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: Integrated Mathematics 4 Language: English
Grade: Grade 9-12
Introduction : Students will be expected to
Understand and use applications and modeling included throughout the course of study.
Use the appropriate technology, from manipulatives to calculators and application software.
Use regularly for instruction and assessment. Introduction
Contents : Integrated Mathematics 4 includes:
An advanced study of trigonometry, functions, analytic geometry, and data analysis with a problem-centered, connected approach in preparation for college-level mathematics. Contents
Pre-requisites : What your student must already know?
Describe phenomena as functions graphically, algebraically and verbally; identify independent and dependent quantities, domain, and range, and input/output.
Translate among graphic, algebraic, numeric, tabular, and verbal representations of relations.
Define and use linear, quadratic, cubic, exponential, rational, absolute value, and radical functions to model and solve problems.
Use systems of two or more equations or inequalities to solve problems.
Use the trigonometric ratios to model and solve problems.
Use logic and deductive reasoning to draw conclusions and solve problems
Pre-requisites
Competency Goal 1 : Competency Goal 1 1.01 Operate with vectors in two dimensions to model and solve problems. Sample Problems:
If the values AB and BC vectors are
5 units and 7 units then find the
value of vector AC. Solution:
According to the triangle law of vector addition AB + BC = AC
Therefore,
AC = 5 units + 7 units
AC = 12 units A B C
Competency Goal 1 : Competency Goal 1 1.02 Define and compute with complex numbers. Sample Problems: Solution: Compute :
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.01 Use the quadratic relations (parabola, circle, ellipse, hyperbola) to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Interpret the constants and coefficients in the context of the problem. A rocket is launched at a contest. The path of the rocket can be modeled by h(t) = -16t 2 - 16t + 480, where t is the time in seconds that the rocket travels and h(t) is the height of the rocket at any time. Graph the equation for the height of the rocket. What is the maximum height of the rocket? Graph the parabola to determine the vertex of the parabola 0 -16(5)2 - 16(5) + 480 5 288 -16(3)2 - 16(3) + 480 3 448 -16(1)2 - 16(1) + 480 1 484 -16(-0.5)2 - 16(-0.5) + 480 -0.5 448 -16(-2)2 - 16(-2) + 480 -2 0 -16(-6)2 - 16(-6) + 480 -6 h(t) -16t 2 - 16t + 480 t Solution:
Competency Goal 2 : Competency Goal 2 Solution: (contd.) The vertex is located on the axis of symmetry. The equation for the axis of symmetry is x = -(b/2a). The highest point the rocket will reach is the y-coordinate of the vertex.
The maximum height of the rocket is 484 feet.
Competency Goal 2 : Competency Goal 2 Sample Problem 2.02 Estimate the area and volume of continuously varying quantities. Find the rate of change of the area of a circle with respect to its radius.
Thus the rate of change of the area of the circle
w.r.t its radius r is
Answer. Solution:
Competency Goal 3 : 3.01 Analyze univariate data to solve problems.
a) Apply and compare methods of data collection.
b) Apply statistical principles and methods in sample surveys.
c) Determine measures of central tendency and spread.
d) Recognize, define, and use the normal distribution curve.
e) Interpret graphical displays of distributions of univariate data.
f) Compare distributions of univariate data. Sample Problem: Competency Goal 3 $ 4260000 $ 5483986 $ 5859691 $ 6931191 $ 8485333 $ 4414285 $ 552250 $ 6020000 $ 694399 $ 8851198 Top ten salaries of Quarterbacks in the NFL in 2001 The following is a list of the top ten salaries of quarterbacks
in the NFL in 2001. Make a spreadsheet of the data for
central tendencies
Competency Goal 3 : Competency Goal 3 Solution: $ 4260000 11 $ 4414285 10 $ 5483986 9 $ 552250 8 $ 5859691 7 $ 6020000 6 $ 6931191 5 $ 694399 4 $ 8485333 3 # N/A 5939846 6280033 $ 8851198 2 MODE MEDIAN MEAN DATA 1 D C B A Use = AVERAGE (A2:A11) to find the mean. Use = MEDIAN (A2:A11) to find the median. Use = MODE (A2:A11) to find the mode. Following is the required spreadsheet
Competency Goal 3 : 3.02 Create and use calculator-generated models of linear, polynomial, exponential, trigonometric, power, logistic, and logarithmic functions of bivariate data to solve problems.
a) Interpret the constants, coefficients, and bases in the context of the data.
b) Check models for goodness-of-fit; use the most appropriate model to draw conclusions or make predictions. Sample Problem: Competency Goal 3 Write an equation for the best fit line which passes through points at
(25, 64) and (35, 58) and the y-intercept is 79. Solution: The line passes through points at (25, 64) and (35, 58). Using these
points the slope of the line is
m = (definition of slope)
or, m =
or, m =
Competency Goal 3 : Competency Goal 3 Solution: (contd.) Using the slope-intercept form of equation y = mx + b
We have m = 3/5 and b = 79
Hence, the equation is y = (3/5) x + 79
Therefore the equation stated above is the best fit
for the given points.
Competency Goal 4 : Competency Goal 4 4.01 Use functions (polynomial, power, rational, exponential, logarithmic, logistic, piecewise-defined, and greatest integer) to model and solve problems; justify results.
a) Solve using graphs and algebraic properties.
b) Interpret the constants, coefficients, and bases in the context of the problem. Sample Problem: Find two numbers whose sum is 6 and whose product is - 55. Solution: Let x = one of the numbers. Then 6 - x = the other number.
Since the product of the two numbers is -55, you know that -55 = x(6 - x). -55 = x(6 - x) or, -55 = 6x - x2 0 = -x2 + 6x + 55 [Adding 55 to each side.]
x2 - 6x - 55 = 0 or, (x-11) (x + 5) = 0
Hence, x = 11 or -5
and, y = -5 or 11
Therefore, the numbers are 11 & -5 Answer.
Competency Goal 4 : Competency Goal 4 4.02 Use recursively-defined functions to model and solve problems.
a) Find the sum of a finite sequence.
b) Find the sum of an infinite sequence.
c) Determine whether a given series converges or diverges.
d) Translate between recursive and explicit representations. Sample Problems: Write the next three terms of the sequence 1, 8, 15, 22, …
Each term is 7 greater than the previous term.
Continue the pattern to find the next three terms.
22 + 7 = 29 29 + 7 = 36 36 + 7 = 43
The next three terms are 29, 36, and 43.
Solution: Write the next three terms of the sequence 5, 20, 80, 320, …
Each term is 4 times the previous term.
Continue the pattern to find the next three terms.
320 4 = 1,280 1,280 4 = 5,120 5,120 4 = 20,480 Sample Problems: Solution: The next three terms are 1,280, 5,120, and 20,480.
Competency Goal 4 : Competency Goal 4 Sample Problems: Solution: Find the sum of 15 terms of the series 3,7,11, 15…….. As the common difference between two consecutive terms of the given series are 4 therefore, the series is in A.P. The first term, a = 3 and d = 4
Competency Goal 4 : Competency Goal 4 4.03 Use the composition and inverse of functions to model and solve problems. Sample Problems: A tomato plant grew 10 centimeters in four weeks. It grew 1 centimeter, 2 centimeters, and 4 centimeters in the first three weeks after it was planted. How much did the plant grow in the fourth week?
Words The sum of the growth for all four weeks is 10 centimeters.
Variable Let g represent the growth during the fourth week.
growth for the growth for the
first three weeks fourth week total growth
Equation 1 + 2 + 4 + g = 10
1 + 2 + 4 + g = 10 Write the equation.
7 + g = 10 1 + 2 + 4 = 7
- 7 = - 7 Subtract 7 from each side.
g = 3 Simplify. Solution:
Competency Goal 4 : Competency Goal 4 4.04 Use trigonometric and inverse trigonometric functions to model and solve problems.
a) Solve using graphs and algebraic properties.
b) Create and identify transformations with respect to period, amplitude, and vertical and horizontal shifts. Sample Problems: 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?
Competency Goal 4 : Competency Goal 4 Solution:
Competency Goal 4 : Competency Goal 4 4.05 Use polar equations to model and solve problems.
a) Solve using graphs and algebraic properties.
b) Interpret the constants and coefficients in the context of the problem. Sample Problem: If we choose t such that cos(t) = 0.5, then r = 0. Thus, the pole is part of C and we don't add a point to C if we multiply both sides of r = 1 + 2.cos(t) by r. C has a polar equation r2 = r + 2.r.cos(t) <=> r = r2 - 2.r.cos(t)
Now take the curve C' : r = - ( r2 - 2.r.cos(t)) .
It is easy to prove that each point of C is on C' and so is the reverse.
Hence, we can write C has a polar equation r = (+1 or -1).( r2 - 2.r.cos(t))
=> C has a polar equation r2 = (r2 - 2 r cos(t))2
As we know that x = r cos(t), y = r sin(t) and x2 + y2 = r2
=> C has a Cartesian equation x2 + y2 = ( x2 + y2 -2 x )2 Suppose a curve C has a polar equation r = 1 + 2.cos(t). Change this equation in Cartesian form Solution:
Competency Goal 4 : Competency Goal 4 4.06 Use parametric equations to model and solve problems. Sample Problems: Convert the following parametric equation into normal Cartesian equation. x = t2 + 1 & y = 2t – 1 Solution: Now, y = 2t – 1
=> y2 = (2t – 1)2 [squaring both sides we get]
=> y2 = 4t2 – 4t + 1
= 4t2 +4 – 4 – 4t + 2 – 2 +1
= 4(t2 +1) – 4 – 2(2t +1) – 2 +1
= 4(t2 +1) – 2(2t +1) – 5
=> y2 = 4x – 2y – 5
=> y2 – 4x + 2y + 5 = 0
This is the required equation.
Competency Goal 4 : Competency Goal 4 4.07 Find the rate of change at any point of a function. Sample Problems: Find the rate of change of y w.r.t. x at x = 2 when y = 5x5 + 4x3 +6 Solution: dx dy