Exponent rules let you simplify expressions with powers quickly, and scientific notation gives us a compact way to write very large or very small numbers.
✖️ Multiplying & dividing powers
Product of powers
am × an = am+n
Same base, multiply → add the exponents.
Quotient of powers
am ÷ an = am−n
Same base, divide → subtract the exponents.
2³ × 2² = 2⁵ = 32
2⁵ ÷ 2² = 2³ = 8
🔁 Power of a power & special cases
(am)n = am×n
(2²)³ = 2⁶ = 64
Zero exponent
a⁰ = 1
Negative exponent
a−n = 1/an
4⁻² = 1/4² = 1/16 = 0.0625
🔭 Scientific notation
Scientific notation writes a number as a × 10n, where 1 ≤ a < 10. It's a fast way to write very large or very small numbers.
45,000 = 4.5 × 10⁴
0.00056 = 5.6 × 10⁻⁴
A positive exponent means a big number; a negative exponent means a small number (less than 1).
✏️ Quick check · 1 of 10
Multiply powers
What is 2³ × 2²?
Same base and you are multiplying, so add the exponents.
✍️ Type your answer · 2 of 10
Evaluate the expression.
What is 5⁰?
Enter an exact answer, without rounding.
Any non-zero number to the power 0 has the same value.
✏️ Quick check · 3 of 10
Power of a power
What is (2²)³?
Power of a power: multiply the exponents.
✍️ Type your answer · 4 of 10
Evaluate the expression.
What is 4⁻²?
Enter an exact answer, without rounding.
A negative exponent means 1 divided by the positive power: 1/4².
✏️ Quick check · 5 of 10
Write in scientific notation
Which is the correct scientific notation for 45,000?
The first number must be at least 1 and less than 10. Count how many places the decimal point moves.
✍️ Type your answer · 6 of 10
Convert from scientific notation.
What is 6 × 10³ written as a standard number?
Enter an exact answer, without rounding.
Multiply 6 by 10 three times, or move the decimal point 3 places right.
✏️ Quick check · 7 of 10
Quotient of powers
What is 3⁴ ÷ 3²?
Same base and you are dividing, so subtract the exponents.
✍️ Type your answer · 8 of 10
Evaluate the expression.
What is 2⁻³?
Enter an exact answer, without rounding.
2⁻³ means 1/2³. Work out 2³ first.
✏️ Quick check · 9 of 10
Compare large numbers
Which number is larger: 5 × 10³ or 4 × 10⁴?
Compare the powers of 10 first: the bigger power wins.
✍️ Type your answer · 10 of 10
Evaluate the expression.
What is 2⁵ ÷ 2³?
Enter an exact answer, without rounding.
Subtract the exponents, then work out the power of 2.
🖼️ See it
Move the decimal point until one non-zero digit is in front of it; the number of places moved is the exponent (positive for big numbers, negative for small ones).
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Simplify 7⁵ × 7³.
Question 2 of 5
What is 10⁻²?
Question 3 of 5
Write 0.00072 in scientific notation.
Question 4 of 5
Simplify (3²)⁴.
Question 5 of 5
What is 9⁰ + 2³?
Lesson 2 · Numbers
√ Square Roots & Irrational Numbers
A square root asks "what number times itself gives this?" Not every number has a neat, whole-number answer — those are called irrational numbers.
📑 Perfect squares
A perfect square is the result of multiplying a whole number by itself.
Square root
Perfect square
√1
1
√4
4
√9
9
√16
16
√25
25
💎 Cube roots
A cube root asks "what number times itself three times gives this?" It's written as ∛x, using a small 3 in the radical's index.
∛8 = 2 (since 2³ = 8)
🌐 Irrational numbers
An irrational number cannot be written as a simple fraction — its decimal goes on forever without repeating.
Rational
√16 = 4, 0.5, 3/4 — terminating or repeating decimals.
Subtract 2x from both sides, then subtract 5 from both sides.
✏️ Quick check · 2 of 10
Solve with distribution
Solve: 4(x − 2) = 20
Multiply out the bracket first: 4 × x and 4 × 2.
✍️ Type your answer · 3 of 10
Solve for x.
5x − 3 = 3x + 9
Enter an exact answer, without rounding.
Get the x-terms on one side: subtract 3x from both sides.
✏️ Quick check · 4 of 10
Recognise no solution
How many solutions does 2x + 3 = 2x + 7 have?
Subtract 2x from both sides. What is left: a true or a false statement?
✍️ Type your answer · 5 of 10
Solve for x.
2(x + 3) = 3x − 1
Enter an exact answer, without rounding.
Expand the bracket first, then collect the x-terms on one side.
✏️ Quick check · 6 of 10
Recognise infinitely many solutions
How many solutions does 3x + 6 = 3(x + 2) have?
Expand 3(x + 2). Are the two sides exactly the same?
✍️ Type your answer · 7 of 10
Solve for x.
7x + 2 = 30
Enter an exact answer, without rounding.
Subtract 2 from both sides, then divide by 7.
✏️ Quick check · 8 of 10
Solve with variables on both sides
Solve: 5x − 2 = 3x + 8
Subtract 3x from both sides, then add 2 to both sides.
✍️ Type your answer · 9 of 10
Solve for x.
9 − 2x = 1
Enter an exact answer, without rounding.
Subtract 9 from both sides, then divide by −2.
✏️ Quick check · 10 of 10
Recognise another no-solution case
How many solutions does 4x − 1 = 4x + 5 have?
Subtract 4x from both sides. Is −1 = 5 ever true?
🖼️ See it
Undo the operations in reverse order and do the same thing to both sides to keep the equation balanced.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Solve 6x − 4 = 2x + 12.
Question 2 of 5
Solve 3(x + 1) = 18.
Question 3 of 5
How many solutions does 5x + 2 = 5x − 3 have?
Question 4 of 5
How many solutions does 2(x + 4) = 2x + 8 have?
Question 5 of 5
Solve 10 − 3x = 1.
Lesson 4 · Algebra
📈 Linear Functions & Gradient
Gradient measures how steep a line is — how much y changes for every step in x. Gradient-intercept form packs both the gradient and starting point into one equation.
📏 Finding gradient
Gradient is "rise over run" — the change in y divided by the change in x between two points.
m = (y₂ − y₁) / (x₂ − x₁)
Points: (2, 3) and (6, 11)
m = (11 − 3) / (6 − 2) = 8/4 = 2
gradient = rise ÷ run = 8 ÷ 4 = 2
✏️ Gradient-intercept form
y = mx + b
m is the gradient, and b is the y-intercept (where the line crosses the y-axis).
y = 5x − 2
gradient (m) = 5
y-intercept (b) = −2
y-interceptgradient = rise/run = 5/1
🧭 Reading the sign of the gradient
Gradient
Line looks like
Positive
Goes up left to right
Negative
Goes down left to right
Zero
Perfectly horizontal (flat)
Undefined
Perfectly vertical
✍️ Type your answer · 1 of 10
Find the gradient of the line through the two points.
Gradient between (2, 3) and (6, 11)?
Enter an exact answer, without rounding.
Gradient = change in y ÷ change in x. Work out 11 − 3 and 6 − 2.
✏️ Quick check · 2 of 10
Read gradient from an equation
What is the gradient of y = −3x + 5?
In y = mx + b the gradient is the number in front of x.
✍️ Type your answer · 3 of 10
Find the gradient of the line through the two points.
Gradient between (1, 4) and (4, 4)?
Enter an exact answer, without rounding.
The y-values are the same. What is the rise?
✏️ Quick check · 4 of 10
Undefined gradient
A line with an undefined gradient is:
Undefined gradient means the run is 0: x never changes.
✍️ Type your answer · 5 of 10
Read the equation.
In y = 5x − 2, what is the y-intercept?
Enter an exact answer, without rounding.
In y = mx + b, the y-intercept is b, including its sign.
✏️ Quick check · 6 of 10
Spot the negative gradient
Which equation has a negative gradient?
A negative gradient has a minus sign in front of the x-term.
✍️ Type your answer · 7 of 10
Find the gradient of the line shown below, passing through (0, 1) and (4, 9).
What is the gradient of this line?
Enter an exact answer, without rounding.
Rise from y = 1 to y = 9, run from x = 0 to x = 4.
✏️ Quick check · 8 of 10
Read another y-intercept
What is the y-intercept of y = −2x + 7?
The y-intercept is the number added on at the end, b.
✍️ Type your answer · 9 of 10
Find the gradient of the line through the two points.
Gradient between (−1, 5) and (3, −3)?
Enter an exact answer, without rounding.
Change in y is −3 − 5; change in x is 3 − (−1).
✏️ Quick check · 10 of 10
Compare steepness
Which line is steeper: one with gradient 3, or one with gradient 1/2?
The bigger the size of the gradient, the steeper the line.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is the gradient between (1, 2) and (3, 8)?
Question 2 of 5
What is the y-intercept of y = 4x − 9?
Question 3 of 5
A horizontal line has a gradient of:
Question 4 of 5
Which line goes down from left to right?
Question 5 of 5
Which equation has gradient 3 and y-intercept −1?
Lesson 5 · Algebra
🎯 Functions: Domain, Range & the Vertical Line Test
A function is one of the most important ideas in algebra — a precise rule that every input maps to exactly one output, and there's a simple visual trick to test it.
📌 What makes a function
A function is a relation where every input (the domain) produces exactly one output (the range).
The vertical line test: if a vertical line ever crosses a graph at more than one point, the graph does NOT represent a function.
🔍 The vertical line test in action
This IS a function
This is NOT a function
✏️ Quick check · 1 of 6
Define a function
A function is a relation where:
Think of a vending machine: one button always gives the same snack.
✏️ Quick check · 2 of 6
Purpose of the vertical line test
The vertical line test checks whether:
If a vertical line hits the graph twice, one input has two outputs.
✏️ Quick check · 3 of 6
Define the domain
The domain of a function refers to:
Domain = what goes in. Is that x or y?
✍️ Type your answer · 4 of 6
A function has inputs {1, 2, 3, 4} and outputs {2, 4, 6, 8}.
How many elements are in the domain?
Enter an exact answer, without rounding.
The domain is the set of inputs. Count them.
✍️ Type your answer · 5 of 6
Same function as above: inputs {1, 2, 3, 4}, outputs {2, 4, 6, 8}.
How many elements are in the range?
Enter an exact answer, without rounding.
The range is the set of outputs. Count them.
✍️ Type your answer · 6 of 6
A graph passes the vertical line test, meaning every vertical line crosses it at exactly one point.
How many outputs correspond to each input on a graph that passes the vertical line test?
Enter an exact answer, without rounding.
Passing the test means a vertical line never crosses the graph twice.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Which set of pairs is a function?
Question 2 of 5
What is the range of {(1, 4), (2, 5), (3, 6)}?
Question 3 of 5
A circle drawn on a grid:
Question 4 of 5
For f(x) = 2x + 1, what is f(3)?
Question 5 of 5
The domain of a function is the set of all:
Lesson 6 · Algebra
🔗 Systems of Equations
A system of equations is a set of two or more equations sharing the same variables. Solving it means finding the values that make every equation true at once.
📍 What is a solution?
Graphically, the solution to a system of two linear equations is the point where their lines cross.
Lines cross once
One solution.
Lines never cross (parallel)
No solution.
Solution: where the lines cross, at (2, 3)
🔄 Solving by substitution
Set the two expressions equal to each other, since both equal y.
y = x + 2
y = 3x − 4
x + 2 = 3x − 4
6 = 2x
x = 3
y = 3 + 2 = 5
➕ Solving by elimination
Add or subtract the equations to cancel out one variable.
x + y = 10
x − y = 2
────── (add the equations)
2x = 12
x = 6
y = 10 − 6 = 4
✍️ Type your answer · 1 of 10
Solve the system by substitution: y = x + 2 and y = 3x − 4.
What is x?
Enter an exact answer, without rounding.
Both equal y, so set x + 2 equal to 3x − 4.
✏️ Quick check · 2 of 10
Parallel lines
A system of two lines that never intersect has:
Parallel lines never meet. How many shared points is that?
✍️ Type your answer · 3 of 10
Solve the system by elimination: x + y = 10 and x − y = 2.
What is x?
Enter an exact answer, without rounding.
Add the two equations: the y-terms cancel.
✏️ Quick check · 4 of 10
Same line twice
If two equations describe the same line, the system has:
Every point on one line is on the other too.
✍️ Type your answer · 5 of 10
Using the same system (x + y = 10, x − y = 2), you found x = 6.
What is y?
Enter an exact answer, without rounding.
Put x = 6 into x + y = 10.
✏️ Quick check · 6 of 10
Graphical meaning of a solution
The solution to a system of equations is the point where the lines:
A solution has to be on both lines at once.
✍️ Type your answer · 7 of 10
Using the graph from the card above, the lines y = x + 1 and y = −x + 5 intersect at one point.
What is the x-coordinate of the solution?
Enter an exact answer, without rounding.
Look where the two lines cross and read across to the x-axis.
✏️ Quick check · 8 of 10
Read the y-coordinate from the graph
Using the same graph, what is the y-coordinate of the solution?
From the crossing point, read across to the y-axis.
✍️ Type your answer · 9 of 10
Solve the system by substitution: y = 2x and y = x + 3.
What is x?
Enter an exact answer, without rounding.
Both equal y, so set 2x equal to x + 3.
✏️ Quick check · 10 of 10
Different gradients always cross
If two equations in a system describe lines with different gradients, how many solutions will the system have?
Lines with different gradients must cross somewhere. How many times?
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Solve y = x + 1 and y = 2x − 3. What is x?
Question 2 of 5
Solve x + y = 8 and x − y = 4. What is y?
Question 3 of 5
y = 3x + 2 and y = 3x − 5 have:
Question 4 of 5
Is (1, 3) a solution of y = 2x + 1 and y = x + 2?
Question 5 of 5
y = 2x + 4 and 2y = 4x + 8 have:
Lesson 7 · Algebra
💰 Systems of Equations Applications: Break-Even, Mixture & Motion Problems
Systems of equations aren't just abstract algebra — they're exactly how businesses find their break-even point, chemists mix solutions, and physicists track two moving objects.
💰 Break-even point
A break-even point is where a cost equation equals a revenue equation.
Mixture problems combine two different quantities (concentrations or prices) to find a total. Motion problems use distance = rate × time, often with two objects moving toward or away from each other.
✏️ Quick check · 1 of 6
Define break-even
A break-even point is where:
At break-even you make no profit and no loss.
✍️ Type your answer · 2 of 6
Cost = 200 + 5x, Revenue = 15x.
At how many units (x) does break-even occur?
Enter an exact answer, without rounding.
Set 200 + 5x equal to 15x and solve.
✏️ Quick check · 3 of 6
Define mixture problems
Mixture problems typically involve combining:
Think of mixing two drinks of different strengths or two foods of different prices.
✍️ Type your answer · 4 of 6
20 kg of $2/kg sweets is mixed with 10 kg of $5/kg sweets.
What is the total cost of both types combined, in dollars? (20×2 + 10×5)
Enter an exact answer, without rounding.
Work out 20 × 2 and 10 × 5, then add.
✏️ Quick check · 5 of 6
Motion problem formula
Motion problems typically use the formula:
Speed is distance per hour. Rearrange that.
✍️ Type your answer · 6 of 6
Two cars start 300 km apart and drive toward each other, one at 40 km/h and one at 60 km/h.
What is their combined closing speed, in km/h?
Enter an exact answer, without rounding.
Moving towards each other, the gap closes by both speeds added together.
🖼️ See it
Where the lines cross, cost equals revenue: 200 + 5x = 15x gives x = 20. Selling more than 20 units makes a profit.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Cost = 100 + 4x and revenue = 9x. How many units to break even?
Question 2 of 5
A car travels at 50 km/h for 3 hours. How far does it go?
Question 3 of 5
Two trains start 240 km apart and drive towards each other at 50 km/h and 70 km/h. When do they meet?
Question 4 of 5
5 kg of $3/kg nuts is mixed with 5 kg of $5/kg nuts. What is the price per kg of the mix?
Question 5 of 5
Past the break-even point, a business:
Lesson 8 · Geometry
🔄 Transformations
A transformation moves or resizes a figure. Translations slide it, reflections flip it, rotations turn it, and dilations resize it.
➔ Translations
A translation slides every point of a figure the same distance in the same direction — the shape and size never change.
Point (3, 5), translated 4 right and 2 down:
new point = (3 + 4, 5 − 2) = (7, 3)
Original pointTranslated point (4 right, 2 down)
🧝 Reflections
Reflect over the x-axis
Flip the sign of the y-coordinate.
Reflect over the y-axis
Flip the sign of the x-coordinate.
(2, −6) reflected over the x-axis → (2, 6)
Reflect over the x-axis
Reflect over the y-axis
🔄 Rotations & 🔍 dilations
A rotation turns a figure around a fixed point. A dilation resizes a figure by a scale factor from a centre point, without changing its shape.
Point (4, 5), dilated by scale factor 3 from the origin:
new point = (4 × 3, 5 × 3) = (12, 15)
A scale factor greater than 1 enlarges the figure; a scale factor between 0 and 1 shrinks it.
Rotation 90° around origin
Dilation, scale factor 3
✍️ Type your answer · 1 of 10
Point (3, 5) is translated 4 units right and 2 units down.
What is the new x-coordinate?
Enter an exact answer, without rounding.
Right means add to x.
✏️ Quick check · 2 of 10
Reflection over the x-axis
Reflecting a point over the x-axis changes the sign of:
Flipping over the x-axis moves a point up or down, not left or right.
✍️ Type your answer · 3 of 10
Point (2, −6) is reflected over the x-axis.
What is the new y-coordinate?
Enter an exact answer, without rounding.
Over the x-axis, the y-coordinate changes sign.
✏️ Quick check · 4 of 10
Effect of a dilation
A dilation with a scale factor greater than 1 makes a figure:
A scale factor bigger than 1 stretches.
✍️ Type your answer · 5 of 10
Point (4, 5) is dilated by a scale factor of 3 from the origin.
What is the new x-coordinate?
Enter an exact answer, without rounding.
Multiply the x-coordinate by the scale factor.
✏️ Quick check · 6 of 10
Name the transformation
Which transformation flips a figure over a line?
Think of a mirror image.
✍️ Type your answer · 7 of 10
Point (5, −2) is translated 3 units left and 4 units up.
What is the new x-coordinate?
Enter an exact answer, without rounding.
Left means subtract from x.
✍️ Type your answer · 8 of 10
Using the same translation (3 left, 4 up) on point (5, −2):
What is the new y-coordinate?
Enter an exact answer, without rounding.
Up means add to y.
✏️ Quick check · 9 of 10
Effect of a small scale factor
A dilation with a scale factor of 1/2 makes a figure:
A scale factor between 0 and 1 shrinks.
✏️ Quick check · 10 of 10
Rotating 180°
Rotating a point (a, b) by 180° about the origin sends it to:
A half turn flips both signs.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
(−2, 4) is translated 5 right and 1 down. Where does it land?
Question 2 of 5
(3, −7) is reflected over the y-axis. Where does it land?
Question 3 of 5
(6, 2) is dilated by scale factor 1/2 from the origin. Where does it land?
Question 4 of 5
Which transformation does NOT always give a congruent image?
Question 5 of 5
(1, 5) is rotated 180° about the origin. Where does it land?
Lesson 9 · Geometry
🔧 Congruence, Similarity & the Triangle Angle Sum Theorem
Some transformations preserve a shape's exact size, while others only preserve its form — and every triangle, no matter how it's transformed, always obeys one unbreakable rule about its angles.
🔄 Congruent vs. similar
Congruent
Exactly the same size and shape. Translations, reflections, and rotations always produce congruent figures.
Similar
Same shape but a different size — corresponding angles are equal and sides are proportional. Dilations produce similar, not congruent, figures.
△ The Triangle Angle Sum Theorem
The three interior angles of ANY triangle always add up to 180°.
a + b + c = 180° — true for every triangle
✏️ Quick check · 1 of 6
Define congruent
Congruent figures are:
Congruent shapes would fit exactly on top of each other.
✏️ Quick check · 2 of 6
Define similar
Similar figures have:
Similar shapes are enlargements of each other.
✏️ Quick check · 3 of 6
Similar-producing transformation
Which transformation creates similar (not congruent) figures?
Which transformation changes the size?
✍️ Type your answer · 4 of 6
A triangle has two angles measuring 50° and 60°.
What is the measure of the third angle, in degrees?
Enter an exact answer, without rounding.
degrees
Add the two angles and take the total from 180°.
✍️ Type your answer · 5 of 6
A triangle has two angles measuring 90° and 40°.
What is the measure of the third angle, in degrees?
Enter an exact answer, without rounding.
degrees
Add the two angles and take the total from 180°.
✍️ Type your answer · 6 of 6
This is true for every triangle, no matter its shape.
What do the three interior angles of any triangle always add up to, in degrees?
Enter an exact answer, without rounding.
degrees
Tear off the corners of a triangle and they make a straight line.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
A triangle has angles of 45° and 75°. What is the third angle?
Question 2 of 5
Two triangles are similar. Their corresponding angles are:
Question 3 of 5
A triangle is reflected and then translated. The image is:
Question 4 of 5
Triangle sides 3, 4, 5 and 6, 8, 10 are:
Question 5 of 5
Can a triangle have two right angles?
Lesson 10 · Geometry
📐 Angle Relationships
When a transversal cuts across two parallel lines, it creates several special angle pairs. Triangles add their own angle rule on top of that.
✂️ Parallel lines & a transversal
Corresponding angles
Same position at each intersection — always equal.
Alternate interior angles
Between the parallel lines, on opposite sides — always equal.
Same-side interior angles
Between the parallel lines, same side — always supplementary (add to 180°).
Square both legs, add them, then take the square root.
✏️ Quick check · 2 of 10
Identify the hypotenuse
In a² + b² = c², which side is c?
It is the longest side, opposite the right angle.
✍️ Type your answer · 3 of 10
A right triangle has a hypotenuse of 13 and one leg of 5.
What is the other leg?
Enter an exact answer, without rounding.
Square the hypotenuse, subtract the leg squared, then take the square root.
✏️ Quick check · 4 of 10
Test for a right triangle
Does a triangle with sides 6, 8, 10 form a right triangle?
Check whether 6² + 8² equals 10².
✍️ Type your answer · 5 of 10
A ladder 10 m long leans against a wall, with its base 6 m from the wall.
How high up the wall does the ladder reach, in metres?
Enter an exact answer, without rounding.
metres
The ladder is the hypotenuse. Work out 10² − 6².
✏️ Quick check · 6 of 10
Find another hypotenuse
A right triangle has legs 9 and 12. What is the hypotenuse?
Work out 9² + 12², then take the square root.
✍️ Type your answer · 7 of 10
A right triangle has legs of 6 and 8.
What is the hypotenuse?
Enter an exact answer, without rounding.
Work out 6² + 8², then take the square root.
✍️ Type your answer · 8 of 10
A right triangle has a hypotenuse of 17 and one leg of 8.
What is the other leg?
Enter an exact answer, without rounding.
Work out 17² − 8², then take the square root.
✏️ Quick check · 9 of 10
Test another right triangle
Does a triangle with sides 5, 12, 13 form a right triangle?
Check whether 5² + 12² equals 13².
✍️ Type your answer · 10 of 10
A right triangle has legs 7 and 24.
What is the hypotenuse?
Enter an exact answer, without rounding.
Work out 7² + 24², then take the square root.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
A right triangle has legs 5 and 12. What is the hypotenuse?
Question 2 of 5
A right triangle has hypotenuse 10 and one leg 8. What is the other leg?
Question 3 of 5
Do sides 4, 5, 6 make a right triangle?
Question 4 of 5
A TV screen is 30 cm wide and 40 cm tall. How long is its diagonal?
Question 5 of 5
In a² + b² = c², the side c is always:
Lesson 12 · Geometry
📏 The Distance Formula & the Converse of Pythagoras’ Theorem
Pythagoras’ theorem does more than find missing triangle sides — it also gives us a way to measure the straight-line distance between any two points on a graph, and even to test whether a triangle is a right triangle at all.
📏 The distance formula
d = √((x₂−x₁)² + (y₂−y₁)²)
The distance formula is derived directly from Pythagoras’ theorem.
Distance between (0,0) and (3,4):
d = √(3²+4²) = √(9+16) = √25 = 5
✅ The converse of Pythagoras’ theorem
If a² + b² = c² for a triangle's three sides, then the triangle IS a right triangle. This lets you test any triangle's sides to check if it's a right triangle.
✏️ Quick check · 1 of 6
Where the distance formula comes from
The distance formula is derived from:
The distance formula is the same idea as a² + b² = c² on a grid.
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Find the distance between (0, 0) and (3, 4) using the distance formula.
What is the distance?
Enter an exact answer, without rounding.
Across 3 and up 4: a right triangle with legs 3 and 4.
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Purpose of the converse
The converse of Pythagoras’ theorem is used to:
The converse runs the theorem backwards: if the sides fit a² + b² = c²…
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A triangle has sides 6, 8, and 10.
What is 6² + 8²?
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36 + 64.
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Find the distance between (1, 1) and (4, 5) using the distance formula.
What is the distance?
Enter an exact answer, without rounding.
Across 4 − 1 and up 5 − 1. Then use a² + b² = c².
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A triangle has sides 5, 12, and 13.
What is 5² + 12²?
Enter an exact answer, without rounding.
25 + 144.
🖼️ See it
The distance between two points is the hypotenuse of a right triangle whose legs are the x- and y-differences. If a² + b² = c² holds, the triangle is right-angled (the converse).
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is the distance between (0, 0) and (6, 8)?
Question 2 of 5
What is the distance between (2, 3) and (5, 7)?
Question 3 of 5
Sides 8, 15, 17: is it a right triangle?
Question 4 of 5
What is the distance between (−1, 2) and (2, 6)?
Question 5 of 5
Sides 7, 8, 12: is it a right triangle?
Lesson 13 · Geometry
🧲 Volume of Cylinders, Cones & Spheres
These three 3D shapes all build on the same circle formula, πr². Each one just scales that base area differently to fill up its volume.
Find the volume of a cylinder with radius 3 and height 10.
Enter an exact answer, without rounding.
V = πr²h. Square the radius first.
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Cone volume formula
The formula for the volume of a cone is:
A cone is a third of a cylinder.
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Use π ≈ 3.14.
Find the volume of a cone with radius 3 and height 6.
Enter an exact answer, without rounding.
Work out πr²h, then divide by 3.
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Sphere volume formula
The formula for the volume of a sphere is:
It uses r cubed.
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Use π ≈ 3.14.
Find the volume of a sphere with radius 3.
Enter an exact answer, without rounding.
V = (4/3)πr³. Work out 3³ first.
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Compare cone and cylinder
A cylinder and a cone have the same radius and height. The cone's volume is what fraction of the cylinder's volume?
Picture filling a cylinder with cones of water.
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Use π ≈ 3.14.
Find the volume of a cylinder with radius 2 and height 5.
Enter an exact answer, without rounding.
V = πr²h with r = 2 and h = 5.
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Use π ≈ 3.14.
Find the volume of a cone with radius 4 and height 9.
Enter an exact answer, without rounding.
Work out 3.14 × 16 × 9, then divide by 3.
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Find another sphere's volume
Use π ≈ 3.14. Find the volume of a sphere with radius 6.
Work out 6³, multiply by 3.14, then by 4/3.
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Doubling the radius of a sphere
If you double the radius of a sphere, by what factor does the volume increase?
The radius is cubed. What is 2³?
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Use π ≈ 3.14. What is the volume of a cylinder with radius 1 and height 10?
Question 2 of 5
A cylinder holds 90 cm³. A cone with the same radius and height holds:
Question 3 of 5
Use π ≈ 3.14. What is the volume of a cone with radius 3 and height 4?
Question 4 of 5
If you double a cylinder's height and keep its radius, its volume:
Question 5 of 5
Use π ≈ 3.14. Which is biggest: a cylinder (radius 3, height 3), a cone (radius 3, height 3) or a sphere (radius 3)?
Lesson 14 · Data
📉 Scatter Plots & Data Modeling
A scatter plot shows the relationship between two related quantities. Reading its shape — direction, strength, outliers, and clusters — helps you spot patterns and make predictions.
⚫ What is a scatter plot?
Each point on a scatter plot represents one individual with two measurements — like a student's weight and their backpack's weight.
Point (50, 6) could mean:
a student who weighs 50 kg
carrying a 6 kg backpack
↗️ Direction & strength of association
Positive association
As x increases, y tends to increase too.
Negative association
As x increases, y tends to decrease.
The strength of the association depends on how closely the points cluster around a line: tightly packed points show a strong association; loosely scattered points show a weak one.
Positive association
Negative association
🔍 Outliers & clusters
Outlier
A single point that doesn't fit the overall pattern.
Cluster
A group of points that sit close together, separate from the rest.
Outlier example
Cluster example
BeginnersExperienced players
📈 Line of best fit
A line of best fit (trend line) models the overall pattern in the data and can be used to make predictions.
Line of best fit: y = 2x + 3
Predict y when x = 5:
y = 2(5) + 3 = 13
Line of best fit: y = 2x + 3
✏️ Quick check · 1 of 10
Identify the direction
As the x-values in a scatter plot increase, the y-values also tend to increase. This is called:
Both go up together.
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Identify the strength
In a scatter plot, points that are close to forming a straight line show a ___ association.
The tighter the points hug a line, the stronger the link.
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Name the odd point out
A data point that doesn't fit the overall pattern of a scatter plot is called a(n):
It is the point that sits far away from the rest.
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Name the grouped points
A group of points that are close together, separate from other points, is called a:
It is a bunch of points grouped together.
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A line of best fit is given by y = 2x + 3.
What is the predicted y-value when x = 5?
Enter an exact answer, without rounding.
Substitute x = 5 into y = 2x + 3.
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A line of best fit is given by y = 0.5x + 10.
What is the predicted y-value when x = 8?
Enter an exact answer, without rounding.
Substitute x = 8: half of 8, plus 10.
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A line of best fit is given by y = 2x + 3.
What is the predicted y-value when x = 10?
Enter an exact answer, without rounding.
Substitute x = 10 into y = 2x + 3.
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Spot the outlier by name
In the "Outlier example" scatter plot above (backpack weight vs. student weight), which labeled point is the outlier?
Find the point sitting far from the general trend.
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A line of best fit is given by y = 0.5x + 10.
What is the predicted y-value when x = 20?
Enter an exact answer, without rounding.
Substitute x = 20: half of 20, plus 10.
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Predict from a strong positive association
In a scatter plot with a strong positive association, if x increases a lot, y most likely:
Strong and positive means y follows x closely upwards.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
Hours of practice against number of mistakes would most likely show:
Question 2 of 5
Using y = 3x + 2, predict y when x = 4.
Question 3 of 5
Shoe size against maths score would most likely show:
Question 4 of 5
A line of best fit should:
Question 5 of 5
Points scattered widely around an upward trend show a:
Lesson 15 · Data
📋 Two-Way Frequency Tables
When you want to compare two different categories of data at once — like grade level and favorite subject — a two-way frequency table lays it all out clearly in rows and columns.
📋 Reading a two-way table
A two-way frequency table organises counts by two categories simultaneously, with row totals and column totals (called marginal frequencies) summarising each category.
Likes Math
Likes Science
Total
Grade 7
15
10
25
Grade 8
12
18
30
Total
27
28
55
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Define a two-way table
A two-way frequency table organises data by:
Two-way means rows for one category and columns for another.
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In the table: 15 Grade 7 students like Math, and 12 Grade 8 students like Math.
What is the total number of students who like Math?
Enter an exact answer, without rounding.
Add the two numbers in the Math column.
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Name the row and column totals
The row and column totals in a two-way table are called:
They sit in the margins of the table.
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In the table: 10 Grade 7 students like Science, and 18 Grade 8 students like Science.
What is the total number of students who like Science?