## 12th Grade Math Skills

Your child’s final year of high school represents a culmination of years of hard work and study. The math curriculum for individual grades will vary from school to school, so please consult our subject-specific math benchmarks.

##### Overview

For high school students, math skills and understandings are organized not by grade level but by concept. In High School Math: Functions, students advance their understanding of functions as relationships between inputs (problems) and outputs (answers), and important tools in building mathematical models to solve real-world problems.

##### Understanding Functions

Understand function as a relationship between quantities: a set of inputs (called the domain) and a set of all possible outputs (called the range or codomain); understand that each input is related to exactly one output. (For example, in x2, the function relates each real number x to its square.)

##### Describing Functions

Understand that a function can be described in various ways, including:

• a graph – for example, a seismograph trace

• an algebraic expression – for example, f(x) = a + bx

• a recursive rule – self-similar repeating items, such as mirror image in a mirror image in a mirror image, or the Fibonacci sequence (each successive number in series is sum of the two numbers before it: 0, 1, 1, 2, 3, 5, 8, 13)

• a verbal rule (for example: “I give you a state, you give me a capital city.” (A state has exactly one capital city.)

##### Constructing Functions

Understand and construct different types of functions based on how they grow:

• linear functions (grow at a constant rate) – for example, if a tree grows 20 cm a year, the height of the tree is related to its age. The function h(age) = age x 20. If the tree is 10 years old, the height is h(10) = 10 x 20 = 200 cm.

• exponential functions (grow at a constant percent rate) – for example, the return on \$10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time the money is invested. V(t) = 10000 x (1.0425)t

##### Comparing Functions

Compare properties of two functions, each represented in a different way: algebraically, graphically, in numeric tables, or in verbal descriptions.

Two students chart their heights over three years. The first student measures 50, 52 and 54 inches; the second student measures 45, 48 , and 51 inches. Which student is growing at the faster rate?

##### Real-World Problems

Solve real-world problems involving exponential growth and exponential decay.

Exponential growth: In 2000, Florida’s population was 16 million. Since 2000, the state’s population has grown about 2% each year. At this rate of growth, find Florida’s population in 2020.

Exponential decay: The factory purchased a new assembly unit for \$45,000. Each year, it depreciates (loses value) at a rate of 5%. What will its approximate value be at the end of the third year after its purchase (rounded to the nearest cent)?

##### Compound Interest

Solve real world problems involving compound interest

When Marta was born, her grandparents deposited \$500 into an account paying 3.5% interest, compounded annually. Find the account balance after 18 years (rounded to the nearest cent).

##### Overview

For high school students, math skills and understandings are organized not by grade level but by concept. In High School Math: Statistics and Probability, students develop their skills in gathering, summarizing, representing, and interpreting data. High School math students work with categorical and quantitative data, work on making inferences and justifying conclusions, and use probability to solve real-world problems and evaluate the outcomes of decisions.

##### Understanding the Difference

Understand the difference between quantitative data (data that are measurable or countable with numbers, such as age, length, weight, etc.) and categorical data (data sorted into categories on the basis of qualities not measured by numbers, such as gender, occupation, political affiliation, etc.).

##### Standard Deviation

Understand standard deviation as a measure of the variability, or spread of a set of numbers or numerical values. Use mean (average) and standard deviation of a data set to solve problems.

College-age women have a mean height of 65 inches with a standard deviation of 2.5 inches. What fraction of college aged women are between 65 and 67.5 inches tall?

##### Randomization

Understand randomization in surveys and research studies: drawing valid conclusions about the whole population by collecting data from a random sample of that population.

##### Understanding Sample Sizes

Understand population proportion, variability, and valid sample size. Calculate margin of error. Identify biased samples.

##### Interpreting Linear Models

Represent, summarize and interpret data on two categorical and quantitative variables. Interpret linear models – the slope (rate of change) and the intercept (constant term).

Compare the number of men and the number of women who earned Engineering degrees in 1962, and in 2012.

##### Correlation and Causation

Understand the concept of correlation – a close relationship or association between two or more variables. Understand the concept of causation – changes in one variable directly caused changes in another variable.

Residents in Chicago spend more money in department stores in the first three weeks of July, when the average temperature is 82°, than in December, when the average temperature is 32°. Do outdoor temperatures cause, or are they correlated to, differences in spending patterns in July and December.

##### Critical Review

Critically review sample surveys, experiments, and observational studies, and news media and organizational reports on surveys, experiments and studies, by:

• evaluating study design (research question, hypothesis)

• evaluating data gathering (sample size, randomization)

• reviewing data summaries and margin of error

• analyzing inferences and conclusions drawn.

##### Understanding Probability

Understand probability as the measure of how likely an event is to occur. Understand conditional probability – the probability of event A, given that event B has already occurred. Use two-way frequency tables to approximate conditional probability when two events are independent.

Example 1: Compare the chance of having lung cancer if you are a smoker with the chance of having lung cancer if you’ve never been a smoker.

Example 2: Collect data from a random sample of students in your high school on favorite subject – math, science or English. Estimate the probability that a randomly-selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

##### Independent and Dependent Events

Understand independent and dependent events. An event is independent if the outcome is not affected by other events. An event is dependent if the outcome can be affected by previous events.

What is the probability of drawing a Jack from a deck of cards given that your first draw was not a Jack?

##### Simple and Compound Events

Use rules of probability to calculate probabilities of simple events (for example, flipping a coin), and compound events (for example, rolling two dice).

##### Making Decisions

Use probability to make decisions, and evaluate a range of decisions and strategies, in business (release of new products, timing of sales), in sports (pitching rotations in baseball, substitutions in football), in health and medicine (likelihood of cancer, effectiveness of one treatment over another), and everyday life (commuting and traffic routes, getting into college, etc.). Weigh possible outcomes of a decision by assigning probabilities to payoff values, and finding expected values.

Example 1: Find the expected value of a state lottery ticket, given its cost of purchase and its payoff amount.

Example 2: Compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.