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Grade 9

What Math Does a 9th Grader Learn? All 31 Topics, in Order

By Knomi Learning · Updated

The short answer

A 9th grader learns 31 math topics, in 7 units: Expressions and Equations, Lines and Systems, Functions, Exponential Functions, Polynomials and Quadratics, Quadratic Functions and Descriptive Statistics. That is 130 smaller skills in all, taught in the order below, each building on earlier ones.

  • The earlier topics 9th grade leans on most: Linear Expressions (Grade 7), Square and Cube Roots (Grade 8), Linear Functions (Grade 8), Integer Exponents (Grade 8) and Exponents (Grade 6).
  • All 130 skills are matched to a Common Core standard.
  • Every one of them has a full lesson on Knomi.

Unit 1: Expressions and Equations

1

Expressions and Structure

The kid learns to read expressions: which parts are terms, factors and coefficients, and when a chunk like (1 + r)^n is one thing. Evaluating and simplifying with negatives and fractions gets fluent here, because every node after it leans on that.

What it covers

  • Evaluating Expressions
  • Combining Like Terms
  • Parts of an Expression
  • Equivalent Expressions

Builds on

  • Grade 8: Multiplying and Dividing Powers
  • Grade 7: Order of Operations with Rationals
  • Grade 7: Signs of Products and Powers
  • Grade 7: Like Terms with Negatives
  • Grade 7: Expanding Expressions
  • Grade 7: Adding Signed Fractions and Decimals
  • and 5 more
Expressions and Structure on the 9th grade map
2

Units and Quantities

Numbers in algebra come with units, and the units do work: they convert, they check a formula, and they decide how many digits are worth reporting. This short node gives those habits a home so every modeling problem can point back to them.

What it covers

  • Rate Conversion
  • Units in Formulas
  • Defining Quantities
  • Level of Accuracy

Builds on

  • Grade 8: Volume of Cylinders
  • Grade 8: Making Scatter Plots
  • Grade 8: Estimating with Powers of Ten
  • Grade 7: Unit Rates with Fractions
  • Grade 7: Constant of Proportionality
  • Grade 6: Converting Units
  • and 3 more
Units and Quantities on the 9th grade map
3

Solving Linear Equations

The kid already solves multi-step equations from Grade 8; now they explain why each step is allowed, solve with letters as coefficients, and build equations from situations. Justifying a step is what later makes the quadratic formula make sense.

What it covers

  • Reasoning Step by Step
  • Equations with Fractions and Decimals
  • Equations with Unknown Coefficients
  • Equations from Situations

Builds on

  • Grade 8: Variables on Both Sides
  • Grade 8: Fraction and Decimal Coefficients
  • Grade 8: Equations with Parentheses
  • Grade 8: One, None or Infinitely Many
  • Grade 8: Multi-Step Equation Word Problems
  • Grade 7: Factoring Expressions
  • and 3 more
Solving Linear Equations on the 9th grade map
4

Literal Equations

Formulas have several letters and no single answer. The kid picks the quantity they want and solves for it with the same moves used on ordinary equations, including formulas with fractions and the letter in two places.

What it covers

  • Solving for a Variable
  • Multi-Step Formulas
  • Formulas in Context

Builds on

  • Grade 8: Rearranging Formulas
  • Grade 8: Solving for y
  • Grade 7: Factoring Expressions
  • Grade 6: Unit Price and Speed Problems
Literal Equations on the 9th grade map
5

Linear Inequalities

Inequalities get the full treatment equations got: multi-step, both sides, and two joined by 'and' or 'or'. The kid reads solution sets on a number line and says which options a budget allows. The flip when multiplying by a negative is the slip to watch.

What it covers

  • Multi-Step Inequalities
  • Compound Inequalities
  • Inequalities in Context

Builds on

  • Grade 7: Two-Step Inequalities
  • Grade 7: Flipping the Inequality Sign
  • Grade 7: Inequality Word Problems
  • Grade 6: Graphing Inequalities
  • Grade 6: Comparing Rational Numbers
Linear Inequalities on the 9th grade map

Unit 2: Lines and Systems

6

Linear Equations and Graphs

Every point on a line is a solution of its equation, and every solution is on the line. The kid reads slope and intercepts as rates and starting amounts in a story, and builds two-variable equations from situations with labelled, scaled axes.

What it covers

  • Graphs as Solution Sets
  • Slope as a Rate of Change
  • Intercepts in Context
  • Two-Variable Equations from Situations

Builds on

  • Grade 8: Solutions of Two-Variable Equations
  • Grade 8: Slope from Two Points
  • Grade 8: Negative, Zero and Undefined Slope
  • Grade 8: Rate of Change and Initial Value
  • Grade 8: Intercepts
  • Grade 8: Interpreting Linear Models
  • and 3 more
Linear Equations and Graphs on the 9th grade map
7

Forms of Linear Equations

One line, three ways to write it: slope-intercept shows rate and start, point-slope a known point, standard form the intercepts. The kid writes a line from whatever is given and picks the form that answers the question. Parallel lines share a slope.

What it covers

  • Slope-Intercept Equations
  • Point-Slope Form
  • Standard Form
  • Choosing a Form

Builds on

  • Grade 8: Equations from Two Points
  • Grade 8: Graphing y = mx + b
  • Grade 7: Expanding Expressions
Forms of Linear Equations on the 9th grade map
8

Solving Linear Systems

The kid gets fluent with graphing, substitution and elimination, including elimination that needs a multiplier first, and learns why adding equations keeps the solutions. The common slip is multiplying one term of an equation but not the rest.

What it covers

  • Solving by Graphing
  • Solving by Substitution
  • Solving by Elimination
  • Why Elimination Works
  • Number of Solutions of a System

Builds on

  • Grade 8: Solving Systems by Graphing
  • Grade 8: Substitution
  • Grade 8: Elimination
  • Grade 8: Number of Solutions of a System
  • Grade 7: Negative Times Negative
Solving Linear Systems on the 9th grade map
9

Systems in Context

Two facts about one situation make a system: tickets and money, a boat against a current, two phone plans. The kid writes both equations, chooses a method, and says what the answer means, including 'never' and 'always'.

What it covers

  • Writing Systems from Situations
  • Interpreting Solutions
  • Choosing a Method

Builds on

  • Grade 8: Systems Word Problems
Systems in Context on the 9th grade map
10

Two-Variable Inequalities

An inequality in x and y is true for a whole region of the plane. The kid shades half-planes with dashed or solid edges, overlaps them for a system, and reads which combinations a budget allows. A test point fixes the wrong-side shading slip.

What it covers

  • Checking Solutions
  • Graphing Half-Planes
  • Systems of Inequalities
  • Constraints in Context

Builds on

  • Grade 6: Graphing Inequalities
Two-Variable Inequalities on the 9th grade map

Unit 3: Functions

11

Functions and Notation

The kid writes functions with f(x) notation, reads f(3) = 12 as a sentence about a situation, works backwards from outputs to inputs, and names domain and range. Reading f(x) as 'f times x' is the misconception the tutor listens for.

What it covers

  • What Makes a Function
  • Evaluating with Function Notation
  • Inputs from Outputs
  • Notation in Context
  • Domain and Range
  • Domain in Context

Builds on

  • Grade 8: Function Rules
  • Grade 8: Functions in Tables
  • Grade 8: Functions on Graphs
  • Grade 8: Solving for y
  • Grade 8: Square Roots of Perfect Squares
  • Grade 6: Independent and Dependent Variables
  • and 1 more
Functions and Notation on the 9th grade map
12

Key Features of Graphs

A graph tells a story: where it crosses zero, where it rises and falls, where it peaks, how fast it changes on average. The kid reads these features, sketches graphs from descriptions, and computes average rate of change.

What it covers

  • Intercepts and Sign
  • Increasing and Decreasing
  • Maxima and Minima
  • Graphs from Stories
  • Average Rate of Change

Builds on

  • Grade 8: Describing Graphs
  • Grade 8: Sketching Graphs from Stories
Key Features of Graphs on the 9th grade map
13

Sequences

A sequence is a function on 1, 2, 3, …: arithmetic ones add the same amount, geometric ones multiply by the same factor. The kid writes both recursively and explicitly, the bridge to linear and exponential functions. The classic slip is a + dn for a + d(n − 1).

What it covers

  • Sequences as Functions
  • Arithmetic Sequences
  • Geometric Sequences
  • Sequence Models

Builds on

  • Grade 8: Functions in Tables
  • Grade 8: Rate of Change and Initial Value
  • Grade 8: Powers with Negative Bases
  • Grade 6: Powers of Fractions
Sequences on the 9th grade map

Unit 4: Exponential Functions

14

Radicals and Exponents

Grade 8's exponent rules take one more step: 5^(1/3) has to be the cube root of 5. The kid reviews integer exponents, simplifies square roots, and moves between radicals and fractional exponents. Reading 8^(1/3) as 8 ÷ 3 is the mistake to catch.

What it covers

  • Exponent Rules Review
  • Roots of Numbers
  • Simplifying Square Roots
  • Rational Exponents
  • Properties with Rational Exponents

Builds on

  • Grade 8: Rules with Integer Exponents
  • Grade 8: Powers of Powers and Products
  • Grade 8: Negative Exponents
  • Grade 8: Square Roots of Perfect Squares
  • Grade 8: Cube Roots
  • Grade 8: Roots of Fractions and Decimals
  • and 3 more
Radicals and Exponents on the 9th grade map
15

Exponential Functions

Linear functions add the same amount each step; exponential functions multiply by the same factor. The kid tells them apart, writes f(x) = a · b^x from a start and a growth factor, and graphs growth and decay. Reading 3 · 2^x as 6^x is the common mix-up.

What it covers

  • Linear or Exponential
  • Initial Value and Growth Factor
  • Graphs of Exponential Growth
  • Exponential Decay
  • Exponential Functions from Data

Builds on

  • Grade 8: Linear or Nonlinear
  • Grade 8: Negative Exponents
  • Grade 8: Equations from Two Points
  • Grade 6: Order with Exponents
  • Grade 6: Powers of Fractions
Exponential Functions on the 9th grade map
16

Exponential Models

Interest, populations and depreciation change by a percent. The kid reads rates from expressions like 1.02^t, compares simple and compound interest, sees exponential growth overtake linear, and solves where two models meet.

What it covers

  • Percent Growth and Decay
  • Simple and Compound Interest
  • Rewriting Exponential Expressions
  • Exponential Overtakes Linear
  • Solving with Graphs and Tables
  • Building Exponential Models

Builds on

  • Grade 7: Percent Change as a Multiplier
  • Grade 7: Commission and Simple Interest
Exponential Models on the 9th grade map

Unit 5: Polynomials and Quadratics

17

Polynomials

Polynomials behave like whole numbers: add, subtract or multiply two and you get another. The kid learns the words, subtracts with care over the minus sign, and multiplies with an area model, special products included. The classic slip is (x + 3)^2 = x^2 + 9.

What it covers

  • Polynomial Vocabulary
  • Adding and Subtracting Polynomials
  • Multiplying by a Monomial
  • Multiplying Binomials
  • Special Products

Builds on

  • Grade 8: Negative Exponents
  • Grade 8: Multiplying and Dividing Powers
  • Grade 7: Adding and Subtracting Expressions
  • Grade 7: Expanding Expressions
  • Grade 7: Negative Times Negative
  • Grade 6: Meaning of Exponents
  • and 1 more
Polynomials on the 9th grade map
18

Factoring Quadratics

Factoring runs multiplication backwards. The kid takes out common factors, factors trinomials with and without a leading coefficient, and spots differences of squares and perfect squares. The usual stall is a sign: two numbers with product −12 and sum 1.

What it covers

  • Greatest Common Factor
  • Factoring x² + bx + c
  • Factoring by Grouping
  • Difference of Squares
  • Perfect Square Trinomials
  • Factoring by Structure

Builds on

  • Grade 8: Square Roots of Perfect Squares
  • Grade 7: Factoring Expressions
  • Grade 7: Positive Times Negative
  • Grade 6: Greatest Common Factor
Factoring Quadratics on the 9th grade map
19

Solving Quadratics

A product is zero only when a factor is zero, and that one fact turns factoring into equation solving. The kid solves by factoring and by square roots and writes quadratics for areas and falling objects, keeping both roots of x^2 = 49.

What it covers

  • Zero Product Property
  • Solving by Factoring
  • Taking Square Roots
  • Quadratic Equations in Context

Builds on

  • Grade 8: Equations with Squares and Cubes
  • Grade 6: Equations and Solutions
Solving Quadratics on the 9th grade map
20

Completing the Square

Any quadratic can be pushed into the form (x − p)^2 = q, which square roots can finish. The kid builds perfect squares with an area model, solves this way, and uses the same move to find a parabola's vertex. Adding to one side only is the usual slip.

What it covers

  • Building a Perfect Square
  • Solving by Completing the Square
  • Vertex by Completing the Square

Builds on

  • Grade 7: Factoring Expressions
  • Grade 6: Powers of Fractions
Completing the Square on the 9th grade map
21

Quadratic Formula

The quadratic formula is completing the square done once for every equation. The kid derives it, uses it, reads the discriminant to count real solutions, and picks the quickest method each time. −b when b is already negative is the slip to watch.

What it covers

  • Deriving the Formula
  • Using the Formula
  • Number of Solutions
  • Choosing a Solving Method
Quadratic Formula on the 9th grade map
22

Linear-Quadratic Systems

A line can cross a parabola, or a circle, twice, once or not at all. The kid solves such systems by graphing and by substitution, which turns them into a quadratic equation. Finding the x-values and forgetting the matching y-values is the common slip.

What it covers

  • Graphing Line and Curve
  • Substituting into the Quadratic
  • How Many Intersections

Builds on

  • Grade 8: Square Roots of Perfect Squares
Linear-Quadratic Systems on the 9th grade map

Unit 6: Quadratic Functions

23

Rational and Irrational

The quadratic formula keeps producing answers like 2 + √5, so the kid learns what kind of number that is: classifying real numbers and reasoning about sums and products. The common guess that two irrationals always make an irrational fails at √2 · √2.

What it covers

  • Classifying Real Numbers
  • Sums and Products
  • Why the Rules Hold

Builds on

  • Grade 8: Rational or Irrational
  • Grade 8: Repeating Decimals as Fractions
  • Grade 7: Multiplying Rational Numbers
  • Grade 7: Adding Signed Fractions and Decimals
Rational and Irrational on the 9th grade map
24

Graphing Quadratics

Every quadratic graph is a parabola, and each form shows a different part: factored form the zeros, vertex form the turning point, standard form the y-intercept. The kid graphs from all three and shifts and stretches y = x^2. The sign of h is the slip.

What it covers

  • Features of a Parabola
  • Graphs in Factored Form
  • Graphs in Vertex Form
  • Graphs in Standard Form
  • Shifting and Stretching Parabolas

Builds on

  • Grade 8: Translations
  • Grade 8: Reflections
Graphing Quadratics on the 9th grade map
25

Quadratic Models

Thrown balls, fenced areas and revenue make parabolas. The kid builds and reads quadratic models, compares functions given in different forms, and decides whether data are linear, quadratic or exponential. Reading the time of the peak as its height is the usual slip.

What it covers

  • Quadratic Word Problems
  • Comparing Functions in Different Forms
  • Rates of Change of Quadratics
  • Choosing a Model

Builds on

  • Grade 8: Comparing Linear Functions
Quadratic Models on the 9th grade map
26

Other Function Families

Some functions follow different rules on different stretches: postage jumps at each ounce and |x| folds a line into a V. The kid graphs absolute value, piecewise, step and root functions, and meets y = k/x, where doubling one quantity halves the other.

What it covers

  • Absolute Value Graphs
  • Piecewise Functions
  • Step Functions
  • Square Root and Cube Root Graphs
  • Inverse Variation

Builds on

  • Grade 8: Graphing y = mx + b
  • Grade 8: Graphing Proportional Relationships
  • Grade 7: Proportional or Not
  • Grade 6: Absolute Value
  • Grade 6: Writing and Testing Inequalities
  • Grade 6: Unit Price and Speed Problems
Other Function Families on the 9th grade map
27

Inverse Functions

An inverse runs a function backwards, Fahrenheit back to Celsius. The kid finds inverses of linear functions, reads inverse values from tables and graphs, and sees why x^2 needs half its domain cut before it can be undone. f^−1(x) is not 1/f(x).

What it covers

  • Undoing a Function
  • Inverses of Linear Functions
  • Restricting the Domain

Builds on

  • Grade 8: Negative Exponents
  • Grade 8: Reflections
  • Grade 8: Equations with Squares and Cubes
Inverse Functions on the 9th grade map

Unit 7: Descriptive Statistics

28

Shape, Center and Spread

The kid chooses the summary that fits the data's shape: mean and standard deviation when it is roughly symmetric, median and IQR when it is skewed or has outliers. They compare groups in context and say what an outlier does to each measure.

What it covers

  • Data Displays
  • Standard Deviation
  • Choosing Center and Spread
  • Comparing Distributions
  • Outliers and Their Effects

Builds on

  • Grade 8: Clusters and Outliers
  • Grade 7: Negative Times Negative
  • Grade 7: Comparing Centers and Spreads
  • Grade 7: Overlap of Distributions
  • Grade 6: Histograms
  • Grade 6: Box Plots
  • and 5 more
Shape, Center and Spread on the 9th grade map
29

Two-Way Frequency Tables

When both variables are categories, a two-way table shows how they relate. The kid computes joint, marginal and conditional relative frequencies and decides whether there is an association. Dividing by the wrong total is the classic mistake.

What it covers

  • Building Two-Way Tables
  • Joint and Marginal Frequencies
  • Conditional Relative Frequencies
  • Association in Categorical Data

Builds on

  • Grade 8: Reading and Building Two-Way Tables
  • Grade 8: Relative Frequencies
  • Grade 8: Association in Two-Way Tables
  • Grade 6: Dot Plots and Frequency Tables
  • Grade 6: Finding the Percent
Two-Way Frequency Tables on the 9th grade map
30

Fitting Lines to Data

Real data never sit exactly on a line. The kid fits lines by eye and with technology, predicts, and checks the fit with residuals; when the residuals curve, they try a quadratic or exponential model instead.

What it covers

  • Describing Scatter Plots
  • Fitting a Linear Function
  • Residuals
  • Fitting Other Functions

Builds on

  • Grade 8: Making Scatter Plots
  • Grade 8: Describing Association
  • Grade 8: Fitting a Line
  • Grade 7: Adding the Opposite
Fitting Lines to Data on the 9th grade map
31

Correlation

The slope and intercept of a fitted line mean something about the data, and r says how tightly the points hug the line. The kid interprets all three and learns the most important caution in statistics: correlation is not causation.

What it covers

  • Slope and Intercept of a Model
  • Correlation Coefficient
  • Correlation and Causation

Builds on

  • Grade 8: Interpreting Slope and Intercept
  • Grade 8: Clusters and Outliers
  • Grade 8: Association in Two-Way Tables
  • Grade 7: Random Sampling
Correlation on the 9th grade map

Questions parents ask

How many math topics are in 9th grade?

On Knomi's map, 9th grade math has 31 topics in 7 units, made of 130 smaller skills.

What should a student know before starting 9th grade math?

The earlier topics 9th grade leans on most are Linear Expressions (Grade 7), Square and Cube Roots (Grade 8), Linear Functions (Grade 8), Integer Exponents (Grade 8) and Exponents (Grade 6). A student who is quick and sure with those is ready. Each topic below lists exactly what it builds on.

Is this 9th grade math list aligned to Common Core?

Common Core is the spine. All 130 skills carry a Common Core reference, and the order follows the usual US course.

Does Knomi teach 9th grade math?

Every one of them has a full lesson on Knomi. The 9th grade map is open to anyone, with no account needed to look around.

See 9th grade math as a map.

Every topic above, with lines to what it needs first. Open it and look around, no account needed.

Open the Grade 9 map

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