Formulae.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Formulae

Algebra

Lesson 4 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Algebra

Lesson 4 of 7

Answer each one, then check.

1

If x = 3, what is 2x + 5?

2

If y = −2, what is y²?

3

What is (−2)³?

4

Last lesson: solve 3x − 4 = 11.

5

Last lesson: solve 2(x + 3) = 14.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Algebra

Lesson 4 of 7

1

If x = 3, what is 2x + 5?

11

2

If y = −2, what is y²?

4

3

What is (−2)³?

−8

4

Last lesson: solve 3x − 4 = 11.

x = 5

5

Last lesson: solve 2(x + 3) = 14.

x = 4

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Algebra

Lesson 4 of 7

1

Tell the difference between an expression, an equation, a formula and an identity.

2

Substitute positive and negative numbers into formulae.

3

Write a formula from a description.

4

Change the subject of a formula.

5

Change the subject when the subject is squared or appears twice. (Higher)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Four Words, Four Meanings

Algebra

Lesson 4 of 7

Expression

Terms with no equals sign: 3x + 2. It cannot be solved.

Equation

Two expressions that are equal for particular values: 3x + 2 = 11 is only true when x = 3.

Formula

A rule linking quantities, each with its own letter: v = u + at, A = πr².

Identity

Two expressions equal for EVERY value of the letter, written with ≡: 2(x + 3) ≡ 2x + 6.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Substituting into a Formula

Algebra

Lesson 4 of 7

Replace each letter with its value, then follow the order of operations.

Use brackets

Put negative numbers in brackets: if a = −2, then a² = (−2)² = 4.

Powers first

In ½at², square t before multiplying.

Show the substitution

Write the formula with the numbers in before working it out: it earns the method mark.

Units

Give the answer in the right units if the question has them.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Substituting Negative Numbers

Algebra

Lesson 4 of 7

s = ut + ½at². Work out s when u = 3, t = 4 and a = −2.

1

Substitute

s = 3 × 4 + ½ × (−2) × 4²

2

Powers first

4² = 16

3

Multiply

3 × 4 = 12 and ½ × (−2) × 16 = −16

4

Add

12 + (−16) = −4

ANSWER

s = −4

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Area of a Trapezium

Algebra

Lesson 4 of 7

A = ½(a + b)h: add the parallel sides, halve, multiply by the height.

The area of a trapezium is A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height. With a = 5 cm, b = 9 cm and h = 4 cm: A = ½ × (5 + 9) × 4 = ½ × 14 × 4 = 28 cm².

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

PART ONE

Changing the Subject

Rearranging a formula to work out a different letter.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

What Is the Subject?

Algebra

Lesson 4 of 7

The subject of a formula is the letter on its own on one side: in v = u + at, the subject is v.

The idea

Rearrange the formula so a different letter is on its own.

The method

Exactly like solving an equation: undo what has been done to the new subject, in reverse order, doing the same to both sides.

Why

To find a when you know v, u and t, it is quicker to rearrange once than to solve a new equation every time.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Changing the Subject

Algebra

Lesson 4 of 7

Make t the subject of v = u + at.

1

Subtract u from both sides

v − u = at

2

Divide both sides by a

(v − u)/a = t

3

Write the subject on the left

t = (v − u)/a

ANSWER

t = (v − u)/a

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Rearranging the Trapezium Formula

Algebra

Lesson 4 of 7

Make h the subject of A = ½(a + b)h.

1

Multiply both sides by 2

2A = (a + b)h

2

Divide both sides by (a + b)

2A/(a + b) = h

ANSWER

h = 2A/(a + b)

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PART TWO · HIGHER

Harder Rearranging

When the subject is squared, or appears twice.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Subject Is Squared

Algebra

Lesson 4 of 7

Make r the subject of A = πr².

1

Divide both sides by π

A/π = r²

2

Square root both sides

√(A/π) = r

ANSWER

r = √(A/π)

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The Subject Appears Twice

Algebra

Lesson 4 of 7

Make x the subject of y = (x + 2)/(x − 3).

1

Multiply both sides by (x − 3)

y(x − 3) = x + 2

2

Expand

xy − 3y = x + 2

3

Collect the x terms on one side

xy − x = 3y + 2

4

Factorise out x

x(y − 1) = 3y + 2

5

Divide by (y − 1)

x = (3y + 2)/(y − 1)

ANSWER

x = (3y + 2)/(y − 1)

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Algebra

Lesson 4 of 7

CASE STUDY

Celsius and Fahrenheit

The Fahrenheit temperature scale, still used in the USA, is linked to Celsius by the formula F = 1.8C + 32. Rearranging it gives C = (F − 32)/1.8, so a weather forecast of 77 °F becomes (77 − 32)/1.8 = 25 °C. There is one temperature that reads the same on both scales: solve C = 1.8C + 32 and you get C = −40. So −40 °C and −40 °F are exactly the same temperature.

−40°

The one temperature that is the same on both scales

100 °C = 212 °F

The boiling point of water

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Key Terms

Algebra

Lesson 4 of 7

Formula

A rule linking two or more quantities, written with letters.

Substitute

Replace the letters in a formula with numbers.

Subject

The letter on its own on one side of a formula.

Change the subject

Rearrange a formula so a different letter is the subject.

Identity

Two expressions that are equal for every value of the letters, written with ≡.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Algebra

Lesson 4 of 7

YOUR TASK

Rearrange It

12 minutes

Make the letter in brackets the subject. (a) P = 4s (s) (b) y = mx + c (x) (c) C = 2πr (r) (d) V = IR (I) (e) A = ½bh (b). Higher: (f) E = ½mv² (v) (g) a(x − 2) = x + 5 (x).

1

Undo operations in reverse order.

2

Do the same to both sides.

3

Higher: collect the subject, then factorise.

WHAT A GOOD ANSWER SHOWS

(a) s = P/4 (b) x = (y − c)/m (c) r = C/2π (d) I = V/R (e) b = 2A/h (f) v = √(2E/m) (g) ax − 2a = x + 5, so ax − x = 5 + 2a, x(a − 1) = 5 + 2a and x = (5 + 2a)/(a − 1)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Algebra

Lesson 4 of 7

Tell a formula from an equation, expression and identity.

Substitute positive numbers into a formula.

Substitute negative numbers into a formula.

Write a formula from words.

Change the subject in one step.

Change the subject in two or more steps.

Change the subject when it is squared. (Higher)

Change the subject when it appears twice. (Higher)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Expression: no equals sign. Equation: true for some values. Formula: links quantities. Identity: true for all values.

Substitute: numbers in, brackets round negatives, powers first.

Change the subject: undo operations in reverse order, doing the same to both sides.

(Higher) Subject twice: collect those terms, factorise, then divide.

EXAM FOCUS

Make t the subject of the formula v = u + at (2 marks)

Write the substitution before calculating, and put negative numbers in brackets. Most lost marks on substitution come from squaring a negative number without brackets.

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