United States · Grade 11 math · Integrated pathway

Integrated Math 3 tutoring: help with Integrated Math III

Integrated Math III is the last course of the integrated pathway and the one that decides what a student can take next: precalculus, a college algebra course for credit, statistics, or calculus later on. It asks a junior to combine everything from the first two years and then go further, into higher-degree polynomials, rational expressions, logarithms, trigonometry for any triangle and inference from data. Here is what the course contains, in a state's own words, with the ideas worked through.

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In short

Integrated Math III is the third course of the integrated sequence, usually taken in grade 11. Utah's Secondary Mathematics III groups it into four areas: inference from data; polynomial, rational and radical functions; trigonometry of general triangles and radians; and modeling with functions, including logarithms. Our High School Mathematics course teaches it live online, to a group of 15 to 20 students or one to one.

Integrated Math III

Three courses for Integrated Math III

The high school course covers the year itself; AP Precalculus and the statistics course carry its two biggest threads further.

High School Mathematics course thumbnail

GAO / 01

High school math

Polynomials, logarithms, trigonometry and modeling.

Open the syllabus →
AP Precalculus course thumbnail

GAO / 02

AP Precalculus

The usual next step for calculus-bound juniors and seniors.

Open the syllabus →
Statistics & Probability course thumbnail

GAO / 03

Statistics and probability

For the inference unit and a statistics senior year.

Open the syllabus →

The course

What is taught in Integrated Math 3?

Integrated Math III gathers up the first two years and adds four new areas. Utah describes it as the course where "students pull together and apply the accumulation of learning that they have from their previous courses."

Utah's description continues in four sentences that serve as a map of the year: "They apply methods from probability and statistics to draw inferences and conclusions from data. Students expand their repertoire of functions to include polynomial, rational, and radical functions. They expand their study of right triangle trigonometry to include general triangles. And, finally, students bring together all of their experience with functions and geometry to create models and solve contextual problems."

Topics new in the third year, from the Utah State Board of Education 2021 pathways chart
Version of the courseListed topics
Secondary Math III (standard)Polynomial, Rational, and Radical Functions; Trigonometry; Logarithms; Data Collection and Analysis
Secondary Math III Plus (extended)Everything above, plus Composition of Functions; Inverse Functions / Trig Functions; Polar Coordinates; Arithmetic and Geometric Series

Sources: Utah State Board of Education Core State Standards for Mathematics, High School and Graduation Pathways. North Carolina's equivalent course is NC Math 3.

Why the year feels heavier

Integrated Math I and II each had one main function family, linear and exponential, then quadratic. Integrated Math III adds several families at once and expects students to choose between them.

A student who is shaky on factoring quadratics or on exponent rules will feel it here first, because polynomial division and logarithms both rest on them.

Why it matters for what comes next

In North Carolina, NC Math 3 is the last of three required integrated courses, and the state then asks for a fourth math course matched to what the student plans after graduation. Here, as in most states, this year decides which fourth course is realistic.

Utah's own overview says that after Secondary Math III, "In precalculus (or college algebra), students will expand their solving to all types of equations including complex number systems, vectors, matrices, and polar coordinates."

Four areas

What are the four units of Integrated Math III?

Utah organizes Secondary Mathematics III into four critical areas. Each line below quotes Utah, then says in our words what the student practices.

Secondary Mathematics III critical areas (from Utah's high school mathematics core, checked 31 October 2026; our practice notes in the right column)
AreaUtah's wording, shortenedPractice in our lessons
1. InferenceWays of collecting data, "including sample surveys, experiments, and simulations", and "the role that randomness and careful design play in the conclusions that can be drawn"Normal-model estimates, study design, simulation
2. Polynomials and rational expressions"structural similarities between the system of polynomials and the system of integers"; "The unit culminates with the fundamental theorem of algebra"Long division, the remainder idea, zeros and factors
3. Trigonometry of general trianglesThe Laws of Sines and Cosines; whether three measures define "0, 1, 2, or infinitely many triangles"; radian measure; simple periodic phenomenaAmbiguous cases, radians, sinusoidal graphs
4. Mathematical modelingSolving exponential equations with logarithms; transformations that have "the same effect regardless of the type of the underlying functions"; choosing and adjusting modelsPick a function family, fit it, judge the fit

Polynomials are a lot like whole numbers

Utah's Area 2 asks students to connect "multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers."

It adds that "A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers." Students who see the parallel stop memorizing steps and start reusing ones they know.

Modeling is the point of the course

Area 4 quotes the definition of modeling as "the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions."

That is why a test question in this course often gives data and asks which kind of function fits, rather than giving the function and asking for a value.

Worked examples

How do you divide polynomials and solve with the Law of Sines?

With the same habits as integer long division, and by checking how many triangles the measurements allow. All examples here are ours, written for this page.

Our side-by-side: integer long division and polynomial long division
Step1573 divided by 11(x^3 + 5x^2 + 7x + 3) divided by (x + 1)
First term11 goes into 15 once: 1, remainder 4x goes into x^3 by x^2; subtract x^3 + x^2, leaving 4x^2 + 7x
Second term11 into 47: 4, remainder 3x into 4x^2 by 4x; subtract 4x^2 + 4x, leaving 3x + 3
Third term11 into 33: 3, remainder 0x into 3x by 3; subtract 3x + 3, leaving 0
Quotient143x^2 + 4x + 3, which factors as (x + 1)(x + 3)
Meaning1573 = 11 x 143The cubic is (x + 1)^2 (x + 3): zeros -1 (twice) and -3, three in all, as the fundamental theorem of algebra predicts for degree 3
The ambiguous case (ours): angle A = 30 degrees, side a = 6 opposite it, side b = 10
StepWorking
1. Find the heightb sin A = 10 x 0.5 = 5
2. Count triangles5 < 6 < 10, so side a can swing to meet the base in two places: two triangles
3. Law of Sinessin B = b sin A / a = 5/6, about 0.833
4. Two anglesB is about 56.4 degrees, or 180 minus that, about 123.6 degrees
5. Check bothBoth leave room for angle C (93.6 or 26.4 degrees), so both triangles exist
More of ours, one per remaining area
ProblemReasoningResult
Simplify (x^2 - 9)/(x^2 + 3x)Factor: (x - 3)(x + 3) over x(x + 3); cancel x + 3, as with a common factor in a fraction(x - 3)/x, with x not 0 or -3
Two sides 7 and 8 meet at 60 degrees. Find the third side.Law of Cosines: c^2 = 49 + 64 - 2 x 7 x 8 x 0.5 = 57c = root 57, about 7.55
Convert 150 degrees to radiansMultiply by pi/1805 pi / 6
Solve 3 x 2^x = 48Divide by 3: 2^x = 16 = 2^4x = 4
Solve 5^x = 20Take logs: x log 5 = log 20x = log 20 / log 5, about 1.861
Scores are roughly normal, mean 70, standard deviation 6. About what share score above 82?82 is two standard deviations above the mean; about 95% lie within two, so about 5% lie outside, split between both tailsAbout 2.5%

Practice

Integrated Math 3 practice problems with answers

A dozen problems our teachers wrote across the four areas. Students looking for Integrated Math 3 help can use them to find which unit needs work first.

Integrated Math III practice set (ours), answers included
#AreaProblemAnswer
1InferenceA school surveys only students in the library. What is the flaw?Not a random sample of the school
2InferenceMean 100, standard deviation 15, roughly normal. About what percent lie between 85 and 115?About 68%
3PolynomialsRemainder when x^3 - 2x + 4 is divided by x - 28
4PolynomialsHow many complex zeros does a degree 5 polynomial have, counted with multiplicity?5
5PolynomialsZeros of (x - 1)(x^2 + 4)1, 2i and -2i
6RationalAdd 1/x + 2/(x + 1)(3x + 1)/(x(x + 1))
7RadicalSolve the square root of (x + 5) = 3x = 4
8TrigonometryConvert pi/4 to degrees45 degrees
9TrigonometryPeriod of y = sin(2x)pi
10TrigonometryA = 40 degrees, B = 60 degrees, a = 10. Find b to one decimal.About 13.5
11LogarithmsSolve log base 2 of x = 5x = 32
12ModelingData double every 3 years. Linear, quadratic or exponential model?Exponential
  1. Fall, first half: statistics and inference. Distributions, the normal model, sampling, experiments and simulation.
  2. Fall, second half: polynomials. Long and synthetic division, zeros and factors, end behavior and the fundamental theorem of algebra.
  3. Winter: rational and radical functions. Simplifying, adding and solving, with attention to excluded values and extraneous roots.
  4. Early spring: trigonometry. Laws of Sines and Cosines, the ambiguous case, radians and the unit circle, and periodic graphs.
  5. Late spring: logarithms and modeling. Inverses of exponentials, solving with logs, transformations, and choosing models for data.
  6. Before the summer: the bridge. A preview of whichever fourth course comes next.

This order is our reading of Utah's four areas; schools often teach trigonometry earlier or split polynomials across the year. We never write a student's graded work for them. We teach the method and leave the work to the student.

After Math III

What math comes after Integrated Math 3?

A fourth course chosen around the student's plans. Utah's pathways chart, from 2021, sets out the options most clearly; your own district will name its courses differently.

Options after Secondary Math III on Utah's pathways chart (Utah State Board of Education, Graduation Pathways PDF, 2021)
OptionWhat Utah says about it
Pre-CalculusA high school class; "doesn't award college credit" but prepares for calculus
Math 1050 and Math 1060Concurrent enrollment; "most of the same topics as Precalculus"
College Statistics or AP StatisticsCollege mathematics credit choices for statistics
Math 1010Does not give the quantitative literacy credit a bachelor's degree needs
An alternative selection insteadAllowed, but "not usually recommended"

Opting out of Math III

Utah answers this directly: "You are allowed to take one of the Alternate Selections in place of Secondary Math III and still receive a high school diploma. This is not usually recommended."

Families weighing that choice should first check what the student's likely college majors expect.

Skipping ahead

Utah is just as direct about skipping: "If you skip a step on the chart you may not be ready for the next class, so it is never advised students to skip classes."

Students who want to move faster can take extended versions or summer classes instead, which Utah also mentions.

The catalogue

Courses for Integrated Math III and after

Live courses for the year itself, the tests juniors sit, and the fourth math course.

I

Shoring up

Skills from Math I and II
II

Junior year

The course and the tests
III

The fourth course

Senior year and beyond

How lessons run

Integrated Math III lessons for busy juniors

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.

Weekdays

Evening slots around sports and jobs.

Weekends

Longer review before unit tests.

1-on-1

Planned around SAT or ACT dates too.

Integer parallels

Polynomial steps tied to number facts.

Triangle counting

Ambiguous cases drawn first.

Logs as inverses

Never a rule without its reason.

Model choice

Which family fits, and why.

Fourth-course plan

Advice on what to take next.

Batch or private

Fifteen to twenty, or one to one.

Student work

Work from our students

Older students fit models to real data in their own code. See examples on student labs.

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Questions families ask

Integrated Math 3: common questions

What is taught in Integrated Math 3?

Integrated Math III covers inference from data, polynomial, rational and radical functions, trigonometry of general triangles with radians and periodic functions, and modeling with function families including logarithms. Utah's Secondary Mathematics III groups these into four critical areas.

Is Integrated Math 3 the same as Algebra 2?

They overlap heavily on polynomials, rational functions and logarithms. Integrated Math III also includes the Laws of Sines and Cosines and a statistics unit, so a student switching from Algebra 2 may need help with those.

Is Integrated Math 3 hard?

It asks for more combining of ideas than the earlier courses, because several function families appear at once. Students with secure factoring and exponent skills usually cope well; gaps from Integrated Math II show up quickly.

What grade takes Integrated Math 3?

Grade 11 for most students, after Integrated Math I and II. Accelerated students may take it in grade 10, which leaves room for precalculus and calculus later.

What comes after Integrated Math 3?

Usually precalculus, a statistics course, or a college algebra course taken for dual or concurrent enrollment credit. Utah's pathways chart lists Pre-Calculus, Math 1050 and 1060, a college Statistics course and AP Statistics among the options.

Can my child skip Integrated Math 3?

Utah allows an alternative course in its place for the diploma but says this is not usually recommended, because the alternatives do not prepare students for college math credit. Ask your school counselor how your district handles it.

Where can I find Integrated Math 3 practice problems?

Use the school textbook first, since tests follow it, then our twelve-problem set on this page. We do not copy state test items or textbook answer keys.

Do you help with modeling projects in Integrated Math 3?

We teach how to choose a model, fit it and judge the fit, and we review the student's reasoning. We do not write or edit the project for a grade.

When are your Integrated Math 3 classes?

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.

Are you affiliated with any state or school?

No. Modern Age Coders runs independently of every state education agency, district and publisher. State documents are quoted here only to describe the course accurately.

Elsewhere on this site

Related pages

More pages for juniors and the courses that follow.

Precalculus tutoring

The most common fourth course.

Algebra 2 tutoring

For the traditional route.

Trigonometry tutoring

The unit circle and beyond.

AP Statistics tutoring

A data-focused senior year.

Eleventh grade math

The usual Integrated Math III year.

Calculus tutoring

Where the pathway is heading.

Related pages in this guide

Part of the guide: US math placement and college readiness tests

High school math courses

School placement and growth tests

PSAT and PreACT

College placement and credit

More guides: US state math tests by state · US gifted and selective school tests · Homeschool and school math curricula guide · US math and coding competitions guide

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