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: Technical Mathematics 1 Language: English
Grade: Grade 9-12
Introduction : Students will be expected to
Understand the different algebraic applications.
Use those representations to solve problems.
Appropriate technology, from manipulatives to calculators and application
Software, should be used regularly for instruction and assessment. Introduction
Contents : Technical Mathematics 1 includes:
Study of algebra and geometry, building upon middle school and Algebra I topics. Problem solving, measurement, special relationships in right triangles, transformations, and geometric applications of algebra Contents
Pre-requisites : What your student must already know?
• Apply geometric properties and relationships to solve problems.
• Use tables, formulas and algebraic expressions to model and solve problems.
• Define and use linear functions to model and solve problems.
• Operate with matrices to model and solve problems. Pre-requisites
Competency Goal 1 : Competency Goal 1 1.01 Apply various techniques and strategies to solve problems.
a) Select or create an appropriate graphical display for a given set of data.
b) Identify and represent patterns using appropriate algebraic notation.
c) Select and apply appropriate formulas.
d) Choose or create appropriate representations of two- and three-dimensional
figures. Sample Problems: Solution: Factorise 4x2 + 12xy +9y2
4x2 + 12xy +9y2 = (2x)2 + 2. (2x). (3y) + (3y)2
= a2 + 2.a.b + b2 [ Putting 2x = a, 3y = b]
= (a + b)2 [ Using the formula (a + b)2 = a2 + 2.a.b + b2 ]
= (2x + 3y)2
= (2x + 3y) (2x + 3y)
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.01 Select and use appropriate tools to measure two- and three-dimensional figures; interpret and communicate results with regard to precision. A cylinder can be completely filled with 46 liters of water. If the capacity of a bottle is 4 liters of water then how many such bottles are required to empty the cylinder? Solution: In the given problem the volume of the containers are measured in lit e r . No. of bottles required = 4 46 = 4 2 12 But number of bottles can not be fraction \ N o. of bottles required = 13.
Competency Goal 2 : Competency Goal 2 Sample Problem 2.02 Interpret and construct maps and scale drawings to solve problems. Dew starts from his home and walks 12 miles in the east then he turns left and walks nearly 5 miles to reach his school. If he would walk along the straight line from his home to the school, find the distance he would travel. Solution:
Competency Goal 2 : Competency Goal 2 Sample Problem 2.03 Use the length, area, and volume of geometric figures to solve problems. Include arc length, area of sectors of circles; lateral area, surface area, and volume of three-dimensional figures; and perimeter, area, and volume of composite figures. A farmer wishes to grow a 100 square meter rectangular vegetable garden. Since he has with him only 30 m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side of the fence. Find the dimensions of his garden. Solution: C D X X A B Let one of the opposite equal sides be x. According to the question, AB + BC + CD = 30 or, x + BC + x = 30 or, 2x + BC = 30 or, 2x +BC – 2x = 30 - 2x or, BC = 30 – 2x Area of rectangle = length x width
Competency Goal 2 : Competency Goal 2 Solution: (contd.) Area of the given rectangle = x (30 – 2x) = 100 [given]
or, 30x – 2x2 = 100
or, 15x – x2 = 50
or, x2 -15x +50 = 0
or, x2 - 5x – 10x + 50 =0
or, (x2 - 5x) – (10x + 50) = 0 or, x(x - 5) – 10(x – 5) = 0 or, (x - 5) (x – 10) = 0
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.04 Use the trigonometric ratios to model and solve problems involving right triangles. Solution: A B C A triangle ABC is drawn so that sin q = 5 3 Thus perpendicular AB and hypotenuse AC are in the ratio 3 : 5 \ let AB = 3u & AC = 5u \ by Pythagorean theorem, BC = 2 2 AB AC - = 2 2 ) u 3 ( ) u 5 ( - = 2 2 u 9 u 25 - = 2 u 16 = 4u [BC cannot be negative] We know tan q = base lar perpendicu = BC AB = u 4 u 3 = 4 3
Competency Goal 3 : 3.01 Apply algebraic and trigonometric concepts to confirm properties of geometric figures in the coordinate plane. Sample Problem: Competency Goal 3 An airplane is sighted at an angle of 28˚ from the airport. It is at a distance of 7000 feet from the airport as shown in the diagram. What is the height h of the airplane from the ground? 7000 ft S 28 deg Solution: We know the length of the hypotenuse and the measure of Ð S . We need to find the measure of the leg opposite Ð S . Use the sine ratio. S in S = PQ PS hypotenuse opposite S in 28? = x 7000 Replace S with 28?, PQ with x, and PS with 7000. 7000 X S in 28? = 7000 ) x 7000 Multiply each side by 7000. 7000 X S in 28? = x 7000 X 0.4695 = 3286. 5 (putting the value of Sine from calculator) Therefore, the airplane is about 3286. 5 feet above the ground.
Competency Goal 3 : 3.02 Describe the transformation (translation, reflection, rotation, dilation) of polygons in the coordinate plane in simple algebraic terms. Sample Problem: Competency Goal 3 Quadrilateral ABCD has vertices A(2, 3), B(3, 5), C(7, 1), and D(5, –2). Find the coordinates of ABCD after a reflection over the y-axis. Then graph the figure and its reflected image. D'(–5, –2) 5 D(5, –2) C'(–7, 1) 7 C(7, 1) B'(–3, 5) 3 B(3, 5) A'(–2, 3) 2 A(2, 3) Vertices of
quad A'B'C'D' Distance
from y-axis Vertices of
quad ABCD Solution:
Competency Goal 3 : Competency Goal 3 Solution: (contd.) Plot the vertices and connect to form quadrilateral ABCD. The y-axis is the line of symmetry. So, the distance from each point on quadrilateral ABCD to the line of symmetry is the same as the distance from the line of symmetry to quadrilateral A'B'C'D'.
Competency Goal 3 : 3.03 Use matrix operations (addition, subtraction, multiplication, scalar multiplication) to describe the transformation of polygons in the coordinate plane. Sample Problem: Competency Goal 3 ABC with vertices A(3, 3), B(3, 6), and C(0, 9). Then graph its image A′B′C′ after a dilation with a scale factor of . Solution: To find the vertices of the dilation, multiply each coordinate in the ordered pairs by 3 1 . Then graph both images on the same axes. The old co - ordinate matrix is ÷ ÷ ÷ ø ö ç ç ç è æ 9 0 6 3 3 3 The old co - ordinate matrix is ÷ ÷ ÷ ø ö ç ç ç è æ = ÷ ÷ ÷ ø ö ç ç ç è æ 3 0 2 1 1 1 9 0 6 3 3 3 3 1
Competency Goal 3 : Competency Goal 3 Solution: (contd.) Check: Draw lines through the origin and each of the vertices of the original figure. The vertices of the dilation should lie on those same lines.