United States · Grade 10 math · Integrated pathway
Integrated Math II is the year quadratics take over. A tenth grader factors, completes the square and graphs parabolas, then finds that some quadratics have no real solutions at all and meets the complex numbers. Between those algebra units sit similarity, the first trigonometry, circle theorems and a probability unit. Students who changed schools between Algebra 1 and Geometry and an integrated course often arrive missing one piece. This page shows what the course holds and teaches its core skill.
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In short
Integrated Math II is the second course of the integrated high school pathway, usually taken in grade 10. Utah, which calls it Secondary Mathematics II, says its focus "is on quadratic expressions, equations, and functions", alongside complex numbers, conditional probability, similarity, right triangle trigonometry and circles. We teach it live online through our High School Mathematics course, either as a 15 to 20 student batch or in private 1-on-1 lessons.
Integrated Math II
The high school course covers every unit of the year; the other two strengthen the algebra below it and the probability inside it.
The focus
Quadratics first, with four other threads tied to them. Utah's course description spells out the links in a single paragraph, quoted here in parts.
| Thread | Utah's own words |
|---|---|
| Quadratics | "THE FOCUS OF SECONDARY MATHEMATICS II is on quadratic expressions, equations, and functions and on comparing their characteristics and behavior to those of linear and exponential relationships" |
| Number system | "real and complex numbers are introduced so that all quadratic equations can be solved" |
| Probability | "The link between probability and data is explored through conditional probability and counting methods" |
| Similarity and trig | "The study of similarity leads to an understanding of right triangle trigonometry and connects to quadratics through Pythagorean relationships" |
| Circles | "Circles, with their quadratic algebraic representations, round out the course" |
Source: Utah Core State Standards for Mathematics, High School (Utah State Board of Education). Other states and districts that teach an integrated pathway use their own course names and unit orders.
Almost every unit returns to a squared term. Right triangles bring a^2 + b^2 = c^2; a circle centered at the origin is x^2 + y^2 = r^2; the complex numbers exist, in this course, to finish off quadratics that have no real roots.
So a student who is weak at factoring and completing the square tends to struggle all year, not just in the first unit. That is where our lessons usually begin.
The Common Core lists an "integrated course sequence (Mathematics 1, Mathematics 2, Mathematics 3)" as one model alongside Algebra I, Geometry and Algebra II.
North Carolina treats the second course as close to essential: even a student exempted from its standard sequence "will be required to pass: NC Math 1 and Math 2 plus two additional courses identified on the NC DPI Math options chart."
Six areas
Utah divides the course into six critical areas. The right-hand column is our summary of what a student actually does in each.
| Area | Utah's description, shortened | What the student does (ours) |
|---|---|---|
| 1. Number systems | Rational exponents; rational and irrational numbers; "equations with no solutions in one number system may have solutions in a larger number system" | Simplify 27^(2/3); meet i, the square root of -1 |
| 2. Quadratic functions | Compare with linear and exponential; read graphs from forms; when there are no real solutions, the graph "does not cross the horizontal axis"; absolute value, step and piecewise functions | Read zeros, vertex and intercept from three forms |
| 3. Expressions and equations | Rewrite expressions to reveal structure; solve equations and systems with exponential and quadratic expressions | Factor, complete the square, use the quadratic formula |
| 4. Probability | Compound events, independence, conditional probability; "Students should make use of geometric probability models wherever possible" | Area models, tree diagrams, combinations |
| 5. Similarity and trig | Similarity criteria, right triangle trigonometry, special right triangles; "It is in this unit that students develop facility with geometric proof" | Prove triangles similar; find sides with sine, cosine, tangent |
| 6. Circles | Circle theorems; equation of a circle from center and radius; a parabola from focus and directrix; intersections of lines and circles | Complete the square to find center and radius |
Utah also offers an extended version, Secondary Math II Plus, which its 2021 pathways chart says adds "Depth in Complex Numbers", "Trigonometric Identities" and "Conic Sections." Students on the extended track need the core skills on this page first, then the extra topics.
Worked examples
Rewrite it in the form that shows what you need. Factored form shows the zeros, vertex form shows the turning point, and standard form shows the y-intercept. Every example below is our own.
| Form | Expression | What it shows at a glance |
|---|---|---|
| Standard | x^2 - 6x + 8 | y-intercept (0, 8) |
| Factored | (x - 2)(x - 4) | Zeros at x = 2 and x = 4 |
| Vertex | (x - 3)^2 - 1 | Vertex (3, -1), a minimum, axis x = 3 |
| Step | Working |
|---|---|
| 1. Complete the square | x^2 + 4x + 4 = -13 + 4, so (x + 2)^2 = -9 |
| 2. Notice the problem | No real number squares to -9, so there are no real roots |
| 3. Extend the number system | x + 2 = 3i or x + 2 = -3i, since (3i)^2 = -9 |
| 4. Solve | x = -2 + 3i or x = -2 - 3i |
| 5. Check with the graph | Vertex form (x + 2)^2 + 9 has vertex (-2, 9), above the x-axis, so the parabola never crosses it |
| Problem | Reasoning | Result |
|---|---|---|
| A 30-60-90 triangle has a short leg of 5. Find the other sides. | Sides are in the ratio 1 : root 3 : 2 | Long leg 5 root 3, hypotenuse 10 |
| Find sin 60 degrees from that triangle | Opposite over hypotenuse: 5 root 3 over 10 | root 3 over 2 |
| Center and radius of x^2 + y^2 - 6x + 4y - 12 = 0 | Complete the square twice: (x - 3)^2 + (y + 2)^2 = 12 + 9 + 4 | Center (3, -2), radius 5 |
| A dart lands at random on a 10 by 10 board with a centered circle of radius 3. P(inside the circle)? | Area of circle over area of board: 9 pi / 100 | About 0.28 |
| Same board: P(inside the circle, given the dart is in the left half)? | Half the circle is in the left half: (9 pi / 2) / 50 | 9 pi / 100 again, so the events are independent |
| How many 3-person teams can be chosen from 7 students? | 7C3 = (7 x 6 x 5) / (3 x 2 x 1) | 35 |
The dartboard pair is our favorite way to teach independence with a geometric model, as Utah suggests. Because the circle is centered, knowing the dart is on the left tells you nothing about whether it hit the circle. Move the circle off center and the two probabilities split apart.
Practice
Twelve problems written by our teachers, two from each area. They are a quick check, not a substitute for the student's own course review.
| # | Area | Problem | Answer |
|---|---|---|---|
| 1 | Number | Simplify 16^(3/4) | 8 |
| 2 | Number | Simplify (2 + i)(2 - i) | 5 |
| 3 | Quadratics | Vertex of y = 2(x + 1)^2 - 7 | (-1, -7) |
| 4 | Quadratics | Zeros of y = x^2 - x - 12 | x = 4 and x = -3 |
| 5 | Equations | Solve x^2 = 6x - 9 | x = 3 (a repeated root) |
| 6 | Equations | Discriminant of 2x^2 + 3x + 5. How many real roots? | -31; none |
| 7 | Probability | Two fair coins: P(at least one head)? | 3/4 |
| 8 | Probability | Ways to arrange the letters of MATH | 24 |
| 9 | Similarity | Triangles with sides 3, 4, 5 and 6, 8, 10: similar? | Yes, scale factor 2 |
| 10 | Trig | Right triangle, angle 30 degrees, hypotenuse 12. Opposite side? | 6 |
| 11 | Circles | Equation of the circle with center (0, 4) and radius 3 | x^2 + (y - 4)^2 = 9 |
| 12 | Circles | Does the line y = 5 meet x^2 + y^2 = 16? | No, 5 is more than the radius 4 |
The quarter-by-quarter order is our reading of Utah's six areas. Your school may teach similarity before probability or start with quadratics; in the first class the teacher maps our plan onto your child's actual calendar.
Moving schools
No. It mixes the quadratic algebra of a traditional Algebra 2 year with part of a traditional Geometry year. Here is what a transferring student usually needs, in our experience.
| Coming from | Going into | Usual gap to close |
|---|---|---|
| Algebra 1 | Integrated Math II | Congruence by transformations and descriptive statistics from Integrated Math I |
| Algebra 1 and Geometry | Integrated Math II | Complex numbers and some probability; much of the geometry repeats |
| Integrated Math I | Geometry | Little; but quadratics will be needed before Algebra 2 |
| Integrated Math II | Algebra 2 | Polynomials beyond degree 2, logarithms, and trigonometry beyond right triangles |
Each district publishes or can share its scope and sequence. Comparing it with the old school's course is the quickest way to find gaps.
A gap of one unit usually closes in three or four lessons when it is found at the start of the year.
Integrated Math III builds on quadratics with polynomial, rational and radical functions, so the forms of a quadratic need to be secure by spring.
Utah lists Secondary Math II as the prerequisite for its alternative math courses, which shows how much later work depends on this year.
The catalogue
Live courses for the skills this year relies on and the tests and courses that follow it.
INT / U / 01
Factoring and exponents.
Open the syllabusINT / U / 02
Pythagoras and ratio.
Open the syllabusINT / U / 03
Graph parabolas in Python.
Open the syllabusINT / Y / 01
Every unit of the year.
Open the syllabusINT / Y / 02
Counting and chance.
Open the syllabusINT / Y / 03
Quadratics on the test.
Open the syllabusINT / N / 01
Functions in depth.
Open the syllabusINT / N / 02
Trig and circles appear.
Open the syllabusINT / N / 03
The long-run goal.
Open the syllabusHow lessons run
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
Short sessions before each school test.
Weekends
Time to rebuild factoring from scratch.
1-on-1
For gaps left by a change of school.
Standard, factored and vertex.
Every root checked on a sketch.
Two-column and paragraph styles.
Probability you can draw.
We follow your district.
A batch of 15 to 20, or one student.
Student work
Parabolas and circles turn up in the games students build. Look through student labs.
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Fees
Lessons carry a monthly fee in US dollars. You can stop at the end of any month, which suits families who want help only through the quadratics units.
Free first class
USD 0
no card required
Group batch
USD 100
a month, billed in US dollars
One to one
USD 150
a month, billed in US dollars
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Questions families ask
Integrated Math II centers on quadratic expressions, equations and functions. It also introduces complex numbers, conditional probability and counting, similarity with right triangle trigonometry, and circles, as Utah's Secondary Mathematics II course description sets out.
Neither. It combines the quadratics usually found in Algebra 1 and Algebra 2 with the similarity, trigonometry and circle work of a Geometry course, so a student switching pathways may meet some topics twice and miss others.
In our teaching, students often find it the most demanding of the three, because so much depends on factoring and completing the square. When those two skills are secure, the rest of the course becomes far more manageable.
Because some quadratic equations have no real solutions. Utah says real and complex numbers are introduced "so that all quadratic equations can be solved", and the graph of such a quadratic never crosses the x-axis.
Usually grade 10, after Integrated Math I in grade 9. Accelerated students may take it in grade 9, and some districts place it in grade 11.
Start with the student's textbook and teacher, since tests follow the school course. This page has twelve practice problems we wrote, with answers. We do not copy end-of-course test items or publisher answer keys.
Yes. Utah says the similarity unit is where students develop facility with geometric proof. We teach how to choose the right criterion and set out each reason, then the student writes their own proof.
No. We teach the method and check the student's reasoning, but graded work has to be their own. That way they can handle the test, where no tutor is present.
Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.
No. We are an independent online school with no link to any state, district or textbook publisher. Course names and documents on this page belong to their owners.
Elsewhere on this site
Other pages for grade 10 math and the integrated pathway.
Grade 10 testing with quadratics.
For the traditional pathway.
Quadratics and beyond.
Right triangles onward.
The usual Integrated Math II year.
A guide to typical tutoring fees.
Part of the guide: US math placement and college readiness tests
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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