Throughout high school, students learn how to apply math skills in six categories. Starting in ninth grade, your student needs to gain solid skills in math. By senior year, your 12th-grader should strive to improve and perfect math skills.

Here are a few of the math concepts high school students will practice, depending on the courses they choose.

### Number and Quantity

- Increase knowledge of numbers to include complex numbers, irrational numbers and imaginary numbers.
**Irrational numbers**: π

**Imaginary numbers:**3*i*;*i*, etc.

**Complex numbers:***a*+*bi*; (ex. 3 + √2*i*or -1 + 5*i*) - Use equations and measurements to answer problems involving batting averages, safety statistics, mortgage rates and other real-world uses of math.
The number of fatalities per year; the number of home runs per year.
- Understand rational numbers and rational expressions.
Pi: (π = 3.14159265359…) is an irrational number because it is an unending decimal. A number such as 1.5 is rational because it can be written as a fraction:

- Solve problems that involve both rational and irrational numbers, including rational exponents.
Q: Solve 3√2 + √16 – 2½

A: = 3√2 + √4×4 – 2½

= 3√2 + (√4 × √4) – √2

= 3√2 + (2×2) -√2

= 3√2 – √2 + 4

=**2√2 + 4** - Use different units (money, distance, size) to solve word problems.

### Algebra

- Understand the structure of algebraic expressions, including quadratic equations, and be able to rewrite them using a different form.
*x*–^{4}*y*could also be written as (^{4}*x*– (^{2})^{2}*y*.^{2})^{2} - Use graphs to solve and represent equations and inequalities.
Q:

A: Solve using the graph:

The two lines meet at the point (2, 1) so*x*= 2 and*y*= 1 is the solution. - Multiply, add and subtract using rational numbers and polynomials; gain an understanding of how zeros and factors of polynomials relate to each other.
**Multiply:**(4*x*+ 3) × (*x*+ 1) = 4x^{2}+ 4*x*+ 3x + 3 =**4***x*^{2}+7*x*+ 3

**Add:**(4*x*+ 3) + (*x*+ 1) = 4*x*+ 3 +*x*+ 1 =**5***x*+ 4

**Subtract:**(4*x*+ 3) – (*x*+ 1) = 4*x*+ 3 – x – 1 =**3***x*+ 2 - Understand algebraic expressions as problem-solving tools; gain knowledge about which equations can answer different types of mathematical questions.
- Write and interpret quadratic equations as needed to solve real-world and numerical problems.
Q: A ball is thrown straight up from 4 meters above the ground with a velocity of 13 m/s. How high is the ball at 2 seconds?

A: Use the equation*h*= 4 + 13*t*– 5*t*^{2}(*h*= height;*t*= time in seconds)

*h*= 4 + 13(2) – 5(2^{2})

*h*= 4 + 26 – (5 × 4)

*h*= 4 + 26 – 20

*h*=**10 m**

### Functions

- Understand what functions are and how they behave
The equation
*ƒ*(*x*) = (*x*+ 1)^{2}– 2 is an upward-facing parabola that shifted left 1 unit and down 2 units. - Gain fluency using two forms of functions to answer a question.
Be able to find the larger number (maximum) when given a graphed quadratic equation and an algebraic expression.
- Understand functions and the relationship between input and output.
In the function
*y*= 3*x*^{2}– 7*x*+ 1 the number you insert into the equation for*x*is called the input, and the value you get out, or the*y*, is the output. - Construct and compare linear and exponential functions and apply them to real-world situations like loan payments or finding distances.
Q: For a new car priced at $22,000, James takes a five-year loan with an interest rate of 5.3%. By the time he owns the car, how much will he have paid including principal (the original cost) and interest?

A: Using the formula*I = P × R × T*, where*I*= Interest,*P*= Principal,*R*= Rate, and*T*= Time, we can find the interest on the loan, then add it to the original payment of the car.

*I*= 22,000 × 0.053 × 5

*I*= $5,830

Total payment = 22,000 + 5,830 =**$27,830** - Understand the various ways a function can be described, including algebra expressions, graphs, or verbal or recursive rules, and construct different types of functions.
Algebra expression: f(x) = a + bx

Graph: the trace of a seismograph

Verbal rule: “I’ll give you the state; you give me the capital city.” - Solve word problems involving simple and compound interest.
Q: Compound Interest: You go to the bank and take out a 5-year, $10,000 loan. How much would you owe after the 5 years if the bank had an 8% interest rate?

A: The formula for compounded interest is*FV*=*PV*× (1 +*r*), where^{n}*FV*is the future value,*PV*is the present value,*r*is the interest rate, and*n*is the number of periods.

*FV*= 10,000 × (1 + 0.08)^{5}

*FV*= 10,000 × (1.08)^{5}

*FV*= 10,000 × 1.469

*FV*=**$14,690**

### Modeling

- Use quantities, measurements and geometry to create mathematical models for real-life situations.
Plan a carnival by calculating how differently sized rides and attractions will fit onto a piece of land.
- Use mathematical modeling to examine real-life situations and formulate solutions or actions.
Plan a new housing community to meet the needs of different population groups.

### Geometry

- Expand the geometry skills learned in eighth grade by developing proofs of theorems and expanding geometric definitions.
Q: Theorem: measures of interior angles of a triangle sum to 180°.

A:

Proof: 60° + 60° + 60° = 180° - Work Euclidean geometry problems with and without coordinates.
- Do problems involving geometric transformation and combination motions like parallel lines, angles and circles.
Q:

A: Create the new circle by 1) reflecting about the y=x line and 2) reflecting about the x-axis. The result of this transformation is circle 2. - Prove geometric theorems using algebraic reasoning.
Q: Prove the two lines are parallel.

Angle m measures 37° and angle p measures 37°. Are the two lines parallel?A: Yes, the two lines are parallel by reasoning of the Alternate Interior Angles Converse. If angle m ≅ angle p, then the two lines are parallel.

### Statistics and Probability

- Apply the basic concepts of probability to real-life scenarios, such as the likelihood of the ace of spades being chosen from a deck of cards.
The likelihood of the ace of spades being chosen from a deck of cards is 1/52 or 1.92% probability.
- Use data from surveys, studies and experiments to make deductions and draw conclusions.
What is the probability a male answered yes to the question in the survey? What is the likelihood of a person over the age of 30 answering yes to the survey? etc.
- Understand how data samples differ, such as random, representative or systematic, and use data distribution — center, spread, overall shape — to answer a statistical question.
“What’s the average number of movies students at my school watch each month?” or “Looking at the overall histogram shape of the data, what is its distribution?”
- Use the rules of probability to evaluate event outcomes; understand the difference between causation and correlation.
Q: A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?

A: P(rolling a 2) = P(rolling a 5) =

Addition rule of probability: P(A or B) = P(A) + P(B)

So, P(rolling a 2 or 5) = P(rolling a 2) + P(rolling a 5)

- "Parents’ Guide to Student Success: High School Math," National PTA
- "Parent Roadmap: Supporting Your Child in High School Mathematics," Council of the Great City Schools
- "Parent Toolkit: Academic Guides," ParentToolkit.com
- Patti Ghezzi, "High School Math: How Parents Can Help Build Skills," SchoolFamily.com