Math · Algebra 1

Algebra 1, explained unit by unit

Linear equations, functions, systems, exponents, polynomials, quadratics and data: here is everything Algebra 1 covers, with worked problems and the mistakes to avoid. Then practice each skill in Mathflow, from intuition to solving.

Start for free See the 7 units

y = x^2 - 2x - 3

: zeros

-1

and

3

, vertex

(1, -4)

, axis of symmetry

x = 1

What you’ll learn

The 7 units of Algebra 1

The course as it is organized in Mathflow, in order. Each unit is split into precise skills that you practice one at a time.

Unit 13 skills
Equations & Inequalities
Isolate a variable step by step, solve formulas for one letter, then solve linear and compound inequalities and graph them on a number line.
- Solve linear and literal equations
- Solve and graph linear inequalities
- Solve and graph compound inequalities
Unit 24 skills
Linear Functions & Graphs
Learn what a function is, graph lines from slope-intercept and standard form, read slope as a rate of change, and write parallel and perpendicular lines.
- Understand the concept of a function
- Graph linear equations in various forms
- Calculate and interpret the slope
- Write equations for parallel and perpendicular lines
Unit 33 skills
Systems of Equations
Find where two lines meet by graphing, substitution and elimination, decide between one, none or infinitely many solutions, and shade the solutions of a system of inequalities.
- Solve systems of equations by graphing
- Solve systems of equations algebraically
- Graph systems of linear inequalities
Unit 43 skills
Exponential Functions
Use the rules of integer exponents, model growth and decay with $f(t) = a \cdot b^t$ , and compare exponential graphs with linear ones.
- Apply properties of integer exponents
- Model exponential growth and decay
- Graph and compare exponential functions
Unit 53 skills
Polynomials & Factoring
Add, subtract and multiply polynomials, factor out the greatest common factor, and factor trinomials and special products such as the difference of squares.
- Perform operations with polynomials
- Factor out the greatest common factor
- Factor trinomials and special products
Unit 64 skills
Quadratic Functions
Read the vertex, axis and intercepts of a parabola, then solve quadratic equations by factoring, with the quadratic formula, and by completing the square.
- Identify characteristics of parabolas
- Solve quadratic equations by factoring
- Use the quadratic formula and the discriminant
- Complete the square to find the vertex form
Unit 73 skills
Data Analysis
Choose and read data displays, analyze two-way frequency tables, and fit a line to a scatter plot, with the correlation coefficient $r$ to measure how strong the link is.
- Interpret and create data displays
- Analyze two-way frequency tables
- Create scatter plots and calculate regressions

Algebra 1 at a glance

Algebra 1 is usually the first high school math course. Many students take it in 9th grade, and some take it in 8th grade.

Before you start. You should be comfortable with negative numbers, fractions, the order of operations, one-variable linear equations and slope: the core of 8th grade math.

After Algebra 1. In the traditional US sequence, Geometry comes next, then Algebra 2. The quadratic and exponential functions you meet here come back in Algebra 2, Pre-Calculus and Calculus.

In Mathflow. The course has $7$ units and $23$ skills. Each skill is a series in three levels, and each unit ends with a Chapter Challenge.

Worked problems

Two problems, solved step by step

A system of equations and a quadratic function: two classics of Algebra 1. Try each one on paper before you read the solution.

Problem 1
Ticket sales: a system of equations
A school play sold $120$ tickets for a total of $720. Student tickets cost $5 and adult tickets cost $8. How many tickets of each kind were sold?
Solution
Call $s$ the number of student tickets and $a$ the number of adult tickets. Write one equation for the tickets and one for the money:
$\begin{cases} s + a = 120 \\ 5s + 8a = 720 \end{cases}$
Use substitution. The first equation gives $s = 120 - a$ . Replace $s$ in the second equation:
$\begin{aligned} 5(120 - a) + 8a &= 720 \\ 600 + 3a &= 720 \\ a &= 40 \end{aligned}$
Then $s = 120 - 40 = 80$ .
Check: $80 + 40 = 120$ tickets, and $5 \times 80 + 8 \times 40 = 400 + 320 = 720$ dollars. The play sold $80$ student tickets and $40$ adult tickets.
Problem 2
A ball in the air: a quadratic function
A ball is thrown upward from a $64$ -foot platform. Its height, in feet, after $t$ seconds is $h(t) = -16t^2 + 48t + 64$ .
1. When does the ball hit the ground?
2. What is its maximum height, and when does it reach it?
Solution
a. Solve $h(t) = 0$ to find when the ball hits the ground. Factor out $-16$ , then factor the trinomial:
$\begin{aligned} h(t) &= -16(t^2 - 3t - 4) \\ &= -16(t - 4)(t + 1) \end{aligned}$
By the zero product property, $t = 4$ or $t = -1$ . A time can’t be negative, so the ball hits the ground after $4$ seconds.
b. The parabola opens down because $a = -16$ is negative, so its vertex is the highest point. The vertex is on the axis of symmetry:
$t = -\dfrac{b}{2a} = -\dfrac{48}{2 \times (-16)} = 1.5$
$\begin{aligned} h(1.5) &= -16(1.5)^2 + 48(1.5) + 64 \\ &= -36 + 72 + 64 = 100 \end{aligned}$
The ball reaches its maximum height, $100$ feet, after $1.5$ seconds. Check: the vertex is halfway between the zeros $-1$ and $4$ , and $(-1 + 4) \div 2 = 1.5$ .

How Mathflow teaches it

Algebra 1 in three levels

Every skill of the course is a series of short missions in three levels. Here is what you actually do in three of them.

Level 1 · Explore
Explore and play with the concepts to build your intuition.
Level 2 · Explain
Turn your intuitions into the right words and uncover hidden mistakes.
Level 3 · Solve
Reason it out, crunch the numbers, and find your way to the answer.

You don’t start with a page of rules. In the first level, you act on the math: you tap operations that apply to both sides of an equation, drag points, move sliders and watch the graph respond. In the second level, you put words on what you saw: you decide which of two students is right, find the first wrong line of a solution, or complete a sentence that explains a method. In the third level, you solve problems on your own, like the ones you meet in class and on tests.

If you get stuck, Alphi, the AI math tutor, gives you a hint instead of the answer, at any hour. When the skills of a unit are done, a Chapter Challenge checks the whole unit.

Unit 1 · Equations & Inequalities
1Solve and graph linear inequalities
- Level 1 · Explore
  Drag $k$ until multiplying both sides of $-2 < 3$ by $k$ turns the symbol around. Then solve $5 - 2x < 11$ with buttons that act on both sides: the symbol turns only when you tap Reverse.
- Level 2 · Explain
  Owen solved $5 - 3(2x - 1) > 2$ and found $x < 0$ . You find the mistake in his work. You also decide whether Emma or Ryan solved $-2(x + 4) < 6$ correctly.
- Level 3 · Solve
  Each ticket costs $4, plus one $6 booking fee, and Ana has $30. You find how many tickets she can buy at most.
Unit 3 · Systems of Equations
2Solve systems of equations algebraically
- Level 1 · Explore
  Swap coins until $30$ coins are worth $4.30. Then choose the multipliers that make $y$ disappear when you add the two equations.
- Level 2 · Explain
  Elimination on $2x - 4y = 6$ and $-x + 2y = -3$ gives $0 = 0$ . You decide who is right about what it means. You also tap the first wrong step in a student’s elimination.
- Level 3 · Solve
  $20$ test questions are worth $2$ or $5$ points each, $76$ points in all. You find how many questions are worth $5$ points.
Unit 6 · Quadratic Functions
3Complete the square to find the vertex form
- Level 1 · Explore
  Add unit tiles to $x^2 + 8x$ until the tiles form one complete square, and slide $h$ until $(x + h)^2$ begins with $x^2 + 10x$ .
- Level 2 · Explain
  Put the steps in order to write $y = 2x^2 + 8x + 1$ in vertex form, and find what went wrong in a classmate’s work.
- Level 3 · Solve
  A garden against a wall has an area of $A = -2x^2 + 16x$ square feet. You find its greatest possible area.

Practice Algebra 1 for free

Common mistakes

Five mistakes to avoid

These slips come up again and again in Algebra 1. Learn to spot them, and you’ll catch them in your own work.

Forgetting to flip the inequality symbol
When you multiply or divide both sides of an inequality by a negative number, the symbol turns around. From $-3x > 12$ , you get $x < -4$ , not $x > -4$ . Test a value to be sure: $x = -5$ gives $15 > 12$ , which is true.
Distributing a minus sign to one term only
$5 - 3(2x - 1)$ equals $5 - 6x + 3$ , not $5 - 6x - 1$ . The factor $-3$ multiplies every term in the parentheses, including $-1$ : $(-3) \times (-1) = 3$ .
Squaring a sum term by term
$(x + 3)^2$ is not $x^2 + 9$ . Expand it: $(x + 3)(x + 3) = x^2 + 6x + 9$ . Try $x = 1$ : $(1 + 3)^2 = 16$ , while $1^2 + 9 = 10$ .
Dividing both sides by $x$
Dividing $x^2 = 4x$ by $x$ gives $x = 4$ and loses the solution $x = 0$ . Move every term to one side instead: $x^2 - 4x = 0$ , so $x(x - 4) = 0$ , and $x = 0$ or $x = 4$ .
Mixing up the order in the slope formula
The slope of the line through $(1, 2)$ and $(4, -4)$ is $\dfrac{-4 - 2}{4 - 1} = -2$ . Subtract in the same order on top and on the bottom: mixing the order flips the sign, and putting the $x$ -values on top gives the reciprocal.

Frequently asked questions

What do you learn in Algebra 1?

Linear equations and inequalities, functions and their graphs, systems of equations, exponents and exponential models, polynomials and factoring, quadratic functions, and data analysis. In Mathflow, that’s 7 units and 23 skills.

What grade is Algebra 1?

Most students take Algebra 1 in 9th grade, as their first high school math course. Some take it in 8th grade.

What should I know before Algebra 1?

Negative numbers, fractions, the order of operations, one-variable linear equations and the idea of slope. These are the main topics of 8th grade math.

Is Algebra 1 hard?

It’s a step up: letters stand for numbers, and one rule describes a whole graph. Gaps in earlier skills, such as negative numbers and fractions, often make it feel harder than it is, so review them first if you feel stuck.

How can I practice Algebra 1 at home?

Short, regular sessions work better than long ones. Redo the worked problems on this page without looking, then practice skill by skill. Mathflow covers the whole course in three levels per skill, with Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

What comes after Algebra 1?

In the traditional US sequence, Geometry comes next, then Algebra 2. Both build on the equations, functions and factoring you learn in Algebra 1.

Start for free

Standards

Algebra 1 and the Common Core

The Common Core State Standards organize high school math by conceptual categories, not by course. An Algebra 1 course draws mostly on three of them: Algebra (seeing structure in expressions, creating equations, reasoning with equations and inequalities), Functions (interpreting functions; linear, quadratic and exponential models) and Statistics and Probability (interpreting categorical and quantitative data).

Appendix A of the standards proposes a traditional US pathway of three courses, Algebra I, Geometry and Algebra II, with some data, probability and statistics in each. Many states use their own version of the standards, so check your school’s course outline for the exact list.

Algebra 1, explained unit by unit

Equations & Inequalities

Linear Functions & Graphs

Systems of Equations

Exponential Functions

Polynomials & Factoring

Quadratic Functions

Data Analysis

Algebra 1 at a glance

Ticket sales: a system of equations

A ball in the air: a quadratic function

Level 1 · Explore

Level 2 · Explain

Level 3 · Solve

1Solve and graph linear inequalities

2Solve systems of equations algebraically

3Complete the square to find the vertex form

Forgetting to flip the inequality symbol

Distributing a minus sign to one term only

Squaring a sum term by term

Dividing both sides by xxx

Mixing up the order in the slope formula

Math is all about flow.

Dividing both sides by $x$