Correlations of Unit Activity to Common Core: Precalculus Vectors & Matrices, Study notes of Technology

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Unit Activity Correlations to

Common Core State Standards

Precalculus

Table of Contents

Number and Quantity 1

Algebra 3

Functions 3

Geometry 5

Statistics and Probability 6

Unit 6, EA 6-1: Parametric Equations and Vectors Unit 6, Unit Practice

Perform operations on vectors.

4. Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of

two vectors is typically not the sum of the magnitudes.

Unit 6, Activity 6-2: Introduction to Vectors Unit 6, Activity 6-3: Vectors in Two and Three Dimensions Unit 6, Activity 6-4: Parametric Equations Revisited

Unit 6, EA 6-1: Parametric Equations and Vectors Unit 6, Unit Practice

b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Unit 6, Activity 6-2: Introduction to Vectors Unit 6, Activity 6-3: Vectors in Two and Three Dimensions Unit 6, Activity 6-4: Parametric Equations Revisited

Unit 6, EA 6-1: Parametric Equations and Vectors Unit 6, Unit Practice

c. Understand vector subtraction v – w as v + (–w), where – w is the additive inverse of w, with the same magnitude as w

and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the

appropriate order, and perform vector subtraction component-wise.

Unit 6, Activity 6-2: Introduction to Vectors Unit 6, Activity 6-3: Vectors in Two and Three Dimensions Unit 6, Activity 6-4: Parametric Equations Revisited

Unit 6, EA 6-1: Parametric Equations and Vectors Unit 6, Unit Practice

5. Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar

multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

Unit 6, Activity 6-3: Vectors in Two and Three Dimensions Unit 6, Activity 6-4: Parametric Equations Revisited

Unit 6, EA 6-1: Parametric Equations and Vectors Unit 6, Unit Practice

b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when

|c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Unit 6, Activity 6-3: Vectors in Two and Three Dimensions Unit 6, Activity 6-4: Parametric Equations Revisited

Unit 6, EA 6-1: Parametric Equations and Vectors Unit 6, Unit Practice

Perform operations on matrices and use matrices in applications.

6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Unit 6, Activity 6-3: Vectors in Two and Three Dimensions

7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Unit 6, Activity 6-3: Vectors in Two and Three Dimensions

8. Add, subtract, and multiply matrices of appropriate dimensions.

Unit 6, Activity 6-3: Vectors in Two and Three Dimensions

9. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative

operation, but still satisfies the associative and distributive properties.

Unit 1, Activity 1-6: Matrix Operations Unit 1, Activity 1-7: Matrix Properties and Equations

10. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role

of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a

multiplicative inverse.

Unit 1, Activity 1-6: Matrix Operations Unit 1, Activity 1-7: Matrix Properties and Equations

Unit 5, EA 5-3: Matrices, Transformations, and Vectors Unit 5, Activity 5-7: Transformations with Matrices

12. Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in

terms of area.

Unit 5, EA 5-3: Matrices, Transformations, and Vectors Unit 5, Activity 5-7: Transformations with Matrices

Algebra

Reasoning with Equations and Inequalities

Solve systems of equations

8. Represent a system of linear equations as a single matrix equation in a vector variable.

Unit 1, Activity 1-6: Matrix Operations Unit 1, Activity 1-7: Matrix Properties and Equations

9. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices

of dimension 3 × 3 or greater).

Unit 1, Activity 1-6: Matrix Operations Unit 1, Activity 1-7: Matrix Properties and Equations

Functions

Interpreting Functions

Analyze functions using different representations

7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using

technology for more complicated cases.

d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing

end behavior.

Unit 2, Unit Overview Unit 2, Activity 2-3: Complex Polynomial Roots and Inequalities Unit 2, Activity 2-4: Rational Functions Unit 2, Activity 2-5: Rational Functions Unit 2, EA 2-2: Rational Functions

Unit 2, Activity 2-8: Transformations of Functions Unit 2, Activity 2-9: Effects of Transformations Unit 2, EA 2-3: Transformed Functions Unit 2, Unit Practice Unit 2, Math Standards Review

4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Unit 3, Activity 3-4: Graphs of the form y = A sin[B(x – C)] + D Unit 3, Activity 3-5: Graphs of Other Trigonometric Functions

Model periodic phenomena with trigonometric functions

6. Understand that restricting a trigonometric function to a domain on which it is always increasing or always

decreasing allows its inverse to be constructed.

Unit 3, Activity 3-6 Inverse Trigonometric Functions Unit 3, Activity 3-7: Solving Simple Trigonometric Equations

Unit 3, EA 3-2: Inverse Trig Functions and Equations Unit 3, Unit Practice

7. Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using

technology, and interpret them in terms of the context.

Unit 3, Activity 3-7: Solving Simple Trigonometric Equations Unit 3, EA 3-2: Inverse Trig Functions and Equations Unit 3, Unit Practice

Prove and apply trigonometric identities

9. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Unit 4, Activity 4-3: Multiple Angle Identities Unit 4, EA 4-1: Trigonometric Equations and Identities

Unit 4, Unit Practice Unit 4, Math Standards Review

Geometry

Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section

3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances

from the foci is constant.

Unit 5, Activity 5-2: Ellipses and Hyperbolas Unit 5, Unit Practice

Geometric Measurement and Dimension

Explain volume formulas and use them to solve problems

2. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid

figures.

Unit 6, Activity 6-3: Volume

Statistics and Probability

Using Probability to Make Decisions

Calculate expected values and use them to solve problems

1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space;

graph the corresponding probability distribution using the same graphical displays as for data distributions.

Unit 6, Activity 6-3: Normal Distribution

2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Unit 6, Activity 6-3: Normal Distribution

3. Develop a probability distribution for a random variable defined for a sample space in which theoretical

probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for

the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each

question has four choices, and find the expected grade under various grading schemes.

Unit 6, Activity 6-3: Normal Distribution

4. Develop a probability distribution for a random variable defined for a sample space in which probabilities are

assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets

per household in the United States, and calculate the expected number of sets per household. How many TV sets

would you expect to find in 100 randomly selected households?

Unit 7, Activity 7-1: Probability Experiments Unit 7, EA 7-1: Counting and Probability Unit 7, Activity 7-2: Dependent and Independent Events Unit 7, EA 7-2: Compound Events, Probability, Simulation

Unit 7, Activity 7-3: Dependent Compound Events Unit 7, Activity 7-4: Geometric Probability Unit 4, Activity 4-7: Binomial Probability

Use probability to evaluate outcomes of decisions

5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or

a game at a fast-food restaurant.

Unit 5, Activity 5-1: Probability

b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a

low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major

accident.

Unit 5, Activity 5-1: Probability