Algebra I: the main ideas and getting unstuck

What Algebra I covers in California

California high schools offer a traditional sequence (Algebra I, Geometry, Algebra II), an integrated sequence (Integrated Math I, II and III), or both. Each is built on the California Common Core State Standards for Mathematics; Algebra I and Integrated Math I overlap heavily but are not identical. A typical Algebra I course includes:

• Expressions and equations: simplifying, solving linear equations and inequalities in one variable, and rearranging formulas, such as solving A = lw for w.
• Linear functions: slope as a rate of change, slope-intercept form y = mx + b, and moving between a table, a graph, an equation, and a story.
• Systems of linear equations: solving by graphing, substitution, and elimination.
• Functions: function notation such as f(3), domain and range, and comparing functions shown in different ways.
• Exponential functions: growth and decay, and how they differ from linear change.
• Polynomials and quadratics: adding, multiplying, and factoring polynomials, then solving quadratic equations by factoring, completing the square, and the quadratic formula.
• Statistics: scatter plots, lines of fit, and interpreting data.

Where Algebra I students get stuck

• Distributing a negative. −2(x − 4) is −2x + 8, and the second sign is the one most often lost.
• Combining unlike terms, such as turning 3x + 2 into 5x.
• Slope from two points. Subtracting in one order on top and the other order on the bottom flips the sign; the order has to match.
• Function notation. f(3) means the output when the input is 3, not f times 3.
• Factoring. Students who are shaky on multiplication facts and negative numbers find factoring x² − 5x + 6 into (x − 2)(x − 3) much harder than it needs to be.
• Word problems. Writing down what the variable stands for, in words, before writing an equation prevents most setup errors.

Worked example: solving a system by elimination

A system of equations asks for values that make two equations true at the same time. On a graph, the solution is the point where the two lines cross. Elimination works well when one variable already has opposite coefficients, as it does here.

Seeing the same answer on a graph

Solving each equation for y gives y = −2x + 7 and y = x − 2. The first line falls 2 units for every step to the right, the second rises 1, and they cross at (3, 1). Sketching the two lines is a good way for students to see that the algebra and the graph tell the same story.

This connection matters beyond systems. Much of Algebra I is about moving between an equation, a table, a graph, and a real situation, and students who can switch views easily tend to find later units, including quadratics and exponential functions, much less mysterious.

Signs a student needs Algebra I help

• They can follow a worked example but freeze when a problem looks slightly different.
• Test scores are much lower than homework scores.
• Sign errors and arithmetic slips cost points on most problems.
• They do not know where to start on word problems.
• They say they stopped understanding after a specific unit. That unit is usually the place to go back to.

How to get unstuck

Algebra builds on itself, so the fastest fix is usually to find the first idea that is shaky, not to redo the current homework.

• Say each step out loud, including why it is allowed: I subtract 6 from both sides to keep the equation balanced.
• Write one step per line and keep the equals signs lined up.
• Check every solution by substituting it into the original equation.
• Connect three views of the same function: equation, table, and graph.
• Practice a few problems every day rather than many problems once a week.
• Ask for help early in a unit, before the next unit builds on it.