A combined solid's volume is the sum of its parts' volumes, and its surface is the sum of the parts' outer surfaces, leaving out the faces where they join
A circus tent is a cylinder with a cone on top. Change the radius, the height of the wall and the height of the cone, and find how much air it holds and how much canvas it needs.
A pyramid holds a third of the prism on the same base and height, V = ⅓a²h; its four triangular faces need the slant height √(h² + (a/2)²)
A temple roof, a tent and the pyramids of Egypt are square pyramids. Change the side of the square base and the height, and find the slant height, the volume and the area of the faces.
The opposite angles of a cyclic quadrilateral add up to 180°, because each is half of the arc it stands on and the two arcs make the whole circle; an exterior angle equals the interior opposite angle
Put the four corners of a quadrilateral ABCD on a circle. Change angles A and B and watch angles C and D, and find the rule for the opposite angles of a cyclic quadrilateral.
A tangent is perpendicular to the radius at the point where it touches, so the two tangents from an outside point are equal, each √(d² − r²)
From a point P outside a circle, draw the two lines that just touch it. Change the radius and how far P is from the centre, and find the length of each tangent and the angle between them.
Compound interest: yearly, half-yearly or quarterly
With interest added k times a year, the compound amount is A = P(1 + R/100k)^(kT); the more often it is added, the more interest earns interest
Put money in a fixed deposit at a bank. Change the principal, the rate, the time and how often interest is added, and compare the compound interest with simple interest.
The distance AB = √((x₂ − x₁)² + (y₂ − y₁)²); the point dividing AB in the ratio m : n is ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)), and the midpoint is the case 1 : 1
Place two points A and B on a coordinate grid, like two shops on a town map. Find the distance between them with Pythagoras, and the point P that divides AB in a ratio such as 1 : 2 or 1 : 1 (the midpoint).
If the rupee is devalued by R%, the rupees for one dollar rise by R% of the old rate; if it is revalued by R%, they fall by R%
A relative abroad sends you US dollars, and you change them at a bank. Change the amount, and devalue or revalue the Nepali rupee, and see how many rupees you get.
Growth by R% a year gives V = V₀(1 + R/100)^T; depreciation by R% a year gives V = V₀(1 − R/100)^T, the compound interest formula with a plus or a minus
A town's population grows by a fixed per cent each year, and a motorbike loses a fixed per cent of its value each year. Change the rate and the years, and see compound growth and compound depreciation.
For independent events, P(A and B) = P(A) × P(B); on a tree diagram you multiply along the branches and add the ends that give the outcome you want
Will it rain tomorrow, and will the school bus be late? Draw both on a probability tree, change their chances, and find the probability of both, neither, at least one and exactly one.
P(A and B) = P(A) × P(B after A): with the card put back the draws are independent; kept out, the second depends on the first
Draw two cards one after the other and find the chance that both are aces, both hearts or both red: once putting the first card back, once keeping it out. A probability tree shows why the second draw changes.
The roots of x² + bx + c = 0 are x = (−b ± √(b² − 4c))/2; the discriminant b² − 4c decides whether the parabola meets the x-axis twice, touches it once or misses it
Change b and c in x² + bx + c = 0. Watch the parabola y = x² + bx + c move, find its roots with the formula, and see from the discriminant whether there are two roots, one or none.
For whole numbers a and b, a = bq + r with 0 ≤ r < b; the HCF of a and b is the HCF of b and r, so dividing again and again until the remainder is 0 leaves the HCF as the last divisor
Divide the bigger number by the smaller, then the smaller by the remainder, again and again. Watch each step a = bq + r until the remainder is 0, and read off the HCF of five pairs of numbers.
An AP adds the same d each time: tₙ = a + (n − 1)d, Sₙ = n/2 × (2a + (n − 1)d); a GP multiplies by the same r: tₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1)/(r − 1)
Put grains of rice on a row of squares: either add the same number each square (an AP) or multiply by the same number (a GP, like the old chessboard story). Find the last term and the total, and see how fast a GP grows.
For three sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)
Students were asked which of three newspapers they read. Change how many read all three, and how many were asked, and watch every region of a three-set Venn diagram change.
For grouped data each class is represented by its class mark x (the middle of the class); the mean is Σfx ÷ Σf, or A + Σfd ÷ Σf with an assumed mean A and d = x − A, and both ways give the same answer
A class's test marks are grouped 0–10, 10–20, …, 40–50. Change how many students scored 20–30 and 40–50, and find the mean from the class marks, directly and with an assumed mean.
For continuous data, Q₁, the median and Q₃ are the values at positions N/4, N/2 and 3N/4 of the cumulative frequency; in the class holding the position, quartile = L + (position − c.f.) ÷ f × h
The marks of 40 students are grouped into classes 10 marks wide. Draw the cumulative frequency curve, read the lower quartile, the median and the upper quartile, and check each one with the formula.
Parallelograms on the same base and between the same parallels are equal in area, and a triangle on the same base and between the same parallels is half of such a parallelogram
Draw a parallelogram ABCD and a triangle ABE on the same base AB, with their tops on a second parallel line. Slide the top along the line and change the base and the gap, and see which areas change and which do not.
If a line parallel to one side of a triangle cuts the other two sides at D and E, it divides them in the same ratio: AD/DB = AE/EC (the basic proportionality, or Thales, theorem)
In triangle ABC, AB is 10 cm long. Slide a line DE parallel to BC up and down, and change the length of AC: AD ÷ DB always equals AE ÷ EC.
With two sides a and b and the angle C between them, the height on a is b sin C, so the area of the triangle is ½ab sin C
A farmer knows two sides of a triangular plot of land and the angle between them, but not its height. Change the sides and the angle and find the area with ½ab sin C.
If the top of a tower is seen at angles α and β from two points d apart on a line towards it, then h = d tan α tan β ÷ (tan β − tan α), and the near point is h ÷ tan β from the tower
You cannot reach the foot of a temple tower, so you sight its top from two points on a straight line towards it. Change the two angles and the distance walked, and find the tower's height and how far away it is.
On a circle of radius 1, a point at angle θ is cos θ across and sin θ up, so by Pythagoras sin²θ + cos²θ = 1 for every angle; tan θ = sin θ ÷ cos θ, which has no value at 90°
Turn a radius of length 1 round a circle. Its height is sin θ and its distance across is cos θ. Change the angle, read sin, cos and tan, and see that sin²θ + cos²θ is always 1.
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A spinner game
On a spinner with equal sectors, P(colour) = sectors of that colour ÷ all sectors; the probabilities add up to 1, and the expected number of times = probability × number of spins
Colour the equal sectors of a spinner red, blue and green, and work out the chance of landing on each colour and how many reds to expect in 60 spins. Then make the game fair.
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Batting average and strike rate
Batting average = runs ÷ times out (runs per dismissal); strike rate = runs ÷ balls faced × 100 (runs per 100 balls)
Change a batter's runs, balls faced and times out, and work out the batting average and the strike rate. Find out why one tells you how reliable a batter is and the other how fast they score.
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Cooking for more people
Amounts in a recipe are in direct proportion to the number of people: each amount × (new number of people ÷ 4), so doubling the people doubles everything
A dal-bhat recipe feeds 4 people. Cook it for 2, 6, 8 or 10 people instead, and see how the rice, dal, water and salt all change by the same ratio.
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Fibonacci and the golden ratio
In the Fibonacci sequence each number is the sum of the two before it (Fₙ₊₁ = Fₙ + Fₙ₋₁); the ratio of neighbours Fₙ₊₁ ÷ Fₙ gets closer and closer to the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618
Build the Fibonacci numbers 1, 1, 2, 3, 5, 8, … as a spiral of squares, and divide each by the one before. Watch the ratio swing above and below and settle on 1.618, the golden ratio found in sunflowers, pine cones and shells.
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How many handshakes?
In a group of n people each person shakes n − 1 hands, but every handshake is shared by two people, so there are n(n − 1) ÷ 2 handshakes: the number grows much faster than the group
At a party everyone shakes hands with everyone else once. Change the number of people from 2 to 12 and count the handshakes, then find the rule that works for any group.
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Pascal's triangle
Each number in Pascal's triangle is the sum of the two above it; the k-th number of row n is C(n, k), the number of ways to choose k things from n, and row n adds up to 2ⁿ
Pick a row n and a place k in Pascal's triangle. Read the number C(n, k), see that it is the sum of the two above it, and find that every row adds up to a power of 2: the ways of choosing, of tossing coins and of walking a grid.
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Paying back a loan
A loan is repaid in equal monthly instalments, EMI = P r (1 + r)ⁿ ÷ ((1 + r)ⁿ − 1), with r the monthly rate and n the months; a longer loan has a smaller EMI but much more interest
Borrow Rs 1 lakh to Rs 10 lakh at 8% to 12% a year and choose to pay it back in 1, 3 or 5 years. See the equal monthly instalment (EMI), the total you pay back and how much of it is interest.
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Symmetry in everyday shapes
A line of symmetry folds a shape into two halves that match exactly; some shapes have one, some have many, and some have none, even if they look balanced when turned
Look at twelve shapes from nature and daily life, from a butterfly and a snowflake to a shoe and a swirl. Find how many lines of symmetry each has, and sort them into one line, more than one, or none.
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The angle between the clock hands
The minute hand turns 6° a minute and the hour hand 30° an hour plus 0.5° a minute, so the angle between them is |30h − 5.5m|, or 360° minus that if it is bigger than 180°
Set a clock to any time and work out the angle between its hands. Find the times when the hands make a right angle or a straight line, and see why the hour hand is never exactly on the number at half past.
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The birthday surprise
With n people there are n(n − 1)/2 pairs, each with a 1 in 365 chance of sharing a birthday; the pairs grow so fast that the chance of a shared birthday passes 50% at 23 people: P ≈ 1 − e^(−n(n − 1)/730)
Fill a room with 5 to 70 people and find the chance that at least two of them share a birthday. See why a class of 23 already has an even chance, and why the number of pairs, not people, is the secret.
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Triangular numbers with dots
The nth triangular number is 1 + 2 + … + n = n(n + 1) ÷ 2, and a triangle with the one before it makes a square: Tₙ + Tₙ₋₁ = n²
Stack dots in rows of 1, 2, 3, 4 and more to build triangles. Count the dots in each triangle, find the rule n(n + 1) ÷ 2, and see how two neighbouring triangles fit together into a square.
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Which unit would you use?
Every measurement has a quantity (length, mass, volume or time) and a unit; we choose a unit that suits the size, so the number is neither huge nor tiny: kilometres for roads, millimetres for an ant
From the distance between two cities to a spoon of cough syrup, decide what each of twelve everyday things measures, length, mass, volume or time, and choose a unit that suits its size.