132 experiments found
Clear filtersArea of a triangle = ½ × base × height, wherever the top corner is; its perimeter is the sum of its three sides
Draw a triangle inside a rectangle with the same base and height, slide its top corner along and see that its area is always half the rectangle, even when the triangle leans over.
Perimeter is the length around a shape and area is the space inside; the same perimeter can hold different areas
Shape a vegetable garden with a 24 m roll of fence and find which rectangle grows the most vegetables.
Split an irregular shape into rectangles and triangles and add their areas; for a curved shape, count the squares at least half inside
Find the area of an L-shaped room, a T-shaped courtyard, a field with a slanted side and a pond with a curved edge, by splitting them into rectangles and triangles or by counting squares on a 1 m grid.
Circumference = 2πr and area = πr²: a circle's slices rearrange into a rectangle πr by r
Cut a round roti-shaped circle into slices and lay them top to tail. With more slices they make a rectangle πr long and r tall, so the area is πr²; then see what doubling the radius does.
A regular polygon with n sides of length s has perimeter n × s and area ½ × perimeter × apothem
Split a regular pentagon, hexagon and polygons with more sides into triangles from the centre, find their perimeter and area, and see them get closer to a circle.
The area of a shaded region is the area of the whole shape minus the area cut out of it
Pave a path round a rectangular lawn and dig a round pond in it, then work out the area of path to pave and the grass left to mow.
A point on a grid is named by an ordered pair (x, y): first how far along the x axis from the origin O, then how far up the y axis; the order matters
A village map is drawn on a grid. Move along the x axis and up the y axis to find the temple, the health post and the chautari, and see why (4, 3) is not the same place as (3, 4).
A scale such as 1 unit = 10 vehicles lets big counts fit on a graph: the height of each bar is the count divided by the scale, and a bar can be part of a unit
Pupils counted the vehicles passing their school in an hour: 75 motorbikes, 50 rickshaws, 45 cars, 30 bicycles and 20 buses. Draw the bar graph with different scales and see how many units tall each bar must be.
In a decimal, each place is ten times smaller than the one to its left; rounding looks at the next place to decide
Put 1 kg bags, 100 g packets and 10 g scoops of rice on a scale to build decimals, then round the weight for the bill.
An equation is a balance: x + 3 = 8 is true only when x = 5; if the left side is heavier, x + 3 > 8, and if it is lighter, x + 3 < 8
A bag of unknown weight x and some 1 kg weights sit on one side of a tarazu, and weights on the other. Try values of x until it balances, and see the inequality when it tips.
The LCM is the first common multiple and the HCF is the biggest common factor; prime factorisation finds both
Find when two buses leave the bus park together again (LCM), and how to cut two poles into the longest equal pieces (HCF).
A fraction, a decimal and a percentage can describe the same share of a whole
Cut a cake into equal pieces, take some, and see the same share as a fraction, a decimal and a percentage.
A function gives exactly one output for each input; a linear rule y = mx + c can be found from its table
Build a machine that turns plates of momo into a bill, then crack secret machines by comparing their inputs and outputs.
A power such as 2³ means the base 2 multiplied by itself 3 times: 2 × 2 × 2 = 8; the index tells how many times; any base to the power 1 is itself, and to the power 0 is 1
Choose a base and an index, see the power written as repeated multiplication, and find its value. Discover what an index of 0 or 1 means, and why powers of 10 make place value.
For the same base, aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ
Multiply, divide and raise powers of 2. See each written out in full, then find the shortcut: add the indices, subtract them or multiply them.
2^m × 2^n = 2^(m + n), 2^m ÷ 2^n = 2^(m − n) and (2^m)^n = 2^(mn); 2^0 = 1 and 2^−n = 1/2^n
Fold, unfold and repeat folds of a sheet of paper to discover the laws of indices, and find out how thick the stack would get.
Adding a positive number moves right on the number line; adding a negative number moves left
Start from a morning temperature, warm up or cool down, and watch each change hop along the number line, including past zero.
The product of two integers with the same sign is positive and of two with different signs is negative: (+) × (+) = +, (−) × (−) = +, (+) × (−) = − and (−) × (+) = −
A lift is at floor 5. It moves up or down a number of floors each minute. Find where it will be in a few minutes, or where it was a few minutes ago, and discover the rules for multiplying positive and negative numbers.
Angles are measured in degrees: acute angles are less than 90°, a right angle is 90°, obtuse angles are between 90° and 180°, a straight angle is 180°, and reflex angles are more than 180°
Measure ten angles, from the hands of a clock to an open laptop and a pizza with a slice gone, compare each with a right angle, and sort them by type.
When a transversal cuts two parallel lines, vertically opposite, corresponding and alternate angles are equal, while a linear pair and co-interior angles add up to 180°
A straight road crosses two parallel canals. Mark one angle where it meets the first canal, then find its vertically opposite angle, its linear pair, and the corresponding, alternate and co-interior angles at the second canal.
Each metric unit is 10 times the next smaller one, so converting means multiplying or dividing by 10, 100 or 1000
Walk up and down a staircase of metric units to convert road lengths, milk and rice between km, m, cm, L, mL, kg and g.
A prime number has exactly two factors, 1 and itself; crossing out the multiples of each prime in turn (the sieve of Eratosthenes) leaves only the primes, and a composite number has a smallest factor other than 1
Write the numbers from 1 to 100 in a grid, cross out the multiples of 2, 3, 5 and 7, and see the primes that are left. Count the primes up to a limit, and test whether numbers such as 91 and 97 are prime.
When each step adds the same amount a, the nth term of the pattern is an + b
Push tables together for a wedding feast, spot the pattern in the number of seats, and write the rule for any number of tables.
A polygon with n sides has angles adding up to (n − 2) × 180°; regular tiles fit round a point only if their angle divides 360°
Cut polygons into triangles to find their angle sums, then test which regular tiles cover a courtyard floor with no gaps.
When the selling price is more than the cost price there is a profit; when it is less there is a loss; profit or loss per cent = profit or loss ÷ cost price × 100
Buy things for your school fair stall and sell them. Change the cost price and the selling price, and find the profit or loss in rupees and per cent.
A quadrilateral has four sides and four angles adding up to 360°; its sides (equal or parallel) and angles (right or not) decide whether it is a square, rectangle, rhombus, parallelogram, trapezium or kite
Look at twelve four-sided shapes around you, from a carrom board to a Dhaka topi and a changa. Check their sides and angles, and sort them into squares, rectangles, rhombuses, parallelograms, trapeziums and kites.
A set is a well-defined collection; a set with no members is empty (∅), one with exactly one member is a singleton, one whose members can be counted to an end is finite, and one whose members never end is infinite
Write twelve sets in set notation, from the provinces of Nepal to the months with 32 days, count their members, and sort them by type.
The universal set U holds every member we are talking about; A is a subset of B (A ⊆ B) when every member of A is also in B, and the complement A′ is everything in U that is not in A
The universal set is the numbers 1 to 12. Choose two sets from it, see their members in a Venn diagram, and find out when one set is a subset of the other.
Two sets can be equal (the same members), one can be a proper subset of the other, they can overlap (some members in common) or be disjoint (none in common); sets with the same number of members are equivalent, even if their members differ
List twelve pairs of sets, from the letters of “follow” and “wolf” to the factors of 12 and 18. Find their members in common, compare how many members each has, and sort the pairs by how the sets are related.
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.
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.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the students in both sets are counted twice
Put every student of a class in a Venn diagram of football and cricket fans, and find how many like both, one or neither.
Volume = length × width × height counts the unit cubes inside; surface area adds up the six faces of the net
Shape a cuboid water tank, count the litres it holds, open it into its net and find the tank that needs the least metal.
distance = speed × time; on a distance–time graph a steeper line means a higher speed
Send a bus and a later motorbike along the 200 km highway, read a distance–time graph and find where the bike catches the bus.
A line of symmetry reflects a shape onto itself; rotational symmetry turns it onto itself before a full turn
Hold a mirror across tiles, a kite and the flag of Nepal, and turn them about their centre to find their lines and order of symmetry.
The three angles of any triangle add up to 180°
Turn the two rafters of a roof frame and see how the angle at the ridge always makes the three angles add up to 180°.
In the unitary method we first find the value of one unit by dividing, then the value of any number of units by multiplying
Rice, sugar, notebooks and eggs come in packs at a price. Find the price of one by dividing, then the price of as many as you want by multiplying.
When the work is fixed, the number of workers and the number of days are inversely proportional: workers × days stays the same, so twice the workers take half the days
A wall needs 60 worker-days of work. Change the number of workers and find how many days the job takes. Then try harvesting a field.
(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b², which can be seen as areas of squares and rectangles
Build a square of side a + b from four tiles and see why (a + b)² = a² + 2ab + b². Then cut strips from a square of side a to see (a − b)² = a² − 2ab + b².
A bearing is a direction measured clockwise from north, written with three figures (045°); on a scale drawing, real distance = distance on the map × the scale
Draw the way from your school to a village on a map. Change the scale, the length on the map and the bearing, and find the real distance and direction.
The back bearing points the opposite way: it is the bearing + 180° if the bearing is less than 180°, and the bearing − 180° otherwise; on a scale drawing, real distance = map distance × scale
Trek from a camp to a village on a bearing read from a map, then find the bearing that brings you back. Change the bearing, the length on the map and the scale, and find the back bearing and the real distance.
The circumference of any circle is π × diameter, and its area is π × radius²
Unroll a wheel's rim along the road to discover π, cut the wheel into slices to find its area, and work out how far it rolls.
The perpendicular from the centre of a circle to a chord bisects the chord, so half the chord is √(r² − d²); the nearer the chord is to the centre, the longer it is, and through the centre it is a diameter
Draw a chord across a round chautari platform at different distances from the centre. See that the perpendicular from the centre cuts it in half, and use Pythagoras to find how long it is.
Congruent figures have the same shape and the same size; similar figures have the same shape (equal angles, sides in the same ratio) but can be different sizes; every congruent pair is also similar
Compare ten pairs of figures, from a page and its photocopy to a small and a large map of Nepal. Check their sides and angles, and sort the pairs into congruent, similar or neither.
Reflections, rotations and translations move every point by a rule on its coordinates and keep the shape congruent
Flip, turn and slide a rangoli motif on a coordinate grid and watch where every corner lands.
An inequality such as x > 2 has many solutions, shown on a number line by a shaded arrow; an open circle means the boundary is not included (> or <), and a filled circle means it is (≥ or ≤)
Choose an inequality such as x > 2, see its solutions shaded on a number line, and test numbers to find out whether they are solutions. Learn when the circle is open and when it is filled.
An equation stays balanced when the same thing is done to both sides
Rice bags of unknown weight sit on a taraju. Take weights off both pans and share them out to find what one bag weighs.
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.
When a transversal crosses parallel lines, corresponding and alternate angles are equal and co-interior angles add up to 180°
Turn a diagonal road across two parallel streets and compare the angles it makes at both crossings.
The volume of a cuboid is length × breadth × height and its total surface area is 2(lb + bh + hl); a cube is a cuboid with all edges equal
Change the length, breadth and height of a cardboard box. Find how much it holds (volume) and how much cardboard it needs (surface area), and see that boxes with the same volume can need different amounts of cardboard.
Profit or loss = SP − CP, and its percentage is worked out on the cost price; commission is a percentage of the selling price
A farmer sells a crate of oranges through a dealer: set the cost, the selling price and the dealer's commission, and see the profit or loss.
Two mixtures taste the same when their ratios are equal, whatever the amount
Pour cups of syrup and water to match a 1 : 3 recipe, then scale it up for more guests without changing the taste.
A rational number is p/q with q ≠ 0; to place it, cut each unit into q equal parts and count p parts from 0, to the right if p/q is positive and to the left if it is negative; equivalent fractions such as 1/2 and 2/4 share one point
Choose a numerator p and a denominator q, positive or negative, and see where p/q sits on the number line: split each unit into q equal parts and count p of them. Find its decimal, the integers it lies between, and fractions equal to it.
A solid has faces (flat surfaces), edges (where faces meet) and vertices (corners); for every solid with flat faces like these, Euler's formula says F + V − E = 2
Build seven solids from straws and clay: a cube, a cuboid, prisms, pyramids, a tetrahedron and an octahedron. Count their faces, vertices and edges, and find the number that F + V − E always gives.
The side of a square is the square root of its area; a square root lies between the roots of the nearest perfect squares
Choose the area of a square carpet and find how long its sides are; estimate square roots that are not whole numbers.
The mean shares the total out equally, the median is the middle value, and one unusual value moves the mean far more than the median
Drag the stacks of tomato crates sold each day and watch the mean, median, mode and range change.
Grouped data are counted in class intervals and shown as a histogram, whose bars touch because the classes follow on without gaps; the tallest bar is the modal class, and the class width decides how much of the pattern you can see
Thirty students got marks out of 50 in a maths test. Group the marks into classes 5, 10 or 25 marks wide, draw a histogram, and see which class width shows the pattern best and where the modal class is.
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.
The interior angles of a polygon with n sides add up to (n − 2) × 180°; a regular polygon tessellates on its own only if its interior angle divides 360° exactly, as for triangles, squares and hexagons
Fit regular triangles, squares, pentagons, hexagons and octagons round a point. Find each polygon's angle sum and interior angle, and discover which ones tessellate with no gaps.
The cube of a number n is n × n × n = n³, the number of unit cubes in a cube with n along each edge; the cube root ∛ undoes it, so ∛64 = 4 because 4³ = 64
Build cubes from small unit cubes, 1, 2, 3 and up to 6 along each edge. Count the cubes, the squares on the outside and the cubes hidden inside, and find cube roots.
(x + a)(x + b) = x² + (a + b)x + ab, so factorising means finding two numbers with the right sum and product
Extend a square courtyard on two sides and see how (x + a)(x + b) breaks into x², x strips and unit squares.
The four walls of a cuboid room unroll into one rectangle 2(l + b) long and h high, so their area is 2h(l + b); the floor is l × b, and the cost is the area × the rate
Your family wants to paint a room before Tihar. Change the room's length, breadth and height and the price of paint, find the area of the four walls and the cost, and see why the walls unroll into one long rectangle.
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.
Each transformation changes coordinates by a rule: reflection in the x-axis (x, y) → (x, −y), in the y-axis (x, y) → (−x, y), rotation of 90° anticlockwise about O (x, y) → (−y, x), of 180° (x, y) → (−x, −y), and translation by (a, b) (x, y) → (x + a, y + b)
Move a point P on a grid by reflecting it in an axis, rotating it about the origin or sliding it. Read the coordinates of its image P′ and find the rule for each transformation.
Simple interest is the same every year; compound interest earns interest on interest; depreciation takes a percentage of the value each year
Save money in a cooperative at simple or compound interest, watch the interest on interest grow, and see how a motorbike loses value.
A linear equation y = mx + c is a straight line; two lines meet at the solution of both equations
Plot two taxi fares as straight lines, read them at any distance and find where they cost the same.
In base b each place is worth b times the place to its right (1, b, b², b³, …), and every digit is less than b; to convert, divide by b again and again and read the remainders upwards
Put a number on a place-value abacus in base 2 or base 5. See how many beads each rod gets, read the number in the new base, and check it by multiplying back.
Discount is a percentage of the marked price; VAT is a percentage of the selling price after the discount
Take a discount off a marked price, add 13% VAT to the selling price, and see when the bill ends up below the price on the tag.
The angle of each slice is its share of the total × 360°
Share out a family's monthly income between food, rent, school fees and savings, and draw the budget as a pie chart.
Probability = number of favourable cards ÷ 52, and P(not E) = 1 − P(E)
Shuffle a pack of 52 playing cards and draw one. Find the probability of a red card, a king, a face card or the ace of spades, and draw many times to see how often each really comes up.
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.
Probability = favourable outcomes ÷ all equally likely outcomes, and the share seen in many trials gets close to it
Fill a lucky-draw bag with coloured balls, place the chance of winning on a scale from impossible to certain, and draw many times to test it.
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.
In a right-angled triangle a² + b² = c²; when c² is not a perfect square, c is an irrational surd that may simplify to k√m
Cut a diagonal brace for a wooden gate: square the sides, add them, and find out when the brace is a whole number and when it is a surd.
A rational number can be written as p/q (q ≠ 0) and its decimal ends or repeats; an irrational number cannot, and its decimal never ends and never repeats; integers are rational too
Write twelve numbers, from √9 and 22/7 to π and √2, as decimals and as p/q, and sort them into integers, rational numbers and irrational numbers.
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.
List every equally likely outcome; P(event) = favourable outcomes ÷ all outcomes
Lay out all 36 ways two dice can land, light up the outcomes you want, and test the probability with many rolls.
Any positive number can be written as a × 10ⁿ with 1 ≤ a < 10; n counts how many places the decimal point moves
Write the sizes of a virus, a hair, Mount Everest and the distance to the Sun in standard form, a × 10ⁿ, on a powers-of-ten ruler.
The three angles of a triangle add up to 180°, and an exterior angle equals the sum of the two interior angles opposite it
Change two angles of a triangle and watch the third. Carry one side on past a corner, and find that the angle outside it equals the two opposite angles inside put together.
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”.
If a job takes a hours, 1/a of it is done each hour; rates add when working together and a drain's rate is subtracted
Open two taps and a drain on a rooftop water tank and find how long it takes to fill, by adding the work done each hour.
A 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.
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.
For ax² + bx + c with zeroes α and β, α + β = −b/a and αβ = c/a; when the discriminant b² − 4ac is negative there are no real zeroes
Change a, b and c in ax² + bx + c and watch where the parabola meets the x-axis. Check that the two zeroes always add up to −b/a and multiply to c/a.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.