Course

Junior Mathematics: arithmetic and geometry foundations

Name: Junior Mathematics: arithmetic and geometry foundations
Price: 999 BDT

The classic word problems (ratio, percentage, speed, mixtures) and the geometry every later course needs: angles, triangles, Pythagoras, polygons, circles, area and volume, learned by doing puzzles.

99 lessons9h 20mEnglish

What you'll learn

Identify which two quantities a ratio actually compares, and avoid the part-to-part versus part-to-whole trap
Share a total in a given ratio by finding the value of one part first
Apply the unitary method to scale a rate to any amount a problem asks for
Tell direct proportion from inverse proportion by checking whether the second quantity grows or shrinks
Find a percent of a whole, and explain why the same percent gives a different amount for a different whole
Compute a percent increase or decrease, and explain why a rise and an equal-sounding fall never fully cancel out
Find profit or loss as a percent of cost price, and follow a price through a mark-up and a discount
Explain why two successive discounts never simply add, and why compound interest outgrows simple interest
Compute a distance from speed and time, and adjust a boat’s speed for a current running with or against it
Find a meeting time by adding two speeds head-on, or a catch-up time by subtracting them from behind
Compute a blended value from a mixing ratio, and explain why a blend never falls outside its two ingredients
Use the cross-difference shortcut to find the ratio that blends two values to a given target
Find any missing angle on lines, in triangles and in polygons, and give the reason for every step
See why Pythagoras' theorem is true, and use it, its triples and the two special triangles to find a side
Find the perimeter, area and volume of everyday shapes, including circles and a triangle given only its sides

Requirements

Comfortable with multiplication, division and simplifying fractions
Has completed Junior Algebra or an equivalent introduction to ratios

Course content

99 lessons · 9h 20m

Ratio, proportion and the unitary method

What a ratio actually compares, part to part or part to whole, then sharing a total fairly, finding the value of one unit and scaling it, and telling direct proportion apart from its opposite.

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Comparing two amounts

A ratio compares two named quantities, in a fixed order.

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Part to part, or part to whole?

Girls to boys and girls to the whole class are two different ratios.

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Sharing a total fairly

Find the value of one part, then multiply each side by its parts.

The unitary method: one first

Find the cost of one unit, then scale it to whatever amount is asked.

Direct proportion, and its opposite

More workers means less time: inverse proportion runs the other way.

Check what you know

Three questions on ratios, the unitary method and proportion type.

Percentage and percent change

A percent as a fraction out of a hundred, then percent increase and decrease, always measured against the original, closing with the classic trap: a rise and an equal-sounding fall never fully cancel out.

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A fraction out of a hundred

Percent means "out of a hundred," nothing more mysterious.

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Percent of a whole

The same percent gives a different amount for every different whole.

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Percent increase

An increase is measured as a percent of the original amount.

Percent decrease

A decrease uses the same rule as an increase, change over original.

Up then down is not even

A rise and an equal-sounding fall are measured against different wholes.

Check what you know

Three questions on percent of a whole and percent change.

Profit, loss, discount and interest

Profit and loss as a percent of cost price, a mark-up followed by a discount, why two successive discounts never simply add, and simple interest against the compound curve that earns interest on its own interest.

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Profit as a percent of cost

Profit is measured against the cost price, never the selling price.

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Loss as a percent of cost

Loss uses the same rule as profit, just pointed the other way.

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Marked price, then a discount

A discount off a marked-up price can still leave a profit.

Two discounts do not add

A second discount acts on the already-reduced price, so 20% then 25% is not 45%.

Simple and compound interest

Compound interest earns interest on its own interest, a curve, not a line.

Check what you know

Three questions on profit, discount and successive percent changes.

Speed, distance and time

Distance as speed repeated for however long you travel, a current that adds to or resists a boat depending on direction, then two objects closing a gap: add the speeds head-on, subtract them chasing from behind.

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Distance equals speed times time

Distance is the same rate-times-count idea as the unitary method.

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Downstream: the current helps

A current flowing the same way adds to the boat’s own speed.

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Upstream: the current resists

The same current subtracts from the boat’s speed going the other way.

Closing the gap head-on

Moving toward each other, the two speeds add to close the gap.

Catching up from behind

Moving the same way, only the speed difference closes a head start.

Check what you know

Three questions on distance, current, and closing a gap.

Mixture and alligation

Blending two prices as a weighted average that always sits between its ingredients, then running that same relationship backward: given a target blend, find the ratio with the cross-difference shortcut.

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Blending two prices

An equal-shares blend lands exactly at the plain average.

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More of the cheaper one

A bigger share pulls the blend toward its own value.

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Alligation: finding the ratio

Given a target blend, drag to find the ratio that produces it.

The cross-difference shortcut

The ratio equals the two distances to the target, swapped.

A real mixture problem

Apply the shortcut to a fresh price-mixing word problem.

Check what you know

Three questions on blending values and the alligation ratio.

Angles and lines

Name every kind of angle, find missing angles on a line, at a point and where roads cross, then chase angles through parallel lines, where big angles are equal and small angles are equal.

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Acute, right, obtuse, straight and reflex angles

Turn the clock hands and name the angle they make, from a small acute angle to a reflex angle past 180°.

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Complementary and supplementary angles

Two angles that make a right angle are complementary; two that make a straight line are supplementary.

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Angles on a straight line and at a point

A straight line holds 180° and a full turn holds 360°. Every missing-angle puzzle starts here.

Vertically opposite angles

Where two roads cross, the angles facing each other are equal. Find out why with one straight line.

Parallel lines: big angles are equal, small angles are equal

A road crossing two parallel rail lines makes only two sizes of angle, and a big one plus a small one is 180°.

Corresponding, alternate and co-interior angles

The F, Z and C shapes give the reasons their names, so every step of a puzzle can say why.

Angle puzzles: a timed round

Zigzags, crossing roads and parallel lines against the clock. Every answer shows its worked reason.

Check what you know: angles and lines

Five questions on angle types, lines, points, crossing lines and parallel lines.

Triangles

The angles of a triangle add to 180°, the exterior angle rule, the kinds of triangle, height and median, which three sticks make a triangle, the longest side and the largest angle, and when two triangles match.

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The angles of a triangle add to 180°

Tear off the three corners and they sit on a straight line. Then find the third angle of any triangle.

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The exterior angle of a triangle

Stretch one side out. The outside angle equals the two far inside angles added: 50° and 80° make 130°.

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Equilateral, isosceles, scalene and right-angled triangles

Equal sides mean equal angles. An isosceles triangle with a 40° top has two 70° base angles.

Height and median: two lines from a corner

A height meets the opposite side at 90°; a median meets it at its middle. In an isosceles triangle they are one line.

Can three sticks make a triangle?

Any two sides must add to more than the third. Sticks of 3, 4 and 8 can never close.

The longest side faces the largest angle

Open an angle wider and the side across from it grows. So you can order the angles from the sides.

Congruent and similar triangles

Same shape and size is congruent (SSS, SAS, ASA, RHS); same shape, any size is similar, with every side scaled.

The midpoint theorem

Join the middles of two sides: the line is parallel to the third side and exactly half as long.

Triangle puzzles: a timed round

Missing angles, isosceles traps and impossible triangles against the clock, each with a worked reason.

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Check what you know: triangles

Five questions on the angle sum, the exterior angle, kinds of triangle and which sides can make one.

Right-angled triangles and Pythagoras' theorem

Move four triangles inside a square to see a² + b² = c², find the long side and a short side, check a corner with a knotted rope, and learn the triples and the two special triangles that admission tests love.

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Pythagoras' theorem by moving four triangles

The same four triangles leave one big square uncovered, or two smaller squares. So the areas must match.

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Finding the longest side

Square, add, square root. A kite frame with 30 cm and 40 cm sticks needs a 50 cm edge.

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Finding a short side

Take away instead of adding. A 13 m bamboo reaching 12 m up a wall stands 5 m out.

The knotted rope: is the corner square?

A rope with 12 equal gaps pulled into 3, 4 and 5 makes a perfect right angle. Test other triangles the same way.

Pythagorean triples: 3-4-5, 5-12-13 and friends

3:4:5, 5:12:13, 7:24:25, 8:15:17, 9:40:41 and their multiples. Spot them and skip the square roots.

Two special triangles: half a square, half an equilateral triangle

Sides in the ratio 1 : 1 : √2 and 1 : √3 : 2. A square pond 10 m wide is about 14.1 m corner to corner.

Pythagoras puzzles: a timed round

Ladders, kites, ponds and a snapped tree against the clock, each with a worked reason.

Check what you know: Pythagoras' theorem

Five questions: the longest side, a short side, a right-angle test, a triple and a special triangle.

Quadrilaterals and polygons

The angles of a quadrilateral add to 360°, the family of special quadrilaterals, the angle sum of any polygon from triangles, the exterior angles that always make 360°, and which regular shapes tile a floor.

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The angles of a quadrilateral add to 360°

One diagonal cuts any quadrilateral into two triangles, so its angles make 2 × 180° = 360°.

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Square, rectangle, parallelogram, rhombus, kite and trapezium

Each name promises equal sides, parallel sides or equal angles. A square keeps every promise.

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The angle sum of any polygon

From one corner, an n-sided polygon splits into n − 2 triangles. A hexagon holds 4 × 180° = 720°.

Exterior angles make a full turn

Walk round any polygon and you turn once: 360°. A regular pentagon turns 72° at each corner.

Regular polygons and floor tiles

Tiles fit round a point only if their angles divide 360°. Triangles, squares and hexagons work; pentagons do not.

Polygon puzzles: a timed round

Missing angles in quadrilaterals and polygons, and how many sides, against the clock.

Check what you know: quadrilaterals and polygons

Five questions on quadrilateral angles, shape names, polygon angle sums and exterior angles.

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Circles

Name the parts of a circle, roll a wheel to find the circumference, cut a circle into slices to find its area, take arcs and sectors as fractions, and meet the centre angle, the semicircle and the tangent.

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Parts of a circle: radius, diameter, chord, arc, sector and tangent

Drag the centre and the rim, and name each line and region. The diameter is the longest chord.

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Circumference: roll a wheel once

One turn of a wheel covers about 3.14 diameters. A 70 cm bicycle wheel rolls 220 cm per turn.

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Area of a circle: slice it into a rectangle

Cut a circle into thin slices and lay them head to tail: a rectangle πr by r. A 7 cm circle covers 154 cm².

Arcs and sectors: slices of a circle

A 90° slice is a quarter of the circle, so it takes a quarter of the area and a quarter of the rim.

Angles in a circle: centre, edge and semicircle

The angle at the centre is double the angle at the edge, so the angle in a semicircle is always 90°.

A tangent touches once, at a right angle

A tangent meets the circle at one point and makes 90° with the radius there.

Circle puzzles: a timed round

Wheels, ponds, cake slices and angles in circles against the clock, each with a worked reason.

Check what you know: circles

Five questions on parts, circumference, area, sectors and the angle in a semicircle.

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Perimeter, area and volume

Perimeter as the distance round, area of rectangles, triangles, parallelograms and trapeziums by cutting and moving, paths and L-shapes, Heron's formula as a challenge, then solids, nets, volume and surface area.

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Perimeter: the distance round

Walk round the school pond path and add every side. A 12 m by 8 m rectangle has a 40 m perimeter.

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Area of rectangles and triangles

A triangle is half of the rectangle around it, so its area is ½ × base × height.

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Area of parallelograms and trapeziums

Cut and slide a parallelogram into a rectangle; two trapeziums make a parallelogram.

Paths, ponds and L-shapes

Split a shape into rectangles, or take the hole away. A 1 m path round a 10 m by 6 m pond covers 36 m².

Challenge: area from three sides with Heron's formula

No height given? Half the perimeter s does it: a 13, 14, 15 triangle has area 84. Equilateral: √3/4 × side².

Solids: faces, edges, corners and nets

Unfold a box into a flat net. A cube has 6 faces, 12 edges and 8 corners.

Volume and surface area of a box

Count the cubes layer by layer: a 10 × 6 × 4 cm box holds 240 cm³ and has 248 cm² of card.

Perimeter, area and volume puzzles: a timed round

Fences, floors, paths and boxes against the clock, each with a worked reason.

Check what you know: perimeter, area and volume

Five questions on perimeter, area of four shapes, a path, and the volume of a box.

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Coordinates: points on a grid

Plot points across then up, use negative numbers in four quadrants, find perimeter and area on a grid, the midpoint of two points, and the distance between them with Pythagoras.

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Plotting points: across, then up

(3, 5) means 3 across and 5 up. Swap them and you land somewhere else.

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Negative coordinates and the four quadrants

Left of zero is negative across, below zero is negative up. (−3, 1) sits in the top-left quadrant.

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Shapes on a grid: perimeter and area

Corners at (0, 0), (6, 0), (6, 4) and (0, 4) make a rectangle with perimeter 20 and area 24.

The midpoint: halfway between two points

Average the across numbers and average the up numbers. Halfway from (1, 2) to (7, 10) is (4, 6).

Distance between two points with Pythagoras

Draw the right-angled triangle under the line: 6 across and 8 up make a 10-unit distance.

Grid puzzles: a timed round

Plot, find the midpoint and measure the distance against the clock, each with a worked reason.

Check what you know: coordinates

Five questions on plotting, quadrants, grid shapes, midpoints and distances.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.