Oct. 2004 Facilitators Notes

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

Number Sense and Operations

Algebra

Geometry

Measurement

Statistics and Probability

Problem Solving Strand

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

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

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

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

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

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)

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:

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)

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.

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

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

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

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

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?

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.

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

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.

Reasoning and Proof Strand

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

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.

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

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.

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

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

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.

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.

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

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?

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

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

Demonstrate that

and

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

Starting with

and using your knowledge of the quotient and reciprocal identities, derive an equivalent identity in terms of

and

. Show all your work.

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

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

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

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.

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.

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?

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

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.

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

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:

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

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.

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

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

Convert the equation

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

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

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