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Algebra (7As)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Algebra - Sequences, functions and graphs
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Algebra (7As) notes

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In this chapter

  1. 1Constructing Algebraic Expressions
  2. 2Simplifying by Collecting Like Terms
  3. 3Expanding Single Brackets
  4. 4Substitution into Expressions and Formulae

1. Constructing Algebraic Expressions

Algebra uses letters, called variables, to represent unknown numbers. Constructing an expression means turning a real-life situation or a description in words into a mathematical statement using these variables. For example, if we don't know how many sweets are in a jar, we can just call the number 's'. If someone adds 5 more sweets, the new amount is 's + 5'. We don't need to know the actual number to describe what happened.

Addition: 'a more than b' is b + a

Subtraction: 'a less than b' is b - a

Multiplication: 'a times b' is ab

Division: 'a divided by b' is a/b

Key term

Expression: A mathematical phrase that combines numbers, variables, and operation symbols, but does not contain an equals sign.

Examiner insight

Examiners want to see that you can correctly translate words into algebraic symbols. Clearly define any letters you choose if they are not given in the question.

Common pitfall

A common mistake is writing subtraction in the wrong order. For example, writing '10 less than x' as '10 - x' instead of the correct 'x - 10'.

Fun fact

The word 'algebra' comes from the Arabic 'al-jabr', which appears in the title of a 9th-century book by the mathematician Al-Khwarizmi and means 'the reunion of broken parts'.

Worked example 13 marks

Yusuf has a bag with 'n' marbles. Write an expression for the number of marbles he has if:a) He adds 7 more marbles.b) He loses 3 marbles.c) He doubles the number of marbles.

  1. 1

    a) 'Adds 7 more' means we use addition. The expression is n + 7.

  2. 2

    b) 'Loses 3' means we use subtraction. The expression is n - 3.

  3. 3

    c) 'Doubles' means multiplying by 2. The expression is 2 × n, which we write as 2n.

Worked example 22 marks

The cost of a drink is 'd' pounds and the cost of a sandwich is 's' pounds. Write an expression for the total cost of 2 drinks and 1 sandwich.

  1. 1

    The cost of 2 drinks is 2 multiplied by the cost of one drink, which is 2 × d or 2d.

  2. 2

    The cost of 1 sandwich is just s.

  3. 3

    The total cost is the sum of the cost of the drinks and the sandwich.

  4. 4

    Total cost = 2d + s.

Recap

  • A letter (variable) is used to stand for a number you don't know.
  • An expression is a mix of numbers and letters connected by operations like +, -, ×, ÷.
  • Key words give clues: 'sum' or 'more than' means add, 'difference' or 'less than' means subtract.
  • 'Product' or 'times' means multiply, and 'quotient' or 'share' means divide.
  • In algebra, we usually omit the multiplication sign, writing 4 × y as 4y.
  • The order is important for subtraction: '5 less than x' is x - 5, not 5 - x.

Quick check

  1. A book has 'p' pages. Write an expression for the number of pages in 5 of these books.1 mark
  2. Write an expression for '4 less than y'.1 mark

2. Simplifying by Collecting Like Terms

Simplifying an expression means making it shorter and easier to work with, without changing its value. One way to do this is by 'collecting like terms'. Like terms are terms that have the exact same variable part (the same letters raised to the same powers). For example, 3x and 5x are like terms, but 3x and 5y are not. Think of it like sorting fruit: you can count all your apples together and all your bananas together, but you can't add apples and bananas to get a single number of 'applenanas'. So, 3x + 5x simplifies to 8x, but 3x + 5y cannot be simplified further.

General Rule: ax + bx = (a + b)x

Key term

Like Terms: Terms in an algebraic expression that have the same variables raised to the same powers.

Examiner insight

Examiners reward clear working. Showing how you have grouped the like terms, for example by rewriting the expression as (5a + 3a) + (2b + 4b), can earn you marks even if you make a small error in the final calculation.

Common pitfall

Forgetting to include the sign in front of a term when rearranging. In '8a + 6b - 4a - 5b', the term is '-4a', not '4a'. If you forget the minus sign, your answer will be wrong.

Worked example 12 marks

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

  1. 1

    Step 1: Identify the like terms. The 'a' terms are 5a and +3a. The 'b' terms are +2b and +4b.

  2. 2

    Step 2: Group the like terms together. (5a + 3a) + (2b + 4b).

  3. 3

    Step 3: Combine the coefficients of the like terms. 5 + 3 = 8, so we have 8a. 2 + 4 = 6, so we have 6b.

  4. 4

    Step 4: Write the final simplified expression: 8a + 6b.

Worked example 22 marks

Simplify the expression: 7x + 5y - 3x - y

  1. 1

    Step 1: Identify the like terms, making sure to include the sign in front of each term. The 'x' terms are 7x and -3x. The 'y' terms are +5y and -y.

  2. 2

    Step 2: Remember that '-y' is the same as '-1y'.

  3. 3

    Step 3: Combine the 'x' terms: 7x - 3x = 4x.

  4. 4

    Step 4: Combine the 'y' terms: 5y - 1y = 4y.

  5. 5

    Step 5: Write the final simplified expression: 4x + 4y.

Recap

  • You can only add or subtract terms that are 'like'.
  • Like terms must have the exact same letter or combination of letters.
  • The number in front of the letter is called the coefficient.
  • When collecting terms, always include the sign (+ or -) that is in front of the term.
  • A letter on its own, like 'x', has a coefficient of 1.
  • It can be helpful to circle, box, or underline like terms to keep track of them.

Quick check

  1. Simplify 10p + 4q - 3p + 5q.2 marks
  2. Are 3xy and 5yx like terms? Explain why.1 mark

3. Expanding Single Brackets

Expanding brackets means to remove them from an expression. We do this by multiplying the term on the outside of the bracket by every single term on the inside of the bracket. A helpful way to remember this is the 'claw method', where you draw lines from the outside term to each inside term to remind you to multiply them all.

a(b + c) = ab + ac

a(b - c) = ab - ac

Key term

Expand: To multiply out the terms in brackets to remove them from an expression.

Common pitfall

The most frequent error is only multiplying the outside term by the first term in the bracket. For example, writing 4(x + 5) as 4x + 5, forgetting to also multiply the 4 by the 5.

Fun fact

Expanding brackets is based on the distributive law of arithmetic, a fundamental rule that connects multiplication and addition. You use it all the time without thinking, for example when calculating 7 × 102 in your head as (7 × 100) + (7 × 2).

Worked example 11 mark

Expand the expression 5(y + 3).

  1. 1

    Step 1: The term outside the bracket is 5. The terms inside are y and +3.

  2. 2

    Step 2: Multiply the outside term by the first term inside: 5 × y = 5y.

  3. 3

    Step 3: Multiply the outside term by the second term inside: 5 × 3 = 15.

  4. 4

    Step 4: Combine the results: 5y + 15.

Worked example 23 marks

Expand and simplify 4(2x - 1) + 3x.

  1. 1

    Step 1: First, expand the bracket. Multiply 4 by each term inside (2x and -1).

  2. 2

    Step 2: 4 × 2x = 8x.

  3. 3

    Step 3: 4 × (-1) = -4.

  4. 4

    Step 4: The expanded part is 8x - 4. Now rewrite the whole expression: 8x - 4 + 3x.

  5. 5

    Step 5: Simplify by collecting like terms. The like terms are 8x and +3x.

  6. 6

    Step 6: 8x + 3x = 11x. The '-4' has no like terms.

  7. 7

    Step 7: The final simplified expression is 11x - 4.

Recap

  • To expand a bracket, multiply the term outside by everything inside.
  • Be very careful with negative signs.
  • A positive times a negative gives a negative result (e.g. 3 × -2 = -6).
  • After expanding, always check if you can simplify the resulting expression by collecting like terms.
  • The grid method can be a useful visual aid for expanding brackets.

Quick check

  1. Expand 7(m - 2).1 mark
  2. Expand 3a(b + 4).1 mark

4. Substitution into Expressions and Formulae

Substitution is the process of replacing the letters (variables) in an expression or formula with specific numbers. Once you've replaced the letters, you just have a numerical calculation to perform. It's crucial to follow the order of operations (BIDMAS/BODMAS) to get the correct answer. BIDMAS stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction.

BIDMAS/BODMAS: The order of operations to perform calculations.

Key term

Substitution: The process of replacing variables in an algebraic expression or formula with their given numerical values.

Examiner insight

Examiners award marks for showing the correct substitution step. Even if your final answer is wrong due to a calculation error, you can still get marks for writing out the expression with the numbers in place, like '3(4) + 2(5)'.

Common pitfall

Incorrectly handling negative numbers, especially when squaring. For example, calculating -3² as -9 instead of the correct (-3)² = 9. Using brackets when substituting, as in (-3)², helps prevent this.

Worked example 12 marks

If x = 4 and y = 5, find the value of the expression 3x + 2y.

  1. 1

    Step 1: Write down the expression: 3x + 2y.

  2. 2

    Step 2: Substitute the given values. Replace 'x' with 4 and 'y' with 5. It's good practice to use brackets: 3(4) + 2(5).

  3. 3

    Step 3: Perform the multiplications first (following BIDMAS): 3 × 4 = 12 and 2 × 5 = 10.

  4. 4

    Step 4: The expression becomes 12 + 10.

  5. 5

    Step 5: Perform the addition: 12 + 10 = 22.

Worked example 22 marks

The formula to convert temperature from Celsius (C) to Fahrenheit (F) is F = 1.8C + 32. What is 20°C in Fahrenheit?

  1. 1

    Step 1: Write down the formula: F = 1.8C + 32.

  2. 2

    Step 2: Substitute the value C = 20 into the formula: F = 1.8(20) + 32.

  3. 3

    Step 3: Calculate the multiplication first: 1.8 × 20 = 36.

  4. 4

    Step 4: The formula becomes F = 36 + 32.

  5. 5

    Step 5: Perform the addition: F = 68.

  6. 6

    Step 6: State the final answer with units: 20°C is 68°F.

Worked example 33 marks

Find the value of 5p - q² when p = 10 and q = -3.

  1. 1

    Step 1: Write the expression: 5p - q².

  2. 2

    Step 2: Substitute p=10 and q=-3. Use brackets, especially for the negative number: 5(10) - (-3)².

  3. 3

    Step 3: Follow BIDMAS. Indices first: (-3)² = (-3) × (-3) = 9. A negative times a negative is a positive.

  4. 4

    Step 4: The expression is now 5(10) - 9.

  5. 5

    Step 5: Next, multiplication: 5(10) = 50.

  6. 6

    Step 6: The expression is now 50 - 9.

  7. 7

    Step 7: Finally, subtraction: 50 - 9 = 41.

Recap

  • Substitution means 'replace the letter with its number'.
  • Always follow the order of operations: BIDMAS/BODMAS.
  • Using brackets when you substitute can help avoid mistakes with signs and powers.
  • Remember that 5x means 5 times x.
  • If a question involves a real-life formula (like area or temperature), give your answer in the correct context with units.

Quick check

  1. If a = 6 and b = 2, what is the value of 4a + b?1 mark
  2. Find the value of 10 - 3x when x = -2.2 marks

End-of-chapter exercise

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

  1. The price of an apple is 'a' pence and the price of a banana is 'b' pence. Write an expression for the total cost of 5 apples and 3 bananas.2 marks
  2. Simplify the expression: 8x + 3y - 5x + 2y.2 marks
  3. Expand 6(2m - 3).2 marks
  4. If p = 5 and q = 3, find the value of the expression 4p - 2q.2 marks
  5. A rectangle has a length of (x + 5) cm and a width of 4 cm. Write a simplified expression for its perimeter.3 marks
  6. Simplify fully: 9ab - 2c + 4ba - 5c.3 marks
  7. Expand and simplify: 3(y + 4) + 2(y - 5).3 marks
  8. Find the value of the expression 2a² + 3b when a = -5 and b = 10.3 marks
  9. In an algebraic pyramid, the expression in each block is found by adding the expressions in the two blocks below it. The expressions in the bottom blocks are (x - 1), (2x + 3), and (5 - x). Find the expression in the top block.4 marks
  10. Zara thinks of a number, n. She multiplies it by 5 and then subtracts 7. The result is 23. Construct an equation and solve it to find the number Zara thought of.3 marks

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