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: Advanced Functions and Modeling
Language: English
Grade: Grade 9-12
Introduction : Students will be expected to
Have in-depth study of modeling and applying functions. Home, work, recreation, consumer issues, public policy, and scientific Investigations
Appropriate technology, from manipulatives to calculators and application
Software, should be used regularly for instruction and assessment. Introduction
Contents : Advanced Function and Modeling includes:
In-depth study of modeling and applying functions. Home work, recreation, consumer issues, public policy, and scientific investigations and many more. 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, and exponential 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 Create and use calculator-generated models of linear, polynomial, exponential, trigonometric, power, 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 and make predictions. Sample Problems: Find the measure of G. N R G 42 21
Competency Goal 1 : Competency Goal 1 Solution: We know the lengths of the side adjacent to Ð G and the hypotenuse. You can use the sine ratio. sin G = NR NG opposite hypotenuse sin G = 21 42 or 1 2 Replace NR with 21 and NG with 42. Now, use a calculator to find the measure of Ð G , an angle whose sine ratio is 1 2 . m Ð G = sin - 1 1 1 2 2 y [SIN - 1 ] 1 ¥ 2 Í 30
Competency Goal 1 : 1.02 Summarize and 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 univariate data.
f) Compare distributions of univariate data. Sample Problem: Competency Goal 1 Use the table to find the mean, median, and mode of the data. 0.6 Vermont 11.4 Ohio 19.0 New York 1.3 Maine 8.2 Georgia 33.9 California 4.4 Alabama Population
(in millions) State
Competency Goal 1 : Competency Goal 1 Solution: Mean =
11.3
The mean is about 11.3 million.
Median Arrange the numbers from least to greatest.
0.6, 1.3, 4.4, 8.2, 11.4, 19.0, 33.9
The median is the middle number or 8.2 million.
Mode Since each number only occurs once, there is no mode.
4.4 + 33.9 + 8.2 + 1.3 + 19.0 + 11.4 + 0.6 7 = 78.8 7
Competency Goal 1 : 1.03 Use theoretical and experimental probability to model and solve problems.
a) Use addition and multiplication principles.
b) Calculate and apply permutations and combinations.
c) Create and use simulations for probability models.
d) Find expected values and determine fairness.
e) Identify and use discrete random variables to solve problems.
f) Apply the Binomial Theorem. Sample Problem: Competency Goal 1 Five points are located on a circle. How many line segments can be drawn with these points as endpoints?
Competency Goal 1 : Solution: Competency Goal 1
Method 1
First, list all of the possible permutations of A, B, C, and D taken two at a time.
Then cross out the segments that are the same as one another.
ED EC EB EA DE DC DB DA CE CD CB CA BE BD BC BA AE AD AC AB AB is the same as BA , so cross off one of them. There are only 10 different segments.
Competency Goal 1 : Solution: Competency Goal 1
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.01 Use logarithmic (common, natural) functions to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Interpret the constants, coefficients, and bases in the context of the problem.
Solve log3 (5x + 7) < log3 (3x + 13). Solution:
Competency Goal 2 : Competency Goal 2 Sample Problem 2.02 Use piecewise-defined functions to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Interpret the constants, coefficients, and bases in the context of the problem. Determine whether the relation in the graph, is a function. Explain your answer. Solution: The graph represents a function. Each element of the domain is paired with exactly one member of the range. Again, it does not matter if two elements of the domain are paired with the same element in the range as is the case with both -4 and -3, -1 and 0, and 1 and 2.
Competency Goal 2 : Competency Goal 2 Solution: If f(x) = - 5x + 1, find the value of f(1)
Sample Problem f(x) = - 5x + 1
f(1) = - 5(1) + 1 Replace x with 1.
= - 5 + 1 Multiply.
= - 4 Add. Sample Problem Solution: If f(x) = - 5x + 1, find the value of f(-2.5) f(x) = - 5x + 1
f(2.5) = - 5(2.5) + 1 Replace x with 2.5.
= - 12.5 + 1 Multiply.
= - 11.5 Add.
Competency Goal 2 : Competency Goal 2 Sample Problem 2.03 Use power functions to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Interpret the constants, coefficients, and bases in the context of the problem. Graph each exponential function. Then state the y-intercept.
y = 4x + 1 17 42 + 1 2 5 41 + 1 1 2 40 + 1 0 1.25 4-1 + 1 -1 1.0625 4-2 + 1 -2 y 4x + 1 x To find the y-intercept, let x = 0 and solve for y.
y = 40 + 1 or 2
In this case, the y-intercept is 2. Solution:
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.04 Use trigonometric (sine, cosine) functions to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Create and identify transformations with respect to period, amplitude, and vertical and horizontal shifts.
c) Develop and use the law of sines and the law of cosines. Find a if c = 7, b = 4, and m A = 52. Solution: Use the Law of Cosines since the measures of two sides and the included are known. a 2 = b 2 + c 2 – 2 bc cos A Law of Cosines a 2 = 4 2 + 7 2 - 2(4)(7) cos 52 ° b = 4, c = 7, and m U A = 52 a 2 = 65 - 56 cos 52 ° Simplify. a = 65 - 56 cos 52 ° Take the square root of each side. a = 5.5 Use a calculator.
Competency Goal 2 : Competency Goal 2 Sample Problem: 2.05 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. When Michael was 13, he received a $10,000 scholarship from a contest. He put all of the money in an investment account with an annual interest rate of 7.2%. He wants to take the money out in 5 years when he is 18. How much money will he have in his account? After the first year, the CD is worth the initial deposit plus interest.
10,000 + 10,000(0.072) = 10,000(1 + 0.072) [Distributive Property]
= 10,720 Solution:
Competency Goal 2 : Competency Goal 2 Solution: (contd.) Michael can make a table to look for patterns in the growth of his CD. Let x represent the number years. Then the function that represents the balance in Michael’s CD is B(x) = 10,000(1.072)x. After five years, it will have a balance of $14,157.09.