Algebra, grade 7

Seventh-grade algebra.Subjects

The topics, in order:

the coordinate system;
signed numbers and the four operations on them;
powers and square roots;
variables and algebraic expressions;
like terms and the distributive law;
linear equations and word problems.
Learning lessons explain a topic with examples and a few questions to try.
Practice lessons are all questions.
Go in order.

How the page works: Using a page.

The coordinate system in the first quadrantLearningA point and its pair of coordinates: reading points, marking points, the length of a segment and the area of a rectangle.
Negative numbersLearningNumbers below zero: on the number line, comparing and ordering, opposite numbers.
Adding and subtracting signed numbersLearningA walk on the number line, adding without the line, and subtracting as adding the opposite.
Practice: adding and subtractingPracticeAdding and subtracting signed numbers – whole, decimal and fractions; temperatures, distances on the number line and sequences.
The extended coordinate systemLearningThe axes go on into the negative numbers: four quadrants, negative coordinates, a segment that crosses an axis.
Multiplying and dividing signed numbersLearningThe sign rule for multiplication and division – and why it holds; a product of several factors; fractions and decimals.
Practice: the four operations on signed numbersPracticeAdding, subtracting, multiplying and dividing signed numbers, the order of operations, fractions and decimals.
Powers and square rootsLearningA power: base and exponent, squares and cubes, powers of a negative number, the order of operations, the square root.
Practice: powers and rootsPracticePowers of positive and negative numbers, fractions and decimals, the order of operations, square roots.
The variableLearningA letter that stands for a number: patterns, an expression for the general case, and substitution.
Algebraic expressionsLearningFrom words and from a story to an expression, how algebra is written, and the value of an expression – with negative numbers too.
Equal expressions – identitiesLearningEquivalent expressions: equal for every substitution. A table of values, identities, and a counterexample that shows two expressions are not equivalent.
Collecting like termsLearningTerms, coefficients and like terms; collecting like terms, with signed numbers too; multiplying a term by a number.
The distributive lawLearningA number times a sum and a difference, with negative numbers too; a minus before brackets; distributing and then collecting like terms.
Practice: algebraic expressionsPracticeExpressions from words, the value of an expression, collecting like terms, the distributive law, identities and counterexamples, a pattern and a perimeter.
An equation and its solutionLearningWhat an equation is, what a solution of an equation is, and how to check by substitution whether a number is a solution. Expression, equation and identity.
Solving a linear equationLearningLike a balance: the same operation on both sides. Equations in one step and in two, with brackets, with the unknown on both sides – and always a check.
Special equationsLearningEquations with no solution, and equations that every number solves – and how they relate to identities.
Practice: equationsPracticeChecking a solution by substitution, equations in one step and in several, with brackets, with the unknown on both sides, fractional solutions, and special equations.
Word problemsLearningFrom a story to an equation: choose the unknown, write an equation, solve, check in the story and answer. Ages, prices, perimeters and motion.
Points on a graph in the first quadrantLearningFrom a table to a graph: each pair of numbers is a point. Reading values off a graph, and when to join the points with a line.
Review: the whole coursePracticeEverything together: signed numbers, powers and roots, the coordinate system, algebraic expressions, identities, equations, word problems and graphs.

Last updated on 18.10.2026

Formulas: algebra, grade 7

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Adding and subtracting signed numbers

The sum of two opposite numbers
$\displaystyle (-a)+a$ $\displaystyle =$ $\displaystyle 0$
Subtracting a number is adding its opposite
$\displaystyle a-b$ $\displaystyle =$ $\displaystyle a+(-b)$
Subtracting a negative number
$\displaystyle a-(-b)$ $\displaystyle =$ $\displaystyle a+b$

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

A power – $\displaystyle a$ the base, $\displaystyle n$ the exponent: how many times $\displaystyle a$ is multiplied
$\displaystyle a^{n}$ $\displaystyle =$ $\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}$
A number to the power 1 is the number itself
$\displaystyle a^{1}$ $\displaystyle =$ $\displaystyle a$
A negative base with an even exponent – the power is positive
$\displaystyle (-a)^{2}$ $\displaystyle =$ $\displaystyle a^{2}$
The square root – the positive number (or 0) whose square is the number under the root
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The laws of arithmetic and like terms

The commutative law of addition
$\displaystyle a+b$ $\displaystyle =$ $\displaystyle b+a$
The commutative law of multiplication
$\displaystyle a\cdot b$ $\displaystyle =$ $\displaystyle b\cdot a$
The associative law of addition
$\displaystyle (a+b)+c$ $\displaystyle =$ $\displaystyle a+(b+c)$
The associative law of multiplication
$\displaystyle (a\cdot b)\cdot c$ $\displaystyle =$ $\displaystyle a\cdot(b\cdot c)$
A number times itself – squared
$\displaystyle x\cdot x$ $\displaystyle =$ $\displaystyle x^{2}$
Collecting like terms – add the coefficients; the letters stay
$\displaystyle ax+bx$ $\displaystyle =$ $\displaystyle (a+b)x$

The distributive law

The distributive law
$\displaystyle a(b+c)$ $\displaystyle =$ $\displaystyle ab+ac$
The distributive law over a difference
$\displaystyle a(b-c)$ $\displaystyle =$ $\displaystyle ab-ac$
A minus before brackets – every term inside changes its sign
$\displaystyle -(a-b)$ $\displaystyle =$ $\displaystyle -a+b$

Distance, speed and time

$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$