Revise the concepts

A pattern is a regularity, something that repeats or grows by a fixed rule. Patterns appear everywhere: the days of the week, the petals of a flower, the tiles on a floor, the beats of a song.

A number sequence is a list of numbers that follows a rule, e.g. 2, 4, 6, 8, … The individual numbers are called terms.

To find the rule of a sequence, look at:

1. Differences between neighbouring terms (1, 3, 6, 10 → differences 2, 3, 4: they grow by 1).
2. Ratios: is each term a fixed number of times the previous one? (1, 3, 9, 27 → × 3 each time).
3. Position: is the 5th term 5 × 5? (1, 4, 9, 16, 25 → squares).
4. Previous terms: is each term made from the ones before it? (1, 2, 3, 5, 8 → sum of the previous two).

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• Triangular numbers: rows of 1, 2, 3, 4 … dots stacked into a triangle.
• Square numbers: dots in an n × n square.
• Cube numbers: small cubes built into an n × n × n big cube (2 × 2 × 2 = 8 cubes).
• Hexagonal numbers 1, 7, 19, 37, 61: dots arranged in a hexagon. Each new ring adds 6, 12, 18, 24 … dots.
• Powers of 2: keep doubling. Fold a sheet of paper and the number of layers goes 1, 2, 4, 8, 16.

(a) Adding odd numbers gives square numbers

Why: each odd number forms an "L" shape that wraps around the previous square to make the next square. 3 dots wrap a 1 × 1 square into a 2 × 2, 5 dots wrap a 2 × 2 into a 3 × 3, and so on.

Rule: sum of the first n odd numbers = n × n. How many odd numbers from 1 to k? (k + 1) ÷ 2. So 1 + 3 + … + 29 has 15 terms and equals 225.

(b) Counting up and down gives square numbers

1 + 2 + 1 = 4, 1 + 2 + 3 + 2 + 1 = 9, 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16. Up to n and back down gives n × n.

(c) Adding counting numbers gives triangular numbers

1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10. Also, adding all 1's gives the counting numbers: 1, 1 + 1 = 2, 1 + 1 + 1 = 3.

(d) Two consecutive triangular numbers make a square

1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25. Why: the two triangles of dots fit together exactly into a square.

(e) Powers of 2 and of 3

• 1 + 2 = 3 = 4 − 1;  1 + 2 + 4 = 7 = 8 − 1;  1 + 2 + 4 + 8 = 15 = 16 − 1. The sum is one less than the next power of 2.
• (1 + 3) × 2 + 1 = 9;  (1 + 3 + 9) × 2 + 1 = 27. Twice the sum, plus 1, gives the next power of 3.

(f) Hexagonal numbers and cubes

• Triangular number × 6 + 1 = hexagonal number: 1 × 6 + 1 = 7, 3 × 6 + 1 = 19, 6 × 6 + 1 = 37.
• Adding hexagonal numbers gives cubes: 1 = 1, 1 + 7 = 8, 1 + 7 + 19 = 27, 1 + 7 + 19 + 37 = 64.

(g) Adding cubes

1 + 8 = 9 = 3 × 3;  1 + 8 + 27 = 36 = 6 × 6;  1 + 8 + 27 + 64 = 100 = 10 × 10. The answers are squares of the triangular numbers 3, 6, 10.

Regular polygons

A polygon is a closed figure made of straight sides. A regular polygon has all sides equal and all angles equal.

Complete graphs

Take some points and join every pair with a line. With 2 points there is 1 line, with 3 points 3 lines, with 4 points 6 lines, with 5 points 10 lines. These are the triangular numbers.

Why: each new point must be joined to every point already there. The 5th point adds 4 new lines, so the total is 1 + 2 + 3 + 4 = 10.

Diagonals: in a polygon, the lines of the complete graph that are not sides are diagonals. A pentagon has 10 − 5 = 5 diagonals, a hexagon 15 − 6 = 9, and an octagon 28 − 8 = 20.

Stacked squares and stacked triangles

A big square split into 1, 2, 3, 4 … rows of small squares contains 1, 4, 9, 16 … squares. A big triangle split into 1, 2, 3, 4 rows of small triangles contains 1, 4, 9, 16 small triangles (rows: 1 + 3 + 5 + 7, the odd numbers again!). Both give the square numbers.

Koch snowflake

Start with an equilateral triangle. At every stage, replace the middle third of each side with a "tent" (two sides of a smaller triangle). Each side becomes 4 sides, each one-third as long.

The sides multiply by 4 at each stage, so the perimeter keeps growing even though each side gets shorter.

• I can write the first 6 terms of all ten sequences and give each rule.
• I can find the sum of the first n odd numbers and the sum 1 + 2 + … + n.
• I can explain with dots why consecutive triangular numbers make a square.
• I can count lines in a complete graph and diagonals of a polygon.
• I can fill in a Koch snowflake table of sides and perimeter.

Point: an exact location. It has no length, breadth or thickness. A dot only shows where it is. Name it with a capital letter: point A.

Line segment: the shortest path between two points. It has two end points and a definite length that can be measured with a ruler. It is written AB or BA (both name the same segment).

Counting line segments

If n points lie on a line, every pair of points makes a segment. The number of segments is 1 + 2 + … + (n − 1):

Line: a line segment extended endlessly in both directions. It has no end points and cannot be measured. It is drawn with arrows at both ends and written as line AB or with a small letter, line l.

Ray: starts at one point and goes on endlessly in one direction. It is written start point first: ray OP starts at O and passes through P. A beam of torchlight or a ray of sunlight is a good picture of a ray.

• Through one point you can draw countless lines (and countless rays).
• Through two points you can draw exactly one line.
• If A, B, C lie on a line in that order, ray AB and ray AC are the same ray: same start point, same direction.

An angle is formed by two rays with a common starting point. The common point is the vertex; the two rays are the arms. The region between the arms is the interior of the angle.

• Name it with three letters, vertex in the middle: ∠ABC or ∠CBA. (∠BAC would be a different angle, with its vertex at A.)
• If only one angle is at a vertex, ∠B is enough.
• An angle shows an amount of turn: opening a door, the blades of scissors, the hands of a clock.
• The length of the arms does not matter. Making the arms longer does not change the angle.

Comparing angles

Place one angle on top of the other (superimposition) so that the vertices and one arm match. The angle whose second arm lies further out is bigger. A tracing-paper copy makes this easy.

Counting angles

If several rays start from the same vertex (all within a straight angle), any two rays make an angle: 3 rays → 3 angles, 4 rays → 6 angles, 5 rays → 10 angles (triangular numbers again).

A full turn is divided into 360 equal parts, each called a degree (1°). People chose 360 because it is close to the number of days in a year and because it has many factors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15 …), so halves, thirds, quarters and sixths of a turn are all whole numbers of degrees.

A protractor is a half circle marked from 0° to 180° with two scales: an outer one and an inner one, running in opposite directions.

To measure an angle

1. Place the protractor's centre point exactly on the vertex.
2. Turn it so the base line (0°) lies along one arm.
3. Find the scale that shows 0° on that arm. Follow that scale up to where the other arm crosses it.
4. Check by eye: if the angle looks smaller than a right angle, the answer must be less than 90°.

To draw an angle, e.g. ∠ABC = 125°

1. Draw ray BC with a ruler.
2. Place the protractor centre on B with the base line along BC.
3. On the scale that starts at 0° on BC, put a dot at 125°. Call it A.
4. Remove the protractor and join BA. Mark the angle with an arc and write 125°.

To draw a reflex angle, e.g. 240°

A protractor only reaches 180°, so draw 360° − 240° = 120°, then mark the arc on the outside and label it 240°.

Clock

• 12 hour marks share 360° → 30° between neighbouring numbers.
• The minute hand turns 360° in 60 minutes → 6° per minute.
• The hour hand turns 30° in 60 minutes → ½° per minute. It moves too!

Compass

N → E → S → W are 90° apart, going clockwise. The in-between directions NE, SE, SW, NW are halfway, so each neighbouring pair of the 8 directions is 45° apart.

• Facing North, turn clockwise to face South: 180°.
• Facing East, turn 135° clockwise: E → S (90°) + 45° → South-West.
• Facing North, turn clockwise to face West: 270° (reflex).

• I can tell a point, segment, ray and line apart and name each correctly.
• I can count the segments on a line and the angles at a vertex.
• I can classify any angle and find its reflex partner.
• I can measure and draw any angle (including reflex) with a protractor.
• I can find the angle between clock hands and between compass directions.

Children stand in a line. Each child calls out how many of their neighbours are taller than them. A child in the middle has 2 neighbours; a child at either end has only 1.

• "0" means the child is taller than every neighbour.
• "2" means both neighbours are taller, so this child is shorter than both.
• A child at the end can only say 0 or 1.

From these numbers alone you can reason about the arrangement. That is the point: numbers carry information.

In a grid of numbers, a cell is a supercell if its number is greater than all its neighbours. Neighbours are only the cells to the left, right, above and below; diagonal cells do not count.

• Corner cells have 2 neighbours, edge cells 3, and inside cells 4.
• The largest number in the grid is always a supercell, because it is bigger than everything.
• The smallest number is never a supercell.
• Two neighbouring cells can never both be supercells, since each would have to be bigger than the other.

The digit sum of a number is the sum of its digits: 4725 → 4 + 7 + 2 + 5 = 18.

Smallest and largest number with a given digit sum

• Smallest: use as few digits as possible and put the big digits on the right. A 3-digit number with digit sum 20 → 299.
• Largest: put the big digits on the left. A 3-digit number with digit sum 20 → 992.

Counting digits

How many times is the digit 7 written from 1 to 100? In the ones place: 7, 17, 27, … 97 = 10 times. In the tens place: 70–79 = 10 times. Total = 20. The number 77 counts twice, since it has two 7s.

The same is true for every digit from 1 to 9.

Parity means whether a number is odd or even. Think of an even number as dots that pair up completely, and an odd number as pairs plus one dot left over.

• Subtraction follows the same rules as addition (odd − even = odd).
• An odd count of odd numbers adds to an odd total; an even count of odd numbers adds to an even total.
• Even numbers can be written 2 × n, odd numbers 2 × n + 1.

A palindrome is a number that reads the same forwards and backwards: 7, 44, 121, 4334, 90709.

Reverse-and-add

Add a number to its reverse; repeat until you get a palindrome:

• 39 + 93 = 132 → 132 + 231 = 363 (2 steps)
• 68 + 86 = 154 → 154 + 451 = 605 → 605 + 506 = 1111 (3 steps)
• 57 + 75 = 132 → 132 + 231 = 363 (2 steps)

Palindromes on clocks and calendars

Times: 1:11, 2:22, 10:01, 12:21. Dates: 12/02/2021 (12022021).

Making palindromes

The smallest 3-digit palindrome is 101 and the largest is 999. A 5-digit palindrome looks like a b c b a, so you only choose the first three digits.

D. R. Kaprekar, an Indian mathematics teacher, discovered this in 1949. Take any 4-digit number whose digits are not all the same, then:

1. Arrange the digits in descending order (largest number).
2. Arrange them in ascending order (smallest number). Keep any zeros at the front.
3. Subtract the smaller from the larger.
4. Repeat with the answer.

You always reach 6174 in at most 7 steps. For 3-digit numbers, the same routine always ends at 495: 352 → 297 → 693 → 594 → 495.

Mental maths strategies

• Round and adjust: 498 + 367 = 500 + 367 − 2 = 865.
• Split by place value: 456 + 327 = (400 + 300) + (50 + 20) + (6 + 7) = 783.
• Make friendly numbers: 25 + 48 + 75 = (25 + 75) + 48 = 148.

The Collatz conjecture (unsolved!)

Rule: if the number is even, halve it; if it is odd, multiply by 3 and add 1. Keep going.

6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1 (8 steps). 7 takes much longer, but it still reaches 1.

Lothar Collatz suggested in 1937 that every starting number reaches 1. Computers have checked enormous numbers, but nobody has proved it. It is one of maths' famous open problems.

Estimation

To estimate something large (students in a school, steps to school), count a small part and multiply. For example: about 40 students per class × 30 classes ≈ 1200 students.

Games and winning strategies

In a game where players take turns adding 1, 2 or 3 to a running total and whoever says 21 wins, work backwards from 21. If you say 17, your opponent can only reach 18–20, and you reach 21. The same reasoning makes 13, 9, 5 and 1 safe. So say 1 first and keep landing on 5, 9, 13, 17. (Your textbook's version of the game may use slightly different rules, but the "work backwards" idea is the same.)

• I can find all the supercells in a grid and explain the rules.
• I can make the smallest or largest number with a given digit sum.
• I can decide odd or even without calculating, and explain why.
• I can run Kaprekar's routine (4- and 3-digit) and reverse-and-add.
• I can describe the Collatz conjecture and why it is "unsolved".

Data is a collection of facts, numbers or observations gathered to give information about a question. Examples: the heights of students, the favourite fruits of a class, the rainfall each month.

1. Decide the question, e.g. "Which fruit does my class like most?"
2. Collect the data by asking each person, observing or measuring. This is raw data, in the order it was collected.
3. Organise it in a frequency table.
4. Present it as a pictograph or bar graph.
5. Interpret it and answer the question.

Tally marks are strokes, one per observation. Every 5th stroke is drawn across the previous four, so you can count quickly in fives: |||| = 5, |||| |||| ||| = 13.

Frequency = the number of times a value occurs.

Method

1. List all the categories in the first column.
2. Read the raw data once, in order. For each value, put a tally stroke against its category and strike it off the list.
3. Count the tallies and write the frequency.
4. Check: the frequencies must add up to the total number of observations.

Grouping data

When the values are spread out (e.g. marks from 8 to 49), tally them in groups of equal size, such as 0–9, 10–19, 20–29, 30–39, 40–49. Each value goes into exactly one group.

A pictograph shows data using pictures or symbols. A key (also called the scale) tells you how much one symbol stands for.

How to draw one

1. Choose a simple symbol connected to the data (📘 for books, 🌳 for trees).
2. Choose a key that divides the values neatly. Look for a number that all the values are multiples of (or multiples of half of it).
3. Number of symbols = value ÷ key. Use a half symbol for half the key.
4. Draw the rows neatly, all symbols the same size, and write the key.

Reading a pictograph

Value = number of symbols × key. 3½ symbols with key 20 → 3 × 20 + 10 = 70.

A bar graph shows data with bars of equal width, with equal gaps between them. The length of each bar shows the value, using a uniform scale.

How to draw one

1. Draw two axes at right angles. The horizontal axis shows the categories; the vertical axis shows the values.
2. Choose a scale: look at the largest value and pick a step so the tallest bar fits your page. Largest 40 → 1 unit = 5 gives 8 units. Largest 840 → 1 unit = 100.
3. Mark the value axis from 0 in equal steps.
4. Draw each bar: bar length = value ÷ scale (85 with 1 unit = 10 → 8.5 units).
5. Keep all bars the same width with equal gaps.
6. Label both axes, give a title and write the scale.

Questions usually asked from a bar graph

• Highest and lowest value.
• "How many more?" (subtract) and "how many times?" (divide).
• Total (add all the bars).
• "How many categories are above 30?" (count the bars).

A graph can mislead if:

• the value axis does not start at 0, so a small difference looks huge;
• the bars or symbols are different widths or sizes;
• the scale is uneven (steps of 10, then 50, then 100).

Always read the scale before believing a graph.

Infographics mix pictures, numbers and charts to present data attractively, as in newspapers and posters. They should still be honest and accurate.

• I can make a frequency table with tally marks, with or without groups, and check the total.
• I can choose a key and draw a pictograph with half symbols.
• I can choose a scale and draw a neat bar graph with a title, labels and the scale written.
• I can answer "how many more / how many times / total" questions from a graph.
• I can explain how a graph can mislead.

A factor of a number divides it exactly (remainder 0). A multiple of a number is that number × 1, × 2, × 3, …

Since 4 × 6 = 24, the numbers 4 and 6 are factors of 24, and 24 is a multiple of 4 and of 6.

Finding all the factors — the pair method

1. Start with 1 × the number.
2. Try 2, 3, 4, … in turn, writing each pair that works.
3. Stop when the pairs meet or cross.

96: 1 × 96, 2 × 48, 3 × 32, 4 × 24, 6 × 16, 8 × 12 (next would be 12 × 8, so stop) → 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.

A square number such as 36 has an odd number of factors, because 6 × 6 pairs with itself.

Perfect numbers

A number that equals the sum of its factors other than itself: 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14.

Common factors are factors of each of the given numbers. 20 → 1, 2, 4, 5, 10, 20; 28 → 1, 2, 4, 7, 14, 28. Common factors: 1, 2, 4.

Common multiples are multiples of each of the given numbers. Multiples of 4 → 4, 8, 12, 16, 20, 24; of 6 → 6, 12, 18, 24. Common multiples: 12, 24, 36 … The smallest is 12.

The Idli-Vada game

Count 1, 2, 3, …; say "idli" for multiples of 3, "vada" for multiples of 5, and "idli-vada" for multiples of both. "Idli-vada" falls on 15, 30, 45, … the common multiples of 3 and 5.

Jumpy and Grumpy

Jumpy jumps 4 units at a time, Grumpy 6 units. Both land on 12, 24, 36, …, the common multiples of 4 and 6. A treasure at 30 can be reached only by Grumpy, because 30 is a multiple of 6 but not of 4.

• Prime number: exactly two factors, 1 and itself (2, 3, 5, 7, 11, …).
• Composite number: more than two factors (4, 6, 8, 9, 10, …).
• 1 is neither, because it has only one factor.
• 2 is the only even prime. Every other even number also has 2 as a factor.

Sieve of Eratosthenes (1 to 100)

1. Write 1 to 100 in a grid and cross out 1.
2. Circle 2 and cross out all its other multiples (4, 6, 8, …).
3. Circle the next uncrossed number, 3, and cross out its multiples. Then do the same for 5, then 7.
4. Everything left uncrossed is prime. You can stop after 7 because 11 × 11 = 121 > 100, so every composite number up to 100 has a factor of 7 or less.

Twin primes are primes that differ by 2: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).

Is a number prime? Quick test

Try dividing by the primes 2, 3, 5, 7, 11, 13, … and stop once the prime × itself is bigger than the number. Example: 107. It is not divisible by 2, 3, 5 or 7, and 11 × 11 = 121 > 107, so 107 is prime.

Two numbers are co-prime if their only common factor is 1. They do not need to be prime themselves: 8 and 15 are both composite, but they are co-prime.

How to check

1. Quick look: if both are even, or both end in 0 or 5, or both have digit sums divisible by 3, they share a factor → not co-prime.
2. Otherwise, write the prime factorisation of each. If no prime appears in both, they are co-prime.

Prime factorisation writes a number as a product of primes only. Every number greater than 1 has exactly one prime factorisation (apart from the order).

Method 1 — Factor tree

Split the number into any two factors, then split each composite factor again, until every branch ends in a prime:

        420
       /   \
     42     10
    /  \   /  \
   6    7 2    5
  / \
 2   3            420 = 2 × 2 × 3 × 5 × 7

Method 2 — Repeated division

1260 = 2 × 2 × 3 × 3 × 5 × 7. Divide by 2 as long as you can, then by 3, then 5, then 7, and so on.

Uses of prime factorisation

• Factorising a product without multiplying: 58 × 30 = (2 × 29) × (2 × 3 × 5) = 2 × 2 × 3 × 5 × 29.
• Checking divisibility: 2520 = 2 × 2 × 2 × 3 × 3 × 5 × 7 contains 2 × 2 × 2 = 8 and 3 × 3 = 9, so it is divisible by 8, 9, 7, 5 and 72, but not by 11.
• Finding common factors: 980 = 2 × 2 × 5 × 7 × 7 and 1155 = 3 × 5 × 7 × 11 share 5 and 7, so their common factors are 1, 5, 7 and 35.
• Checking co-primes (see 5.4).

The textbook chapter covers the tests for 2, 4, 5, 8 and 10. The school worksheet also asks for 3, 6, 9 and 11, so learn all nine.

• I can list all the factors of a number using pairs, and the first few multiples.
• I can find common factors and the smallest common multiple.
• I can write all 25 primes up to 100 and the twin primes.
• I can test whether two numbers are co-prime.
• I can prime-factorise by a factor tree and by division.
• I can apply all nine divisibility tests and give the reason.

Perimeter = the total length of the boundary. Imagine walking once around the edge: the distance you walk is the perimeter.

Finding a missing side (working backwards)

• Rectangle with perimeter 25 cm and breadth 5 cm: l + b = 25 ÷ 2 = 12.5 → l = 12.5 − 5 = 7.5 cm.
• Square with perimeter 84 cm: side = 84 ÷ 4 = 21 cm.
• Regular pentagon with perimeter 100 cm: side = 100 ÷ 5 = 20 cm.

• Running laps: distance = number of rounds × perimeter. Akshi runs 4 rounds of a 60 m × 40 m track: 4 × 200 = 800 m.
• Fencing: length of fence = perimeter − any gate left open. Cost = length × rate. A 24 m × 15 m garden with a 3 m gate: (78 − 3) × ₹35 = ₹2,625.
• Lace or border: the length needed = perimeter of the cloth, photo frame or table.
• A wire bent into a new shape: the length of the wire (the perimeter) stays the same. A square of side 15 cm (60 cm of wire) bent into a rectangle 18 cm long → breadth = 30 − 18 = 12 cm.
• "At least" questions: round up. To cover at least 1 km on a 140 m track: 1000 ÷ 140 ≈ 7.1 → 8 rounds.

Area is measured in square units: 1 cm² is the area of a square of side 1 cm.

Counting squares on squared paper

• A full square counts as 1.
• More than half a square counts as 1.
• Exactly half counts as ½ (two halves make 1).
• Less than half: ignore it.

Formulas

Rectangle: area = length × breadth. A 5 × 3 rectangle has 5 squares in each of 3 rows = 15 cm². Square: area = side × side.

Tiling problems

Irregular shapes

Split the shape into rectangles, find each area, and add them. Or take the big rectangle and subtract the missing part. An 8 × 6 rectangle with a 3 × 2 corner cut out has area 48 − 6 = 42 cm².

A diagonal cuts a rectangle into two identical right triangles, so each triangle is half the rectangle: area of a right triangle = ½ × base × height. A 10 cm × 6 cm rectangle → each triangle is 30 cm².

A triangle drawn in a rectangle on the same base, with its top corner on the opposite side, also has half the rectangle's area.

Tangram

A tangram is a square cut into 7 pieces: 2 large triangles, 1 medium triangle, 2 small triangles, 1 square and 1 parallelogram. As fractions of the whole square:

Check: ¼ + ¼ + ⅛ + ⅛ + ⅛ + 1⁄16 + 1⁄16 = 1. Any shape made from all 7 pieces has the same area as the original square.

• With the same perimeter, the square has the largest area.
• With the same area, the square has the smallest perimeter.
• Cutting a notch from a corner: the area decreases, but the perimeter stays the same, because the two new inner edges exactly replace the two pieces removed.
• Cutting a square into 4 equal squares: the total area is unchanged, but the total perimeter doubles (side 12 → four squares of side 6 → 4 × 24 = 96 cm, against 48 cm before).
• A path inside a garden (1 m wide): the inner rectangle is 2 m shorter in each direction. Path area = outer area − inner area.

• I can find the perimeter and area of rectangles, squares and regular polygons, and work backwards to a missing side.
• I can solve track, fencing, wire and tiling problems with the correct units.
• I can find the area of a right triangle as half a rectangle.
• I can find areas of L-shapes and paths by adding or subtracting.
• I can explain why a notched shape keeps the same perimeter.