Works each step out as the key is pressed. 2 + 3 is already 5 by the time × is pressed, and 5 × 4 is 20. A printing adding machine has always worked this way, and the tape prints the route it took. Open the adding machine.
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- 3
- 4
- 20
Math, explained
one step at a time.
Free lessons that explain why each rule works before asking you to use it, and tools that write out every step so you can check your own. No account, no sign-up.
Looking for one answer? Start with your question.
- Start with the equation as it is written.3x + 5 = 20
- Take 5 from both sides, so the x term is on its own.3x = 15
- Divide both sides by 3.x = 5
x = 5
Start with your question
Each question leads to the page on this site that answers it. Nothing here is a quiz and nothing is saved.
- I have been away from math for years. Where do I start?Chapter 1, Introducing pre-algebraPre-algebra connects the math you already know with the algebra you are about to learn. This is a good place to begin if you are new to pre-algebra or returning to math after some time away.
- What is a negative times a negative?Chapter 2, Integers and negative numbersWhole numbers and their opposites, and what happens to addition, subtraction and multiplication below zero.
- Why is 2 + 3 × 4 equal to 14 and not 20?Chapter 3, Order of operationsThe agreed order for working through an expression, so that two people reading it arrive at the same answer.
- How do I solve for x?Chapter 9, Inequalities and one-step equationsUndoing the one thing that was done to the letter, and what changes when the equals sign becomes a less-than sign.
- How do I check my working on an equation?Equation solverSolves an equation with one unknown in exact fractions and writes out every step, so the working can be compared with your own.
- What is the difference between an expression and an equation?Chapter 8, Expressions, variables and equationsThe letter that stands for a number, the difference between an expression and an equation, and the one rule that solves both.
- How do I work out a percent?Chapter 6, Fractions, decimals and percentsThree ways of writing part of a whole, how to move between them, and the money formulas built on percents.
- How do I find an area, a perimeter or a missing side?Chapter 12, Geometry and measurementArea, perimeter and the missing side of a right triangle, and the formulas each one needs.
- How do I add up a long list of numbers and check it?Adding machineA calculator that prints every entry to a tape, so a long column of figures can be checked line by line after it is added.
- What does an average or a median tell me?Chapter 11, Probability and statisticsMeasuring how likely something is, and describing a whole set of numbers with one number.
A course and three tools that keep their working in view
Pre-algebra course
Written for a student meeting each topic for the first time. Every chapter states the rule, shows the mechanism behind it, then works examples through line by line.
12
chapters, free
- 01 Introducing pre-algebra
- 12 Geometry and measurement
Adding-machine tape for 125 + 48 - 19 = - 125
- 48
- 19
- 154
Adding machine
A calculator that prints every entry to a tape, so a long column of figures can be checked line by line after it is added.
Scientific calculator window for √(3²+4²) √(3²+4²)
5
Scientific calculator
Multiplication and division are done before addition and subtraction, the way written math works.
Solver working for 4x = 18 4x = 18
- Start with the equation as it is written.4x = 18
x = 9/2
Equation solver
Nothing is rounded. A third stays 1/3 rather than becoming 0.333.
How a chapter teaches
Every chapter page is laid out the same way, so the layout is familiar by the second one and the attention can go to the math.
Read chapter 1From chapter 3, order of operations The order of operations ranks the four operations, so that an expression is worked through the same way by everyone who reads it.
The reason comes with the rule
A rule on its own is something to memorise. A rule with its reason is something you can rebuild when you forget it, so each topic opens by saying what happens and why.
From chapter 2, integers and negative numbers Work out −6 + 10.
- Start at −6 on the number line.
- Adding 10 moves 10 places to the right.
- Six of those places reach zero, and four more go past it.
4
Examples are worked one line at a time
Where a chapter has worked examples, each is written the way it would be on paper: the question, each move with what it does, then the answer.
Solver working for 5x - 7 = 2x + 8 5x - 7 = 2x + 8
- Start with the equation as it is written.5x − 7 = 2x + 8
- Take 2x from both sides, so every x is on the left.3x − 7 = 8
- Add 7 to both sides, so the x term is on its own.3x = 15
- Divide both sides by 3.x = 5
x = 5
The solver checks your working, not only your answer
Type an equation into the solver and it writes out each move it makes to both sides. Lay that beside your own working and the first line that differs is where to look.
Each chapter leads into the next
The chapters run in the order a class takes them, and the foot of each page links to the previous and the next, so a chapter can be read on its own or as part of the whole.
Same keys, different answers, and why
Key 2 + 3 × 4 into two calculators and you can get two answers. Both are correct for the rule each one follows, and knowing the rule is what makes an answer trustworthy.
Reads the whole line first, then multiplies before it adds: 3 × 4 is 12, and 2 + 12 is 14. This is the order of operations, and it is the rule a math class uses. Open the scientific calculator.
2+3×4
14
Keeps every number as an exact fraction. In x + 1/3 = 1, the answer is x = 2/3, written as a fraction rather than a decimal that has to stop somewhere. A rounded decimal is marked wrong on a test where a fraction is expected. Open the equation solver.
x + 1/3 = 1
- Start with the equation as it is written.x + 1/3 = 1
- Take 1/3 from both sides, so the x term is on its own.x = 2/3
x = 2/3
The route through math
Math is learned in a sequence, and each level uses the one before it. This is the route the site is built to follow. Only pre-algebra is open today. The other levels are planned and have no pages yet.
MathPencil is for learners of every age: a child meeting a topic for the first time, an adult coming back to math after years away. There are no accounts and no email collection. The site is free, and advertising pays for it.
- 01Grade-school mathPlanned. Not available yet.
- 02Pre-algebraOpen now, 12 chapters
- 03High school mathPlanned. Not available yet.
- 04College mathPlanned. Not available yet.
Start with the introduction.
Pre-algebra connects the math you already know with the algebra you are about to learn. This is a good place to begin if you are new to pre-algebra or returning to math after some time away. No account, no sign-up, and nothing to pay.