The State Education Department / The University oF the
       State of New York / Albany, NY 12234

                                        Curriculum, Instruction, and Instructional Technology Team - Room 320 EB

                                        www.emsc.nysed.gov/ciai

         email: emscnysmath@mail.nysed.gov

 

Algebra 2 and Trigonometry

 

 

Sample Tasks for Integrated Algebra, developed by New York State teachers, are clarifications, further explaining the language and intent of the associated Performance Indicators. These tasks are not test items, nor are they meant for students' use.

Note: There are no Sample Tasks for the Geometry Strand. Although there are no Performance Indicators for this strand in this section of the core curriculum, this strand is still part of instruction within the other strands as an ongoing continuum and building process of mathematical knowledge for all students.

 

 

 

 

 

Strands

Process

Content

 

Problem Solving

 

 

Reasoning and Proof

 

 

Communication

 

 

Connections

 

 

Representation

 

 

 

 

Number Sense and Operations

 

 

Algebra

 

 

Geometry

 

 

Measurement

 

 

Statistics and Probability

 

 

 

 

 

 

 

 

 

Problem Solving Strand

 

Students will build new mathematical knowledge through problem solving.

 

A2.PS.1            Use a variety of problem solving strategies to understand new mathematical content

 

A2.PS.1a

For each of the following:

    Sketch a graph of the function.

    Set the function equal to 0 and solve.

    Find the discriminant.

    What connections can you make between the discriminant, the solution(s), and the graph of the function?

A2.PS.2            Recognize and understand equivalent representations of a problem situation or a mathematical concept

 

A2.PS.2a

Given the following equations, determine the amplitude, period, frequency, and phase shift of each equation.

              

 

Two students, Anthony and Chris, can be overheard discussing these equations.  Anthony is certain these equations are equivalent, while Chris insists that they are different.  Which student is correct? Explain your answer fully with graphs, tables, and a carefully written paragraph supporting your position.

 

Students will solve problems that arise in mathematics and in other contexts.

 

A2.PS.3            Observe and explain patterns to formulate generalizations and conjectures

 

A2.PS.3a

Simplify each of the following:

                                                                                 

                                                       

                                                                  

                                                                  

Can you see a pattern?  If so, what conjecture can you make about the powers of ?

 

               Based on your conjecture, simplify the following:

                                                                             

 

A2.PS.3b

On the same set of axes, use a graphing package or graphics calculator to graph the following functions:

The functions above are all members of the family.

               What effect does changing  values have on the shape of the graph?

               What is the y-intercept of each graph?

               What is the horizontal asymptote of each graph?

 

A2.PS.3c

Sketch one cycle of each of the following equations. Carefully label each graph.

              

Carefully graph and  at the same time on your calculator with a window of . What conclusions can you make? Precisely describe the similarities between the 2 functions.

 

Now, carefully graph and  at the same time on your calculator with a window of .  What conclusions can you make?  Precisely describe the similarities between the 2 functions.

 

A2.PS.4            Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)

 

A2.PS.4a

For each of the following values of the discriminant, state the number of  x-intercepts the graph would have, and sketch a graph of a parabola that would satisfy these conditions:

                              144

                               – 6

                              27

                              0

Is it possible to find more than one graph with each discriminant? If so, describe how you obtained each graph and how each discriminant affects the graph.

 

Students will apply and adapt a variety of appropriate strategies to solve problems.

 

A2.PS.5            Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)

 

A2.PS.5a

A team of biologists have discovered a new creature in the rain forest. They note the temperature of the animal appears to vary sinusoidally over time. A maximum temperature of 125 occurs 15 minutes after they start their examination.  A minimum temperature of  99 occurs 28 minutes later. The team would like to find a way to predict the animal’s temperature over time in minutes. Your task is to help them by creating a graph of one full period, an equation of temperature as a function over time in minutes, and a table of maximum, minimum, and average temperatures for the first 3 hours.

Discuss the advantages and disadvantages of each representation.

 

A2.PS.6            Use a variety of strategies to extend solution methods to other problems

 

A2.PS.6a

Use the strategies learned in solving quadratic equations to solve the following equations. Express any irrational solutions in simplest radical form.

 

                             

                             

                             

                               =  0

 

A2.PS.7            Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving

A2.PS.7a

With a partner, research to find three situations that can be modeled by exponential equations. Write a description of each situation and write a problem based on each of the three situations. Solve each of the problems in a different way and be prepared to present the solutions to the class.

 

Students will monitor and reflect on the process of mathematical problem solving.

 

A2.PS.8            Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions

 

A2.PS.8a

Brianna decided to invest her $500 tax refund rather than spending it. She found a bank account that would pay her 4% interest compounded quarterly.  If she deposits the entire $500 and does not deposit or withdraw any other amount, how long will it take her to double her money in the account?

 

A2.PS.8b

Dante has to travel from Cambridge, NY to Buffalo, NY, a distance of approximately 333 miles. He estimates that he can average 20 mph faster during the 273 miles that he will be driving on the New York State Thruway than he can when he drives on other roads.  If he wants to complete the trip in eight hours, find, to the nearest integer, the rate that Dante must travel on the Thruway.

 

A2.PS.9            Interpret solutions within the given constraints of a problem

 

A2.PS.9a

Last year’s senior class spent $23.95 for each prom favor.  This year’s prom committee knows that their prom favor must be within $5.50 of last year’s favor.  Write an absolute value inequality that could be used to model the acceptable price range for this year’s prom favor, and then solve the inequality to find the range of acceptable prices for a favor. Explain you answer.

 

A2.PS.10          Evaluate the relative efficiency of different representations and solution methods of a problem

 

A2.S.PS.10a

A bag contains three chocolate, four sugar, and five lemon cookies. Greg takes two cookies from the bag for a snack. Find the probability that Greg did not take two chocolate cookies from the bag. Explain why using the complement of the event of not choosing two chocolate cookies might be an easier approach to solving this problem.

 

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Reasoning and Proof Strand

 

Students will recognize reasoning and proof as fundamental aspects of mathematics.

 

A2.RP.1           Support mathematical ideas using a variety of strategies

 

A2.RP.1a

Read through the experiment described below. Before beginning this experiment, think about what is happening. What type of function will you expect to see?  Why would you expect to see this type of function to model the data?

Based on your answer, count the number of coins that are in the cup initially and write an equation that could be used to model the number of coins remaining each time.  Explain how you determined this equation. Use this equation to predict the number of times the experiment would have to be repeated until there is one coin remaining.

 

The Experiment

Take a handful of coins and put them into a cup.  Shake the cup and pour the coins onto the desk. 

Count the total number of coins, and record this number.

Remove all of the coins that are face up and record the total number left.

Using the new total of coins each time, repeat the procedure until there are no coins left.  When the number of coins reaches zero, the experiment is over and you should not use zero as part of your data.

Write an appropriate regression equation to model your data.  What do the variables represent?

Compare this equation to the one that you wrote before the experiment began. Explain any differences between the two equations and any errors that you made in the equation that you wrote.

 

How many trials did it actually take until there was one coin remaining?  Compare the actual number to the number predicted by the regression equation and the equation that you wrote.

 

Students will make and investigate mathematical conjectures.

 

A2.RP.2           Investigate and evaluate conjectures in mathematical terms, using mathematical strategies to reach a conclusion

 

A2.RP.2a

Lauren and Diana disagree about one of the rules for simplifying logarithms. 

 

Lauren says that because you can factor out the log.

 

Diana says that, because you add exponents when you are multiplying.

Which student is correct? 

Explain your answer using two different strategies such as a table, graph, algebraic proof, etc.

 

A2.RP.3           Evaluate conjectures and recognize when an estimate or approximation is more appropriate than an exact answer

 

A2.RP.3a

Based on census data, the U.S. Census Bureau has projected the population of the United States until 2050.  The table below contains these predictions.

 

Year

2000

2010

2020

2030

2040

2050

Population

(in thousands)

282,125

308,936

335,805

363,584

391,946

419,854

 

Make a scatter plot of the data.  Why do you think that the U.S. Census Bureau expressed the population in thousands?

Determine a regression equation that could be used to model the data.  Use this equation to determine the number of people expected in the United States this year. 

How should you round your answer? Why?

Research the current population of the United States. How does your estimate compare to the actual population?  Explain why your answer is different from the actual population.

 

A2.RP.4           Recognize when an approximation is more appropriate than an exact answer

 

A2.RP.4a

Fill in the blanks in the following chart. 

 

Trigonometric Function

Exact Value

Approximate Value

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Under what circumstances would you use an approximation for each of these values, rather than giving an exact answer? 

 

 A2.RP.4b

Give an example of an experiment where it is appropriate to use a normal distribution as an approximation for a binomial probability. Explain why in this example an approximation of the probability is a better approach than finding the exact probability.

 

Students will develop and evaluate mathematical arguments and proofs.

 

A2.RP.5           Develop, verify, and explain an argument, using appropriate mathematical ideas and language

 

A2.RP.5a

What is the range of the function   Based on your answer, what is the range of the function   Explain your answer.

 

A2.RP.6           Construct logical arguments that verify claims or counterexamples that refute claims

 

A2.RP.6a

Find a counterexample to refute each of the following claims:

All functions of the form are one-to-one.

All one-to-one functions are onto.

 

A2.RP.7           Present correct mathematical arguments in a variety of forms

 

A2.RP.7a

Demonstrate that  and  are inverses using at least two different strategies (numeric, graphic, algebraic).

 

A2.RP.7b

Starting with  and using your knowledge of the quotient and reciprocal identities, derive an equivalent identity in terms of  and . Show all your work.

 

A2.RP.8           Evaluate written arguments for validity

 

A2.RP.8a

Liza was absent from school and emailed two of her friends to help her understand how to decide if a relation is a function.  

Mike said “Make a table, and see if you get two of the same y-values.”

John said “Look at the graph.  See if a vertical line crosses the graph in more than one place.  If it does, then we have a function.”

 

Which student is correct? Why? Provide a counterexample to explain any errors made by either Mike or John. 

 

 Students will select and use various types of reasoning and methods of proof.

 

A2.RP.9           Support an argument by using a systematic approach to test more than one case

 

A2.RP.9a

Sketch right triangle LMD,  Write your favorite number as the length of one of the sides.  Using this information, find the lengths of the other two sides.  Write the lengths as exact lengths. Do not use decimal approximations.  Express the value of each of the following as the ratio of the sides of the triangle.

                                            

Reduce all of the fractions to lowest terms and compare your answers to another student’s answers.  What pattern do you see?  What conclusions can you reach?  

 

Fill in another number as the length of one of the sides. Compare your answers again.  Based on what you have found, determine the exact values of each of the following:

                                            

 

A2.RP.10             Devise ways to verify results, using counterexamples and informal indirect proof

 

A2.RP.10a

Jenna’s teacher has asked the class to find the exact value of sin(105). Jenna’s work is as follows.

Jenna’s teacher has marked this as incorrect. Using counterexamples or an indirect proof, demonstrate why Jenna’s work is not correct.

  

A2.RP.11             Extend specific results to more general cases

 

A2.RP.11a

Let be a point in quadrant one on the unit circle,  Draw the line segment   Let  be the angle formed by  and the positive portion of the x-axis. Now draw the perpendicular from P to meet the x-axis at  point M.

State the ratio of  in terms of

               State the ratio of  in terms of .

               State the coordinates of point P in terms of .

 

               Substitute your coordinates into the unit circle equation to verify one of the Pythagorean identities.

 

               Now choose P in a different quadrant and repeat the process. Does the identity continue to be true?

 

A2.RP.11c

Sketch a scatter plot whose regression model could be logarithmic and one whose regression model could be exponential.  Use your knowledge of exponential and logarithmic functions to explain the differences and similarities in the two scatter plots.

 

A2.RP.12             Apply inductive reasoning in making and supporting mathematical conjectures

 

A2.RP.12a

Given the sequence , Paul notices a pattern and finds a formula he believes will find the sum of the first n terms.  His formula is .  Show that Paul’s formula is correct.

 

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Communication Strand

 

A2.CM.1          Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

 

A2.CM.1a

One of the students in class was absent the day the class learned the technique of completing the square.  Using the technique of completing the square, write an explanation of how to solve the following equation that you could give to the student who had been absent: 

                             

 

A2.CM.1b

Simplify each of the following.  Assuming your friend is absent; write him a complete description on the steps necessary to simplify these problems.  Include as much information as possible.

 

 

 

A2.CM.2          Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams

 

A2.CM.2a

The Art Club has purchased flat sheets of cardboard to make storage boxes for the club’s art supplies.  They will make boxes by cutting a square from each corner of the 24 inch by 36 inch sheet of cardboard. 

Draw a diagram to illustrate the given information.

Express the volume of the box as a function of the side of the square cut from the cardboard. 

Make a table for the function.  Use the table to find the volume of a box formed by removing a square which has a 10 inch side, and the length of the side of a square that would produce a volume of 1792 cubic inches. 

Sketch a graph of the function, and use the graph to find the side of the square that would produce the maximum volume. Find the maximum volume of the box that can be made.

Make a chart containing the side of the square removed, length, width, height and volume of the box created that the club could use for quick reference to make boxes of appropriate size for the supplies. Be sure to determine an appropriate degree of accuracy for the entries in the chart.

 

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

 

A2.CM.3          Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols  and other representations when sharing an idea in verbal and written form

 

A2.CM.3a

What is the domain and range of the function shown below? Express your answer in standard mathematical notation, and explain this notation.

 

A2.CM.4          Explain relationships among different representations of a problem

 

A2.CM.4a

Convert the equation  into center-radius form. When is this form of the equation more useful?

Explain how to convert from center-radius form to standard form. 

 

 

 

A2.CM.5          Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

 

A2.CM.5a

Sally’s math teacher said you could use the conjugate of a complex number to rationalize the denominator of  a fraction which contains a complex number.  Sally is trying to rationalize the denominator of   but cannot remember if  or  is the conjugate.  Which one is the conjugate and why?

 

A2.CM.5b

A group of eight students decided that they wanted to lose weight. Four of them decided to walk a mile each school day before school. The other four of them decided to walk a mile each school day after school. All eight weigh themselves each Wednesday and report their weight to their math teacher, who is keeping it confidential.

One student in the class says this is an experiment. A second student disagrees and says this is an observational study. A third student thinks this is just a survey.

After discussing the information with your partner, write a paragraph to explain why you believe the study is an experiment, an observational study, or a survey. Be clear and concise.

 

A2.CM.6          Support or reject arguments or questions raised by others about the correctness of mathematical work

 

A2.CM.6a

Two students in Ms. Baum’s class were working on a problem and got different solutions.  The students each were certain they had the correct solution. Who had the correct answer?  Write a detailed paragraph to explain whose work was correct and why.

Given triangle PQR, find the largest angle to the nearest degree.

Student #1                                                                                      Student #2

                                               

 

 

 

 

 

 

A2.CM.6b

It has been decided that 78 people need to be surveyed to decide the public’s opinion on a school building project. A student suggests that they survey the first 78 people who enter the school. 

Do you think that this proposed way of sampling is an unbiased way to perform the survey or can you describe a better way to achieve a fair and accurate response?

 

Students will analyze and evaluate the mathematical thinking and strategies of others.

 

A2.CM.7          Read and listen for logical understanding of mathematical thinking shared by other students

 

A2.CM.7a

Which of the following two graphs represents a function? Write an explanation of the reasoning used in obtaining this answer. Read the explanation provided by another student, and compare solutions.

 

 

 

 

 

 

 

 

                                                                                                   

 

 

 

 

 

 

 

 

 

 

A2.CM.8          Reflect on strategies of others in relation to one’s own strategy

 

A2.CM.8a

Factor each of the expressions completely. Compare your answers with others in the class, discussing the process you followed to obtain the answers.

                                                 

                                        

 

A2.CM.9          Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others

 

A2.CM.9a

Describe a situation that would require calculating a permutation, and another situation that would require calculating a combination.  Share your thoughts with a classmate and discuss the strategies used to determine when to calculate a permutation and when to calculate a combination. Write a brief summary of your conclusions and be prepared to discuss your ideas with the class.

 

Students will use the language of mathematics to express mathematical ideas precisely.

 

A2.CM.10        Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students’ conjectures

 

A2.CM.10a

Solve each quadratic equation and use the roots to complete the table:

Quadratic Equation

Roots

Sum of Roots

Product of Roots

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use the information in the table to make a conjecture about the relationship between a quadratic equation and the sum and product of its roots. Use correct mathematical language to write your conjecture.

When finished writing, exchange the paper with another student. Read the conjecture and decide whether the conjecture is valid.

If it is valid, write questions that will help the student prove that the conjecture is valid. 

If the conjecture is invalid, write questions that challenge the conjecture based upon correct mathematics and mathematical reasoning.

Return the paper. Respond to the student’s questions using correct mathematical language and reasoning to prove or disprove the validity of the conjecture.

 

A2.CM.11        Represent word problems using standard mathematical notation

 

A2.CM.11a

Maya has decided to train for a marathon (26 miles) and decides to set up a practice schedule to build her stamina.  When she begins, she can only run 3 miles, but she intends to train every day and increase her run by 2 miles each week  Find a pattern and write a formula that will give the number of miles Maya can run in week n.

Using the formula, how many weeks will Maya need to train in order to be ready for the marathon?

 

A2.CM.12        Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale

 

A2.CM.12a

Write a word problem that illustrates an application of the Law of Sines, Law of Cosines, or a combination of both. Be creative, but be certain to use appropriate language and mathematical terminology when describing the problem situation.  Also prepare a complete solution of the problem including a correctly labeled diagram and full mathematical explanation of how to solve the problem. 

 

A2.CM.13        Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing

 

A2.CM.13a

Research the topic of “The Set of Complex Numbers” to answer the following:

Define:  complex number, imaginary number, real number, imaginary part, real part, conjugate

When speaking of a complex number what do a, b, and i represent or mean?

Give an example and draw a graphical representation of a complex number that fits each description: 

and

and

 and

Write the conjugate of each of the complex numbers, and draw the graphical representation of each conjugate.

How are the graphical representations of real numbers and complex numbers the same and how are they different?

How are the graphical representations of a complex number and its conjugate the same and different?

 

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Connections Strand

 

Students will recognize and use connections among mathematical ideas.

 

A2.CN.1           Understand and make connections among multiple representations of the same mathematical idea

 

A2.CN.1a

Describe the circle whose graph is shown below. Write an equation for the circle and name six points that lie on the circle.

A2.CN.1b

In the accompanying diagram, unit circle O has radii and  is tangent to circle O at B, and is tangent to circle O at E.  Points O, F, D, and C are collinear, and  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If  name the line segment whose measure is each of the following:

 

                                              

 

A2.CN.1c

Write an equation for a trigonometric function that matches each of the following graphs.  Check your answers with a partner.  If different equations have been obtained, confirm by graph or table, the accuracy of each equation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.CN.2               Understand the corresponding procedures for similar problems or mathematical concepts

 

A2.CN.2a

Solve the following inequalities graphically. Express the solutions correct to the nearest hundredth.  Explain how the solutions to the two problems are related.

                             

                             

 

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

 

A2.CN.3           Model situations mathematically, using representations to draw conclusions and formulate new situations

 

A2.CN.3a

Stretch your arms out as wide as you can.  Have someone measure across your back the length (in inches) of your arm span, from the tip of your fingers on one hand to the tip of your fingers on your other hand.  Have the person also measure your height in inches.  Share this information with your classmates and aggregate all of the data. 

Enter the information in two lists on your calculator.  In the first list, enter the heights, in inches, of all of the participants.  In the second list, enter the arm span, in inches, of all of the participants.

Make a scatter plot, where the independent variable is height, in inches, and the dependent variable is arm span, in inches.  What type of regression function would best model this data?  Why?

Write the regression equation with all constants rounded to the nearest tenth.

Using this equation, determine the arm span of a person who is 5 feet tall.

Research to see what other measurements would form similar relationships.

 

A2.CN.4           Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

 

A2.CN.4a

Anthony’s teacher told the class that a unit circle has a circumference of .  This confused him because he thought a circle has. Since Anthony is your friend, you would like to help him understand what the teacher means. Write a detailed explanation that compares degrees to radians.  Be as thorough as possible in order to help Anthony understand the connection.  Include anything that might make this clear, such as diagrams, equations, and so forth.

 

A2.CN.5           Understand how quantitative models connect to various physical models and representations

 

A2.CN.5a

The angle of inclination of the sun changes throughout the year. This changing angle affects the heating and cooling of buildings. The overhang of the roof of a house is designed to shade the windows for cooling in the summer and allow the sun’s rays to enter the house for heating in the winter.

The sun’s angle of inclination at noon can be modeled by the formula:

              

for Albany, NY, where x is the number of days elapsed in the day of the  year, with January first represented by  January second represented by  and so on.

Find the sun’s angle of inclination at noon on Valentine’s Day.

Sketch a graph illustrating the changes in the sun’s angle of inclination throughout the year.

On what date of the year is the angle of inclination at noon the greatest in Albany, NY?

 

A2.CN.5b

Mrs. Frost has 1000 bushels of corn to sell. Presently the market price for corn is  a bushel. She expects the market price to increase by  per week. For each week she waits to sell, she loses ten bushels due to spoilage.

Determine a function to express Mrs. Frost’s total income from selling the corn.  What is a reasonable domain and range for this function?

Graph the function and determine when Mrs. Frost should sell the corn to maximize her income.  Show your answer is correct by computing the income one week earlier and one week later and showing these values would result in less income. 

 

Students will recognize and apply mathematics in contexts outside of mathematics.

 

A2.CN.6           Recognize and apply mathematics to situations in the outside world

 

A2.CN.6a

With a partner, research to find three situations that can be modeled by exponential equations.  Write a description of each situation, write and solve one problem based on each of the three situations, and prepare to present the solution to the problem.

 

A2.CN.6b

You just decided to save money each week in order to buy an expensive cell phone. You decide to start by saving $15 from your paycheck and then save $22 the second week and $29 the third week.  You continue this pattern until you have saved enough money.

Use sigma notation to write the indicated sum of this series.

If the cell phone you want to purchase costs $585, how many weeks will you need to save enough money?

Show or explain fully how you arrived at your answer.

 

A.2.CN.6c

Madison was determined to help clean her local park.  She collected one bag of trash the first week, 2 bags the second week, 3 bags the third week, and so on. 

Assuming she continues this process, how many bags of trash will she collect in 26 weeks?

How many bags of trash will she collect in n weeks?

 

A2.CN.7           Recognize and apply mathematical ideas to problem situations that develop outside of mathematics

 

A2.CN.7a

Use two thermometers, one Celsius and the other Fahrenheit, to measure the temperature of your hand.  Place both thermometers in the palm of your hand and gently squeeze your hand shut.  (If you have a temperature probe for your calculator or computer, it may be used instead.)

Share this information with your classmates, and aggregate all of the data. 

Enter the information in two lists on your calculator.  In the first list, enter the Celsius temperatures.  In the second list, enter the Fahrenheit temperatures.

Make a scatter plot, where the independent variable is degrees Celsius, and the dependent variable is degrees Fahrenheit. 

Find the linear regression equation that best fits the data. 

What is the correlation coefficient? Why would you expect its value to be close to one?

 

A2.CN.7b

Determine whether each of the following situations would require calculating a permutation or a combination. Explain your answer.

Selecting a lead and an understudy for a school play

Assigning students to their seats on the first day of school

Selecting three students to represent the school at an Honor Society conference in Washington, D.C.

 

2A.CN.8           Develop an appreciation for the historical development of mathematics

 

A2.CN.8a

Research to find out how calculations with logarithms were done before the invention of calculators. 

 

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Representation Strand

 

Students will create and use representations to organize, record, and communicate mathematical ideas.

 

A2R.1               Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

 

A2.R.1a

Take a cup of very hot water, and measure the temperature of the water.  Place the cup in a safe place, and measure the temperature every five seconds for a total of one minute.  Record this information in your notebook.

Enter the information in two lists on your calculator. In the first list, enter the time, in seconds, since the experiment began. In the second list, enter the temperature of the water for each indicated time.

Make a scatter plot, where the independent variable is time, in seconds, and the dependent variable is temperature. What type of regression function would best model this data? Why?

Write the regression equation with all constants rounded to the nearest thousandth.

Using this equation, determine the temperature of the water after 25 seconds has passed.  Determine the temperature of the water after two hours has passed. Do both of these answers seem logical to you?  Why?

(N.B. If you have a temperature probe for your calculator or computer, it may be used to conduct this experiment.)

  

A2.R.2              Recognize, compare, and use an array of representational forms

 

A2.R.2a

Explain clearly in words and with a pair of graphs, the differences between the function and each of the following functions.

                             

                             

                             

After examining the four sets of graphs, graph one sine function containing all four transformations.  Write the equation of the new graph.

 

A2.R.2b

Give the definition of a circle.

Consider the general equation of a circle,

State the circle in the form  with center  and radius 7.

Start with the circle,  Find its center and radius and graph the circle.

Start with the circle. Find its center and radius and graph the circle.

How do the coordinates of the center of a circle relate to C and D when the equation of the circle is in the form ?

 

A2.R.3              Use representation as a tool for exploring and understanding mathematical ideas

 

A2.R.3a

Use the technique of completing the square to write the equation  in an appropriate form to find the transformations that  produce the graph of the given equation from the graph of the equation .

Students will select, apply, and translate among mathematical representations to solve problems.

 

A2.R.4              Select appropriate representations to solve problem situations

 

A2.R.4a

Solve the following problems using direct or inverse variation as appropriate:

Carlos drove from his home to Union College in 4 hours at 55 mph. How long would it take him if he had traveled at 65 mph? 

Keisha earned $44.70 for working 6 hours. How much does she earn for working 8.5 hours? 

 

A2.R.5              Investigate relationships between different representations and their impact on a given problem

 

A2.R.5a

A company offers its employees a choice of two salary schemes (A and B) over a period of ten years. Scheme A offers a starting salary of $33,000 in the first year and an annual increase of $1200 per year. Scheme B offers a starting salary of $30,000 in the first year and an annual increase of 7% of the previous year’s salary. An employee about to join this company hires you to examine the two schemes. Decide which scheme is better financially for the employee. Prepare a variety of ways to present the information to the client, such as tables and graphs.

  

Students will use representations to model and interpret physical, social, and mathematical phenomena.

 

A2.R.6              Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)

 

A2.R.6a

The table shows the amount of medicine for treating a disease in the bloodstream over the 9 hours following a dose of 10 mg. It seems that the rate of decrease of medicine is approximately proportional to the amount remaining.

 

Time(hours)

0

1

2

3

4

5

6

7

8

Drug Amount (mg)

10

8.3

7.2

6.0

5.0

4.4

3.7

2.8

2.5

Use this information to find a suitable function to model this data.

Using your model, when will there be less than 1 mg. of medicine in the patient’s blood stream?

If the initial dose was 15 mg, when would the amount of medicine in the bloodstream, fall below 5 mg?

 

A2.R.6b

According to the National Weather Service, the 2005 average monthly temperature, in degrees Fahrenheit, at Central Park in New York City is given below:

 

Month

Jan.

Feb.

March

April

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Average Temperature

(oF)

 

31.3

 

 36.5

 

39.4

 

55.1

 

58.9

 

74.0

 

77.5

 

79.7

 

73.3

 

57.9

 

49.6

 

35.3

 

Write a sinusoidal function that models the average monthly temperature, using t = 1 to represent January. According to your model, what is the average temperature in December? Explain the discrepancy from your model to the average monthly temperature in December. How could the model be improved?

 

A2.R.7              Use mathematics to show and understand social phenomena (e.g., interpret the results of an opinion poll)

 

A2.R.7a

Search in newspapers and magazines and then write a brief but thorough description of a study that could be characterized as:

a survey

an observational study

a designed experiment

 

A2.R.8              Use mathematics to show and understand mathematical phenomena (e.g., use random number generator to simulate a coin toss)

 

A2.R.8a

Use a random number generator to generate thirty random numbers that will simulate tossing a fair coin thirty times. Let an even number represent the result of the coin coming up heads, and an odd number represent the result of tails. Generate a second list of thirty random numbers to represent the second toss of a fair coin. Use the two lists of random numbers to compute the following empirical probabilities: P(head, tail), P(both the same), P(both different).  Compute the following theoretical probabilities for the same experiment: P(head, tail), P(both the same), P(both different). How do the empirical and theoretical probabilities compare?

 

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Number Sense and Operations

 

Students will understand meanings of operations and procedures, and how they relate to one another.

 

Operations

A2.N.1              Evaluate numerical expressions with negative and/or fractional exponents, without the aid of a calculator (when the answers are rational numbers)

 

A2.N.1a

Find the value of  when  25.

 

A2.N.1b

Evaluate when .

A2.N.1c

If, find

 

A2.N.2              Perform arithmetic operations (addition, subtraction, multiplication, division) with expressions containing irrational numbers in radical form

 

A2.N.2a

Find the sum of  and  in simplest form.

 

A2.N.2b

Express in terms of

 2 - .

 

 A2.N.3              Perform arithmetic operations with polynomial expressions containing rational coefficients

 

A2.N.3a

Find the difference:

 

A2.N.3b

Find the product: 

 

A2.N.4              Perform arithmetic operations on irrational expressions

 

A2.N.4a

Expand the expression ( π + )

 

A2.N.4b

Express as a single term of :

A2.N.5              Rationalize a denominator containing a radical expression

 

A2.N.5a

Given , write an equivalent expression with a rational denominator. 

 

A2.N.6              Write square roots of negative numbers in terms of i

 

A2.N.6a

Simplify the problems below.  Your friend was absent.  Write your friend a complete description of the steps necessary to simplify the problems. 

 

 

 

A2.N.7              Simplify powers of i

 

A2.N.7a

Simplify each of the following:

                                                                                 

                                                       

                                                                  

                                                                  

               Can you see a pattern?  If so, what conjecture can you make about the powers of ?

               Based on your conjecture, simplify the following:

                                                                             

 

A2.N.8              Determine the conjugate of a complex number

 

A2.N.8a

Sally’s math teacher said you could use the conjugate of a complex number to rationalize the denominator of  a fraction which contains a complex number.  Sally is trying to rationalize the denominator of   but cannot remember if  or  is the conjugate.  Which one is the conjugate and why?

 

A2.N.8b

Research and define: complex number, imaginary number, real number, imaginary part, real part, conjugate

When speaking of a complex number what do a, b, and i represent or mean?

Give an example and draw a graphical representation of a complex number that fits each description: 

and

and

 and

Write the conjugate of each of the complex numbers, and draw the graphical representation of each conjugate.

How are the graphical representations of real numbers and complex numbers the same and how are they different?

How are the graphical representations of a complex number and its conjugate the same and different?

 

A2.N.9              Perform arithmetic operations on complex numbers and write the answer in the form   Note: This includes simplifying expressions with complex denominators.         

 

A2.N.9aSimplify the following:

 

A2.N.10            Know and apply sigma notation

 

A2.N.10a

Find the value of

 

A2.N.10b

Adrianna decided to save money each week in order to buy an expensive sound system.  She decide to start by saving $15 from her first paycheck, and then save $22 the second week and $29 the third week.  She continued this pattern until she had saved enough money. 

Use sigma notation to write the indicated sum of this series.

If the sound system she wanted to purchase costs $585, how many weeks will she need to save enough money?

Explain how you arrived at your answer.

 

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Algebra

 

Students will represent and analyze algebraically a wide variety of problem solving situations.

 

Equations and Inequalities           

 

A2.A.1              Solve absolute value equations and inequalities involving linear expressions in one variable

 

A2.A.1a

Solve and check:

                             

                             

                             

  

A2.A.1b

Last year’s senior class spent $23.95 for each yearbook.  This year’s yearbook committee knows that their yearbook must be within $5.50 of last year’s favor.  Write an absolute value inequality that could be used to model the acceptable price range for this year’s yearbook, and then solve the inequality to find the range of acceptable prices for the yearbook.

 

A2.A.2              Use the discriminant to determine the nature of the roots of a quadratic equation

 

A2.A.2a

Describe the nature of the roots of the quadratic equation whose discriminant is:

144

                              – 36

                              27

                              0

 

A2.A.2b

For each of the following values of the discriminant, state the number of

x-intercepts the graph would have, and sketch a graph of the parabola:

                              144

                               – 6

                              27

                              0

 

A2.A.2c

For each of the following:

Sketch a graph of the function.

Set the function equal to 0 and solve.

Find the discriminant.

What connections can you make between the discriminant, the solution(s), and the graph of the function?

 

A2.A.3              Solve systems of equations involving one linear equation and one quadratic equation algebraically  Note: This includes rational equations that result in linear equations with extraneous roots.

 

A2.A.3a

Solve the following systems of equations algebraically and check all solutions:

1.)         

 

 

                              2.)         

 

                              3.)         

 

 

A2.A.4              Solve quadratic inequalities in one and two variables, algebraically and graphically

 

A2.A.4a

Solve the following inequalities algebraically, and then check the solutions graphically:

                              1.)         

                              2.)         

                              3.)           

 

A2.A.4b

Solve the following inequalities graphically.  Express the solutions rounded to the nearest hundredth.  Explain how the solutions to the two problems are related.

                             

                               

 

A2.A.5              Use direct and inverse variation to solve for unknown values

 

A2.A.5a

Use direct or inverse variation to solve for the unknown values:

 

If  p varies directly as q, and p = 7 when q = 9, find p when q = 12.

If m varies inversely as t, and m = 5 when t = 6, find t when m = 10.

 

A2.A.5b

Solve the following problems using direct or inverse variation as appropriate:

1.)  Carlos drove from his home to Union College in 4 hours at 55 mph. How long would it have taken him if he had traveled at 65 mph? 

2.)  Keisha earned $44.70 for working 6 hours. How much will she earn for working 8 hours? 

 

A2.A.6              Solve an application which results in an exponential function

 

A2.A.6a

Brianna decided to invest her $500 tax refund rather than spending it.  She found a bank that would pay her 4% interest, compounded quarterly.  If she deposits the entire $500 and does not deposit or withdraw any other amount, how long will it take her to double her money in the account?

 

Students will perform algebraic procedures accurately.

 

Variables and Expressions

 

A2.A.7              Factor polynomial expressions completely, using any combination of the following techniques: common factor extraction, difference of two perfect squares, quadratic trinomials

 

A2.A.7a

Factor each of the expressions completely. Discuss the process that you followed to obtain the answers.

A2.A.8              Apply the rules of exponents to simplify expressions involving negative and/or fractional exponents

 

 

A2.A.8a

Simplify:

 

 

A2.A.8b

Simplify:

 

 

A2.A.9              Rewrite algebraic expressions that contain negative exponents using only positive exponents

 

A2.A.9a

Rewrite each of the following expressions, using only positive exponents, with none of the variables equal to zero:

                             

 

 

                             

 

A2.A.10            Rewrite algebraic expressions with fractional exponents as radical expressions

 

A2.A.10a

Rewrite the three expressions below, using radicals.  All of the variables are positive numbers.

                             

                             

                               

 

A2.A.11            Rewrite algebraic expressions in radical form as expressions with fractional exponents

 

A2.A.11a

Rewrite the radical expressions below, using exponents.  All of the variables are positive numbers.

                             

                             

                             

  

A2.A.12            Evaluate exponential expressions, including those with base e

 

A2.A.12a

Find the exact value of 81.

 

A2.A.12b

Evaluate 

 

A2.A.12c

Find the value of  9x if  x = 4.

 

A2.A.13            Simplify radical expressions

 

A2.A.13a

Simplify:

 

A2.A.13b

Simplify:

 .

 

A2.A.14            Perform addition, subtraction, multiplication, and division of radical expressions

 

A2.A.14a

Find the sum of  and  in simplest form.

 

A2.A.14b

Express in terms of :

2 - .

 

A2.A.14c

Perform the indicated operations, and express your answer in simplest form:

                             

 

                             

 

A2.A.15            Rationalize denominators involving algebraic radical expressions

 

A2.A.15a

Express as a fraction with a rational denominator:

 

A2.A.15b

Express as a single term of :

A2.A.16            Perform arithmetic operations with rational expressions and rename to lowest terms

 

A2.A.16a

Perform the indicated operation and express in lowest terms:

                             

 

A2.A.16b

Perform the indicated operation and express in lowest terms:

                             

 

A2.A.16c

Perform the indicated operation and express in lowest terms:

 

A2.A.17            Simplify complex fractional expressions

 

A2.A.17a

Simplify:

 

 

A2.A.18            Evaluate logarithmic expressions in any base

 

A2.A.18a

Evaluate each of the following without using a calculator:

                                

 

                             

 

                             

 

A2.A.18b

Evaluate each of the following to the nearest hundredth:

                             

 

 

 

A2.A.19            Apply the properties of logarithms to rewrite logarithmic expressions in equivalent forms

A2.A.19a

Express as a single logarithm:

               

 

A2.A.19b

Expand each of the following:

 

 

 

 

A2.A.19c

Lauren and Diana disagree about one of the rules for simplifying logarithms. 

Lauren says that because you can factor out the log.

Diana says that, because you add exponents when you are multiplying.

Which student is correct? 

Explain your answer using two different strategies ( e.g., use a table, a graph, an algebraic proof, etc.).

 

Equations and Inequalities                   

 

A2.A.20            Determine the sum and product of the roots of a quadratic equation by examining its coefficients

 

A2.A.20a

Find the sum of the roots and the product of the roots for the following quadratic equations:

                             

                             

                             

                             

 

A2.A.20b

Given the quadratic equation:

What is the sum of the roots?

What is the product of the roots?

If one of the roots of the equation is -3, what is the other root?

 

A2.A.21            Determine the quadratic equation, given the sum and product of its roots

 

A2.A.21a

Write a quadratic equation whose roots have a sum of -4 and a product of 4.

 

A2.A.21b

Write a quadratic equation whose roots have a sum of 4 and a product of -4, and find the roots of this equation.

 

A2.A.21c

If a quadratic equation has the roots given below, find the sum of the roots, find  the product of the roots, and write a quadratic equation with the given roots.

 

                               3 and -6

                             

                             

                             

 

A2.A.21d

Solve each quadratic equation and use the roots to complete the table:

 

Quadratic Equation

Roots

Sum of Roots

Product of Roots

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Use the information in the table to make a conjecture about the relationship between a quadratic equation and the sum and product of its roots. Use correct mathematical language to write your conjecture.

When finished writing, exchange the paper with another student.  Read the conjecture and decide whether the conjecture is valid.

If it is valid, write questions that will help the student prove that the conjecture is valid. 

If the conjecture is invalid, write questions that challenge the conjecture based upon correct mathematics and mathematical reasoning.

Return the paper. Respond to the student’s questions using correct mathematical language and reasoning to prove or disprove the validity of the conjecture.

 

A2.A.22            Solve radical equations

 

A2.A.22a

Solve and check: 

.

 

A2.A.22b

Is {1, -2} the solution set of  the equation  Explain your answer.

 

A2.A.22c

Determine the number of solutions for each of the following two equations: 

                             

 

                             

 

Explain how to determine the number of solutions to a radical equation without actually solving an equation.

 

A2.A.22d

The sum of a number and its square root is equal to six. Find the number.

 

A2.A.23            Solve rational equations and inequalities

 

A2.A.23a

Solve and check: 

 

A2.A.23b

Solve and check: 

 

A2.A.23c

Solve the inequality: 

 

A2.A.23d

Dante has to travel from Cambridge, New York to Buffalo, New York, a distance of approximately 333 miles. He estimates that he can average 20 mph faster during the 273 miles that he will be driving on a main highway than he can when he drives on back roads.  If he wants to complete the trip in eight hours, find, to the nearest integer, the rate that Dante must travel on the main highway.

 

A2.A.24            Know and apply the technique of completing the square

 

A2.A.24a

Solve the following equations by completing the square:

                             

                             

                             

 

A2.A.24b

One of the students in class was absent the day the class learned the technique of completing the square.  Write an explanation of how to solve the following equation using the technique of completing the square that you could give to the student who had been absent: 

                             

 

A2.A.24c

Use the technique of completing the square to write the equation  in an appropriate form to find the transformations that  produce the graph of the given equation from the graph of the equation .

 

A2.A.25            Solve quadratic equations, using the quadratic formula

 

A2.A.25

Solve each of the following equations using the quadratic formula:

                             

                             

                             

                             

 

A2.A.26            Find the solution to polynomial equations of higher degree that can be solved using factoring and/or the quadratic formula

 

A2.A.26a

Solve the following equations. Express any irrational solutions in simplest radical form.

                             

                             

                               =  0

 

A2.A.27            Solve exponential equations with and without common bases

 

A2.A.27a 

Solve the following equations:

                             

 

                             

                             

 

A2.A.27b

Solve the following equations. Express solutions correct to the nearest hundredth.

                             

                             

 

A2.A.28            Solve a logarithmic equation by rewriting as an exponential equation

 

A2.A.28a

Solve the following equations:

                             

                             

                             

                             

 

Students will recognize, use, and represent algebraically patterns, relations, and functions.

 

Patterns, Relations, and Functions

 

A2.A.29            Identify an arithmetic or geometric sequence and find the formula for its nth term

 

A2.A.29a

Maya has decided to train for a marathon (26 miles) and has set up a practice schedule to build her stamina. When she began she was able to run 3 miles, but she intends to train every day and increase her run by 2 miles each week. Find a pattern and write a formula that will give the number of miles Maya can run in week n.

Using the formula, how many weeks will Maya need to train in order to be ready for the marathon?

 

A2.A.29b

Dahar is taking a class in word processing and is trying to increase his typing speed.  When he began, he could type 20 words per minute. He practiced faithfully every day and noticed an increase in his speed of 10% per week. Find a pattern and write a formula that will give the number of words per minute Dahar can type in n weeks.

Using the formula, how many weeks will Dahar need to practice in order to type 60 words per minute?

 

A2.A.30            Determine the common difference in an arithmetic sequence

 

A2.A.30a

What is the common difference in the following arithmetic sequence?

               5, 9, 13, 17,…

              

 

A2.A.30b

What is the common difference in the arithmetic sequence defined by the following formula?

              

 

A2.A.31            Determine the common ratio in a geometric sequence

 

A2.A.31a

What is the common ratio in the following geometric sequence?

              

 

A2.A.31b

What is the common ratio in the geometric sequence defined by the following formula?

              

 

A2.A.32            Determine a specified term of an arithmetic or geometric sequence

 

A2.A.32a

Find the specified term of each of the following arithmetic sequences.

               :

               :

 

:

A2.A.32b

Find the specified term of each of the following geometric sequences.

              

0

 

              

 

A2.A.33            Specify terms of a sequence, given its recursive definition

 

A2.A.33a

Use the recursive rule given to write the first four terms of each sequence.

 

              

 

              

 

A2.A.34            Represent the sum of a series, using sigma notation

 

A2.A.34a

Use sigma notation to represent the sum of the following series.

 for the first 33 terms.

 

 for the first 50 terms.

 

 for n terms.

 

A2.A.35            Determine the sum of the first n terms of an arithmetic or geometric series

 

A.2.A.35a

Madison was determined to help clean her local park. She collected one bag of trash the first week, 2 bags the second week, 3 bags the third week, and continued at the same rate.

Assuming she continues this process, how many bags of trash will she collect in 26 weeks?

How many bags of trash will she collect in n weeks?

If Madison collected one bag of trash the first week, 2 bags the second week, 4 bags the third week, and continued at the same rate, how many bags of trash would she collect in 26 weeks?

 

A2.A.35b

Find the indicated sum of the series.

 

              

 

A2.A.35c

Given the sequence , Paul notices a pattern and finds a formula he believes will find the sum of the first n terms. His formula is .  Show that Paul’s formula is correct.

 

A2.A.35d

A company offers its employees a choice of two salary schemes (A and B) over a period of ten years. Scheme A offers a starting salary of $33,000 in the first year and an annual increase of $1200 per year. Scheme B offers a starting salary of $30,000 in the first year and an annual increase of 7% of the previous year’s salary. An employee about to join this company hires you to examine the two schemes. Decide which scheme is better financially for the employee.  Prepare a variety of ways to present the information to the client, such as tables and graphs.

 

A2.A.36            Apply the binomial theorem to expand a binomial and determine a specific term of a binomial expansion

 

A2.A.36a

Use the binomial theorem to write the expansion of the following:

              

 

A2.A.36b

Find the sixth term of the binomial expansion of the following:

              

 

A2.A.37            Define a relation and function

 

A2.A.37a

Use the definition of a relation to explain why the set {(-3,5), (4,7), (4,5)} is a relation.

 

A2.A.37b

Use the definition of a function to explain why the set {(-3,5), (4,7), (0,5)} is a function.

 

A2.A.37c

Which of the following sets are relations? Explain your answer.

                              {2,4,6, …}

                              {students}

                              {(1,2),(3,7)}

                             

                             

 

A2.A.37d

Which of the following sets are functions? Explain your answer.

                              {(-3,7),(-5,9),(0,0),(8,9)}

                              {(-4,7),(-4,8),(3,4),(11,8)}

                             

                              a list which matches students with their ages

  

A2.A.38            Determine when a relation is a function

 

A2.A.38a

Which of the following relations are functions? Explain your answer.

                             

                             

                             

                             

                             

 

A2.A.38b

Liza was absent from school and emailed two of her friends to help her understand how to decide if a relation is a function.  

Mike said: Make a table, and see if you get two of the same y-values.

John said: Look at the graph.  See if a vertical line crosses the graph in more than one place.  If it does, then we have a function.

Which student is correct? Why? Provide a counterexample to explain any errors made by either Mike or John. 

 

A2.A.39            Determine the domain and range of a function from its equation

 

A2.A.39a

Find the domain and range of each of the following functions:

                             

                             

                             

                             

                             

                             

 

A2.A.39b

Mrs. Frost has 1000 bushels of corn to sell. Presently the market price for corn is  a bushel. She expects the market price to increase by  per week. For each week she waits to sell, she loses ten bushels due to spoilage.

Determine a function to express Mrs. Frost’s total income from selling the corn.  What is a reasonable domain and range for this function?

Graph the function and determine when Mrs. Frost should sell the corn to maximize her income.  Show your answer is correct by computing the income one week earlier and one week later and showing these values would result in less income. 

 

A2.A.40            Write functions in functional notation

 

A2.A.40a

Use function notation to describe each of the following functions:

x

y

-3

9

-2

4

-1

1

0

0

1

1

2

4

3

9

 

 

 

 

 

 

 

 

 

 

                             

                             

                             

 

A2.A.41            Use functional notation to evaluate functions for given values in the domain

 

A2.A.41a

Given The Function: 

Evaluate:  , , :

Given The Function: 

 

Evaluate: , , :

Given The Function: 

Evaluate: , , :

Given The Function: 

Evaluate:  , , :

 

A2.A.42            Find the composition of functions

 

A2.A.42a

Evaluate each of the following pairs of functions:

:

 

A2.A.42b

Is the composition of two functions ever commutative? Explain your answer.

 

A2.A.42c

John claims that he performs the composition of functions each morning when he puts on his shoes and socks. Is he correct? Explain.

 

A2.A.43            Determine if a function is one-to-one, onto, or both

 

A2.A.43a

For each of the following functions, state whether the function is one-to-one, onto, neither, or both:

 

A2.A.43b

Find a counterexample to refute each of the following claims:

All functions of the form are one-to-one.

All one-to-one functions are onto.

 

A2.A.44            Define the inverse of a function

 

A2.A.44a

Use the definition of the inverse of a function to explain why  is the inverse of the function .

 

A2.A.44b

Find .

 

A2.A.45            Determine the inverse of a function and use composition to justify the result

 

A2.A.45a

Find the inverse of the following functions.  Use composition to show your result is the inverse of the given function.

 

A2.A.45b

Demonstrate that  and  are inverses using at least two different strategies (numeric, graphic or algebraic).

 

A2.A.46            Perform transformations with functions and relations: , , , ,  

 

A2.A.46a

On the same set of axes, graph and . What transformation describes the relationship between  and ? What transformation describes the relationship between  and  ? Use the answers to these questions to predict what the graph of and will look like, then use a graphing calculator to check your prediction. Write a description of how the graph of and the graph of are related.

 

A2.A.46b

Write an equation for the graph of the function, obtained by shifting the graph of three units to the left, stretching the graph vertically by a factor of  two, reflecting that result over the x – axis, and then translating the graph up four units.

 

A2.A.46c

Describe the transformations that would produce the graph of  the second function from the graph of the first function:

                        

                  

                      

 

A2.A.46d

Given the graph of  the function f(x), sketch the graphs of :

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.46e

Explain clearly in words and with a pair of graphs, the differences between the function and each of the following functions.

                             

                             

                             

After examining the four sets of graphs, graph one sine function containing all four transformations.  Write the equation of the new graph.

Coordinate Geometry

 

A2.A.47            Determine the center-radius form for the equation of a circle in standard form

 

A2.A.47a

Use the technique of completing the square to convert the equation into center-radius form. What is the center and what is the radius of this equation?

 

A2.A.47b

Convert the equation  into center-radius form. When is this form of the equation more useful?

Explain how to convert from center-radius form to standard form. 

 

A2.A.47c

Give the definition of a circle.

Consider the general equation of a circle,

State the circle in the form  with center  and radius 7.

Start with the circle,  Find its center and radius and graph the circle.

Start with the circle.  Find its center and radius and graph the circle.

How do the coordinates of the center of a circle relate to C and D when the equation of the circle is in the form ?

 

A2.A.48            Write the equation of a circle, given its center and a point on the circle

 

A2.A.48a

Write an equation of a circle with a center that is (3, -2) and passes through the point (-5, 8).

 

A2.A.48b

Write an equation of a circle whose diameter has endpoints (4, -1) and (-6, 7).

 

A2.A.48c

Write an equation of a circle that is tangent to the x-axis, with a center that is (-2, 4).

 

A2.A.49            Write the equation of a circle from its graph

 

A2.A.49a

Write an equation for the circle whose graph is shown below. 

Write an equation for the circle if it were shifted two units to the left and three units up.

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.49b

Describe the circle whose graph is shown below. Write an equation for the circle and name six points that lie on the circle.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.50            Approximate the solution to polynomial equations of higher degree by inspecting the graph

 

A2.A.50a

The function  is graphed below. Use the graph to approximate the solutions to the equation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.50b

Use the graph below to approximate the solutions to the equation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.50c

The Art Club has purchased flat sheets of cardboard to make storage boxes for the club’s art supplies.  They will make boxes by cutting a square from each corner of the 24 inch by 36 inch sheet of cardboard. 

Draw a diagram to illustrate the given information.

Express the volume of the box as a function of the side of the square cut from the cardboard. 

Make a table for the function.  Use the table to find the volume of a box formed by removing a square which has a 10 inch side, and the length of the side of a square that would produce a volume of 1792 cubic inches. 

Sketch a graph of the function, and use the graph to find the side of the square that would produce the maximum volume. Find the maximum volume of the box that can be made.

Make a chart containing the side of the square removed, length, width, height and volume of the box created that the club could use for quick reference to make boxes of appropriate size for the supplies. Be sure to determine an appropriate degree of accuracy for the entries in the chart.

 

A2.A.51            Determine the domain and range of a function from its graph

 

A2.A.51a

What is the domain and range of the function shown below? Express your answer in standard mathematical notation, and explain the notation.

 

A2.A.51b

Determine the domain and range of the function  by looking at its graph, shown below.

 

Explain how to determine the domain and range by examining the equation of the function.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.52            Identify relations and functions, using graphs

 

A2.A.52a

Which of the following two graphs represents a function?  Explain your answer.

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.52b

Which of the following two graphs represents a function? Explain how you determined your answer. Read explanations provided by other students, and compare solutions.

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.53            Graph exponential functions of the form  for positive values of b, including

 

A2.A.53a

Sketch and label the following graphs:

 .

When  is a positive number, what similarities and differences do the graphs of all forms ofhave?

 

A2.A.53b

On the same set of axes, use a graphing package or graphics calculator to graph the following functions:

The functions above are all of the form.

What effect does changing  values have on the shape of the graph?

What is the y-intercept of each graph?

What is the horizontal asymptote of each graph?

 

A2.A.54            Graph logarithmic functions, using the inverse of the related exponential function

 

A2.A.54a

Sketch the graph of where.

On the same axes, sketch the graph of, where g is the inverse function of  f.

State the equation for

 

Trigonometric Functions

 

A2.A.55            Express and apply the six trigonometric functions as ratios of the sides of a right triangle

 

A2.A.55a

Sketch right triangle ABC,   Express the value of each of the following as the ratio of the sides of the triangle.

 

                                                              

 

A2.A.55b

Sketch right triangle LMD,  Write your favorite number as the length of one of the sides.  Using this information, find the lengths of the other two sides.  Write the lengths as exact lengths.  Do not use decimal approximations.  Express the value of each of the following as the ratio of the sides of the triangle:

 

                             

 

Reduce all of the fractions to lowest terms and compare your answers to another student’s answers.  What pattern do you see? What conclusions can you reach?

Fill in another number as the length of one of the sides. Compare your answers again. Based on what you have found, determine the exact values of each of the following:

                             

 

A2.A.56            Know the exact and approximate values of the sine, cosine, and tangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles

 

A2.A.56a

Fill in the blanks in the following chart, giving exact numerical values.

 

 

 (in radians)

 

 

0

 

 

 

 

 

 

 

(in degrees)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.56b

Fill in the blanks in the following chart. 

 

Trigonometric Function

Exact Value

Approximate Value

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Under what circumstances would you prefer to find an approximation for each of these values, rather than giving an exact answer?

 

A2.A.57            Sketch and use the reference angle for angles in standard position

 

A2.A.57a

For each of the following angles, sketch the angle in standard position, then sketch its reference angle. Label the reference angle as , and determine its measure.

 

                             

 

A2.A.57b

For each of the following, use a reference angle to find the exact function value.

 Include a sketch to show how you determined your answer.

 

                                            

 

A2.A.58            Know and apply the co-function and reciprocal relationships between trigonometric ratios

 

A2.A.58a

If find

 

A2.A.58b

If find

 

A2.A.58c

What is the range of the function  Based on your answer, what is the range of the function   Explain your answer.

 

A2.A.59            Use the reciprocal and co-function relationships to find the value of the secant, cosecant, and cotangent of 0º, 30º, 45º, 60º, 90º, 180º, and 270º angles

 

A2.A.59a  Fill in the blanks in the following chart, giving exact numerical values.

 

 

 (in radians)

 

 

 

 

 

 

 

 

 

 

 

 

(in degrees)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.59b

Rewrite each of the following functions in terms of sine, cosine or tangent, and find the exact value:

                             

                             

                             

 

A2.A.60            Sketch the unit circle and represent angles in standard position

 

A2.A.60a

Each of the following graphs contains a graph of a portion of the unit circle and the corresponding portion of the curve  Utilize the two curves to determine each of the indicated function values. 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                                          

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.A.60b

In the accompanying diagram, unit circle O has radii and  is tangent to circle O at B, and is tangent to circle O at E.  Points O, F, D, and C are collinear, and  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If  name the line segment whose measure is each of the following:

 

                             

 

A2.A.61            Determine the length of an arc of a circle, given its radius and the measure of its central angle

 

A2.A.61a

Find in centimeters the length of the arc intercepted by a central angle of 4 radians in a circle with a radius of 3.5 centimeters.

 

A2.A.61b

A sector has radius of 12 cm and angle  To the nearest tenth of a centimeter, find its arc length.

 

A2.A.61c

A wheel has a radius of 3 feet. As the wheel turns, a rope connected to a five kilogram weight winds onto the wheel, causing the weight to move. If the wheel turns  to the nearest foot, how far does the weight move?

 

A2.A.62            Find the value of trigonometric functions, if given a point on the terminal side of angle

 

A2.A.62a

Find the exact value of each of the following for angle  in standard position if the point (4, -1) lies on its terminal side:

 

A2.A.62b

Find the exact value of  for angle in standard position if the point 

(-5, -4) lies on its terminal side. 

 

A2.A.62c

The line  creates an acute angle  when it crosses the x-axis. Find the exact value of

 

A2.A.63            Restrict the domain of the sine, cosine, and tangent functions to ensure the existence of an inverse function

 

A2.A.63a

In groups, sketch the graph of  from  to

 

Does the graph pass the vertical line test?

Now reflect over the line

Is this a function? Why?

Find a section of the sine graph with a range of  and is a one-to-one function.

Reflect only the section with the restricted domain over the line

Does the new image pass the vertical line test?

State the domain and range of the image.

 

A2.A.64            Use inverse functions to find the measure of an angle, given its sine, cosine, or tangent

 

A2.A.64a

If the and , find the exact value of .

 

A2.A.64b

Assuming an angle lies in Quadrant I, evaluate  as a fraction in lowest terms.

 

A2.A.65            Sketch the graph of the inverses of the sine, cosine, and tangent functions

 

A2.A.65a

Sketch the graph ofover the interval .

Reflect the graph over the line

How would you restrict the domain to make the image a function?

A2.A.66            Determine the trigonometric functions of any angle, using technology

 

A2.A.66a

Using your calculator, find the value of each of the following to three decimal places:

                             

 

A2.A.67            Justify the Pythagorean identities

 

A2.A.67a

Starting with  and using your knowledge of the quotient and reciprocal identities, derive an equivalent identity in terms of  and . Show all your work.

 

A2.A.67b

Let be a point in quadrant one on the unit circle,  Let point O be the origin (0,0).  Draw the line segment   Let  be the angle formed by  and the positive portion of the x-axis. Now draw the perpendicular from P to meet the x-axis at point M.

 

State the ratio of  in terms of

State the ratio of  in terms of .

State the coordinates of point P in terms of .

Substitute your coordinates into the unit circle equation to verify one of the Pythagorean identities.

Now choose P in a different quadrant and repeat the process. Does the identity continue to be true?

 

A2.A.68            Solve trigonometric equations for all values of the variable from 0º to 360º

 

A2.A.68a

Solve for , to the nearest tenth of a degree, in the interval

 

 

 

 

A2.A.69            Determine amplitude, period, frequency, and phase shift, given the graph or equation of a periodic function

 

A2.A.69a

A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally with respect to time, in days.  He carefully collected data and graphed his resulting equation.  From the graph, determine amplitude, period, frequency, and phase shift.

              
 

A2.A.69b

Given the following equations, determine the amplitude, period, frequency, and phase shift of each equation.

 

              

Two students, Anthony and Chris, can be overheard discussing these equations. Anthony is certain that these equations are equivalent, while Chris is insisting they are different. Which student is correct? Explain your answer fully with graphs, tables, and a written paragraph supporting your position.

 

A2.A.70            Sketch and recognize one cycle of a function of the form  or

 

A2.A.70a

Sketch one complete cycle of each of the following.

              

              

A2.A.70b

Write both a sine and a cosine equation for the following graph.

 

A2. A.70c

The angle of inclination of the sun changes throughout the year. This changing angle affects the heating and cooling of buildings. The overhang of the roof of a house is designed to shade the windows for cooling in the summer and allow the sun’s rays to enter the house for heating in the winter.

The sun’s angle of inclination at noon in Albany, New York can be modeled by the formula:

                       

where x is the number of days elapsed in the day of the  year, with January first represented by  January second represented by  and so on.

Find the sun’s angle of inclination at noon on Valentine’s Day.

Sketch a graph illustrating the changes in the sun’s angle of inclination throughout the year.

On what date of the year is the angle of inclination at noon the greatest in Albany, New York?

 

A2.A.71            Sketch and recognize the graphs of the functions , , , and

 

A2.A.71a

Sketch one cycle of each of the following equations.  Carefully label each graph.

              

Graph and  at the same time on your calculator with a window of .  What conclusions can you make?  Describe the similarities between the 2 functions.

 

Now, graph and  at the same time on your calculator with a window of  .  What conclusions can you make?  Describe the similarities between the 2 functions.

 

A2.A.71b

Write an equation for each of the following graphs. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

A2.A.72            Write the trigonometric function that is represented by a given periodic graph

 

A2.A.72a

Write a trigonometric function that matches each of the following graphs. Check your answers with a partner. If different equations have been obtained, confirm by graph or table, the accuracy of each equation.

A2.A.72b

A team of biologists have discovered a new creature in the rain forest.  They note the temperature of the animal appears to vary sinusoidally over time.  A maximum temperature of 125 occurs 15 minutes after they start their examination.  A minimum temperature of  99 occurs 28 minutes later.  The team would like to find a way to predict the animal’s temperature over time in minutes.  Your task is to help them by creating a graph of one full period, an equation of temperature as a function over time in minutes, and a table of maximum, minimum, and average temperatures for the first 3 hours.

Discuss the advantages and disadvantages of each representation.

 

A2.A.73            Solve for an unknown side or angle, using the Law of Sines or the Law of Cosines

 

A2.A.73a

Chris is planning to build a new roof on his garage.  He decides to pitch the sides of the roof at an angle of , and the width of the garage is 30 ft.  Find the length of the sides of the roof, to the nearest tenth of a foot.

A2.A.73b

A surveyor measures the sides of a triangular plot of swamp land and labels the diagram.  What is the largest angle in the triangle, to the nearest degree?

A2.A.73c

Two students in Ms. Baum’s class worked on the problem below and got different solutions. Who got the correct answer? Explain your answer in detail.

Given triangle PQR, find the largest angle to the nearest degree.

Student #1                                                                                      Student #2

 

                                                               

 

A2.A.73d

Write a word problem that illustrates an application of the Law of Sines, Law of Cosines, or a combination of both. Be creative, but be certain to use appropriate language and mathematical terminology when describing the problem situation.  Also prepare a complete solution of the problem including a correctly labeled diagram and full mathematical explanation of how to solve the problem. 

 

A2.A.74            Determine the area of a triangle or a parallelogram, given the measure of two sides and the included angle

 

A2.A.74a

Since Meisha’s family has 3 people, her mother decided to make a triangular dinner table for them.  She wanted to make sure that there was enough room for each of them to eat comfortably and wanted to determine the area of the table. Meisha’s mother drew the design below to fit into their kitchen.  Find the area of the table to the nearest square foot.

How long will the third side of the table be?

 

A2.A.74b

Samuel has decided to use stamped concrete for his backyard patio.  He has chosen to make his design out of parallelograms.  He has created a pattern and needs to find the area of his design.  Using the following diagram, find the area to the nearest tenth of a square foot.

 

A2.A.75            Determine the solution(s) from the SSA situation (ambiguous case)

 

A2.A.75a

In the triangle below, find the missing angles and sides.  Indicate the number of triangles which can be drawn with the given information.  Round each measure to the nearest whole number.

A2.A.75b

Given triangle DEF, where , find the missing angles and sides.  Indicate the number of triangles which can be drawn with the given information.  Round each measure to the nearest whole number.

 

 

A2.A.75c

Given triangle ABC, where , find the missing angles and sides.  Indicate the number of triangles which can be drawn with the given information.  Round each measure to the nearest whole number.

 

 

A2.A.76            Apply the angle sum and difference formulas for trigonometric functions

 

A2.A.76a

If , and x is a positive acute angle, find .

 

A2.A.76b

If  and   where  and  find:

 

 

 

tan

 

A2.A.76c

Jenna’s teacher has asked the class to find the exact value of sin(105). Jenna’s work is as follows.

Jenna’s teacher has marked this as incorrect. Using counterexamples or an indirect proof, demonstrate why Jenna’s work is not correct.

 

A2.A.77            Apply the double-angle and half-angle formulas for trigonometric functions

 

A2.A.77a

If , what is

 

A2,A.77b

Using the half-angle formula, find the exact value of

 

A2.A.77c

Given that , and evaluate in terms of a and b:

                                        

                               tan

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Measurement

 

Students will determine what can be measured and how, using appropriate methods and formulas.

 

Units of Measurement

 

A2.M.1             Define radian measure

 

A2.M1a

Anthony’s teacher told the class that a unit circle has a circumference of . This confuses him because he thought a circle has. Write a detailed explanation that compares degrees to radians. Be as thorough as possible in order to help Anthony understand the connection, including diagrams and equations.

 

A2.M.2             Convert between radian and degree measures

 

A2.M2a

Convert each radian measure to degree measure.

 radians,  radians,  4 radians

 

A2.M2b

Convert each degree measure to radian measure.

 

A2.M2c

In which quadrant, or on what axis, does the terminal side of each angle lie?

              

              

              

 

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Statistics and Probability

 

Students will collect, organize, display, and analyze data.

 

Collection of Data

 

A2.S.1               Understand the differences among various kinds of studies (e.g., survey, observation, controlled experiment)

 

A2.S.1a

Search in newspapers and magazines and then write a brief but thorough description of a study  that could be characterized as:

               a survey

               an observational study

               a designed experiment

 

A2.S.2               Determine factors which may affect the outcome of a survey

 

A2.S.2a

Randomization is an important characteristic of a well-designed survey. Name two sources of bias that can be eliminated by randomization.

 

A2.S.2b

It has been decided that 78 people need to be surveyed to decide the public’s opinion on a school building project. A student suggests that they survey the first 78 people who enter the school.

Do you think that this proposed way of sampling is an unbiased way to perform the survey or can you describe a better way to achieve a fair and accurate response?

 

Organization and Display of Data

 

A2.S.3               Calculate measures of central tendency with group frequency distributions

 

A2.S.3a

The table displays the frequency of scores on a twenty point quiz. The mean of the quiz scores is 18.

Score

15

16

17

18

19

20

Frequency

2

4

7

13

k

5

Find the value of k in the table. Find the mode and the median of all the quiz scores .

 

A2.S.3b

The table displays the number of uncles of each student in a class of Algebra 2 .

 

uncles

0

1

2

3

4

5

students

2

5

4

6

10

8

Find the mean, median, and mode of the uncles per student  for this data set.

 

A2.S. 3c

The average earning of 110 juniors for a week was $35, while during the same week 90 seniors averaged
$ 50. What were the average earnings for that week for the combined group?

 

A2.S.4               Calculate measures of dispersion (range, quartiles, interquartile range, standard deviation, variance) for both samples and populations

 

A2.S.4a

For the data set: {5, 4, 2, 5, 9, 3, 4, 5, 3, 1, 6, 7, 5, 8, 3, 7} construct the five-number summary. Find the range and the interquartile range of the data set.

 

A2.S.4b

Find to the nearest tenth the standard deviation of the distribution:

score

1

2

3

4

5

frequency

14

15

14

17

10

 

A2.S.4c

If the five numbers { 3, 4, 7, x, y} have a mean of 5 and a standard deviation of .  Find x and y given that y > x.

 

A2.S.5               Know and apply the characteristics of the normal distribution

 

A2.S.5a

Five hundred values are normally distributed with a mean of 125 and a standard deviation of 10.

Answer the following questions:

                              What percent of the values is in the interval 115-135?

                              What percent of the values is in the interval 100-150?

                              What interval about the mean includes 50% of the data?

                              What interval about the mean includes 95% of the data?

 

A2.S.5b

A machine is used to fill plastic soda bottles. The amount of soda dispensed into each bottle varies slightly. Suppose the amount of soda dispensed into the bottles is normally distributed. If at least 99% of the bottles must have between 585 and 595 milliliters of soda, find the greatest standard deviation, to the nearest hundredth, that can be allowed.

 

A2.S.5c

The lifetime of a battery is normally distributed with a mean life of 40 hours and a standard deviation of 1.2 hours. Find the probability that a randomly selected battery lasts longer than 42 hours.

 

Students will make predictions that are based upon data analysis.

 

Predictions from Data

 

A2.S.6               Determine from a scatter plot whether a linear, logarithmic, exponential, or power regression model is most appropriate

 

A2.S.6a

Which type of function (linear, exponential, logarithmic, or power) would best model the data in each of the scatter plots shown below? Explain your reasoning.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.S.6b

Sketch a scatter plot whose regression model could be logarithmic and one whose regression model could be exponential. Use your knowledge of exponential and logarithmic functions to explain the differences and similarities in the two scatter plots.

 

A2.S.7               Determine the function for the regression model, using appropriate technology, and use the regression function to interpolate and extrapolate from the data

 

A2.S.7a

Take a cup of very hot water, and measure the temperature of the water.  Place the cup in a safe place, and measure the temperature every five seconds for a total of one minute.  Record this information in your notebook.

Enter the information in two lists on your calculator.  In the first list, enter the time, in seconds, since the experiment began. In the second list, enter the temperature of the water for each indicated time.

Make a scatter plot, where the independent variable is time, in seconds, and the dependent variable is temperature. What type of regression function would best model this data? Why?

Write the regression equation with all constants rounded to the nearest thousandth.

Using this equation, determine the temperature of the water after 25 seconds has passed.  Determine the temperature of the water after two hours has passed.  Do both of these answers seem logical to you?  Why?

(N.B. If you have a temperature probe for your calculator or computer, it may be used to conduct this experiment.)

 

A2.S.7b

Stretch your arms out as wide as you can.  Have someone measure the length, in inches, of your arm span, from the tip of your fingers on one hand, across your back, to the tip of your fingers on your other hand.  Have the person also measure your height, in inches. Share this information with your classmates, and aggregate all of the data.

Enter the information in two lists on your calculator.  In the first list, enter the heights, in inches, of all of the participants.  In the second list, enter the arm span, in inches, of all of the participants.

Make a scatter plot, where the independent variable is height, in inches, and the dependent variable is arm span, in inches.  What type of regression function would best model this data?  Why?

Write the regression equation with all constants rounded to the nearest tenth.

Using this equation, determine the arm span of a person who is 5 feet tall.

Research to see what other measurements would form similar relationships.

 

A2.S.7c

The table shows the amount of a medicine for treating a disease in the bloodstream over the 9 hours following a dose of 10 mg. It seems that the rate of decrease of the drug is approximately proportional to the amount remaining.

 

Time(hours)

0

1

2

3

4

5

6

7

8

Drug Amount (mg)

10

8.3

7.2

6.0

5.0

4.4

3.7

2.8

2.5

 

Use this information to find a suitable function to model this data.

Using your model, when will there be less than 1 mg. of the medicine in the patient’s blood stream?

If the initial dose was 15 mg, when would the amount of the medicine in the bloodstream fall below 5 mg?

 

A2.S.7d

According to the National Weather Service, the 2005 average monthly temperature in degrees Fahrenheit at Central Park in New York City is given below:

 

Month

Jan.

Feb.

March

April

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Average Temperature

(oF)

 

31.3

 

 36.5

 

39.4

 

55.1

 

58.9

 

74.0

 

77.5

 

79.7

 

73.3

 

57.9

 

49.6

 

35.3

 

Write a sinusoidal function that models the average monthly temperature, using t = 1 to represent January. According to your model, what is the average temperature in December?

Explain the discrepancy from your model to the average monthly temperature in December.

How could the model be improved?

 

A2.S.8               Interpret within the linear regression model the value of the correlation coefficient as a measure of the strength of the relationship  

 

A2.S.8a

Which of the following scatter plots would have a positive linear correlation coefficient? 

Which would have a negative linear correlation coefficient?

Which would have a linear correlation coefficient closest to one?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A2.S.8b

Use two thermometers, one Celsius and the other Fahrenheit, to measure the temperature of your hand.  Place both thermometers in the palm of your hand and gently squeeze your hand shut.  (If you have a temperature probe for your calculator or computer, it may be used instead.)

Gather this information with your classmates, and aggregate all of the data. 

Enter the information in two lists on your calculator.  In the first list, enter the Celsius temperatures.  In the second list, enter the Fahrenheit temperatures.

Make a scatter plot, where the independent variable is degrees Celsius, and the dependent variable is degrees Fahrenheit. 

Find the linear regression equation that best fits the data. 

What is the correlation coefficient? Why would you expect its value to be close to one?

 

Students will understand and apply concepts of probability.

 

Probability

 

A2.S.9               Differentiate between situations requiring permutations and those requiring combinations

 

A2.S.9a

Determine whether each of the following situations would require calculating a permutation or a combination.  Explain your answer.

Selecting a lead and an understudy for a school play

Assigning students to their seats on the first day of school

Selecting three students to represent the school at an Honor Society conference in Washington, D.C.

 

A2.S.9b

Describe a situation that would require calculating a permutation, and another situation that would require calculating a combination.  Share your thoughts with a classmate and discuss the strategies used to determine when to calculate a permutation and when to calculate a combination.  Write a brief summary of your conclusions and be prepared to discuss your ideas with the class.

 

A2.S.10                               Calculate the number of possible permutations of n items taken r at a time

 

A2.S.10a

Evaluate each of the following

                             

 

A2.S.10b

A teacher is making a multiple-choice quiz.  She wants to give each student the same questions, but have each student’s questions in a different order.  If there are twenty-seven students in the class, what is the least number of questions the quiz must contain?

 

A2.S.11            Calculate the number of possible combinations of n items taken r at a time

 

A2.S.11a

Calculate each of the following:

                    

                   

                 

                 

            

 

A2.S.11b

A coach must choose five starters from her team of twelve players.  How many different ways can she choose the starters?

 

A2.S.12            Use permutations, combinations, and the Fundamental Principle of Counting to determine the number of elements in a sample space and a specific subset (event)     

 

A2.S.12a

The local Family Restaurant has a daily breakfast special in which the customer may choose one item from each of the following groups:

 

Breakfast Sandwiches

Accompaniments

Juice

egg and ham

egg and bacon

egg and cheese

breakfast potatoes

apple slices

fresh fruit cup

pastry

orange

cranberry

tomato

apple

grape

 

How many different breakfast specials are possible?  How many different breakfast specials without meat are possible?

 

A2.S.12b

How many three-digit numbers can be formed from the digits 0,1,2, …, 9?  How many of these numbers would be multiples of five?

A2.S.12c

There are fourteen juniors and twenty-three seniors in the Service Club. The club is to send four representatives to the State Conference. How many different ways are there to select a group of four students to attend the conference? If the members of the club decide to send two juniors and two seniors, how many different groupings are possible?

 

A2.S.12d  Find the number of arrangements that can be made from the letters of each of the following words:

               THINK

               MATHEMATICS

How many arrangements would there be if only consonants are used?

 

A2.S.13            Calculate theoretical probabilities, including geometric applications

 

A2.S.13a

Each of the five members of the Music Committee for the spring dance is allowed to select one song to be played by the DJ at the dance from a list of forty approved songs. What is the probability that at least two committee members select the same song?

  

A2.S.13b

A bag contains three chocolate, four sugar, and five lemon cookies.  Greg takes two cookies from the bag for a snack. Find the probability that Greg did not take two chocolate cookies from the bag.  Explain why using the complement of the event of not choosing two chocolate cookies might be an easier approach to solving this problem.

 

A2.S.13c

Find the probability that a point selected at random will lie inside the triangle below:

A2.S.14            Calculate empirical probabilities

 

A2.S.14a

When a paper cup is tossed onto a table, it can land one of three ways: open end up, bottom up, or on its side. Give each student in class a paper cup. Have each student toss their cup onto a table thirty times, and record the result of each of their tosses. Then tally the results of the entire class’s tosses.

Using the class’s results of the experiment, compute the following empirical probabilities: P(open end up), P(bottom up), P(landed on side).

Compute the following theoretical probabilities: P(open end up), P(bottom up), P(landed on side). How do the empirical probabilities found compare to the theoretical probabilities?  Tape a penny to the bottom of the cup. Repeat the experiment. What effect do you think this will have on the outcomes of the tosses? Compute the same empirical and theoretical probabilities as you did the first time. How do these result compare to your prediction?

 

A2.S.14b

Use a random number generator to generate five hundred random numbers which will simulate tossing a fair coin thirty times. Let an even number represent the result of the coin coming up heads, and an odd number represent the result of tails. 

Generate a second list of five hundred random numbers to represent the second toss of a fair coin. 

Use the two lists of random numbers to compute the following empirical probabilities: P(head, tail), P(both the same), P(both different). 

Compute the following theoretical probabilities for the same experiment: P(head, tail), P(both the same), P(both different). How do the empirical and theoretical probabilities compare?

A2.S.15            Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most

 

A2.S.15a

If a binomial experiment has seven trials in which the probability of success is p and the probability of failure is q, write an expression that could be used to compute each of the following probabilities:  P(exactly five successes), P(at least five successes), P(at most five successes).

 

A2.S.15b

Experience has shown that  of all CDs produced by a certain machine are defective.  If a quality control technician randomly tests twenty CDs, compute each of the following probabilities:  P(exactly one is defective), P(half are defective), P(no more than two are defective).

 

AS.S.15c

If a fair coin is tossed ten times, what is the probability that it lands heads up:  exactly six times, at most three times, at least seven times?

 

A2.S.16            Use the normal distribution as an approximation for binomial probabilities

 

A2.S.16a

Give an example of an experiment where it is appropriate to use a normal distribution as an approximation for a binomial probability.  Explain why in this example an approximation of the probability is a better approach than finding the exact probability.

 

A2.S.16b

Estimate the probability of getting at least fifty-four heads in one-hundred tosses of a fair coin.

 

A2.S.16c

A fifty-question multiple-choice test consists of questions which each have four possible choices for the answer to each question. Estimate the probability of answering at least 70% of the questions correctly if all of the answers are random guesses.

 

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