34 experiments found
Clear filters(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.
Area 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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.