36 experiments found
Clear filters(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.
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 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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.