Starter - Pascal’s Triangle

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Pascal’s Triangle

A Level Maths Induction

Course Structure

• At Hardenhuish School we follow the Edexcel Advanced GCE in Mathematics Syllabus.

• This is a linear course with a mixture of Pure and Applied Mathematics

• Two thirds of the course is Pure Mathematics and one third is Applied Mathematics.

• Three 2 hour exams at the end of Year 13

Lesson Structure

• You will have five 50 minute lessons each week, plus one lesson of independent study.

• You will have two teachers who teach different parts of the course.

• It is your responsibility to catch up with any work you may miss.

• You will be set homework by each teacher each week.

Equipment

• You should have an A4 folder for your notes

• There are 43 chapters over the 2 years (21 in Year 1, 22 in Year 2)

• You will have a login for electronic resources

• You will need a Casio Classwiz calculator

Summer Work

• During the summer you will be expected to complete a small booklet of work to prepare yourself for September.

• All the topics in the booklet you will have covered for your GCSE exams.

• It will be important that you are confident and comfortable with these topics.

• There will be a short test in your second lesson in September to ensure you have prepared yourself for starting the A-Level work.

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Binomial Expansion

We will introduce a pure topic which extends what you know about expanding brackets

from GCSE

Binomial Expansion

• Expand and simplify the following

(𝑎 + 𝑏)0

(𝑎 + 𝑏)1

(𝑎 + 𝑏)2

(𝑎 + 𝑏)3

(𝑎 + 𝑏)4

1

1𝑎 + 1𝑏

1𝑎2 + 2𝑎𝑏 + 1𝑏2

1𝑎3 + 3𝑎2𝑏 + 3𝑎𝑏2 + 1𝑏3

1𝑎4 + 4𝑎3𝑏 + 6𝑎2𝑏2 + 4𝑎𝑏3 + 1𝑏4

Binomial Expansion

• Expand and simplify the following

(𝑎 + 𝑏)0

(𝑎 + 𝑏)1

(𝑎 + 𝑏)2

(𝑎 + 𝑏)3

(𝑎 + 𝑏)4

𝟏

𝟏𝑎 + 𝟏𝑏

𝟏𝑎2 + 𝟐𝑎𝑏 + 𝟏𝑏2

𝟏𝑎3 + 𝟑𝑎2𝑏 + 𝟑𝑎𝑏2 + 𝟏𝑏3

𝟏𝑎4 + 𝟒𝑎3𝑏 + 𝟔𝑎2𝑏2 + 𝟒𝑎𝑏3 + 𝟏𝑏4

(𝑎 + 𝑏)5

1𝑎5

Binomial Expansion

+5𝑎4𝑏 +10𝑎3𝑏2+10𝑎2𝑏3 +5𝑎𝑏4+1𝑏5

(𝑎 + 𝑏)5

1𝑎5

(𝑥 + 3)4

Binomial Expansion

1

4

6

4

1

+5𝑎4𝑏 +10𝑎3𝑏2+10𝑎2𝑏3 +5𝑎𝑏4+1𝑏5

81 + 108𝑥 + 54𝑥2 + 12𝑥3 + 𝑥4

(𝑥)0

(𝑥)1

(𝑥)2

(𝑥)3

(𝑥)4

(3)4

(3)3

(3)2

(3)1

(3)0

Exam Question• Find the binomial expansion of (𝑥 + 2)6,

giving each term in its simplest form.(Total 4 marks)

• Challenge, find the expansion of 2𝑥 − 3 4

Exam Question• Find the binomial expansion of (𝑥 + 2)6,

giving each term in its simplest form.(Total 4 marks)

64 + 192𝑥 + 240𝑥2 + 160𝑥3 + 60𝑥4 + 12𝑥5 + 𝑥6

• Challenge, find the expansion of 2𝑥 − 3 4

Binomial Expansion

• Where do we go from here?• Year 1 pure using a formula instead of drawing

the triangle – very useful for things like 2 + 𝑥 25 and uses factorial (!)

• Year 2 binomial expansion with fractional & negative powers and estimating square roots

• Year 1 stats binomial distribution to find probabilities

• Further Maths links with complex numbers and trigonometry

Summer Work

• During the summer you will be expected to complete a small booklet of work to prepare yourself for September.

• All the topics in the booklet you will have covered for your GCSE exams.

• It will be important that you are confident and comfortable with these topics.

• There will be a short test in your second lesson in September to ensure you have prepared yourself for starting the A-Level work.

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