C4 Hexagon TopicsC4 Hexagon Topics

What is Esmee up

to?

Our C4 Hexagon Our C4 Hexagon

Esmee is just about to complete Esmee is just about to complete her last A Level maths topics. her last A Level maths topics. Here they are:Here they are:

Partial fractions

Coordinate Geometry

Binomial Expansion

Differentiation

Integration

Vectors

The C4 Mind mapThe C4 Mind map

Click hereClick here

A nice recap of our A nice recap of our primary yearsprimary years

Esmee’s younger brother Ernest is Esmee’s younger brother Ernest is in P5. He is a maths genius and can in P5. He is a maths genius and can easily add the following fractions:easily add the following fractions:

His teacher has given him a His teacher has given him a challenge to add algebraic fractions challenge to add algebraic fractions together: together:

4

1

3

1)2

8

3

3

1)1

Ernest’s challenge Ernest’s challenge questionsquestions

12

1

2

2

3

4)3

2

3

1

2)2

1

1

3

2)1

xxx

xx

xx

How would we work How would we work backwards? backwards?

)1)(3(

1

1

1

3

2)1

xx

x

xx

Partial fractionsPartial fractions

Linear factorsLinear factors

The cover up methodThe cover up method

The cover up method The cover up method

)1)(3(

1

1

1

3

2)1

xx

x

xx

)1()3()1)(3(

1

x

b

x

a

xx

x

)3()1(1 xbxax

How do we find the values of a and b?

Another example proving Another example proving our theory:our theory:

)2)(1(

15

2

3

1

2)2

xx

x

xx

Last one:Last one:

)12)(2(3

824)3

2

xxx

xx

•Express the following fraction as the sum/difference of three fractions. In other words express as partial fractions.

““I like pictures!” says I like pictures!” says ErnestErnest

Ernest is a visual learner and Ernest is a visual learner and learns best when given a learns best when given a pictorial representation of pictorial representation of mathematical concepts. mathematical concepts.

Click on this applet to see the Click on this applet to see the graphs of partial fractions:graphs of partial fractions:

Your investigationYour investigationPartial fractionsPartial fractions

Quadratic factors- is this really in our Quadratic factors- is this really in our syllabus? syllabus?

Shall we do it? Shall we do it? Use mathsnet.net to investigate how to Use mathsnet.net to investigate how to

write an algebraic fraction with a write an algebraic fraction with a quadratic factor in the denominator quadratic factor in the denominator into partial fractions.into partial fractions.

)3)(23(

1222

xxxx

x

The steps The steps

More steps More steps

Quadratic factors in the Quadratic factors in the denominatordenominator

What is the rule? What are the steps?What is the rule? What are the steps? So if there is a quadratic factor in the So if there is a quadratic factor in the

denominator then the numerator of denominator then the numerator of the partial fraction will be Ax+B. But the partial fraction will be Ax+B. But remember factorize first if possible!remember factorize first if possible!

Example 1:Example 1:

)3)(23(

1222

xxxx

x

Repeated Linear FactorsRepeated Linear Factors

Look at the graphical Look at the graphical representation of repeated linear representation of repeated linear factors:factors:

Partial fractionsPartial fractions

Repeated Linear FactorsRepeated Linear Factors

12

3

)1(

2

)1(

4)1

2 xxx

So how do you work So how do you work backwards?backwards?

)12()1(

514112

2

xx

xxSplit

Arthur Arthur

A quick quizA quick quiz

Five questions:Five questions:

A true or false quizA true or false quiz

What is an improper What is an improper fraction?fraction?

Give Ernest some examples of Give Ernest some examples of some improper fractions?some improper fractions?

What about some examples of What about some examples of some algebraic improper some algebraic improper fractions?fractions?

Partial fractionsPartial fractions

Improper algebraic fractions Improper algebraic fractions Use the FAMILY division methodUse the FAMILY division method DadDad MumMum SisterSister BrotherBrother

An example An example

Split the following Split the following into partial into partial fractions:fractions:

)2)(1(

233 2

xx

xx

Kelvin Kelvin

)2)(1(

233 2

xx

xx

Flow chart Flow chart

Make a flow chart of the steps you Make a flow chart of the steps you would take to split an algebraic would take to split an algebraic fraction into partial fractions. fraction into partial fractions.

The flow chart The flow chart

Are there repeated linearfactors in theDenominator?

Is there a quadratic factor in theDenominator?

Are there only linear factors in the denominator?

Is it an improper fraction?

Why do we want to find Why do we want to find partial fractions?partial fractions?

When we split a fraction to When we split a fraction to partial fractions it helps us to partial fractions it helps us to find the integral of the function find the integral of the function using a standard result? What is using a standard result? What is this standard result? this standard result?

Why do we want to find the Why do we want to find the integral of a function? integral of a function?