Polynomial functions and equations
Students start the year working with polynomials, the family of expressions built from powers of x. They factor, combine, and solve them, and study what their graphs look like and where they cross zero.
What does a student learn in
This is the year math stretches beyond straight lines and simple curves into the wider family of functions students will see in calculus and statistics. Students graph and solve with polynomials, rational expressions, exponents, logarithms, and trigonometry, learning how each one bends and where it breaks. Sine and cosine show up as tools for anything that repeats, like tides or sound waves. By spring, students can take a real-world pattern, pick the right function to model it, and use a sample to make a fair claim about a larger group.
- Polynomials
- Rational expressions
- Exponents and logarithms
- Trigonometry
- Function graphs
- Modeling with data
- Statistical inference
Year at a glance
How the year usually goes. Every school and district set their own curriculum, so treat this as a guide, not official pacing.
- 1
- 2
Rational and radical expressions
Students move into expressions with variables in denominators and under roots. They learn to simplify these, solve equations that include them, and spot when an answer breaks the original problem.
- 3
Exponential and logarithmic functions
Students study growth and decay patterns, like compound interest or a population doubling. Logarithms come in as the tool that undoes exponents and lets students solve for time or rate.
- 4
Trigonometry and periodic models
Students extend trig beyond triangles to describe anything that repeats, such as tides, sound waves, or daylight hours through the year. They graph sine and cosine and use identities to rewrite expressions.
- 5
Statistics and inference
The year closes with sampling and inference. Students use data from a smaller group to draw careful conclusions about a larger one, and judge how much confidence the sample actually supports.
Mastery Learning Standards
The required skills a student should display by the end of Grade 12.
Standards for Mathematical Practice
Standards for Mathematical Practice
- Algebra II
Make Sense of Problems
When a problem looks complicated, students slow down, figure out what it's actually asking, and keep working through it even when the path isn't obvious.
- Algebra II
Reason Abstractly
Students take a real situation, strip it down to an equation or expression, solve it, then translate the answer back into what it means in the original problem.
- Algebra II
Construct Arguments
Students explain why a math solution works, using examples or logical steps to back up their thinking. They also listen to a classmate's reasoning and point out what holds up or where the logic breaks down.
- Algebra II
Model with Mathematics
Students take a real situation, like figuring out loan payments or predicting population growth, and write equations or draw graphs to make sense of it. Math becomes a tool for answering questions that actually come up outside school.
- Algebra II
Use Tools Strategically
Students choose the right tool for the problem, whether that means a calculator, a quick estimate, or working it out by hand. Knowing when to use each one is part of the skill.
- Algebra II
Attend to Precision
Students use math terms correctly, label answers with the right units, and check their arithmetic. A solution that drops a unit or misuses a term is incomplete.
- Algebra II
Use Structure
Students spot patterns and hidden structure in equations and graphs, then use what they notice to solve problems faster. Recognizing that a complicated expression has a familiar shape is often the shortcut to the answer.
- Algebra II
Express Regularity
Students notice when the same pattern keeps showing up in a calculation and use that pattern as a shortcut or rule. Instead of solving each problem from scratch, they spot the repetition and generalize it.
| Standard | Definition | Code |
|---|---|---|
| Make Sense of Problems Algebra II | When a problem looks complicated, students slow down, figure out what it's actually asking, and keep working through it even when the path isn't obvious. | NH-MATH.MP.hs-algebra-2.1 |
| Reason Abstractly Algebra II | Students take a real situation, strip it down to an equation or expression, solve it, then translate the answer back into what it means in the original problem. | NH-MATH.MP.hs-algebra-2.2 |
| Construct Arguments Algebra II | Students explain why a math solution works, using examples or logical steps to back up their thinking. They also listen to a classmate's reasoning and point out what holds up or where the logic breaks down. | NH-MATH.MP.hs-algebra-2.3 |
| Model with Mathematics Algebra II | Students take a real situation, like figuring out loan payments or predicting population growth, and write equations or draw graphs to make sense of it. Math becomes a tool for answering questions that actually come up outside school. | NH-MATH.MP.hs-algebra-2.4 |
| Use Tools Strategically Algebra II | Students choose the right tool for the problem, whether that means a calculator, a quick estimate, or working it out by hand. Knowing when to use each one is part of the skill. | NH-MATH.MP.hs-algebra-2.5 |
| Attend to Precision Algebra II | Students use math terms correctly, label answers with the right units, and check their arithmetic. A solution that drops a unit or misuses a term is incomplete. | NH-MATH.MP.hs-algebra-2.6 |
| Use Structure Algebra II | Students spot patterns and hidden structure in equations and graphs, then use what they notice to solve problems faster. Recognizing that a complicated expression has a familiar shape is often the shortcut to the answer. | NH-MATH.MP.hs-algebra-2.7 |
| Express Regularity Algebra II | Students notice when the same pattern keeps showing up in a calculation and use that pattern as a shortcut or rule. Instead of solving each problem from scratch, they spot the repetition and generalize it. | NH-MATH.MP.hs-algebra-2.8 |
Algebra II
Algebra II
- Algebra II
Functions and Graphs
Students read graphs of advanced functions, including curved and wave-shaped lines, to identify patterns like peaks, valleys, and where the graph crosses zero. This builds the graph-reading skills used in science, economics, and higher math.
- Algebra II
Polynomial and Rational
Adding, subtracting, multiplying, and dividing polynomials and fractions with variables, then solving equations built from those expressions.
- Algebra II
Trigonometry
Students use sine and cosine functions to describe things that repeat on a cycle, like a pendulum's swing or a wave's height. They connect the math to real patterns and use identities to simplify or solve the equations that describe them.
- Algebra II
Statistics and Probability
Students use data from a sample group to draw conclusions about a larger population. This is the math behind surveys and polls.
| Standard | Definition | Code |
|---|---|---|
| Functions and Graphs Algebra II | Students read graphs of advanced functions, including curved and wave-shaped lines, to identify patterns like peaks, valleys, and where the graph crosses zero. This builds the graph-reading skills used in science, economics, and higher math. | NH-MATH.A2.hs-algebra-2.1 |
| Polynomial and Rational Algebra II | Adding, subtracting, multiplying, and dividing polynomials and fractions with variables, then solving equations built from those expressions. | NH-MATH.A2.hs-algebra-2.2 |
| Trigonometry Algebra II | Students use sine and cosine functions to describe things that repeat on a cycle, like a pendulum's swing or a wave's height. They connect the math to real patterns and use identities to simplify or solve the equations that describe them. | NH-MATH.A2.hs-algebra-2.3 |
| Statistics and Probability Algebra II | Students use data from a sample group to draw conclusions about a larger population. This is the math behind surveys and polls. | NH-MATH.A2.hs-algebra-2.4 |
Assessments
The state tests students at this grade and subject take.
NAEP (National Assessment of Educational Progress)
Federally administered sample-based assessment in reading, mathematics, science, and writing. NAEP results inform state-by-state comparisons rather than individual student or school accountability.
- When given:
- biennial in winter
- Frequency:
- every two years
Common Questions
What does Algebra II actually cover this year?
Students study families of functions and learn how each one behaves: polynomials, rationals, exponentials, logarithms, and trig. They also solve equations built from those functions and use sample data to make claims about a larger group.
How can I help at home if my student is stuck on a problem?
Ask them to explain the problem out loud and sketch a quick graph or table before reaching for a formula. Most stuck moments come from skipping that step. If the numbers look messy, have them check units and signs before redoing the work.
Do students still need to do work by hand if they have a graphing calculator?
Yes. Calculators and graphing tools are part of the course, but students are expected to choose when to use them. Hand work builds the structure sense that makes the calculator answers meaningful.
How should I sequence the function families across the year?
A common path is polynomials first, then rationals, then exponentials and logs as inverse pairs, then trig last so periodic behavior gets real time. Statistics fits well as a unit between two function families, since it gives a break from algebraic manipulation.
Which topics usually need the most reteaching?
Rational expressions and logarithm rules tend to take the longest, because they pull in factoring, domain, and inverse thinking all at once. Building in short review of factoring and exponent rules before those units pays off later.
What does mastery look like by the end of the year?
Students should recognize a function from its graph, equation, or table, and switch between those forms. They should solve polynomial and rational equations cleanly, model a repeating real-world pattern with a trig function, and use a sample to draw a reasonable conclusion about a population.
How do I know if a student is ready for precalculus or college math?
Watch for fluency with function families and comfort moving between graphs, equations, and word problems. A student who can set up a model from a described situation, solve it, and explain whether the answer makes sense is ready for the next course.
What is the statistics part of Algebra II really asking for?
Students learn to treat a sample as evidence about a larger population, not as the whole story. They estimate, compare results to what chance alone would produce, and decide how confident to be in a claim.
My student says they will never use this. What can I say?
The specific equations may not show up again, but the habits do: reading a graph, checking if an answer is reasonable, and arguing from data. Point out interest rates, loan payments, sound waves, and polling results as everyday places this thinking lives.