Mathematical Skills (Cambridge International IGCSE Geography)

Mathematical skills in geography

Spec point — 3.7.6

• Some aspects of geography require a range of mathematical skills

• These skills are used for:

  • Calculations
  • Analysing data

Percentages

• Percentages are frequently used in geography

  • 'Percent' simply means 'out of one hundred'

  • For example, 25 of 360 homes in a town were burgled. What is the percentage (to the nearest whole number)?

    • $25 \div 360 \times 100 = 6.94 [ to nearest whole number ] = 7 \%$

• Percentages can be used in many ways, for example, literacy rates or the numbers of people in different age groups in a country

• Percentages can be converted into fractions and back again

  • If 20% of the population cannot read or write, this is 1/5th of the population

Worked Example

• 28 of 180 homes in a village do not have internet access; convert into a fraction.

  • Simplify 28/180 by dividing by 4 = 7/45

• Calculate the fraction of homes to a percentage (to the nearest whole number)

  • 7 divided by 45 = 0.1555
  • Round to two decimal places 0.16
  • To turn into a percentage 0.16 x 100 = 16%
  • 16% of homes do not have internet access

Percentage change

• A percentage change shows by how much something has either increased or decreased

• A percentage change shows by how much something has either increased or decreased

• $Percentage change=\frac{final value-original value}{original value}\times100$

  • In 2020, 25 out of 360 homes in a town were burgled. In 2021, 21 houses were burgled. What is the percentage change?
  • $\frac{21 - 25}{25} \times 100 = - 16 \%$
  • There has been a decrease of 16% in the rate of burglaries in the town

• Do remember that a positive figure shows an increase but a negative is a decrease

Exam Hint Always check when making calculations what the question has asked you to do. Does the question ask you to state or calculate units to the nearest whole number or quote to two decimal places?

Rounding

• In the example above the answer to 7 divided by 45 is 0.1555

• In geography significant figures can be used to make the numbers easier to work with

• In the example 0.15555 was rounded up to 0.16 to make the calculations easier

• To round up:

  • Identify the digit in the required place value

  • Circle the number to the right of the required place value

    • If the circled number is 5 or more, then you round to the bigger number
    • If the circled number is less than 5, then you round to the smaller number
    • Put a zero in any following place values before the decimal point

Significant figures

• To find the first significant figure when reading from left to right, find the biggest place value that has a non-zero digit

  • The first significant figure of 3097 is 3

  • The first significant figure of 0.0062070 is 6

    • The zeros before the 6 are not significant
    • The zero after the 2 but before the 7 is significant
    • The zero **after **the 7 is not significant

• Count along to the** right from the first significant figure to identify** the position of the required significant figure

  • Do count zeros that are between other non-zero digits

    • E.g. 0 is the second significant figure of 3097
    • 9 is the third significant figure of 3097

  • Use the normal rules for rounding

  • For large numbers, complete places up to the decimal point with zeros

    • E.g. 34 568 to 2 significant figures is 35 000

  • For decimals, complete places between the decimal point and the first significant figure with zeros

    • E.g. 0.003 435 to 3 significant figures is 0.003 44

Proportion

Spec point — 3.7.7 · Candidates should be able to: • demonstrate an understanding of number (add, subtract, multiply and divide), area and scale • use standard notation, including positive and negative indices • understand and use significant figures • understand and use proportion, ratio, magnitude and frequency • understand and use mean, mode and median, range, decimals, fractions, percentages, and ratios • design fieldwork data collection sheets and collect data with an understanding of accuracy, sample size and procedures, control groups and reliability • complete, interpret and evaluate tables of data • identify and describe relationships between two or more sets of data • identify and interpret trends or relationships over time and draw line of best fit • make predictions, interpolate and extrapolate trends from data • identify weaknesses and limitations in statistical presentations of data • draw and justify conclusions from numerical and statistical data.

Proportion

• Direct proportion

  • As one quantity increases/decreases by a certain rate (factor)
  • The other quantity will increase/decrease by the same rate

• The ratio of the two quantities is constant

  • A map has a scale of 1:25,000 this means that 1 unit on the map (e.g. 1 cm) represents 25,000 units (25,000 cm or 0.25km) in real life

• An inverse proportion means that as one variable increases, the other decreases by a proportionate amount

  • As urban populations increase, rural populations decrease

Ratio

Spec point — 3.7.7 · Candidates should be able to: • demonstrate an understanding of number (add, subtract, multiply and divide), area and scale • use standard notation, including positive and negative indices • understand and use significant figures • understand and use proportion, ratio, magnitude and frequency • understand and use mean, mode and median, range, decimals, fractions, percentages, and ratios • design fieldwork data collection sheets and collect data with an understanding of accuracy, sample size and procedures, control groups and reliability • complete, interpret and evaluate tables of data • identify and describe relationships between two or more sets of data • identify and interpret trends or relationships over time and draw line of best fit • make predictions, interpolate and extrapolate trends from data • identify weaknesses and limitations in statistical presentations of data • draw and justify conclusions from numerical and statistical data.

Ratio

• A ratio is a way of comparing one part of a whole to another

  • Ratios are used to compare one part to another part

What do ratios look like?

• Ratios involve two or three different numbers separated using a colon

  • E.g. 2 : 5, 3 : 1, 4 : 2 : 3

• Dependency ratios compare the number of dependants (individuals aged 0-14 and over 65) to the working-age population (aged 15-64)

  • In the ratio 2:1, when referring to the dependency ratio this means that for every 2 working-age people there is one dependent person

Magnitude

Spec point — 3.7.7 · Candidates should be able to: • demonstrate an understanding of number (add, subtract, multiply and divide), area and scale • use standard notation, including positive and negative indices • understand and use significant figures • understand and use proportion, ratio, magnitude and frequency • understand and use mean, mode and median, range, decimals, fractions, percentages, and ratios • design fieldwork data collection sheets and collect data with an understanding of accuracy, sample size and procedures, control groups and reliability • complete, interpret and evaluate tables of data • identify and describe relationships between two or more sets of data • identify and interpret trends or relationships over time and draw line of best fit • make predictions, interpolate and extrapolate trends from data • identify weaknesses and limitations in statistical presentations of data • draw and justify conclusions from numerical and statistical data.

Magnitude

• In geography the term 'magnitude' has two meanings

• In mathematical skills it is the relative size or scale of a quantity when comparing different geographical data

  • For example, if Country A has a population of 40 million and Country B has a population of 10 million, we can say that Country A's population is four times greater than Country B's

    • This means Country A's population is greater by a magnitude of 4.

• Magnitude can also refer to the amount of energy released in an earthquake

Frequency

Spec point — 3.7.7 · Candidates should be able to: • demonstrate an understanding of number (add, subtract, multiply and divide), area and scale • use standard notation, including positive and negative indices • understand and use significant figures • understand and use proportion, ratio, magnitude and frequency • understand and use mean, mode and median, range, decimals, fractions, percentages, and ratios • design fieldwork data collection sheets and collect data with an understanding of accuracy, sample size and procedures, control groups and reliability • complete, interpret and evaluate tables of data • identify and describe relationships between two or more sets of data • identify and interpret trends or relationships over time and draw line of best fit • make predictions, interpolate and extrapolate trends from data • identify weaknesses and limitations in statistical presentations of data • draw and justify conclusions from numerical and statistical data.

Frequency

• Frequency refers to how often a particular value or category appears within a set of data

• In a traffic survey the number of times each type of vehicle (car, bus, bicycle) passes by is recorded

  • The count of each vehicle type represents its frequency.

• To organise and interpret this data effectively, geographers use frequency tables

• These tables list each category alongside its corresponding frequency, making it easier to identify patterns and trends

  • For example, a frequency table can help to identify the most common mode of transport used in a particular area

Mean, median, mode & range

Spec point — 3.7.7 · Candidates should be able to: • demonstrate an understanding of number (add, subtract, multiply and divide), area and scale • use standard notation, including positive and negative indices • understand and use significant figures • understand and use proportion, ratio, magnitude and frequency • understand and use mean, mode and median, range, decimals, fractions, percentages, and ratios • design fieldwork data collection sheets and collect data with an understanding of accuracy, sample size and procedures, control groups and reliability • complete, interpret and evaluate tables of data • identify and describe relationships between two or more sets of data • identify and interpret trends or relationships over time and draw line of best fit • make predictions, interpolate and extrapolate trends from data • identify weaknesses and limitations in statistical presentations of data • draw and justify conclusions from numerical and statistical data.

Mean, median, mode & range

Mean, median and mode

• These are measures of central tendency

  • Mean = average value

    • The mean is calculated by adding up all of the values in the data set and then dividing by the total number of values in the data set

  • The median is the middle value of a set of data. The numbers are arranged in rank order and then the middle value selected

  • The mode is the value which occurs most frequently in a set of data

Range

• A measure of dispersion: the spread of data around the average

  • Range is the distance between the highest and lowest value

• Interquartile range is the part of the range that covers the middle 50% of the data

Worked Example

Site	1	2	3	4	5	6	7
No. pebbles	184	90	159	142	64	64	95

• Mean:$\frac{184+90+159+142+64+64+95}{7}=\frac{798}{7}=114$
• Median: reordering by size =* *$64 64 90 [ 95 ] 142 159 184 = 95$ is the middle value
• Mode: only $64$ appears more than once
• Range =$184-64=120$