Curriculum for the high school grades (9-12) for the state of North Carolina (US) : Curriculum for the high school grades (9-12) for the state of North Carolina (US)
Objectives : Objectives This module explains the curriculum for the high school grades (9-12) for the state of North Carolina (US) Country: United States
State: North Carolina
Time Zone: Eastern Standard Time (GMT-5) till 11th March 2007
Subject: Mathematics
Topic: Algebra 1
Language: English
Grade: Grade 9-12
Introduction : Students will be expected to
Describe and translate among graphic, algebraic, numeric, and verbal representations of relations and use those representations to solve problems
Extend their use of symbols to include vectors and matrices
Use technology to assist in developing models and analytical solutions. use appropriate terminology and notation to define function, domain, range, composition, and inverses of functions
Expand their understanding of functions to include power, polynomial, exponential, periodic, piece-wise, and recursively defined functions
Solve equations, inequalities, and systems using algebraic, tabular, numerical, and graphical methods Introduction
Contents : Algebra 1 includes:
The study of algebraic concepts
Operations with polynomials and matrices,
Creation and application of linear functions and relations,
Algebraic representations of geometric relationships, and
An introduction to nonlinear functions
Contents
Pre-requisites : What your student must already know?
Operate with the real numbers to solve problems.
Find, identify, and interpret the slope and intercepts of a linear relation.
Visually determine a line of best fit for a given scatter-plot; explain the meaning of the line; and make predictions using the line.
Collect, organize, analyze, and display data to solve problems.
Apply the Pythagorean Theorem to solve problems. Pre-requisites
Competency Goal 1 : Competency Goal 1 1.01 Write equivalent forms of algebraic expressions to solve problems
Apply the laws of exponents
Operate with polynomials
Factor polynomials Sample Problems: Write an algebraic expression for each verbal expression.
a. a number k minus six
The word minus suggests subtraction.
Thus, the algebraic expression is k - 6.
b. the sum of two times a number b and four
Sum implies addition, and times implies multiplication. So the expression can be written as 2b+4.
Competency Goal 1 : Competency Goal 1 A ball is thrown from the top of a building that is 34 feet above ground. How long
until the ball is 10 feet above the ground?
Use the model for vertical motion. Let s = 34, v = 40, and h = 10.
h = -16t 2 + 40t + s Vertical motion model
10 = -16t 2 + 40t + 34 Substitute.
0 = -16t 2 + 40t + 24 Subtract 10 from each side.
0 = -8(2t 2 - 5t + 3) Factor out –8.
0 = (2t 2 - 5t + 3) Divide each side by –8.
0 = (2t + 1)(t – 3) Factor 2t 2 - 5t + 3.
2t + 1 = 0 or t – 3 = 0 Zero Product Property
2t = -1 t = 3 Solve each equation
t = -1/2
The solutions are -1/2 and 3 seconds. The only reasonable solution is the positive 3
seconds, therefore, the ball will reach a height of 10 feet after 3 seconds.
Competency Goal 1 : Competency Goal 1 Example: Power of a Quotient
Simplify -3m2n 2
5p
= (-3m2n)2
(5p) 2 Example: Quotient of Powers
Simplify 63x6y2
6x4y
= 63 x6 y2
6 x4 y
= (63-1) (x6-4) (y2-1 )
= 62 x2 y
= 36x2y
= (-3) 2 (m2) 2 n2 52p2 = 9m4n2
25p2
Competency Goal 1 : 1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems Sample Problem: A student strained her knee in an intramural volleyball game, and her doctor has prescribed an anti-inflammatory drug to reduce the swelling. She is to take two 220-mg tablets every 8 hours for 10 days. Her kidneys eliminate 60% of this drug from her body every 8 hours. Assume she faithfully takes the correct dosage at the prescribed regular intervals. Start with the initial dose (440), the elimination rate (0.60), and the recurring dose (440). Calculate values for the amount of medicine in her body just after taking each dose of medicine. Competency Goal 1
Competency Goal 1 : Sample Problem: Jane wants to make a doll that is a replica of a friend. The scale that she will use is 1 centimeter on the doll will represent 2 inches on her friend. Her friend’s foot measures 8.5 inches. What will be the size of the doll’s foot?
Let s represent the size of the doll’s foot.
scale scale
1 s
2 = 8.5
actual actual
1 (8.5) = 2 (s) Find the cross products.
8.5 = 2s Simplify.
8.5 = 2s
2 2 Divide each side by 2.
4.25 = s Simplify.
The size of the doll’s foot will be 4.25 centimeters. Competency Goal 1
Competency Goal 2 : Competency Goal 2 Sample Problem: Write the ordered pair for point D.
• Follow along a vertical line through the point to find the x-coordinate on the x-axis.
The x-coordinate is 2.
• Follow along a horizontal line through the
point to find the y-coordinate on the y-axis.
The y-coordinate is –3.
So, the ordered pair for the point D is (2, -3). This can also be written as D(2, -3).
Competency Goal 2 : Competency Goal 2 Sample Problem:
Competency Goal 2 : Competency Goal 2 Sample Problem 2.02 Use the parallelism or perpendicularity of lines and segments to solve problems Parallelogram QRST has vertices Q(-2, 1), R(1, 1), S(3, 3), and T(0, 3). Find the coordinates translated 1 unit to the right and 3 units up.
To translate the parallelogram 1 unit to the right, add 1 to the x-coordinate of each vertex. To translate the parallelogram 3 units up, add 3 to the y-coordinate of each vertex.
(x, y)→(x + 1, y + 3)
Q(-2, 1)→Q’(-2 + 1, 1 + 3) → Q’(-1, 4)
R(1, 1)→R’(1 + 1, 1 + 3) → R’(2, 4)
S(3, 3)→S’ (3 + 1, 3 + 3) → S’ (4, 6)
T(0, 3)→T’(0 + 1, 3 + 3) → T’(1, 6)
The coordinates of the vertices of the image are Q’(-1, 4), R’(2, 4), S’(4, 6), and T’(1, 6).
Competency Goal 3 : Sample Problem: Walter, Emily, and Ryan are swimmers. Walter swam 12 freestyle laps of the swimming pool and 30 backstroke laps, Emily swam 20 freestyle laps and 20 backstroke laps, and Ryan swam 18 freestyle laps and 20 backstroke laps. Write a matrix that could represent this information?
The matrix that represent the data is
12 20 18
30 20 20
Competency Goal 3
Competency Goal 3 : Sample Problem: Perform Scalar Multiplication If J= 4 -10 , find -2J
14 17
-2J = -2 4 -10
14 17
= -2 (4) -2(-10)
-2 (14) -2(17)
= -8 20
-28 -34 Competency Goal 3
Competency Goal 3 : 3.03 Create linear models for sets of data to solve problems.
a) Interpret constants and coefficients in the context of the data.
b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions. Sample Problem: The table shows the percent of U.S. workers in Farm Occupations.
Years since 1900 0 20 40 60 80 94
Farm Workers 37.5 27 17.4 6.1 2.7 2.5
The line of fit is y = - 49 x + 1244. Check the solution.
148 37 Check: Check your result by substituting (94, 2.5) into y = - 49 x + 1244
148 37
y = - 49x + 1244 Line of fit equation
148 37
2.5 = - 49 (94) + 1244 Replace x with 94 and y with 2.5.
148 37
2.5 = - 2303 + 1244 Multiply.
74 37
2.5 = 5 0r 2.5 The solution checks.
2 Competency Goal 3
Competency Goal 3 : Competency Goal 3 Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points.
The constant of variation is - 1
2
m = (y2-y1) Slope Formula
(x2 - x1)
m = 2 - 0 (x1 , y1 ) = (0, 0)
- 4 - 0
(x 2 , y 2 ) = (-4, 2)
m = - 1
2
The slope is -1
2
Competency Goal 4 : Competency Goal 4 4.01 Use linear functions or inequalities to model and solve problems; justify results.
Solve using tables, graphs, and algebraic properties.
b) Interpret constants and coefficients in the context of the problem. If g(x) = -3x – 4, find each value.
a. g(-3) –2
g(-3) – 2 = [-3(-3) – 4] - 2 Replace x with –3.
= 5 - 2 Simplify.
= 3 Subtract.
b. g(2x – 1)
g(2x – 1) = -3(2x – 1) – 4 Replace x with 2x – 1.
= -6x +3 – 4 Distributive Property
= -6x – 1 Simplify. Sample Problems:
Competency Goal 4 : Competency Goal 4 Sample Problem:
Competency Goal 4 : Sample Problems: 4.02 Graph, factor, and evaluate quadratic functions to solve problems Use a table of values to graph y = x2 + 6x + 8. Graph these ordered pairs and connect them with a smooth curve. Competency Goal 4
Competency Goal 4 : Use quadratic method to solve x 2 + 12x +20 = 0
Method 1 For this equation, a = 1, b = 12, and c = 20
x= -b+ b2-4ac Quadratic Formula
2a
= -12 + 122-4(1)(20) a = 1, b = 12, and c = 20
2(1)
= -12 + 144 - 80 Multiply.
2
= -12 + 64 Subtract
2
= -12 + 8 Take the square root of 64.
2
x = -12 - 8 or x = -12 + 8
2 2
= -10 = -2 Competency Goal 4
Competency Goal 4 : Method 2 Factoring
x 2 + 12x + 20 = 0 Original equation
(x + 10)(x + 2) = 0 Factor x 2 + 12x + 20.
x + 10 = 0 or x + 2 = 0 Zero Product Property
= -10 = -2 Solve.
The solution set is {-10, -2}. Competency Goal 4
Competency Goal 4 : Sample Problems: 4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results. Graph each system of equations. Determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
a. y = 1 x – 3
3
x – 3y = - 3
The graphs of the equations are parallel lines. Since
they do not intersect, there are no solutions to this
system of equations. Notice that the lines have the
same slope but different y-intercepts. Recall that a
system of equations that has no solution is said to be
inconsistent. Competency Goal 4
Competency Goal 4 : Alexis bought pizza and soda for the ski club meeting. For one meeting she bought 4 pizzas and 10 sodas for $63. The next meeting she bought 3 pizzas and 8 sodas for $48. What is the cost of one pizza?
Let p = the cost of one pizza, and let s = the cost of one soda. Write a system of equations to represent the situation.
Total cost of pizzas plus total cost of sodas equals total cost
4p + 10s = 63
3p + 8s = 48
Graph the equations 4p + 10s = 63 and 3p + 8s = 48.
The graphs appear to intersect at the point with
coordinates (12, 1.5). Check this estimate by replacing
p with 12 and s with 1.5 in each equation.
Competency Goal 4
Competency Goal 4 : Sample Problem: 4.04 Graph and evaluate exponential functions to solve problems Graph an Exponential Function with x > 1 Graph y = 3x . State the y-intercept.
Graph the ordered pairs and connect the points with a smooth curve.
The y-intercept is 1.
Notice that they-values change little for small values of x, but they increase quickly as the values of x becomes greater. Competency Goal 4 1 2 3
Competency Goal 4 : Connor has 4 weeks before his math final exam. He plans to study for 3 hours the first week and increase the time he will study S(x) in hours according to the function S(x) = 3(1.7)x , where x represents the number of weeks of studying.
a) How many hours did he study the second week?
S(x) = 3(1.7)x Original equation
S(2) = 3(1.7)2 x = 2
S(2) = 8.67 Use a calculator.
He studied 8.67 hours during the second week.
b) Connor has scheduled 20 hours to study during the fourth week. According to the function, has he scheduled enough time?
S(x) = 3(1.7)x Original equation
S(4) = 3(1.7)4 x = 4
S(4) = 25.0563 Use a calculator.
According to the function, he should schedule 25.0563 hours. He has not scheduled enough time.
Competency Goal 4