Cambridge Lower Secondary CheckpointStage 7

Number (7Np)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Number - Place value, ordering and rounding
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Number (7Np) notes

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In this chapter

  1. 1Working with Integers
  2. 2Factors, Multiples, HCF and LCM
  3. 3Squares, Cubes and Their Roots
  4. 4Order of Operations (BIDMAS)
  5. 5Converting Between Forms
  6. 6Percentage Calculations

1. Working with Integers

Integers are whole numbers, which can be positive, negative, or zero (e.g., ...-3, -2, -1, 0, 1, 2, 3...). Understanding how to work with them is a fundamental skill. When adding and subtracting, a number line is a great visual tool. For multiplication and division, a simple set of rules for signs will guide you.

a - (-b) = a + b

(+) × (+) = (+)

(-) × (-) = (+)

(+) × (-) = (-)

(-) × (+) = (-)

(+) ÷ (+) = (+)

(-) ÷ (-) = (+)

(+) ÷ (-) = (-)

(-) ÷ (+) = (-)

Key term

Integer: A whole number that can be positive, negative, or zero.

Examiner insight

Examiners look for correct application of the 'add the opposite' rule for subtracting negative numbers. Clearly writing 5 - (-3) = 5 + 3 = 8 shows your understanding.

Common pitfall

Confusing the rules for addition with multiplication. For example, thinking that -3 + -4 is +7 because 'two negatives make a positive'. The correct answer is -7.

Worked example 13 marks

Calculate:a) -6 + 11b) 4 - 9c) 5 - (-3)

  1. 1

    a) Start at -6 on a number line. Move 11 places in the positive direction (to the right). You will land on 5. So, -6 + 11 = 5.

  2. 2

    b) Start at 4 on a number line. Move 9 places in the negative direction (to the left). You will land on -5. So, 4 - 9 = -5.

  3. 3

    c) Subtracting a negative number is the same as adding its positive counterpart. So, 5 - (-3) becomes 5 + 3. 5 + 3 = 8.

Worked example 22 marks

Calculate:a) -7 × -8b) 48 ÷ -6

  1. 1

    a) First, multiply the numbers: 7 × 8 = 56. Then, apply the sign rule: a negative times a negative gives a positive. So, -7 × -8 = 56.

  2. 2

    b) First, divide the numbers: 48 ÷ 6 = 8. Then, apply the sign rule: a positive divided by a negative gives a negative. So, 48 ÷ -6 = -8.

Recap

  • Integers are all positive and negative whole numbers, including zero.
  • Use a number line to visualise adding (move right) and subtracting (move left).
  • Subtracting a negative is the same as adding a positive (e.g., 2 - (-5) = 2 + 5).
  • For multiplication/division, if the signs are the same the result is positive.
  • For multiplication/division, if the signs are different the result is negative.

Quick check

  1. What is the value of -15 - (-8)?1 mark
  2. Calculate -5 × 12.1 mark

2. Factors, Multiples, HCF and LCM

Factors are numbers that divide exactly into another number. Multiples are found by multiplying a number by an integer. The Highest Common Factor (HCF) is the largest number that is a factor of two or more numbers. The Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. Prime factorisation is a powerful tool for finding the HCF and LCM of larger numbers.

Key term

Prime Factorisation: The process of writing a number as a product of its prime factors.

Examiner insight

Clear working using prime factor trees and Venn diagrams is often rewarded as it demonstrates a systematic and reliable method.

Common pitfall

Mixing up the methods for finding HCF and LCM from a prime factor Venn diagram. Remember HCF is the intersection (multiply numbers in the middle) and LCM is the union (multiply all numbers in the circles).

Worked example 14 marks

Find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of 24 and 36.

  1. 1

    Step 1: Find the prime factors of each number using factor trees. 24 = 2 × 2 × 2 × 3. 36 = 2 × 2 × 3 × 3.

  2. 2

    Step 2: Draw a Venn diagram. The intersection will contain the common prime factors. Both numbers share two 2s and one 3.

  3. 3

    Step 3: Place the shared factors (2, 2, 3) in the intersection. Place the remaining factors in the correct circles (an extra 2 for 24, an extra 3 for 36).

  4. 4

    Step 4: To find the HCF, multiply the numbers in the intersection: HCF = 2 × 2 × 3 = 12.

  5. 5

    Step 5: To find the LCM, multiply all the numbers in the Venn diagram: LCM = 2 × (2 × 2 × 3) × 3 = 72.

Recap

  • A factor divides a number exactly, with no remainder.
  • A multiple is the result of multiplying a number by an integer.
  • HCF is the biggest factor shared by two or more numbers.
  • LCM is the smallest multiple shared by two or more numbers.
  • Use prime factorisation and Venn diagrams to find the HCF and LCM of larger numbers.
  • HCF is the product of common prime factors (the intersection of the Venn diagram).
  • LCM is the product of all prime factors from both numbers (the union of the Venn diagram).

Quick check

  1. List all the factors of 30.2 marks
  2. What is the LCM of 9 and 12?1 mark

3. Squares, Cubes and Their Roots

Squaring a number means multiplying it by itself (e.g., 5² = 5 × 5 = 25). Cubing a number means multiplying it by itself twice (e.g., 5³ = 5 × 5 × 5 = 125). Square roots and cube roots are the inverse operations. The square root of a number is a value that, when multiplied by itself, gives the original number (e.g., √25 = 5). The cube root is the value that gives the original number when cubed (e.g., ∛125 = 5).

a² = a × a

a³ = a × a × a

√x = a, where a × a = x

∛x = a, where a × a × a = x

Key term

Inverse Operation: An operation that reverses the effect of another operation, such as square root being the inverse of squaring.

Common pitfall

Mistaking 'squaring' for 'multiplying by 2', or 'cubing' for 'multiplying by 3'. For example, calculating 5² as 10 instead of 25.

Fun fact

The sum of the first 'n' odd numbers is always a perfect square, n². For example, 1 + 3 + 5 + 7 = 16, which is 4².

Worked example 12 marks

Calculate the value ofa) 13²b) 4³

  1. 1

    a) 13² means 13 multiplied by itself. 13 × 13 = 169.

  2. 2

    b) 4³ means 4 multiplied by itself twice. 4 × 4 × 4 = 16 × 4 = 64.

Worked example 22 marks

Work out the value of √81 + ∛27.

  1. 1

    Step 1: Find the square root of 81. We need a number that multiplies by itself to make 81. 9 × 9 = 81, so √81 = 9.

  2. 2

    Step 2: Find the cube root of 27. We need a number that cubes to make 27. 3 × 3 × 3 = 27, so ∛27 = 3.

  3. 3

    Step 3: Add the results together. 9 + 3 = 12.

Recap

  • To square a number, multiply it by itself.
  • To cube a number, multiply it by itself, and then by itself again.
  • A square root is the inverse of squaring a number.
  • A cube root is the inverse of cubing a number.
  • It is useful to memorise squares up to 15² and cubes up to 5³.

Quick check

  1. What is the value of 15²?1 mark
  2. Find ∛64.1 mark

4. Order of Operations (BIDMAS)

When a calculation has multiple operations, we need a standard order to follow so that everyone gets the same answer. This order is often remembered by the acronym BIDMAS (or BODMAS). It stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction. It's crucial to remember that Division and Multiplication have equal priority (work left to right), and so do Addition and Subtraction.

Key term

BIDMAS: An acronym for the order of operations: Brackets, Indices, Division, Multiplication, Addition, Subtraction.

Examiner insight

Examiners often use questions with multiple operations to test BIDMAS. Showing each step of simplification clearly, line by line, can earn partial marks even if the final answer is wrong.

Common pitfall

Incorrectly applying BIDMAS by always doing division before multiplication, or addition before subtraction, instead of working from left to right for operations with equal priority. For example, in 10 ÷ 2 × 5, the correct answer is 25 (10÷2=5, then 5×5=25), not 1 (2×5=10, then 10÷10=1).

Worked example 12 marks

Calculate 7 + (18 - 3) ÷ 5.

  1. 1

    Step 1 (Brackets): First, calculate the value inside the brackets. (18 - 3) = 15.

  2. 2

    The calculation becomes: 7 + 15 ÷ 5.

  3. 3

    Step 2 (Division): Next, perform the division. 15 ÷ 5 = 3.

  4. 4

    The calculation becomes: 7 + 3.

  5. 5

    Step 3 (Addition): Finally, perform the addition. 7 + 3 = 10.

Worked example 22 marks

Calculate 5 × 4² - 8.

  1. 1

    Step 1 (Indices): There are no brackets, so we do indices first. 4² = 16.

  2. 2

    The calculation becomes: 5 × 16 - 8.

  3. 3

    Step 2 (Multiplication): Next, perform the multiplication. 5 × 16 = 80.

  4. 4

    The calculation becomes: 80 - 8.

  5. 5

    Step 3 (Subtraction): Finally, perform the subtraction. 80 - 8 = 72.

Recap

  • Follow the BIDMAS order: Brackets, Indices, Division/Multiplication, Addition/Subtraction.
  • Brackets are always the highest priority.
  • Indices (powers and roots) are second.
  • Division and Multiplication have equal priority; work from left to right.
  • Addition and Subtraction have equal priority; work from left to right.

Quick check

  1. Calculate 20 - 6 × 2.1 mark
  2. Calculate (3 + 5)² ÷ 4.2 marks

5. Converting Between Forms

Fractions, decimals, and percentages are three different ways of showing the same value. Being able to convert fluently between them is essential. For example, 1/2 is the same as 0.5, which is the same as 50%. The methods are simple and just require practice.

Fraction to Decimal: Divide the numerator by the denominator.

Decimal to Percentage: Multiply by 100.

Percentage to Decimal: Divide by 100.

Percentage to Fraction: Write the percentage over 100 and simplify.

Decimal to Fraction: Use place value (e.g., 0.75 = 75/100) and simplify.

Key term

Equivalence: The state of being equal in value or amount, even if expressed in a different form.

Fun fact

The '%' symbol evolved from an Italian term 'per cento', meaning 'for a hundred'. Early scribes wrote 'p c', which over time morphed into the symbol we use today.

Worked example 12 marks

Convert the fraction 3/8 into a decimal and a percentage.

  1. 1

    Step 1 (Fraction to Decimal): Divide the numerator by the denominator. 3 ÷ 8 = 0.375.

  2. 2

    Step 2 (Decimal to Percentage): Multiply the decimal by 100. 0.375 × 100 = 37.5%.

  3. 3

    So, 3/8 = 0.375 = 37.5%.

Worked example 23 marks

Convert 65% into a decimal and a simplified fraction.

  1. 1

    Step 1 (Percentage to Decimal): Divide the percentage by 100. 65 ÷ 100 = 0.65.

  2. 2

    Step 2 (Percentage to Fraction): Write the percentage value over 100. 65/100.

  3. 3

    Step 3 (Simplify Fraction): Find the highest common factor of 65 and 100, which is 5. Divide the numerator and denominator by 5. 65 ÷ 5 = 13; 100 ÷ 5 = 20. The simplified fraction is 13/20.

  4. 4

    So, 65% = 0.65 = 13/20.

Recap

  • Fractions, decimals and percentages are different ways to show parts of a whole.
  • To convert a fraction to a decimal, divide the top by the bottom.
  • To convert a decimal to a percentage, multiply by 100.
  • To convert a percentage to a decimal, divide by 100.
  • To convert a percentage to a fraction, put it over 100 and simplify.
  • Memorise key equivalences: 1/2=0.5=50%, 1/4=0.25=25%, 1/10=0.1=10%.

Quick check

  1. Write 0.4 as a percentage.1 mark
  2. Write 3/5 as a decimal.1 mark

6. Percentage Calculations

Percentages are used everywhere, from shopping discounts to test scores. The two most common calculations are finding a percentage of an amount, and expressing one number as a percentage of another. A key tip is to remember that 'of' in mathematics usually means 'multiply'.

Percentage of an Amount = (Percentage / 100) × Amount

Express A as a percentage of B = (A / B) × 100

Key term

Percentage: A number or ratio expressed as a fraction of 100.

Examiner insight

For 'percentage of an amount' questions, using a decimal multiplier (e.g., multiplying by 0.35 for 35%) is efficient and less prone to error than 'find 10% then 1%' methods, especially with a calculator.

Common pitfall

When expressing A as a percentage of B, students sometimes divide the wrong way around (B/A). Always put the 'part' over the 'whole total'.

Worked example 13 marks

A t-shirt costing £40 is in a sale with 30% off. How much is the discount and what is the sale price?

  1. 1

    Step 1: Find the discount amount. We need to calculate 30% of £40.

  2. 2

    Convert 30% to a decimal: 30 / 100 = 0.3.

  3. 3

    Multiply by the amount: 0.3 × £40 = £12. The discount is £12.

  4. 4

    Step 2: Find the sale price. Subtract the discount from the original price.

  5. 5

    £40 - £12 = £28. The sale price is £28.

Worked example 22 marks

Ayan scores 39 marks out of 60 in a science test. What is his score as a percentage?

  1. 1

    Step 1: Write the score as a fraction: 39 out of 60 is 39/60.

  2. 2

    Step 2: To turn this fraction into a percentage, multiply by 100.

  3. 3

    Calculation: (39 / 60) × 100.

  4. 4

    Step 3: Simplify the calculation. (39 / 60) = 0.65.

  5. 5

    Step 4: 0.65 × 100 = 65. Ayan's score is 65%.

Recap

  • To find a percentage of an amount, convert the percentage to a decimal or fraction and multiply.
  • The word 'of' means multiply.
  • To express a value as a percentage of a total, use the formula (Value / Total) × 100.
  • For non-calculator work, use building blocks like 10% (divide by 10) and 1% (divide by 100).
  • For calculator work, the quickest method is to use a decimal multiplier (e.g., for 45%, multiply by 0.45).

Quick check

  1. Find 15% of 200.1 mark
  2. Express 12 as a percentage of 20.1 mark

End-of-chapter exercise

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

  1. Calculate the value of (-5) + (-10) - (-4).2 marks
  2. Find the Highest Common Factor (HCF) of 42 and 70.2 marks
  3. Work out the value of 30 - 2 × 3².3 marks
  4. Arrange these values in order from smallest to largest: 0.65, 2/3, 60%. Show your working.3 marks
  5. A cinema has 250 seats. On Tuesday evening, 80% of the seats were sold. How many seats were empty?3 marks
  6. In a bag of 40 sweets, 14 are red. What percentage of the sweets are red?2 marks
  7. Two buses, Route A and Route B, leave a bus station at 8:00 am. Route A takes 12 minutes to complete its circuit and Route B takes 15 minutes. At what time will they next arrive back at the station at the same time?3 marks
  8. Place brackets in the following calculation to make it correct: 5 + 4 × 10 - 7 = 272 marks
  9. Calculate √121 + ∛1000 ÷ 5.4 marks
  10. A number 'n' is a whole number such that 50 < n < 70. 'n' is a multiple of 3 and a factor of 126. Find the value of 'n'.3 marks

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