Surds - Teacher Slides.pptx

EDEXCEL GCSE MATHS · HIGHER TIER

Surds

Number

Lesson 7 of 7

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EDEXCEL GCSE MATHS · HIGHER TIER

Last Lesson and Before

Number

Lesson 7 of 7

Answer each one, then check.

1

What is √36?

2

What is √4 × √9?

3

Expand 3(x + 2).

4

Last lesson: write 4 × 10³ as an ordinary number.

5

Which of these is a whole number: √16 or √17?

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EDEXCEL GCSE MATHS · HIGHER TIER

Last Lesson and Before - Answers

Number

Lesson 7 of 7

1

What is √36?

6

2

What is √4 × √9?

2 × 3 = 6 - the same as √36.

3

Expand 3(x + 2).

3x + 6

4

Last lesson: write 4 × 10³ as an ordinary number.

4000

5

Which of these is a whole number: √16 or √17?

√16 = 4. √17 is not a whole number.

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EDEXCEL GCSE MATHS · HIGHER TIER

Learning Objectives

Number

Lesson 7 of 7

1

Know what rational and irrational numbers are, and what a surd is.

2

Simplify surds such as √72.

3

Multiply, divide, add and subtract surds.

4

Expand brackets containing surds.

5

Rationalise the denominator of a fraction such as 6/(√3).

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EDEXCEL GCSE MATHS · HIGHER TIER

What Is a Surd?

Number

Lesson 7 of 7

A surd is a root that cannot be written exactly as a fraction or a terminating or recurring decimal.

Rational numbers

Can be written as a fraction of two integers: ¾, −7, 0.6, √25 = 5.

Irrational numbers

Cannot: √2 = 1.414 213 56…, π= 3.141 592 65… The decimals never end and never repeat.

Surds

Roots that are irrational: √2, √10, ∛5. √9 is not a surd, because √9 = 3.

Why use them

√2 is exact; 1.414 is not. When a question says "give your answer in surd form", it wants the exact answer.

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EDEXCEL GCSE MATHS · HIGHER TIER

Rational or Irrational?

Number

Lesson 7 of 7

RATIONAL

• ¾

• 0.666…= ⅔

• √25 = 5

• −7

• √(4/9) = ⅔

IRRATIONAL

• √2

• √10

• π

• ∛5

• 1 + √3

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The Rules for Surds

Number

Lesson 7 of 7

Roots can be split up and combined when multiplying and dividing - but not when adding.

Multiplying

√a × √b = √ab. √2 × √8 = √16 = 4.

Dividing

√a/(√b) = √(a/b). √50/(√2) = √25 = 5.

A surd times itself

√a × √a = a. √7 × √7 = 7.

Not for adding

√(a + b) is NOT √a + √b: √(9 + 16) = √25 = 5, but √9 + √16 = 7.

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Surds in Right-Angled Triangles

Number

Lesson 7 of 7

The long side of a right-angled triangle is often a surd.

Surds turn up all the time in geometry. A right-angled triangle with shorter sides 1 and 1 has a hypotenuse of exactly √2 - a length you can draw perfectly but never write exactly as a decimal.

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EDEXCEL GCSE MATHS · HIGHER TIER

PART ONE

Simplifying Surds

Take out the biggest square factor.

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EDEXCEL GCSE MATHS · HIGHER TIER

Simplifying a Surd

Number

Lesson 7 of 7

Simplify √72

1

Find the largest square number that is a factor of 72

Square numbers: 4, 9, 16, 25, 36 ... and 36 × 2 = 72

2

Split the root

√72 = √36 × √2

3

Work out the square root

√36 = 6

ANSWER

√72 = 6√2

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Adding Surds

Number

Lesson 7 of 7

Simplify √50 + √18

1

Simplify each surd

√50 = √25 × √2 = 5√2 and √18 = √9 × √2 = 3√2

2

They are now "like surds" - both multiples of √2

5√2 + 3√2

3

Add them like algebra: 5x + 3x = 8x

8√2

ANSWER

8√2

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Expanding Brackets with Surds

Number

Lesson 7 of 7

Expand and simplify (3 + √2)(5 − √2)

1

First times first

3 × 5 = 15

2

Outer and inner

3 × (−√2) = −3√2 and √2 × 5 = 5√2

3

Last times last

√2 × (−√2) = −2

4

Collect like terms

15 − 2 = 13 and −3√2 + 5√2 = 2√2

ANSWER

13 + 2√2

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PART TWO

Rationalising the Denominator

Getting the surd out of the bottom of a fraction.

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Rationalising the Denominator

Number

Lesson 7 of 7

To rationalise a fraction like a/(√b), multiply the top AND the bottom by √b.

Why it works

Multiplying top and bottom by the same number does not change a fraction's value, and √b × √b = b removes the surd from the bottom.

Example

6/(√3) = (6 × √3)/(√3 × √3) = 6√3/3 = 2√3.

Always simplify

Cancel any common factors, and simplify any surd left on top.

Exam wording

"Rationalise the denominator" and "give your answer in the form a√b" both mean do this.

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Rationalising

Number

Lesson 7 of 7

Rationalise the denominator of 10/(√5). Give your answer in its simplest form.

1

Multiply the top and the bottom by √5

(10 × √5)/(√5 × √5)

2

The bottom becomes a whole number

√5 × √5 = 5

3

So

10√5/5

4

Simplify

10 ÷ 5 = 2

ANSWER

2√5

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Number

Lesson 7 of 7

CASE STUDY

A4 Paper and the Square Root of 2

Every A-size sheet of paper - A3, A4, A5 - has its long side √2 times its short side. That is the one shape that stays exactly the same when you fold it in half: halve the long side of a sheet with sides 1 and √2 and you get sides √2/2 and 1, which is the same shape again. It is why an A4 page shrinks perfectly onto A5 on a photocopier. An A4 sheet is 210 mm by 297 mm, and 297 ÷ 210 = 1.414…, which is √2 to the nearest millimetre.

210 × 297 mm

The size of an A4 sheet

1 : √2

The ratio of the sides of every A-size sheet

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Folding A-Size Paper

Number

Lesson 7 of 7

Fold A3 in half and you get A4; fold A4 in half and you get A5.

Halving an A-size sheet always makes the next size down, with exactly the same shape. The ratio of the sides is 1 : √2 at every size - a surd you use every day.

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Key Terms

Number

Lesson 7 of 7

Rational number

A number that can be written as a fraction of two integers.

Irrational number

A number that cannot be written as a fraction; its decimal never ends or repeats.

Surd

A root that is irrational, such as √2 or √10.

Like surds

Multiples of the same surd, such as 5√2 and 3√2, which can be added and subtracted.

Simplest form

A surd with the largest possible square number taken out, e.g. 6√2.

Rationalise the denominator

Rewrite a fraction so there is no surd on the bottom.

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Number

Lesson 7 of 7

YOUR TASK

Surd Simplifying Race

12 minutes

Simplify each. (a) √12 (b) √45 (c) √200 (d) √8 + √32 (e) √75 − √27 (f) √3 × √15 (g) (1 + √3)² (h) 12/(√6)

1

Look for the largest square factor.

2

Simplify before adding or subtracting.

3

Rationalise any surd in the denominator.

WHAT A GOOD ANSWER SHOWS

(a) 2√3 (b) 3√5 (c) 10√2 (d) 2√2 + 4√2 = 6√2 (e) 5√3 − 3√3 = 2√3 (f) √45 = 3√5 (g) 1 + 2√3 + 3 = 4 + 2√3 (h) 12√6/6 = 2√6

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Can I...?

Number

Lesson 7 of 7

Explain what a surd is.

Say whether a number is rational or irrational.

Simplify a surd such as √72.

Multiply and divide surds.

Add and subtract like surds.

Expand brackets containing surds.

Rationalise a denominator like a/(√b).

Give an exact answer in surd form.

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Summary & Exam Focus

A surd is an irrational root, and gives an exact value.

√ab = √a × √b: take out the largest square factor to simplify.

Only like surds can be added or subtracted.

Rationalise a/(√b) by multiplying the top and the bottom by √b.

EXAM FOCUS

Rationalise the denominator of 12/(√3). Give your answer in its simplest form. (2 marks)

"Exact" or "surd form" means do not use a decimal. Leave the root in the answer and simplify it as far as it goes.

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Exam Practice: Surds

Number

Lesson 7 of 7

Answer all questions. Show your working. All questions are non-calculator. · 25 minutes

Question 1 · 1 mark · Non-calculator

Write √45 in the form k√5, where k is an integer.

Question 2 · 2 marks · Non-calculator

Rationalise the denominator of 12/(√3). Give your answer in its simplest form.

Question 3 · 3 marks · Non-calculator

Expand and simplify (2 + √3)(4 − √3). Give your answer in the form a + b√3, where a and b are integers.

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Exam Practice: Surds (cont.)

Number

Lesson 7 of 7

Question 4 · 3 marks · Non-calculator · Show that

Show that √98 + √8 = 9√2

Question 5 · 3 marks · Non-calculator

The diagram shows the right-angled triangle ABC. AB = √5 cm and BC = √11 cm. Work out the length of AC.

Question 6 · 3 marks · Non-calculator

A rectangle has length (3 + √5) cm and width (3 − √5) cm. Work out the area of the rectangle.

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Question 1 · 1 mark · Non-calculator

Number

Lesson 7 of 7

“

Write √45 in the form k√5, where k is an integer.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 1 mark, so plan before writing.

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Question 1 · mark scheme

Number

Lesson 7 of 7

1 mark available. Award a mark for each point made.

3√5

B1

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Question 2 · 2 marks · Non-calculator

Number

Lesson 7 of 7

“

Rationalise the denominator of 12/(√3). Give your answer in its simplest form.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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Question 2 · mark scheme

Number

Lesson 7 of 7

2 marks available. Award a mark for each point made.

Multiplying the top and the bottom by √3

M1

4√3

A1

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Question 3 · 3 marks · Non-calculator

Number

Lesson 7 of 7

“

Expand and simplify (2 + √3)(4 − √3). Give your answer in the form a + b√3, where a and b are integers.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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Question 3 · mark scheme

Number

Lesson 7 of 7

3 marks available. Award a mark for each point made.

At least 3 of the 4 terms correct: 8, −2√3, 4√3, −3

M1

All 4 terms correct

M1

5 + 2√3

A1

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Question 4 · 3 marks · Non-calculator · Show that

Number

Lesson 7 of 7

“

Show that √98 + √8 = 9√2

HOW TO ANSWER IT

Command word: Non-calculator · Show that. Worth 3 marks, so plan before writing.

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Question 4 · mark scheme

Number

Lesson 7 of 7

3 marks available. Award a mark for each point made.

√98 = 7√2 or √8 = 2√2

M1

Both simplified correctly

M1

Correctly added to reach 9√2

A1

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Question 5 · 3 marks · Non-calculator

Number

Lesson 7 of 7

The diagram shows the right-angled triangle ABC. AB = √5 cm and BC = √11 cm. Work out the length of AC. (3 marks)

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Question 5 · mark scheme

Number

Lesson 7 of 7

3 marks available. Award a mark for each point made.

(√5)² + (√11)² or 5 + 11

M1

√16

M1

4 (cm)

A1

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Question 6 · 3 marks · Non-calculator

Number

Lesson 7 of 7

“

A rectangle has length (3 + √5) cm and width (3 − √5) cm. Work out the area of the rectangle.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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Question 6 · mark scheme

Number

Lesson 7 of 7

3 marks available. Award a mark for each point made.

Area = (3 + √5)(3 − √5)

M1

At least 3 of the 4 terms correct: 9, −3√5, +3√5, −5

M1

4 (cm²)

A1

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