38 experiments found
Clear filtersA sector with angle θ is θ/360 of the circle: area = θ/360 × πr², arc length = θ/360 × 2πr
Set how far a sprinkler turns and how far it sprays, and work out the area it waters and the length of its curved edge.
Two crossing paths w wide cover w × l + w × b − w², because the square where they meet would otherwise be counted twice; the lawn is the rest
A school garden has two paths crossing in the middle, one along it and one across. Change the garden's size, the paths' width and the paving price, and find the area of the paths, the lawn left and the cost.
The angle at the centre is twice the angle at the circumference standing on the same arc
Walk a photographer round the edge of a circular park and compare the angle at which the gate is seen with the angle at the centre.
A dividend is declared as a per cent of the face value of a share: a cash dividend is paid in rupees (with 5% tax taken off), and a bonus (stock) dividend is paid in new shares
Your family owns shares in a hydropower company. Change the number of shares and the cash and bonus dividends the company declares, and find the cash you receive after tax and the new shares you get.
The bank buys foreign money at its buying rate and sells it at its higher selling rate; the difference is its margin
Change money sent home from abroad into rupees, buy Indian rupees for a trip, and see why the bank's buying and selling rates differ.
For grouped data the mean is estimated as Σfx ÷ Σf using mid-values; cumulative frequency finds the median class
Group the heights of a class into 10 cm classes, draw the histogram, and estimate the mean, the modal class and the median class.
Units used = new reading − last reading; the tariff charges each tier of units at its own rate, plus a service charge, so a big user pays more per unit
Read the meter, find the units used this month, and work out the bill tier by tier. Change the units from 15 to 200 and see why using more electricity costs more for each unit.
log_b x is the power of b that gives x (b^y = x means log_b x = y); multiplying numbers adds their logarithms: log (xy) = log x + log y
A logarithm asks how many times you multiply the base to reach a number. Change the base and two numbers, and see on a slide-rule scale that multiplying numbers adds their logarithms.
The solution of two linear equations is the pair (x, y) that makes both true at once; on a graph it is the point where the two lines cross
Try values of x and y in two equations at once, see each one checked, and watch the guess move on a graph of both lines until it sits where they cross.
For discrete data in a frequency table, the mean is Σfx ÷ N, the median is the ((N + 1)/2)th value and the mode is the value with the biggest frequency; new data move the mean every time, but the median and the mode only when enough values change
A survey counts the members in each family of a tole. Add more families of five to the frequency table, and watch how the mean, the median and the mode each change, or stay put.
The area of a triangle with sides a, b and c is √(s(s − a)(s − b)(s − c)), where s = (a + b + c) ÷ 2 is the semi-perimeter; any two sides together must be longer than the third
A farmer can measure the three sides of a triangular field but not its height. Change the sides, find the semi-perimeter and the area with Heron's formula, and see when three lengths cannot make a triangle.
When p(x) is divided by (x − a), the remainder is p(a); if p(a) = 0, then (x − a) is a factor of p(x) and a is a zero, where the graph meets the x-axis
Divide a cubic polynomial p(x) by (x − a) without long division: the remainder is just p(a). Change a and the polynomial, and find the values of a that make (x − a) a factor.
After cards are removed, probability = favourable cards left ÷ cards left
Take all the kings, the clubs, the face cards or the aces out of the pack before you draw. See how the number of cards left and the favourable cards both change, and how that changes the probability.
P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events the overlap is 0
Draw one card and win if it is a king or a queen, then a king or a heart. Count the cards that win and find out when you can just add the probabilities and when you must take away the overlap.
For independent events, multiply along the branches: P(A and B) = P(A) × P(B); P(at least one) = 1 − P(none)
Set how often each traffic light is green and use a tree diagram and an area model to find the chance of a trip with no stops.
Solving −x² + bx + c = h finds where a parabola meets a line; the discriminant tells whether there are two, one or no real solutions
Shape a fountain jet y = −x² + bx + c and find where it reaches a balcony, or prove with the discriminant that it never can.
In any triangle ABC, the line joining the mid-points D of AB and E of AC is parallel to BC and DE = ½ BC, whatever the shape of the triangle
Mark the mid-points D and E of two sides of a triangle and join them. Change the base, move the top corner and make it taller or lower: DE always stays parallel to BC and exactly half as long.
In an arithmetic sequence the same number d is added each time, so the nth term is a + (n − 1)d, and the sum of n terms is n/2 × (2a + (n − 1)d)
The school hall has rows of seats: a few at the front and more in each row behind. Change the seats in the first row, the extra seats in each row and the number of rows, and find the seats in the last row and in the whole hall.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the students in both are counted twice; A − B is A without B, and the complement (A ∪ B)′ is everyone in U outside both sets
Sixty students say whether they like football, cricket, both or neither. Change the numbers, watch the Venn diagram fill in, and find n(A ∪ B), n(A − B) and the students who like neither.
Triangles with the same angles are similar: their sides are in the same ratio, the scale factor
Use a stick and two shadows to find the height of a tree, a pole or a temple, whatever the height of the sun.
A prism has the same cross-section all along it, so its volume is the area of its end × its length; its surface is the two ends plus the rectangles round it
A trekking group is choosing a tent. Change the width and height of its triangular end and its length, and find how much air it holds (volume) and how much canvas it needs (surface area).
Income tax is charged in slabs: each part of the income is taxed at its own rate, so the total tax is the sum of the slabs, and the average rate is less than the rate on the last rupee
Change a person's yearly income from 4 lakh to 25 lakh rupees. See the income cut into slabs, the tax on each slab, the total tax, and why the tax as a per cent of income is always less than the top rate.
Two triangles are congruent if three sides match (SSS), two sides and the angle between them match (SAS), two angles and a side match (ASA or AAS), or a right angle, the hypotenuse and a side match (RHS); equal angles alone, or two sides with an angle not between them, do not fix one triangle
Twelve pairs of triangles, from a bridge truss to two ladders against a wall. Check what is given and which parts match, and sort each pair by the test that proves it congruent, or by “not enough”.
For an angle θ in a right-angled triangle, sin θ = opp ÷ hyp, cos θ = adj ÷ hyp and tan θ = opp ÷ adj; they depend only on the angle
Lean a ladder against a wall at different angles and see how sin, cos and tan compare its sides; find the safe angle.
In a right-angled triangle, tan θ = opposite ÷ adjacent, so height = distance × tan(angle of elevation) + eye height
Stand back from a temple tower, aim a clinometer at its top and use tan of the angle of elevation to find how tall it is.
A cone holds a third of the cylinder with the same base and height; a sphere holds two thirds of the cylinder that just fits round it
Fill a cone or a ball with water and pour it into a drum of the same size to discover the volume formulas.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.