Cambridge Lower Secondary CheckpointStage 6

Algebra (6As)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Algebra — Sequences, functions and graphs
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Algebra (6As) notes

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In this chapter

  1. 1Letters as Unknown Numbers
  2. 2From Words to Algebra
  3. 3Simplifying by Collecting Like Terms
  4. 4Substitution: Finding the Value
  5. 5Solving Equations: Finding the Unknown

1. Letters as Unknown Numbers

In algebra, we use letters to stand in for numbers. Think of them as mystery boxes. If you have a box 'x' and someone gives you 3 more apples, you now have 'x + 3' apples. We don't know the exact number until we're told what 'x' is. These letters, like x, y, a, or n, are called variables because their value can change or vary.

a + 5 (a number 'a' plus 5)

y - 2 (a number 'y' minus 2)

3n (3 multiplied by a number 'n')

c / 4 (a number 'c' divided by 4)

Key term

Variable: A letter or symbol that represents an unknown number or a value that can change.

Fun fact

The word 'algebra' comes from the Arabic title of a 9th-century book, 'Al-Jabr wa'l-muqabala', written by the mathematician Al-Khwarizmi. 'Al-Jabr' means 'reunion of broken parts'.

Worked example 11 mark

Asha has 'p' pencils. She gives one to a friend. Write an expression for the number of pencils Asha has now.

  1. 1

    Start with the number of pencils Asha has: p.

  2. 2

    She gives one away, which means we need to subtract 1.

  3. 3

    The expression is p - 1.

Worked example 22 marks

If n = 10, what is the value of n + 5?

  1. 1

    The expression is n + 5.

  2. 2

    We are told that n = 10.

  3. 3

    Replace 'n' with 10: 10 + 5.

  4. 4

    Calculate the answer: 10 + 5 = 15.

Recap

  • Letters in algebra stand for unknown numbers and are called variables.
  • An expression is a mix of numbers, variables, and operation signs like +, -, ×, ÷.
  • We don't use a multiplication sign (×) in algebra to avoid confusion with the letter 'x'.
  • Writing a number next to a letter, like '4a', means multiplication (4 times a).

Quick check

  1. What does the expression '7b' mean?1 mark

2. From Words to Algebra

A key skill in algebra is turning word problems into algebraic expressions. You need to look for keywords that tell you which operation to use. For example, 'more than' or 'sum' means add (+). 'Less than' or 'difference' means subtract (-). 'Product' or 'times' means multiply. 'Shared' or 'quotient' means divide.

Key term

Expression: A mathematical phrase made up of numbers, variables, and operation signs, but no equals sign.

Common pitfall

A common mistake is mixing up '5 less than n' and '5 minus n'. '5 less than n' means you start with n and take 5 away, so it's 'n - 5'. '5 minus n' is '5 - n'.

Worked example 11 mark

I am 'y' years old. My brother is 3 years older than me. Write an expression for my brother's age.

  1. 1

    My age is y.

  2. 2

    My brother is '3 years older', which means we need to add 3.

  3. 3

    The expression for my brother's age is y + 3.

Worked example 22 marks

A cup of tea costs 't' pence. A cake costs 40 pence more than two cups of tea. Write an expression for the cost of the cake.

  1. 1

    First, find the cost of two cups of tea. One cup is 't', so two cups is 2 × t, which we write as 2t.

  2. 2

    The cake costs '40 pence more than' this amount.

  3. 3

    'More than' means we add 40.

  4. 4

    The final expression is 2t + 40.

Recap

  • 'More than' or 'sum' suggests addition (+).
  • 'Less than' or 'difference' suggests subtraction (-).
  • 'Times' or 'product' suggests multiplication.
  • 'Shared' or 'quotient' suggests division.
  • Remember to write the number before the variable, like '5x', not 'x5'.

Quick check

  1. Write an expression for 'a number n multiplied by 5, then 2 is subtracted'.1 mark

3. Simplifying by Collecting Like Terms

Simplifying an expression means making it shorter and tidier without changing its value. We do this by 'collecting like terms'. Like terms are terms that have the exact same variable part. For example, 3a and 5a are like terms, but 3a and 5b are not. Think of it like fruit: you can add 3 apples and 5 apples to get 8 apples, but you can't add 3 apples and 5 bananas to get 8 'apple-bananas'!

Key term

Like Terms: Terms that have the exact same variable part, which can be collected together through addition or subtraction.

Examiner insight

Examiners appreciate it when you show which terms you are grouping. You can do this by underlining or circling the like terms in the original expression before you combine them.

Common pitfall

Forgetting to take the sign in front of a term. In '7x - 4x', the second term is '-4x', not '4x'.

Worked example 12 marks

Simplify the expression: 5a + 2b + 3a + 4b

  1. 1

    Identify the like terms. The 'a' terms are 5a and 3a. The 'b' terms are 2b and 4b.

  2. 2

    Collect the 'a' terms: 5a + 3a = 8a.

  3. 3

    Collect the 'b' terms: 2b + 4b = 6b.

  4. 4

    Combine them into the final simplified expression: 8a + 6b.

Worked example 22 marks

Simplify: 7x + 8 - 4x - 3

  1. 1

    Identify the like terms. The 'x' terms are 7x and -4x. The number terms are +8 and -3.

  2. 2

    Be careful with the signs! Collect the 'x' terms: 7x - 4x = 3x.

  3. 3

    Collect the number terms: 8 - 3 = 5.

  4. 4

    The simplified expression is 3x + 5.

Recap

  • You can only add or subtract terms that are 'alike'.
  • Like terms must have the same letter.
  • When collecting terms, make sure you include the sign (+ or -) in front of the term.
  • Remember that a letter on its own, like 'b', is the same as '1b'.
  • Numbers on their own are also like terms and can be collected.

Quick check

  1. Simplify 10p + 4q - 3p + 5q.2 marks

4. Substitution: Finding the Value

Substitution is when we replace the letters in an expression or formula with specific numbers. Once you've swapped the letters for numbers, you just have a normal calculation to do. Remember to use the correct order of operations (BODMAS/PEMDAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction) to find the final value.

BODMAS/PEMDAS (Order of Operations)

Key term

Substitution: Replacing a variable in an algebraic expression with a specific numerical value.

Common pitfall

Ignoring the order of operations. In '3x + y' with x=4, y=5, some students might add 3+4 first. You must do the multiplication 3×4 before the addition.

Worked example 12 marks

If x = 4 and y = 5, find the value of the expression 3x + y.

  1. 1

    Write down the expression: 3x + y.

  2. 2

    Substitute x = 4 and y = 5 into the expression. Remember 3x means 3 × x.

  3. 3

    The expression becomes (3 × 4) + 5.

  4. 4

    Calculate the multiplication first (BODMAS): 12 + 5.

  5. 5

    Calculate the final answer: 12 + 5 = 17.

Worked example 22 marks

The formula for the area of a rectangle is A = lw. Find the area (A) if the length(l) is 8 cm and the width(w) is 5 cm.

  1. 1

    Write down the formula: A = lw.

  2. 2

    Substitute the given values: l = 8 and w = 5.

  3. 3

    A = 8 × 5.

  4. 4

    Calculate the answer: A = 40.

  5. 5

    Include the units in your final answer: The area is 40 cm².

Recap

  • Substitution means swapping letters for their given number values.
  • Rewrite the expression with the numbers in place of the letters.
  • Always follow the order of operations (BODMAS/PEMDAS) to calculate the answer.
  • Remember that a number next to a letter means multiplication.
  • If a question involves units (like cm), remember to include them in your answer.

Quick check

  1. If m = 6, what is the value of 10 - m?1 mark
  2. If a = 5, find the value of 4a + 2.2 marks

5. Solving Equations: Finding the Unknown

An equation is different from an expression because it has an equals sign (=). An equation is like a balanced scale. If you do something to one side, you must do the exact same thing to the other to keep it balanced. Our goal is to find the value of the variable that makes the equation true. We do this by getting the letter by itself on one side of the equals sign, using 'inverse' (opposite) operations.

To undo '+ a', you '- a'

To undo '- a', you '+ a'

To undo '× a', you '÷ a'

To undo '÷ a', you '× a'

Key term

Equation: A mathematical statement that two expressions are equal, indicated by an equals sign (=).

Examiner insight

Examiners want to see your method. Simply writing the answer might not get you full marks. Show each step of you balancing the equation clearly.

Common pitfall

In a two-step equation like 4y - 5 = 15, some students divide by 4 first. This is much harder. It's almost always easier to undo the addition or subtraction part first.

Worked example 12 marks

Solve the equation: x + 6 = 14.

  1. 1

    We want to get 'x' by itself. Currently, 6 is being added to it.

  2. 2

    The inverse of adding 6 is subtracting 6.

  3. 3

    Subtract 6 from both sides of the equation: x + 6 - 6 = 14 - 6.

  4. 4

    This simplifies to x = 8.

  5. 5

    Check your answer: 8 + 6 = 14. It's correct.

Worked example 23 marks

Solve the equation: 4y - 5 = 15.

  1. 1

    This is a two-step equation. Deal with the addition/subtraction first.

  2. 2

    To get rid of '- 5', do the inverse: add 5 to both sides. 4y - 5 + 5 = 15 + 5.

  3. 3

    This simplifies to 4y = 20.

  4. 4

    Now, to get 'y' by itself, we need to undo the '× 4'. The inverse is '÷ 4'.

  5. 5

    Divide both sides by 4: 4y / 4 = 20 / 4.

  6. 6

    This gives the final answer: y = 5.

Recap

  • An equation must always be balanced; what you do to one side, you do to the other.
  • To solve, use inverse operations to isolate the variable.
  • The inverse of + is -, and the inverse of × is ÷.
  • In two-step equations, deal with any adding or subtracting first, then the multiplying or dividing.
  • You can always check your answer by substituting it back into the original equation.

Quick check

  1. Solve for n: 3n = 18.1 mark
  2. What is the first step to solve the equation 2a + 7 = 13?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write an expression for the following: 'A number k is divided by 4, and then 3 is added'.2 marks
  2. Simplify the expression: 8x + 3y - 5x + 7y.2 marks
  3. Solve the equation: a - 9 = 11.1 mark
  4. If p = 5 and q = 3, find the value of 4p + 2q.2 marks
  5. Solve the equation: 5n + 4 = 29.2 marks
  6. A shop sells bags of marbles. Each bag contains 'm' marbles. I buy 3 bags and find a loose marble on the floor. Write an expression for the total number of marbles I have.2 marks
  7. Simplify fully: 9 + 4a + 2b - 6 + a - b.3 marks
  8. I think of a number, x. I multiply it by 3 and then subtract 5. The result is 16. Write an equation to represent this and solve it to find my number.3 marks
  9. The formula for the cost (C) in pounds of hiring a carpet cleaner is C = 15 + 8h, where 'h' is the number of hours it is hired for. How much would it cost to hire the cleaner for 4 hours?2 marks
  10. An expression is written as 10 - 2x. What is the value of the expression when x = 3? What about when x = 5?3 marks

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