Download Whitman-Hanson Regional High School provides all students with a

Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned
citizens and contributing members of the global community.
Course Number
430
Title
Algebra II A
Grade 9-12
# of Days 120
Course Description
This course emphasizes facility with algebraic expressions and forms. Functions based on quadratic forms,
powers and roots, logarithms, trigonometry, and polynomials are studied for their abstract properties and as tools
for modeling real-world situations. Student owned graphing calculators are necessary for this course. This
course is recommended for students with a good background in algebra, and is mandatory for students applying
to Massachusetts State Colleges and Universities. This course addresses Whitman-Hanson Student Learning
Expectations 1-6.
Instructional Strategies
Instructional Strategies include but may not be limited to the following:
1. Whole class instruction
2. Individual work: homework, classwork, assessments
3. Group work: activities, problem solving
4. Video presentations
5. Use of technology (graphing calculators and computers)
6. Projects
7. Daily quizzes
8. Homework check
9. Test corrections
10. Games
11. Websites
Student Learning
Expectations
1.
2.
3.
4.
5.
6.
Read, write and communicate effectively.
Utilize technologies appropriately and effectively.
Apply critical thinking skills.
Explore and express ideas creatively.
Participate in learning both individually and collaboratively.
Demonstrate personal, social, and civic responsibility.
1
Unit of Study
Matrices
MA Standard/Strands:
Time Frame:
Text
(Chapter/Pages)
Other Resources:
Advanced Algebra Tools for a Changing World
Chapter 3
Essential Questions
Concepts, Content:
1. How is a matrix defined?
A matrix is defined by the number of rows and columns.
2. How is data organized into a matrix?
Each column represents a different idea and each column represents another idea.
3. How are the dimensions of a matrix determined?
The dimensions of a matrix are determined by the number of rows and the number of columns in the
matrix.
4. How is a matrix element identified?
A matrix element is identified by its position. Write the row number first and then the column
number.
5. How is it determined whether two matrices are equal?
Two matrices are equal if an only if they have the same dimensions and their corresponding elements
are equal.
6. What is meant by corresponding elements?
Corresponding elements are found in the same row and same column of matrices.
7. When can two matrices be added together?
Two matrices may be added together if the dimensions are the same.
8. What is a matrix equation?
A matrix equation is an equation in which the variable is a matrix.
9. How is scalar multiplication performed?
A scalar is a number that multiplies a matrix. To perform scalar multiplication multiply every element
in the matrix by the scalar.
10. How is it determined whether two matrices can be multiplied?
A product matrix is defined if the number of columns in the first matrix equals the number of rows in
the second matrix.
2
11. How are the dimensions of the product matrix determined?
If matrix A is an m x n matrix and matrix B is an n x p matrix, then the product matrix AB is an
m x p matrix.
12. How is matrix multiplication performed?
To perform matrix multiplication, multiply the elements of each row of the first matrix by the
elements of each column of the second matrix. Add the products.
13. How are matrices entered into a graphing calculator?
To enter a matrix into a graphing calculator press the matrix key. Arrow over to the edit key and
press enter. In the next screen enter the dimensions and the elements for the matrix.
14. How are matrices multiplied on a graphing calculator?
Press the matrix key and highlight the desired matrix and press enter. Press the multiplication key.
Press the matrix key again and highlight the other desired matrix and press enter. Press the enter key
again.
15. What is a transformation?
A transformation is a change made to a figure.
16. What is a preimage?
A preimage is the original figure.
17. What is an image?
An image is the transformed figure.
18. What is a dilation?
A dilation is a transformation that changes the size of a figure.
19. What is a finite graph?
A finite graph is a set of points connected by paths. The points are called vertices.
20. What is a directed graph?
Directed graphs are finite graphs that indicate the direction of a path.
21. How is it determined that a matrix has an inverse?
A square matrix is a matrix with the same number of columns as rows. Every square matrix with real
number elements has a real number associated with it called its determinant. If the determinant ≠ 0,
the matrix has an inverse.
22. How is the determinant of a matrix found?
The determinant if a 2 x2 square matrix is the real number ad – bc.
23. How is the inverse matrix found on a graphing calculator?
Enter the matrix on the graphing calculator. To find the inverse use the 𝐴−1 key on the calculator.
24. How is an inverse matrix used to solve a matrix equation?
To solve the equation 𝐴𝑋 = 𝐵, where A and B are matrices, 𝑋 = 𝐴−1 𝐵
3
Targeted Skill(s):
Writing:
Assessment Practices:
4
Unit of Study Linear Systems in two or three variables
MA Standard/Strands:
Time Frame:
Text
(Chapter/Pages)
Other Resources:
Chapter 4
Essential Questions
Concepts, Content:
1. After graphing two lines and finding the point of intersection, how is the point of intersection
written?
The point of intersection is written as (x, y).
2. How is it determined that a system has one solution?
Graphically, the point of intersection represents a solution.
3. How is it determined that a system has no solution?
If the equations have the same slope the lines will not intersect. Therefore there is no solution.
Graphically, if the lines do not intersect there is no solution.
4. How is it determined that a system has an infinite number of solutions?
If the lines coincide there are infinitely many solutions.
5. How it the substitution method used to solve a system of equations?
In one of the equations, solve for one of the variables. Substitute the expression for that variable into
the other equation. Solve. Then substitute the salved value into one of the equations. This will give
you both values of the variables.
6.
How it the elimination method used to solve a system of equations?
In solving a system, a variable can be eliminated by adding the equations if like terms have opposite
coefficients. In solving a system, a variable can be eliminated by subtracting the equations if like
terms have the same coefficients.
7.
What if the two equations do not have like terms with the same or opposite coefficients?
If the two equations do not have like terms with the same or opposite coefficients, either one or both
equations must be multiplied to create opposite coefficients. Then the equations can be added to
eliminate a variable.
8. How do linear programming problems find the values of x and y that maximize the objective
function?
After graphing the restrictions, find the coordinates of each vertex. Evaluate the objective function at
each vertex.
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10.
11.
12.
13.
How are (x , y, z ), ordered triples plotted?
The first coordinate tells whether to go forward or backward, the second coordinate tells whether to
go left or right, and the third coordinate tells whether to go up or down.
How are equations in three dimensions graphed?
Find each intercept by replacing the other two variables with zero. Graph each of the intercepts.
Draw the traces. Shade the plane.
How is the elimination method used to solve equations with three variables?
Pair two equations to eliminate one variable. Pair two other equations to eliminate the same
variable. Write the two new equations as a system and solve for both variables. Substitute the
values for both variables into one of the original equations and solve for the first variable.
How is the substitution method used to solve equations with three variables?
Substitute the value of one of the variables into another equation. Then substitute the new value
into the third equation and solve for the value of one of the variables. Then repeat the process.
How can a system of equations be solved using inverse matrices?
First write the system as a matrix equation. Using a graphing calculator, enter the coefficient matrix
as matrix A and the constant matrix as matrix B. Then calculate 𝐴−1 𝐵.
Targeted Skill(s):
Writing:
Assessment Practices:
Unit of Study Quadratic Equations and Functions
MA Standard/Strands:
Time Frame:
Text
(Chapter/Pages)
Other Resources:
Chapter 5
6
Essential Questions
Concepts, Content:
1. What is a Quadratic Function?
A quadratic function is one that can be written in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 .
2. How is it decided whether to use a linear or a quadratic model?
The graph of a quadratic function is a parabola. If the data appear to fit the shape of a parabola you can
use a quadratic equation to model the data.
3. How is a quadratic model found?
To find a quadratic model, you can either solve a system of equations or use the quadratic regression
feature of a graphing calculator.
4. What is the vertex of a parabola?
The highest (or lowest) point of a parabola that is the graph of a function is its vertex.
5. What is the axis of symmetry?
The axis of symmetry is the imaginary line that divides the parabola into two parts that are mirror
images.
6. What is vertex form?
An equation in the form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 is said to be in vertex form. This is a translation of
𝑦 = 𝑎𝑥 2 .
7. When the function is written in standard form, how is the vertex found?
𝑏
The equation of the axis of symmetry of a function while written in standard form is 𝑥 = − 2𝑎.
8. If an equation is given in vertex form how can it be changed to standard form?
First, the binomial must be squared. Second, the distributive property used must be used. Third, the
expression must be simplified and set equal to 𝑦.
9. If an equation is given in standard form how can it be changed to vertex form?
𝑏
Find the x-coordinate of the vertex by using 𝑥 = − 2𝑎. This is the value of ℎ. Substitute the found
value into the original equation for 𝑥 and find the value of 𝑦. This is the value of 𝑘. Substitute the
values of 𝑎, ℎ, and 𝑘 into the equation 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘.
10. How is the inverse of a function found?
Switch 𝑥 and 𝑦 and solve for 𝑦.
11. What is a square root function?
A square root function is a function in the form 𝑦 = √𝑥.
12. In what ways can a quadratic equation be solved?
A quadratic equation can be solved by factoring, finding square roots, and by graphing.
13. What is the imaginary number ?
The imaginary number 𝑖 is defined as √−1 .
14. What is a complex number?
A complex number is a number that can be written in the form 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are real
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15.
16.
17.
18.
19.
20.
21.
22.
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numbers (including 0).
What is the absolute value of a complex number?
The absolute value of a complex number is its distance from the origin on the complex number
plane.
How are complex numbers added and subtracted?
To add or subtract complex numbers, combine the real parts and the imaginary parts separately.
What does 𝒊 ∙ 𝒊 equal?
𝑖 ∙ 𝑖 = 𝑖 2 = −1
How are complex numbers multiplied?
Multiply two imaginary numbers of the form 𝑎 + 𝑏𝑖 by using the procedure for multiplying
binomials.
What is the process for completing the square?
Write all of the terms containing an 𝑥 on one side of the equation. Take half of the coefficient of the
𝑥 - term and square it. Add this to both sides of the equation. Write the trinomial as a binomial
squared.
How is a quadratic equation solved by completing the square?
Write all of the terms containing an 𝑥 on one side of the equation. Take half of the coefficient of the
𝑥 - term and square it. Add this to both sides of the equation. Write the trinomial as a binomial
squared. Find the square root of each side. Solve for 𝑥 and simpllify.
What is the quadratic formula?
Given a quadratic equation written in standard form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, the quadratic formula is
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
What is the discriminant?
In the quadratic formula, the value of the expression 𝑏 2 − 4𝑎𝑐 is the discriminant.
What does the value of the discriminant determine?
The value of the discriminant determines the type and number of solutions the quadratic equation has.
If 𝑏 2 − 4𝑎𝑐 > 0 there are two real solutions. If 𝑏 2 − 4𝑎𝑐 = 0 there is one real solution.
If 𝑏 2 − 4𝑎𝑐 < 0 there is no real solution, there are two imaginary solutions.
Targeted Skill(s):
8
Writing:
Assessment Practices:
9