Integrated Algebra

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.

Problem Solving Strand

Students will build new mathematical knowledge through problem solving.

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

A.PS.1a

Using one inch grid paper, draw a rectangle with a length of 3 inches and a width of 5 inches and find the area of the rectangle.  Next, using cubes or stackable blocks, build a tower or rectangular prism with a height of 4 inches on the rectangle that you drew. How can you use the area of a rectangle to help find the volume of the rectangular prism? 

On the one inch grid paper draw a circle with a diameter of 4 inches and find the area of the circle.  How can you use the area of a circle to determine the volume of a cylinder with diameter of 4 inches and height of 5 inches?

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

A.PS.2a

Sabrina was examining three different data sets using her computer spreadsheet

program. Below are the screen shots of what she was examining.

Which of the following terms best describe each correlation?

positive, negative, or does not exist?

              Data Set #1

   _______________________

               Data Set #2

   _______________________

                Data Set #3

   _____________________

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

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

A.PS.3a

In 2001 the city of Bedford Falls recorded the following number of sunny days in nine consecutive months:

5, 8, 15, 14, 11, 14, 8, 12, 8.

Find the mean, median, mode and range for this set of data.

In 2002 a weather forecaster for Bedford Falls predicted that the number of sunny days would double for each month from the previous year. What would be the mean, median, mode and range for the 2002 data set, based on the weather forecaster’s prediction?  Based on this prediction, how would these four measures change from 2001 to 2002?

Another weather forecaster predicted that 2002 would result in five more sunny days per month than in 2001. What would be the mean, median, mode and range for the 2002 data set, based on the forecaster’s prediction? Based on this prediction, how would these four measures change from 2001 to 2002?

A.PS.3b

The chart below shows the prices of gasoline and milk at a local convenience store, over a 3-week period.

Price of Gasoline and Milk in March 2006

What type of correlation, if any, during this three week period existed between the price of gasoline and the price of milk?

Could either of these events cause the other? Explain your answer.

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

A.PS.4a

The cost of CDs at a discount store is recorded in the following table.  Describe verbally and symbolically the relationship between the number of CDs purchased and the total cost. If the relationship were represented as a graph, what would be the slope of the resulting line?

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

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

A.PS.5a

Solve the following system of equations graphically or algebraically.  Explain why you chose the method that you did.

y = 2x

y = 3x -3

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

A.PS.6a

In Triangle ABC, AB = 4 and BC = 5.  If the perimeter of triangle ABC is 3 times AB, find AC.

A.PS.6b

If , and x is an integer, then find all possible values of x.

A.PS.6c

Solve the following system of equations graphically: 

y = 2x + 6 and y = 3x + 2. 

Now solve the equation 2x + 6 = 3x + 2.   

State a similarity between the graphs for the system of equations and the algebraic solution to the equation. Provide an explanation for this similarity.

A.PS.6d

A rectangle, ABCD, has the length, AB, of 12 and width, BC, of 4.  Draw a rectangle, KLMN, so that rectangle KLMN is similar to rectangle ABCD.  If AB:KL = 3:2, find the dimensions of the new rectangle. 

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

A.PS.7a

Work with two other students to solve the following problem:

Cameron received a set of four grades. If the average of the first two grades is 50, the average of the second and third grades is 75, and the average of the third and fourth grades is 70, then what is the average of the first and fourth grades?

Be prepared to present your solution to the class. The other groups in the class will also present their solutions. You will evaluate each other’s solutions according to the following criteria: accuracy of the solution, clarity of the explanation, efficiency of the solution method, and creativity.

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

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

A.PS.8a

Determine if the graph of each of the relations is a function. Justify your answer.

A.PS.8b

Determine if each relation is a function.  Justify your answer.

A.PS.8c

The number of e-mails 20 different students sent in a week varied from 35 to 90, as seen in the box-and-whisker graph below:

What is the minimum number of e-mails sent?

What is the number at the 25^th percentile?

What is the number at the 50^th percentile?

What is the number of e-mails sent at the 75^th percentile?

What is the maximum number sent?

A.PS.8d

Mathematics Test Scores of Mr. Smith’s Class

How many students in Mr. Smith’s class scored greater than an 85?

How many students scored at least a 66, but no more than an 85?

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

A.PS.9a

Mario has a cookie shop. He has 30 pounds of butter and 60 pounds of sugar.  A batch of butter cookies takes 6 pounds of butter and 10 pounds of sugar. A batch of chocolate chip cookies takes 5 pounds of butter and 12 pounds of sugar.

Determine which combinations of cookie batches he can make. Verify your answer.

3 batches of butter cookies and 2 batches of Chocolate Chip cookies

3 batches of butter cookies and 3 batches of Chocolate Chip cookies

6 batches of Chocolate Chip cookies

5 batches of Butter Cookies

1 of Butter and 5 of Chocolate Chip

What is the maximum amount of batches of Chocolate Chip cookies that Mario could make?

A.PS.9b

Ahmed wants to build a rectangular garden with an area between 100 square feet and 150 square feet. The length must be exactly two feet longer than the width. Find all possible integral dimensions for the garden. Determine the actual minimum and maximum amount of fencing Ahmed might use.

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

A.PS.10a

Three different students were presented with the following problem:

Find the values of y for the equation y = 3x +13 when x = 3, 10, 18, 23, and 31.

Student 1 solved the equation by substituting the given values for x in the equation and then solving for y each time.

Student 2 correctly keyed this equation into a graphing calculator and used the “table” function for finding what the value of y would be for each of the values of x given in the problem.

Student 3 solved the equation by graphing it on graph paper and using knowledge of y=mx+b.

Which solution method will lead to finding the answers more efficiently? Justify your answer.

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Reasoning and Proof Strand

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

A.RP.1             Recognize that mathematical ideas can be supported by a variety of strategies

A.RP.1a

Sabrina was examining three different data sets using her computer spreadsheet program. Below are the screen shots of what she was examining.

Which of the following terms best describe each correlation?        positive, negative or does not exist?

              Data Set #1

   _______________________

               Data Set #2

   _______________________

                Data Set #3

   _____________________

Students will make and investigate mathematical conjectures.

A.RP.2             Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture

A.RP.2a

Graph the following equations on a single set of coordinate axes:

Describe how the changes in the coefficient affect the graphs.

A.RP.2b

Graph the following equations on a single set of axes.

y = x²  

y = 2x²

y = 0.5x²

Describe how the changes in the coefficient affect the graphs.

A.RP.2c

The chart below shows the prices of gasoline and milk at a local convenience store, over a 3-week period.

Price of Gasoline and Milk in March 2006

What type of correlation, if any, during this three week period existed between the price of gasoline and the price of milk?

Could either of these events cause the other? Explain your answer.

A.RP.3             Recognize when an approximation is more appropriate than an exact answer

A.RP.3a

The retail price of various diamonds by size was recorded at a local jewelry store, as seen in the graph below.

Size of a Diamond (Carats) vs. Price (Dollars)

On the graph determine the line of best fit.

Which is the best estimate of the price of a diamond that is 0.31 carats?

Students will develop and evaluate mathematical arguments and proofs.

A.RP.4             Develop, verify, and explain an argument, using appropriate mathematical ideas and language

A.RP.4a

The formula,, is used for finding the sum of the first n consecutive numbers, where n represents the amount of numbers to be counted.  Use the formula to find the sum of the first ten counting numbers. Add the first ten counting numbers to verify your answer.

A.RP.4b

The sides of a triangle are 5,  and 10.  Do these sides form a right triangle?  Justify your answer.

A.RP.5             Construct logical arguments that verify claims or counterexamples that refute them

A.RP.5a

Three friends (Mark, Oman and Patty) agreed to buy lunch for each other on Monday, Tuesday and Wednesday. They decided to eat pizza, tacos or burgers on each of the three days. They also agreed that:

Mark would not pay on Monday.

Oman would pay before Mark paid.

Patty would pay the first day.

Determine if this statement is correct, and justify your answer:

“Mark paid on Wednesday.”

If Patty paid on the day they ate pizza, and Mark paid on a day they did not eat burgers, did they eat tacos on Tuesday? Justify your answer.

A.RP.5b

A math class practiced multiplying numbers written in scientific notation. The students were asked to devise a rule that could be used for this operation. Stephen made the following conjecture:

Stephen believes that you should multiply the leading numbers between 1 and 10 (N and M) and you should also multiply the exponents (x and y). Do you agree with Stephen’s conjecture? If you agree, justify the conjecture. If you disagree, provide a counterexample.

A.RP.6             Present correct mathematical arguments in a variety of forms

A.RP.6a

Having just purchased a DVD player, Dean is comparing the costs of renting DVDs from Lucas DVDs and Dynamite DVDs. Lucas charges a monthly membership fee of $50.00 and rents DVDs in multiples of 5.  For each multiple of 5 DVDs, there is a $7.00 charge. Dynamite DVDs does not have a membership fee and also rents DVDs in multiples of 5.  For each multiple of 5 DVDs, there is a charge of $30.00 

Determine which DVD rental company offers a more cost-efficient offer by constructing a table and graphing the results for each company. For every multiple of 5 DVDs rented, would it always be better to use one company over the other? Justify your answer.

A.RP.7             Evaluate written arguments for validity

A.RP.7a

Explain why LINE #3 is not a causal relationship to Lines #1 and #2.

LINE #1: As children progress through elementary school, they often get more tooth fillings.

LINE #2: As children progress through elementary school, they often increase their vocabulary.

LINE #3: Therefore, if children get more cavities, they will increase their vocabulary.

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

A.RP.8             Support an argument by using a systematic approach to test more than one case

A.RP.8a

Given is an even integer and is an odd integer, is + always even or always odd? Explain.

A.RP.9             Devise ways to verify results or use counterexamples to refute incorrect statements  

A.RP.9a

Given the following statement:  If a and b are integers, then . 

Determine whether this statement is true or false. If true, justify. If false, give a counterexample.

A.RP.9b

In math class, Joseph did the following problem. Is Joseph correct?  If not, what is the correct sum and how would you show Joseph how to get that answer.

A.RP.9c

Tom stated that if n is a positive integer, then  is prime. Sue stated that Tom was incorrect.  Show how Sue was able to prove her case.

A.RP.10           Extend specific results to more general cases

A.RP.10a

In how many ways can each of the following collections of different objects be arranged?

Five flowers

The six letters in the word HOCKEY

10 books

n candy bars

The flags of r countries

y playing cards from a standard deck

A.RP.10b

Write an equation for the line that passes through each of the following pairs of points:

(1, 5) and (-7, 5)                                              (12, 8) and (12, 1)

(3, -2) and (-6, -2)                                           (-9, -4) and (-9, 0)

(0, 4.5) and (11, 4.5)                                       and

(8, a) and (-3, a)                                               (p, -1) and (p, 5)

(m, q) and (-2n, q)                                            (c, 2b) and (c, 3b)

A.RP.11           Use a Venn diagram to support a logical argument

A.RP.11a

Mr. Johnson has 30 students in his first period math class.  Of those 30 students, 18 students passed their first exam, 21 students passed the second exam and 14 students passed both exams.  Construct a Venn diagram to represent this information and determine how many students did not pass either exam.

A.RP.11b

100 children attended summer camp. 46 children played soccer.  25 children played soccer and basketball. 24 children played soccer and baseball.  50 children played basketball.  19 children played baseball and basketball.  46 children played baseball.  13 children played all three sports.  Construct a Venn diagram to represent how many children played each sport.

How many children did not play any of these sports?  How many children played only one sport?  How many children played baseball and soccer but not basketball?

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

A.RP.12a

In Mr. Smith’s classroom this school year, the students have studied the following quadrilaterals: parallelograms, rectangles, and squares. Based on the properties of these quadrilaterals, make at least 2 conjectures about the properties of quadrilaterals in general. When Mr. Smith’s students study the trapezoid, will your conjectures still be true? Explain your answer.

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Communication Strand

Students will organize and consolidate their mathematical thinking through communication.

A.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

A.CM.1a

A research company wanted to obtain data on what is watched on television by community members who are 18 years old and older. The research company made random telephone calls to homes in the community. The telephone calls resulted in:

An inability to reach a person in 53% of the homes called.

The exclusion of non-telephone homes in the community.

Those surveyed were 72% male and 28% female.

Explain how each of the three factors above could create a bias in the survey results.

A.CM.1b

A restaurant owner wanted to determine what her customers consider the most appealing quality of her restaurant. A brief survey card was placed on each table before customers were seated. A portion of the customers voluntarily completed the survey card.  As an incentive, those who completed the survey card were entered in a random drawing for a new skateboard.  The chart below displays the ages of those who completed the survey card:

What biases might exist as a result of the design of this survey?

The results of the survey are summarized in the chart below:

What conclusions might the restaurant owner draw from this survey?

What biases might exist in this data?

Do you believe the results of this survey are valid?

Explain your answer.

A.CM.1c

When A = {3, 4, 5} and B = {4, 5, 6, 7}, find:

                    and

A.CM.1d

A police officer was studying traffic patterns in one part of town. In this study

he recorded the speeds of 17 cars traveling in a 40 mph zone.

Here are the speeds in miles per hour:

35, 40, 42, 43, 45, 46, 37, 38, 52, 39, 47, 42, 41, 39, 54, 52, and 25.

Construct a histogram and a cumulative frequency histogram for this data.

(For the histogram, use the intervals 21-25, 26-30, 31-35, 36-40, 41-45, 46-50 and

51-55. For the cumulative frequency histogram, use the intervals 21-25, 21-30, 21-35,

21-40, 21-45, 21-50 and 21-55.)

A.CM.1e

Draw a box-and-whisker plot for the ages of 13 employees at a local store:

21, 28, 29, 30, 31, 33, 34, 34, 37, 39, 40, 43, 57

A.CM.1f

Given the following ages: 20, 27, 28, 29, 30, 31, 33, 33, 37, 39 and 55:

What is the minimum age?

What is the lower quartile age?

What is the median age?

What is the upper quartile age?

What is the maximum age?

Construct a box-and-whisker plot for this data.

A.CM.1g

A student surveyed ten of his friends on how much time they spent studying for a recent math test and their grade on the test.

                                         Time Spent Studying vs. Test Grade