Mathematics for Kids

How the course is built

• It climbs. Unit 1 is what maths even is. Unit 13 is functions and rates of change, the doorway to calculus. Each rung assumes the one below it.
• The worksheets never run out. Every unit generates new numbers on every press, and the answers are computed rather than stored, so the answer key cannot drift.
• Every formula is checked. All 45 of them are verified numerically against a value worked out another way, so a typo in a formula fails our tests rather than reaching a child.
• Every unit has a note for the grown-up, usually naming the misconception to listen for. In maths these are very predictable: that a bigger bottom number means a bigger fraction, that the equals sign means the answer comes next, that a ratio is a fraction.

The thirteen units

Every formula, in one place

All 45 formulas taught in the course, from place value to the quadratic formula. Print this page for a reference sheet.

🔎 1 · What Is Mathematics?

Kindergarten to Grade 2

Mathematics is the study of number, shape and pattern. It is not really about getting sums right quickly. It is about noticing that things follow rules, and then using those rules to work out something you did not already know.

What it teaches

• Maths is about patterns. Once you spot a pattern you can say what comes next without being told.
• Numbers are one kind of pattern. Shapes are another. So are the rules for sharing things out fairly.
• Every piece of maths can be checked. If you are not sure, you can test it, which is not true of most subjects.
• Being stuck is normal and is where the learning happens. Mathematicians are stuck most of the time.
• There is almost always more than one way to get an answer. Finding a second way is the best check there is.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

🔢 2 · Numbers and Counting

Kindergarten to Grade 1

Every number you will ever meet is built from ten digits, 0 to 9. What makes a number big or small is not the digits themselves but where they sit.

What it teaches

• Counting is matching one number word to one object. Say it and touch it.
• Zero means none. It is a real number, and it took people a very long time to invent it.
• Where a digit sits tells you what it is worth. In 23, the 2 means two tens, not two.
• Counting in twos, fives and tens is faster than counting in ones, and it is the beginning of times tables.
• The number line puts numbers in order. Further right means bigger, and it never runs out.

Formulas

• Place value: a two digit number = tens x 10 + ones. 23 means 2 tens and 3 ones, which is 20 + 3.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

➕ 3 · Adding and Subtracting

Kindergarten to Grade 2

Adding is putting together. Subtracting is taking away, or finding the difference. They are two sides of the same fact, which means learning one gives you the other for free.

What it teaches

• Adding can be done in any order. 6 + 4 is the same as 4 + 6, so always start with the bigger number.
• Subtracting cannot be done in any order. 10 - 4 is not the same as 4 - 10.
• Every adding fact hides a subtracting fact. If 6 + 4 = 10, then 10 - 4 = 6 and 10 - 6 = 4.
• Number bonds to ten are worth knowing by heart: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5.
• A number line helps: adding jumps right and subtracting jumps left.

Formulas

• Adding in any order: a + b = b + a. Swapping the two numbers does not change the answer. This is called being commutative.
• Subtraction undoes addition: if a + b = c then c - b = a. Every adding fact gives you a subtracting fact for free.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

✖️ 4 · Times Tables and Division

Grades 2 to 4

Multiplying is adding the same number over and over, done quickly. Dividing is sharing out, or finding how many groups fit. Like adding and subtracting, they undo each other.

What it teaches

• 4 x 6 means four lots of six. Drawn as a rectangle of dots, that is 4 rows of 6.
• Turn the rectangle sideways and it becomes 6 rows of 4. The total is the same, so 4 x 6 = 6 x 4. That halves what you have to learn.
• Dividing undoes multiplying. If 4 x 6 = 24, then 24 / 6 = 4 and 24 / 4 = 6.
• Look for patterns rather than memorising blindly. The tens all end in 0, the fives end in 5 or 0, and the nines' digits add up to 9.
• The square numbers run down the diagonal of the times table: 1, 4, 9, 16, 25, because they are 1x1, 2x2, 3x3 and so on.

Formulas

• Multiplying in any order: a x b = b x a. 4 rows of 6 is the same total as 6 rows of 4.
• Division undoes multiplication: if a x b = c then c / b = a. Every times fact gives you a division fact for free.
• Sharing out: total / groups = size of each group. 24 shared between 6 people is 4 each.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

🍕 5 · Fractions

Grades 3 to 5

A fraction is a part of a whole. The bottom number says how many equal parts the whole was cut into, and the top number says how many of them you have.

What it teaches

• In 3/4, the 4 says the whole was cut into four equal parts and the 3 says you have three of them.
• The parts must be equal. Four uneven pieces of cake are not quarters.
• The bigger the bottom number, the smaller each piece. An eighth is smaller than a quarter, which surprises almost everybody at first.
• Different fractions can be worth the same. 1/2, 2/4 and 4/8 are all the same amount.
• To add fractions the bottom numbers must match. 1/8 + 3/8 = 4/8, which is also 1/2.
• To find a fraction of an amount, divide by the bottom then multiply by the top. Three quarters of 20 is 20 / 4 x 3 = 15.

Formulas

• Equivalent fractions: a/b = (a x n)/(b x n). Multiply top and bottom by the same number and the fraction is worth the same.
• Adding fractions: a/c + b/c = (a + b)/c. With the same bottom number, just add the tops.
• Multiplying fractions: a/b x c/d = (a x c)/(b x d). Multiply the tops together and the bottoms together.
• Fraction of an amount: a/b of n = n / b x a. Three quarters of 20 is 20 divided by 4, then times 3, which is 15.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

💷 6 · Decimals, Percentages and Money

Grades 4 to 6

Fractions, decimals and percentages are three ways of writing the same idea. Money is where you meet all three at once, which makes it the best place to practise.

What it teaches

• A decimal point separates whole ones from parts. The first place after it is tenths, the next is hundredths.
• Percent means out of a hundred. 50% is 50 out of 100, which is one half, which is 0.5.
• The useful ones to know by heart: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 1/10 = 0.1 = 10%, 3/4 = 0.75 = 75%.
• To find a percentage of an amount, divide by 100 and multiply by the percentage. 25% of 80 is 80 / 100 x 25 = 20.
• 10% is easy: move the decimal point one place left. Once you have 10% you can build 20%, 5% and 30% from it in your head.
• Money is decimals with two places, because there are 100 pennies in a pound and 100 cents in a dollar.

Formulas

• Percent to decimal: p% = p / 100. Percent means out of a hundred, so 50% is 0.5.
• Percentage of an amount: p% of n = n x p / 100. 25% of 80 is 80 x 25 / 100, which is 20.
• Percentage change: change % = (new - old) / old x 100. Going from 40 to 50 is a rise of 25%.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

📐 7 · Shape and Space

Grades 3 to 6

Geometry is the maths of shape, size and position. Two ideas do most of the work: area, which is how much is inside, and perimeter, which is how far it is around the edge.

What it teaches

• Area is measured in square units, because you are counting squares. Perimeter is measured in plain units, because you are measuring a length.
• Area of a rectangle is length times width. A rectangle 5 by 3 holds 15 squares.
• A triangle is exactly half the rectangle drawn around it, so its area is base times height, divided by two.
• The angles inside any triangle add up to 180 degrees. Inside any four sided shape they add up to 360.
• A circle's distance around is its circumference, and it is always a little more than three times across. That number is pi, about 3.14159.
• Note for readers of both British and American books: a trapezium in Britain is a trapezoid in America, and the two words mean opposite things. Check which book you are in.

Formulas

• Area of a rectangle: A = l x w. Count the squares inside: 5 across by 3 down is 15.
• Perimeter of a rectangle: P = 2 x (l + w). All the way round the edge: 5 + 3 + 5 + 3 is 16.
• Area of a triangle: A = (b x h) / 2. A triangle is exactly half of the rectangle around it.
• Area of a circle: A = pi x r x r. For a circle of radius 1 the area is pi, about 3.14159.
• Circumference of a circle: C = 2 x pi x r. All the way round the outside. For radius 1 that is about 6.283.
• Angles in a triangle: a + b + c = 180 degrees. Tear the three corners off a paper triangle and they fit on a straight line.
• Angles in a quadrilateral: a + b + c + d = 360 degrees. Any four sided shape. A rectangle is four right angles: 4 x 90 = 360.
• Volume of a cuboid: V = l x w x h. How many unit cubes fit inside.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

📏 8 · Measurement

Grades 3 to 6

Measuring turns the world into numbers you can calculate with. The metric system makes it easy, because every step is ten, a hundred or a thousand.

What it teaches

• Length goes millimetres, centimetres, metres, kilometres. 10 mm in a cm, 100 cm in a metre, 1000 m in a km.
• Mass goes grams and kilograms, with 1000 grams in a kilogram.
• Capacity goes millilitres and litres, with 1000 ml in a litre.
• Always write the unit. 5 on its own is not an answer; 5 cm is.
• Time is the odd one out, because it does not go in tens. 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day.
• Speed links distance and time: speed is distance divided by time. 120 km in 2 hours is 60 km per hour.

Formulas

• Length: 1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m. Each step up is ten, a hundred or a thousand times bigger.
• Mass: 1 kg = 1000 g. A bag of sugar is about 1 kg.
• Capacity: 1 litre = 1000 ml. A large bottle of water is about 1 litre.
• Time: 1 minute = 60 s, 1 hour = 60 min, 1 day = 24 h. So a day is 24 x 60 x 60 = 86400 seconds.
• Speed: speed = distance / time. 120 km in 2 hours is 60 km per hour.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

⚖️ 9 · Ratio, Proportion and Rates

Grades 6 to 8

A ratio compares two amounts. It is how recipes scale, how maps work, and how you tell whether a bigger box is actually better value.

What it teaches

• A ratio of 2 : 3 means two parts of one thing for every three of the other, so five parts in all.
• A ratio compares part to part. A fraction compares part to whole. In a 2 : 3 mix, orange is 2/5 of the total, not 2/3.
• Scaling keeps a ratio the same. 2 : 3 is the same mix as 4 : 6 and 20 : 30.
• To share an amount in a ratio, add the parts, divide the total by that, then multiply out. Share 20 in the ratio 2 : 3: five parts, so one part is 4, giving 8 and 12.
• A rate compares two different units, such as km per hour or pence per gram.
• Unit price is the practical use: divide price by amount and compare. It is the only reliable way to tell which packet is better value.

Formulas

• Sharing in a ratio: each share = total / (sum of the parts). Share 20 in the ratio 2 : 3. There are 5 parts, so one part is 4, giving 8 and 12.
• Scaling a ratio: a : b = (a x n) : (b x n). 2 : 3 is the same mix as 4 : 6.
• Unit rate: rate = amount / number of units. If 4 pens cost 60p, one pen costs 15p.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

🧮 10 · Introducing Algebra

Grades 6 to 9

Algebra is arithmetic with a letter standing in for a number you do not know yet. The letter is not mysterious. It is a box waiting to be filled.

What it teaches

• In x + 3 = 10, the x is simply the number that makes the sentence true. Here it is 7.
• Think of the equals sign as a balance. Whatever you do to one side you must do to the other, or it tips.
• To solve, undo what was done, in reverse order. For 2x + 3 = 11, undo the +3 first, then the x2.
• Always check by putting your answer back in. 2 x 4 + 3 = 11, so x = 4 is right. This takes ten seconds and catches almost every mistake.
• Brackets mean do this first, and a(b + c) = ab + ac lets you take them apart.
• When multiplying powers of the same number, add the powers: 2 cubed times 2 squared is 2 to the fifth.

Formulas

• Solving a one step equation: if x + b = c then x = c - b. Do the same to both sides. x + 3 = 10 gives x = 7.
• Solving a two step equation: if a x + b = c then x = (c - b) / a. Undo the adding first, then the multiplying. 2x + 3 = 11 gives x = 4.
• Expanding brackets: a(b + c) = ab + ac. 3(4 + 5) is the same as 12 + 15, which is 27.
• Powers: a^m x a^n = a^(m + n). Add the powers when multiplying. 2 cubed x 2 squared is 2 to the fifth, which is 32.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

📊 11 · Data, Averages and Chance

Grades 5 to 9

Statistics is how you make sense of a pile of numbers, and probability is how you talk about things that have not happened yet. Both are everywhere in the news, which is reason enough to understand them.

What it teaches

• The mean is the total shared out evenly: add everything up and divide by how many there are.
• The median is the middle value once they are in order. The mode is the one that appears most often.
• The mean can be pulled a long way by one unusual value. For 2, 3, 3, 6, 11 the mean is 5 but four of the five numbers are below it.
• The range is the biggest take away the smallest, and it tells you how spread out the numbers are.
• Probability runs from 0, impossible, to 1, certain. It is the number of ways something can happen divided by the total number of ways.
• A fair dice has no memory. Five sixes in a row does not make a six less likely next time, and believing otherwise is a genuinely expensive mistake.

Formulas

• Mean (the average): mean = sum of values / how many values. For 2, 3, 3, 6, 11 the total is 25 and there are 5, so the mean is 5.
• Median (the middle one): put them in order, take the middle value. 2, 3, 3, 6, 11 in order has 3 in the middle.
• Range: range = biggest - smallest. 11 take away 2 is 9.
• Probability: P = ways it can happen / total possible ways. One 6 on a dice out of six faces is 1/6. Something certain is 1, impossible is 0.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

📐 12 · Pythagoras and Trigonometry

Grades 8 to 11

Right angled triangles have a rule so useful that it is over two thousand years old and still used every day by builders, sailors and game programmers.

What it teaches

• Pythagoras' theorem: in a right angled triangle, the two short sides squared add up to the longest side squared. a squared plus b squared equals c squared.
• The longest side is called the hypotenuse and it is always opposite the right angle.
• The 3, 4, 5 triangle is the one to remember: 9 + 16 = 25. Builders still use it to check a corner is square.
• To find the long side, add the squares and take the square root. To find a short side, subtract instead.
• Trigonometry links the angles to the sides. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
• One worth memorising: the sine of 30 degrees is exactly 0.5. It also tells you instantly whether your calculator is set to degrees.

Formulas

• Pythagoras' theorem: a^2 + b^2 = c^2. In a right angled triangle, the two short sides squared add up to the long side squared. 3, 4, 5 works.
• Finding the long side: c = square root of (a^2 + b^2). With sides 3 and 4 the hypotenuse is 5.
• Sine, cosine, tangent: sin = opp/hyp, cos = adj/hyp, tan = opp/adj. In the 3, 4, 5 triangle the sine of the angle opposite the 3 is 3/5, which is 0.6.
• A known angle: sin 30 degrees = 0.5. Worth knowing by heart, and a good check that your calculator is in degrees.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

📈 13 · Functions, Graphs and Change

Grades 9 to 12

A function is a machine: put a number in, get exactly one number out. Drawing a function as a graph turns a rule into a picture, and the shape of that picture tells you how things change.

What it teaches

• A function gives exactly one output for each input. Put 3 into y = 2x + 1 and you always get 7.
• A straight line graph has the form y = mx + c, where m is the gradient and c is where it crosses the y axis.
• The gradient is rise over run: the change in y divided by the change in x. A gradient of 2 means y goes up 2 every time x goes up 1.
• y = x squared is not a straight line. It curves upwards, getting steeper as x grows.
• The quadratic formula solves any equation of the form ax squared plus bx plus c equals zero. For x squared minus 5x plus 6, the answers are 2 and 3.
• The average rate of change between two points is the gradient of the line joining them. Squeezing those two points together is the idea that becomes calculus.

Formulas

• A straight line: y = m x + c. m is the slope and c is where it crosses the y axis. y = 2x + 1 passes through (0, 1) and (3, 7).
• Gradient of a line: m = (y2 - y1) / (x2 - x1). Rise over run. From (0, 1) to (3, 7) the gradient is 6/3 = 2.
• The quadratic formula: x = (-b +/- square root of (b^2 - 4ac)) / 2a. Solves any equation of the form ax^2 + bx + c = 0. For x^2 - 5x + 6 the answers are 2 and 3.
• Average rate of change: rate = (change in y) / (change in x). For y = x squared between x = 1 and x = 3, y goes 1 to 9, so the rate is 8/2 = 4. This idea becomes calculus.

Sample questions and answers

Worksheets in this course are generated fresh, with new numbers every time you press New Worksheet, and every answer is computed rather than stored.

Browse by grade