Patterns, and data handling — tables, graphs and interpretation

What this heading covers

Two topics the syllabus names together and the papers ask separately: finding the rule behind a sequence, and reading what a table or graph is telling you.

Neither needs much formula. Patterns need one method applied patiently; data handling needs one step performed in the right order.

Mensuration and measurement have a lesson of their own.

Patterns

The question is always what comes next or what is the rule, and the method is the same: take the difference between consecutive terms. If that is not constant, take the difference of the differences.

• A constant first difference means the rule is linear — the same amount is added each time. 2, 5, 8, 11 … differences 3, 3, 3 → add 3.
• A constant second difference means it is quadratic, built on squares. 1, 4, 9, 16 … first differences 3, 5, 7; second differences 2, 2.
• A constant ratio rather than a constant difference means it is geometric — each term multiplied by the same number. 3, 6, 12, 24 … × 2 each time.

Try differences first, then ratios. Most school patterns are one or the other, and the few that are neither are the named sequences below.

Sequences worth recognising on sight

Sequence	Rule
1, 3, 6, 10, 15 …	Triangular numbers — differences 2, 3, 4, 5; the nth is n(n+1)/2
1, 4, 9, 16, 25 …	Square numbers — also the running totals of the odd numbers
1, 8, 27, 64 …	Cube numbers
1, 1, 2, 3, 5, 8, 13 …	Fibonacci — each term the sum of the two before it
2, 3, 5, 7, 11, 13 …	Primes — no rule of difference at all
1, 3, 5, 7 … and 2, 4, 6, 8 …	Odd and even

Three identities the paper uses directly:

• The sum of the first n natural numbers is n(n + 1)/2.
• The sum of the first n odd numbers is n² — 1, 1+3 = 4, 1+3+5 = 9.
• The sum of the first n even numbers is n(n + 1).

Fibonacci is the one the difference method cannot find, which is why it is worth knowing by sight.

Shape patterns

A shape pattern usually hides a number pattern. Count the matchsticks, tiles or dots at each stage, write the counts in a row, and the sequence appears — then treat it as above.

A standard example: a row of n squares made of matchsticks needs 3n + 1 sticks, because the first square takes 4 and every square after it adds 3. The question asks for stage 10 or stage 50 precisely so that drawing it is not an option and the rule must be found.

Why patterns are in the syllabus at all

Because they are where generalisation starts. A child who sees that "add 3 each time" describes an unlimited sequence is doing algebra before meeting a letter, and the pedagogy papers treat patterns as the bridge from arithmetic to algebra.

Data handling

The words first

• Data — collected facts or figures. Raw data is as collected; organised data has been arranged.
• Primary data is collected by the investigator; secondary data comes from someone else's records.
• Frequency — how many times a value occurs. A frequency table lists values against their frequencies.
• Range — the largest value minus the smallest.
• Class interval — a group such as 10–20 when values are grouped rather than listed singly.

Tally marks

Group in fives, the fifth struck across the other four, which makes a block of five countable at a glance. Counting tallies in fours is a standard wrong option.

The graphs

Graph	Shows	Watch for
Pictograph	Counts, using a symbol	Read the key first — the symbol rarely stands for one, and half-symbols are common
Bar graph	Comparison between categories	Where the axis starts. A bar twice as tall is twice the value only if the axis starts at zero. Bars are separated by gaps
Histogram	Frequency over continuous class intervals	Bars touch, because the intervals are continuous — this is the difference from a bar graph
Pie chart	Parts of a whole	The whole circle is 360°
Line graph	Change over time	The trend, not the single points

Pie chart arithmetic, which is asked both ways:

• A category's angle = (its value ÷ total) × 360°.
• Going back: a sector of 90° is a quarter of the total; 180° is a half; 36° is a tenth.
• A category's percentage = (its angle ÷ 360) × 100.

Bar graph against histogram is the distinction most often tested: a bar graph has gaps between the bars and compares separate categories; a histogram has no gaps and shows a continuous range.

The three averages

• Mean = sum ÷ number of values.
• Median = the middle value after sorting. With an even count, the mean of the two middle ones.
• Mode = the value occurring most often. A set can have two modes, or none at all.

Sorting first is the step candidates skip, and a median question is usually built to punish exactly that: the values are given out of order, and the unsorted middle is offered as a wrong option.

Which average a question wants

This is the part that carries the reasoning marks.

• The mean uses every value, and is therefore pulled by an extreme one. A few very large salaries drag the mean above what most people earn.
• The median ignores how extreme the extremes are, and so represents the typical case better when the data is skewed. That is why income is usually quoted as a median.
• The mode is the only one that works for data that is not numerical — the most common shoe size, the most popular subject, the colour most children chose.

A question giving a small set with one enormous value and asking which average best represents it wants the median, and it wants the reason.

Questions of the kind the paper sets

What comes next: 2, 6, 12, 20, 30 …? (a) 38 (b) 42 (c) 40 (d) 36

Differences are 4, 6, 8, 10 — rising by 2 — so the next difference is 12 and the next term is 42.

A row of n squares is made with matchsticks. How many sticks are needed for 10 squares? (a) 40 (b) 31 (c) 30 (d) 33

4 for the first square, 3 for each of the other nine: 4 + 27 = 31, which is 3n + 1.

The median of 7, 3, 9, 5, 11 is — (a) 9 (b) 7 (c) 5 (d) 7.5

Sorted: 3, 5, 7, 9, 11. The middle is 7. The unsorted middle is 9, which is option (a).

In a pie chart, a category making up 25 % of the total is drawn with an angle of — (a) 25° (b) 90° (c) 45° (d) 120°

A quarter of 360°.

Which average can be used for the most popular colour in a class? (a) Mean (b) Median (c) Mode (d) Range

Colour is not numerical; only the mode applies.

The marks of five pupils are 20, 22, 24, 25 and 99. Which average best represents the group? (a) Mean (b) Median (c) Mode (d) Range

The 99 pulls the mean to 38, above four of the five values. The median, 24, is typical of the group.

In a histogram, the bars touch one another because — (a) it looks neater (b) the class intervals are continuous (c) the data is categorical (d) the frequencies are equal

Mistakes that cost marks here

• Finding the median without sorting first.
• Reading a pictograph without its key, or forgetting that a half-symbol is half the value.
• Reading a bar graph without checking where the axis starts.
• Confusing a bar graph with a histogram. Gaps against no gaps; categories against intervals.
• Taking a pie chart's sector as a percentage of 100 rather than of 360°.
• Choosing the mean where the data has an extreme value and the question asks what is typical.
• Trying to use a mean or median for non-numerical data, where only the mode works.
• Assuming a set has exactly one mode. It can have two, or none.
• Giving up on a sequence because the first differences are not constant — take the second differences, then the ratios.

In one line

For a pattern, difference the terms and then difference again before trying ratios; for data, sort before taking a median, read the key and the axis before reading anything off a graph, and choose the average that the shape of the data actually supports.

For the classroom

Data handling is the easiest topic in the syllabus to make real, because the class is itself a data set. Counting birthdays by month, or how everyone travels to school, produces a tally, a bar graph and a mode from the children themselves — and a mean they can see need not be one of the values they collected.

Patterns work the same way with matchsticks or tiles. Build stage one, stage two and stage three, write the counts in a row, and ask for stage ten — large enough that building it is unattractive and the rule becomes the easier route. That moment, when a child would rather find a rule than keep counting, is the whole purpose of the topic.