the HCF for 4x2 and 6xy 2x

2. HCF goes outside the bracket 2x( )

3. To see what goes inside the bracketdivide each term by HCF

4x2 ÷ 2x =2x 6xy ÷ 2x = 3y 2x( 2x- 3y )

Factorise 4x2 – 6xy

Check by multiplying out the bracket to get back to where

you started

10 Apr 2023

Algebra

Simply find the HCF for a given set of data and write the data using brackets :-

Factorising – The Common Factor

6 12x

HCF ?6

6( )x 2

8 12a b

HCF ?4

4( )2a 3b

10 Apr 2023

Algebra

Simply find the HCF for a given set of data and write the data using brackets :-

Factorising – The Common Factor

am an

HCF ?a

( )a m n

29 6x x

HCF ?3x

3 ( )x 3 2x

10 Apr 2023

Factorising

Factorise the following :

(a) 3x + 6

(b) 4xy – 2x

(c) 6a + 7a2

(d) y2 - y

3(x + 2)

2x(2y – 1)

a(6 + 7a)

y(y – 1)

Be careful !

10 Apr 2023

10 Apr 2023

Now try Exercise 6iQuestion 1 – 17

(page 246)

AlgebraFactorising – The Common Factor

10 Apr 2023

Learning Intention Objective

1. To show how to factorise expression by grouping

1. Factorise expression by grouping

AlgebraEvaluating Expressions – number for letter

10 Apr 2023

Factorising

Example Factorise x2 + 3x +2x + 6

1. Can you see two groups? x2 + 3x+2x + 6

2. Find the HCF of both groupsx(x + 3) +2(x + 3)

3. Take out (x + 3) since it is a common factor

(x+3) (x+2)

Check by multiplying out the bracket to get back to where

you started

10 Apr 2023

Algebra

Simply group term according to common factor then find HCF for a given set of data

and write the data using brackets :-

Factorising – Factorise by grouping

Be careful

with signs x2 + 5x + 2x -

10= x(x-5) + 2(x-5)

(x+2)(x-5)

x2 + 4x - x + 4 =

x(x+4)-1(x+4)

(x-1)(x+4)

10 Apr 2023

Algebra

Simply group term according to common factor then find HCF for a given set of data

and write the data using brackets :-

Factorising – Factorise by grouping

Be careful

with signs x2 - x – 5x + 5= x(x-1) - 5(x-1)

(x-1)(x-5)

10 Apr 2023

Now try Exercise 13g

(page 770 Question 1, 10,

18 )

AlgebraFactorising – Factorise by grouping

10 Apr 2023

Learning Intention Success Criteria

1. Recognise when we have a difference of two squares.

1. To show how to factorise the special case of the difference of two squares.

2. Factorise the difference of two squares.

Difference of Two Squares

10 Apr 2023

When we have the special case that an expression is made up of

the difference of two squares then it is simple to factorise

The format for the difference of two squares

a2 – b2

First square term

Secondsquare term

Difference

Difference of Two SquaresInt 2

10 Apr 2023

a2 – b2

First square term

Secondsquare term

Difference

This factorises to

( a + b )( a – b )

Two brackets the same except for + and a -

Check by multiplying out the bracket to get back to where

you started

Difference of Two Squares

10 Apr 2023

Keypoints

Format a2 – b2

Always the difference sign -

( a + b )( a – b )

Difference of Two Squares

10 Apr 2023

Factorise using the difference of two squares

(a) x2 – y2

(b) w2 – z2

(c) 9a2 – b2

(d) 16y2 – 100k2

(x + y )( x – y )

( w + z )( w – z )

( 3a + b )( 3a – b )

( 4y + 10k )( 4y – 10k )

Difference of Two Squares

10 Apr 2023

Trickier type of questions to factorise.Sometimes we need to take out a commonAnd the use the difference of two squares.

Example Factorise 2a2 - 18

2( a + 3 )( a – 3 )

Difference of Two Squares

First take out common factor 2(a2 - 9)

Now apply the difference of two squares

10 Apr 2023

Factorise these trickier expressions.

(a) 6x2 – 24

(b) 3w2 – 3

(c) 8 – 2b2

(d) 27w2 – 12

6(x + 2 )( x – 2 )

3( w + 1 )( w – 1 )

2( 2 + b )( 2 – b )

3(3 w + 2 )( 3w – 2 )

Difference of Two Squares

10 Apr 2023

Now try Exercise 5

Ch5 (page 54)

Difference of Two Squares

10 Apr 2023

Learning Intention Success Criteria

1. To show how to factorise trinomial by rewriting to form two groups

2. Be able to factorise a trinomial (Quadratic Expression)

Algebra

1. Be able to rewrite a trinomial to form two groups

Evaluating Expressions – number for letter

Factoring ChartThis chart will help you to determine which method of factoring to use.Type Number of Terms

1. GCF 2 or more

2. Diff. Of Squares 23. Trinomials 3

First terms:Outer terms:Inner terms:Last terms: Combine like terms.

y2 + 6y + 8

y +2

y

+4

y2

+4y

+2y

+8

y2

+4y

+2y

+8

Review: (y + 2)(y + 4)

In this lesson, we will begin with y2 + 6y + 8 as our problem and finish with (y + 2)(y + 4) as our

answer.

Here we go! 1) Factor y2 + 6y + 8Use your factoring chart.

Do we have a GCF?Is it a Diff. of Squares problem?Now we will learn Trinomials! The general

form of a Quadratic equation is ax2 +bx + c.

Nope!No way! 3 terms!

Product of the first(a)and last coefficients (c)

Sum to give the middle

coefficient

The goal is to find two factors of ac in the first column that add up to the middle term (b) in the second

column.We’ll work it out in the next few slides.

1) Factor y2 + 6y + 8Create your MAMA table.

Multiply Add+8 +6

Product of the first and

last coefficients

Middlecoefficient

Here’s your task…What numbers multiply to +8 and add to +6? If you cannot figure it out right away, write the combinations.

MA

1) Factor y2 + 6y + 8Place the factors in the table.

+1, +8

-1, -8+2,

+4 -2, -4

Multiply Add+8 +6

Which has a sum of +6?

+9, NO-9, NO+6, YES!!

-6, NOWe are going to use these numbers in the next step!

1) Factor y2 + 6y + 8

+2, +4

Multiply Add+8 +6

+6, YES!!Hang with me now! Replace the middle

number of the trinomial with our working numbers from the MAMA table

y2 + 6y + 8

y2 + 2y + 4y + 8Now, group the first two terms and the

last two terms.

2) Factor x2 – 2x – 63Create your MAMA table.

Multiply Add-63 -2

Product of the first and last

coefficients

Middlecoefficient

-63, 1-1, 63-21, 3-3, 21-9, 7-7, 9

-6262-1818-2 2

Signs need to be

different since

number is negative.

MA

x2 – 9x + 7x – 63 x2 – 9x + 7x – 63 x(x – 9)+7(x – 9)

(x + 7)(x – 9)

Replace the middle term with our working numbers.

x2 – 2x – 63

2) Factor 5x2 - 17x + 14 Create your MAMA table.

Multiply Add+70 -17

Product of the first and last

coefficients

Middlecoefficient

-1, -70-2, -35-7, -10

-71-37-17

Signs need to be the same as

the middle sign since

the product is positive.

Replace the middle term.5x2 – 7x – 10x + 14Group the terms.

MA

5x2 - 17x + 14 5x2 – 7x – 10x + 14

x(5x – 7) -2(5x – 7)(x – 2)(5x – 7)

10+ 3x-x2

10 + 5x –2x + x 5(2+x)- x(2 + x) (2 + x)(5 – x)

Factor x2 + 3x + 21. (x + 2)(x + 1)2. (x – 2)(x + 1)3. (x + 2)(x – 1)4. (x – 2)(x – 1)

Factor 2x2 + 9x + 101. (2x + 10)(x +

1)2. (2x + 5)(x + 2)3. (2x + 2)(x + 5)4. (2x + 1)(x +

10)

Factor 6y2 – 13y – 51. (6y2 – 15y)(+2y –

5)2. (2y – 1)(3y – 5)3. (2y + 1)(3y – 5)4. (2y – 5)(3y + 1)

2) Factor 2x2 - 14x + 12

Multiply Add+6 -7

Find the HCF!2(x2 – 7x + 6)Now do the MAMA table!

-7-5

Signs need to be the same as

the middle sign since

the product is positive.

Replace the middle term.2[x2 – x – 6x + 6]Group the terms.

-1, -6-2, -3

2[x2 – x– 6x + 6]

2[x(x – 1) -6(x – 1)]2(x – 6)(x – 1)

10 Apr 2023

Learning Intention Success Criteria

1. To show how to reverse the process of removing bracket ‘factorising’.

1. To understand a perfect square trinomials.

AlgebraFactorising – Perfect Square

2. Factorize trinomial as the a perfect square .

Factoring ChartThis chart will help you to determine which method of factoring to use.Type Number of Terms

1. GCF 2 or more2. Diff. Of Squares 23. Trinomials 3

First terms: Outer terms:Inner terms:Last terms: Combine like terms.

y2 + 2y + 2y+ 4y2 + 4y + 4

y2

+2y

+2y

+4

Review: Multiply (y + 2)2

(y + 2)(y + 2)Do you remember

these?(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Using the formula, (y + 2)2 = (y)2 + 2(y)(2) +

(2)2

(y + 2)2 = y2 + 4y + 4

Which one is quicker?

1) Factor x2 + 6x + 9Does this fit the form of our

perfect square trinomial?1) Is the first term a perfect

square?Yes, a = x

2) Is the last term a perfect square?

Yes, b = 33) Is the middle term twice

the product of the a and b?Yes, 2ab = 2(x)(3) = 6x

Perfect Square Trinomials

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your answer!

(x + 3)2= (x+3)(x+3)

You can still factor the other way but this is

quicker!

2) Factor y2 – 16y + 64 Does this fit the form of

our perfect square trinomial?

Is the first term a perfect square?

Yes, a = y 2) Is the last term a

perfect square? Yes, b = 8

Is the middle term twice the product of the a and b?

Yes, 2ab = 2(y)(8) = 16y

Perfect Square Trinomials

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your

answer!(y – 8)2=(y-8)(y-8)

Factor m2 – 12m + 36

1. (m – 6)(m + 6)2. (m – 6)2

3. (m + 6)2

4. (m – 18)2

3) Factor 4p2 + 4p + 1Does this fit the form of our

perfect square trinomial?1) Is the first term a perfect

square?Yes, a = 2p

2) Is the last term a perfect square?

Yes, b = 13) Is the middle term twice

the product of the a and b?

Yes, 2ab = 2(2p)(1) = 4p

Perfect Square Trinomials

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your

answer!(2p + 1)2

Does this fit the form of our perfect square trinomial?

1) Is the first term a perfect square?

Yes, a = 5x2) Is the last term a perfect

square?Yes, b = 11y3) Is the middle term twice

the product of the a and b?

Yes, 2ab = 2(5x)(11y) = 110xy

4) Factor 25x2 – 110xy + 121y2

Perfect Square Trinomials

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 – 2ab + b2

Since all three are true, write your

answer! (5x – 11y)2=(5x – 11y)(5x –

11y)2

Factor 9k2 + 12k + 4

1. (3k + 2)2

2. (3k – 2)2

3. (3k + 2)(3k – 2)4. I’ve got no

clue…I’m lost!

Factor 2r2 + 12r + 18

1. prime2. 2(r2 + 6r + 9)3. 2(r – 3)2

4. 2(r + 3)2

5. 2(r – 3)(r + 3)

Don’t forget to factor the GCF first!