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REFINED Resources. The resources on this page have been updated and revised to align with the refined K-12 mathematics TEKS. These refined TEKS were adopted by the Texas State Board of Education in 2005–06 and implemented in 2006–07.

(A.1) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.

(A.1.A) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to describe independent and dependent quantities in functional relationships.

Clarifying Activity

Students are given a set of functional relationships on index cards. Some written following the form, "________ depends on _________." and others written in the form, "__________ is a function of ____________." Students identify the independent and dependent quantities. The key idea for students to demonstrate is that a functional relationship is the dependency between two quantities in which one quantity depends, in a systematic way, on another quantity. Independent and dependent quantities are often associated with “input-output” or “if-then” statements.

An example of a functional relationship:

The math club wants to register for a math competition. The total cost of registration is a function of the number of club members who register for the competition.

Independent quantity: the number of math club members registering for the competition

Dependent quantity: the total cost for registration

Assessment Connections

Ask . . .

Start with . . .

• Tell me about the quantities in the relationships.

Probe further with . . .

• What is the independent quantity? How do you know?
• What is the dependent quantity? How do you know?
• How can your function sentence be written as an "if-then" or "input-output" statement?
• How do you know that this relationship is a function?

Listen for . . .

• Does the student appropriately identify independent and dependent quantities for the functional relationship described?
• Can the student clearly explain why, in a particular situation, a quantity is independent or dependent?
• Does the student use appropriate language when describing the independent and dependent quantities for the functional relationship?
• Does the student describe a function as representing a dependence of one quantity on another?

Look for . . .

• How effectively does the student identify the independent and dependent variables?
• Does the student write an "if-then" or "input-output" sentence that accurately describes a functional relationship?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• April 2006, grade 9, item 41
• Spring 2003, grade 10, item 19
• April 2006, grade 11, item 13

(A.1.B) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to gather and record data and use data sets to determine functional relationships between quantities.

Clarifying Activity

Teams of students are assigned a task that asks them to gather and record data sets of related quantities. They devise a plan and then collect and organize data. They use the data sets to determine whether or not there is a functional relationship between the quantities and explain why. Finally, each team presents their task, their plan, and the data they collected to the class.

Example tasks:

• Students gather data about the lengths of the first name and last name for each member of their class. They record the information in a table. They then determine that the relationship between the length of a student’s first name and last name is not a functional relationship. Two students with a first name of equal length may have last names of a different length, so the length of a last name cannot be predicted using the length of the first name.
• Students gather data on exam scores and the number of hours students studied for the exam. While these should correlate (the more time a student studies the more likely his/her score is high), the relationship may not be functional. Data may reveal that two students studying the same amount of time earned different scores.
• Students gather data relating the height of a stack of identical cups to the number of cups in the stack. They record the information in a table. They then determine that the relationship between the height of a stack of cups and the number of cups in the stack is a functional relationship; that is, stack height depends on the number of cups in the stack.

Assessment Connections

Ask . . .

Start with . . .

• Tell me about the data you collected.

Probe further with . . .

• How did you organize the data? Why did you choose to organize it in this way? What units of measure did you use? Do the units of measure make sense?
• What is a function?
• Is there a dependency relationship between the quantities? Does one quantity depend on the other in a systematic way? How do you know?
• Describe the relationship between the variables using a function statement.
• What is the independent/dependent quantity? How do you know?

Listen for . . .

• Can the student determine whether or not a relationship is a functional relationship?
• Does the student clearly and accurately describe what it means for a relationship between two quantities to be a functional relationship?
• Does the student use appropriate language when describing the functional relationship?
• Does the student appropriately identify independent and dependent quantities for the functional relationship described?

Look for . . .

• Does the student demonstrate that he/she understands that a function represents a dependence of one quantity on another?
• Can the student use data sets to determine whether or not a relationship is a function?
• Can the student organize and record the collected data in an efficient and useful way (for example, length of name and month of birth of same person)?
• Does the student accurately represent the data? Check for reasonable units of measure.
• Does the student label the data correctly?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 49
• April 2006, grade 9, item 25
• Spring 2004, grade 10, item 35
• April 2006, grade 10, item 14
• Spring 2003, grade 11, item 45

(A.1.C) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations.

Clarifying Activity

Students are given a problem situation that can be described using a functional relationship. They determine that the relationship is a function and represent the function using a variety of ways, including a function rule. They answer questions arising from the situation by writing and solving equations or inequalities.

Two example situations . . .

The functional relationship:

Scott lives in Texas. His friend lives in Michigan. Scott has to make a long distance call to talk to his friend. He plans to use his calling card. The card charges $0.75 for access to long distance plus $0.10 for each minute. Determine the functional relationship that describes how the total charge depends on the length of a call in minutes.

Answering questions by writing and solving equations or inequalities:

Scott lives in Texas. His friend lives in Michigan. Scott has to make a long distance call to talk to his friend. He plans to use his calling card. The card charges $0.75 for access to long distance plus $0.10 for each minute. He has $3.00 left on his card. Investigate whether or not Scott can talk with his friend for at least 15 minutes.

Assessment Connections

Ask . . .

Start with . . .

• What factor(s) determine the total cost of a call Scott makes?
• Develop a function rule or statement that shows how the charge depends on the length of the call.

Probe further with . . .

• What function rule can you use to describe the functional relationship between the cost and the length of the call?
• How can you use the function rule to find out how much a 15-minute call costs?
• What is the rate of change in the cost with respect to the length of the call?
• Can you use a table, graph, or formula to find how long he can talk? How are the different representations connected? (For example, “Where is this number in the table shown on the graph?”)
• What is the cost of a call that lasts 1 minute? 2 minutes? 3 minutes? x minutes? Write equations that allow you to answer these questions.
• How many minutes can Scott talk for $3.00? How do you know?
• What inequality can be used to identify the lengths of time that Scott could talk for $3.00?
• Can Scott talk for 5 minutes? 10 minutes? 20 minutes? 30 minutes? How do you know?

Listen for . . .

• Can the student verbalize the pattern of the functional relationship arising from the situation?
• Does the student clearly explain and justify his/her strategies for developing and using equalities and inequalities to answer questions arising from the situation?
• Can the student explain how he/she uses various representations of the functional relationship (table, graph, description, picture, function rule) to answer questions arising from the situation?
• Can the student verbally describe the connections between two or more representations? (For example, "The numbers in this column of the table correspond to the first coordinate of the points on the graph of this function.")

Look for . . .

• Can the student write a function rule to describe the relationship between the quantities described in the situation?
• Does the student write and solve equations or inequalities to answer questions arising from the situations?
• Can the student represent the function describing the relationship between the two quantities using a variety of ways, including a table, graph, description, picture, and function rule?
• Can the student use various representations of the functional relationship (table, graph, verbal description, diagram, function rule) to answer questions arising from a situation?
• Can the student support the reasonableness of his/her answer?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 45
• April 2006, grade 9, item 30
• Spring 2003, grade 10, item 7
• Spring 2004, grade 10, item 49
• Spring 2004, grade 11, item 1

(A.1.D) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

Clarifying Activity

When given a series of buildings made of interlocking cubes that follow a pattern, students determine the relationship between the building number and the number of cubes needed to construct that building. Students represent this relationship in a variety of ways.

To help develop their representations, students use a table like the one shown below. The table describes the building pattern by providing pictorial representations of the buildings for the first three building numbers. Students construct buildings, complete the table, write a function rule that expresses the relationship between the total number of interlocking cubes needed to construct the building and the building number, and graph the data from the relationship by hand or using a graphing calculator. Students also write the equations and inequalities used to answer questions arising from the situation.

An example pattern:

Students construct buildings consisting of 2-cube bases and chimneys of increasing heights.

Assessment Connections

Ask . . .

Start with . . .

• How did you represent the relationship between the building number and the number of cubes in the building?

Probe further with . . .

• What patterns do you observe in your concrete model? In the table? In the graph?
• Describe the dependency using symbols and then words.
• What equation can be used to determine the number of cubes in a building with a chimney that is exactly 12 cubes tall? What inequality can be used to determine the number of cubes in a building with a chimney that is up to 12 cubes tall?
• How many cubes are needed to make the fourth building? How do you know? For the nth building?
• Does the ordered pair (30, 32) belong to the graph? How do you know? Is there another way you could determine this?
• How would the function rule change if each building has a 4-cube base?

Listen for . . .

• Can the student verbalize the pattern of the building number and the number of cubes in the building?
• Does the student recognize that the relationship between the quantities in the situation is a function?
• Can the student verbally describe the connections between two or more representations? (For example, “The column in this table is represented by the axis in this graph or this ordered pair is located here on the graph.”)
• Can the student explain how he/she uses various representations of the functional relationship (table, graph, description, picture, rule) to answer questions arising from the situation? (For example, I see from the picture that a building with a 7-cube chimney has 9 cubes. I see the point (7, 9) on the graph. I see the numbers 7 and 9 in a row in the table, and this also fits the description of a 2-cube base and a 7-cube chimney.)
• Does the student self-monitor and self-correct, noticing that the different representations should result in the same answers to questions?

Look for . . .

• Can the student represent the function describing the relationship between the two quantities using a variety of ways, including concrete models, tables, graphs, verbal descriptions, diagrams, and function rules?
• Can the student use various representations of the functional relationship (table, graph, verbal description, diagram, rule) to answer questions arising from the situation?
• Can the student use symbols to represent unknowns and variables?
• Is the student organizing and recording his/her findings on the table given?
• Does the student appropriately use technology to graphically represent the function and answer questions arising from the situation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 24
• Spring 2004, grade 10, item 20
• April 2006, grade 11, item 54

(A.1.E) Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to interpret and make decisions, predictions, and critical judgments from functional relationships.

Clarifying Activity

Students determine the mailing weight of a box containing multiple copies of the same item after weighing the empty box and the one item.

An example situation and possible situations:

Andrea needs to mail 5,000 pamphlets. Each pamphlet weighs 8 grams. She has two different boxes to choose from. Box 1 has a mass of 20 grams when empty. Box 1 can be mailed for $20 with a maximum mailing weight of 7 kilograms. Box 2 weighs 30 grams when empty. Box 2 can be mailed for $22 with a maximum mailing weight of 10 kilograms. How many of each box should Andrea use to mail the 5,000 pamphlets for as cheap as possible?

Students interpret and make inferences and critical judgments using at least three representations of the relationship between the mailing weight of each box and the number of pamphlets in the box. They may build tables, draw graphs, write algebraic formulas, or give verbal descriptions to help solve the problem.

Assessment Connections

Ask . . .

Start with . . .

• What are the reasons for and against using Box 1 to ship the pamphlets? Box 2?

Probe further with . . .

• What is the weight of Box 1 with 1 pamphlet? 2 pamphlets? 3 pamphlets? 25 pamphlets? n pamphlets? What patterns do you notice? What about for Box 2?
• What is the function rule relating the mailing weight of Box 1 containing pamphlets and the number of pamphlets in the box? How did you figure this out? What about for Box 2?
• How can you use the function rule to determine the weight of Box 1 with 25 pamphlets inside? What is the equation that you solved? What about for Box 2?
• How could you use a graph to determine the number of pamphlets in Box 1 with a mailing weight of 1 kilogram? What is the ordered pair on the graph showing this? How could you use a table to do the same thing? What about for Box 2?

Listen for . . .

• Can the student correctly make predictions based on patterns?
• Can the student explain how he/she uses various representations of the functional relationship (table, graph, description, diagram, function rule) to answer questions arising from the situation?
• Does the student use appropriate vocabulary (for example, relationship, trend, pattern, prediction, table, graph, plot, scatterplot)?
• Does the student recognize that the relationship between the quantities in the situation is a function?
• Can the student justify his/her work?

Look for . . .

• Can the student create and use various representations of the functional relationship (table, graph, description, diagram, function rule) to make predictions and answer questions arising from the situation?
• Does the student use different representations to justify conclusions about the relationship between the data?
• Can the student write and solve equations or inequalities to answer questions arising from the situations?
• Can the student use the graphing calculator to create lists, tables, scatterplots, and so on?
• Can the student evaluate reasonableness of predictions made?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 25
• Spring 2004, grade 9, item 46
• April 2006, grade 9, item 37
• Spring 2004, grade 10, item 51
• April 2006, grade 11, item 59

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(A.2) Foundations for functions. The student uses the properties and attributes of functions.

(A.2.A) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to identify and sketch the general forms of linear (y = x) and quadratic (y = x²) parent functions.

Clarifying Activity

When given a variety of linear and quadratic functions represented in an assortment of ways (function rule, graph, concrete model, description, table), students identify and sketch the parent function for each.

Assessment Connections

Ask . . .

Start with . . .

• What is the parent function for this functional relationship? How do you know?

Probe further with . . .

• What type of function is described by this representation? Linear? Quadratic? How do you know?
• Are the function’s values changing at a constant rate? How do you know?
• What is the parent function of a linear function?
• What is the parent function of a quadratic function?
• What does the sketch of the parent function look like? How does the behavior of the parent function relate to the behavior of this function?

Listen for . . .

• Does the student know what is meant by parent function?
• Can the student describe the parent function of a linear or quadratic function?

Look for . . .

• Can the student sketch the parent functions for linear and quadratic functions?
• Does the student know what is meant by parent function?
• Can the student identify the parent function for linear functions that are represented in a variety of ways (table, graph, verbal description, diagram, function rule)?
• Can the student identify the parent function for quadratic functions that are represented in a variety of ways (table, graph, verbal description, diagram, function rule)?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• April 2006, grade 9, item 22
• Spring 2004, grade 11, item 34
• April 2006, grade 11, item 57

(A.2.B) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete.

Clarifying Activity

Groups of students are given a set of cards that describe situations involving functional relationships. Students take turns selecting a card and writing a function rule to model the situation. Next they state the mathematical domain and range and describe a reasonable domain and range for the situation. Students identify the situation as either continuous or discrete.

An example situation:

Chris rents roller blades for $12 a day. His total cost is a function of the number of days he rents the roller blades. The function rule for this situation is c = 12t, where c is the cost in dollars and t the number of days rented.

An example solution:

The independent variable is the number of days Chris rents the roller blades, and the dependent variable is the total cost. The set of all possible values for an independent variable is the domain. The mathematical domain is the set of possible inputs to the function rule, c = 12t. So the mathematical domain is all real numbers. In the problem situation, the number of days he rents the roller blades can only be a whole number. This means that the domain of the problem situation is the set of whole numbers: 0, 1, 2, 3, 4, . . . .

The set of all possible values for the dependent variable is the range. The mathematical range is the set of possible outputs to the function rule, c = 12t. So the mathematical range is all real numbers. Because the cost increases $12 for each day Chris rents, the range of the problem situation is a subset of the natural numbers: $0, $12, $24, . . . .

This situation is modeled discretely.

Assessment Connections

Ask . . .

Start with . . .

• What is the domain and the range for this function rule? What is the domain and the range for this problem situation?

Probe further with . . .

• What do we mean by the mathematical domain (range)? How do you find it?
• What are the independent and dependent variables?
• What are reasonable domain (range) values for this situation?
• What values for the domain (range) do not make sense? What do you know about the values for this situation?
• How does the mathematical domain compare to the domain of the problem situation?

Listen for . . .

• Can the student identify independent and dependent variables?
• Can the student describe the mathematical domain and range for this functional relationship?
• Can the student describe reasonable domain and range values for this situation?
• Can the student compare the mathematical and situational domains?

Look for . . .

• Can the student write the domain and the range for this situation?
• Can the student write reasonable domain and range values for the problem situation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 10, item 14
• Spring 2004, grade 11, item 27
• April 2006, grade 11, item 12

(A.2.C) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to interpret situations in terms of given graphs or creates situations that fit given graphs.

Clarifying Activity

Groups of students are given a set of graphs. For each graph, the groups create and write a situation in which the value of one variable depends on the value of the other as represented by the graph.

An example graph and situation:

Mary began to walk to school. She stopped at the stoplight and then ran the rest of the way.

Assessment Connections

Ask . . .

Start with . . .

• Tell me about a situation that could be modeled using this graph.

Probe further with . . .

• Tell me about the graph. How are the values changing?
• What type of function is the graph representing?
• Where is the graph increasing? Decreasing? Neither increasing nor decreasing?
• How do the various parts of the graph of the function relate to your situation?
• How can you use this information to help you describe a situation that the graph could represent?

Listen for . . .

• Does the explanation of the situation match the graph?
• Can the student verbally describe the connections between the graph and the situation that they developed?

Look for . . .

• Can the student interpret a given graph?
• Can the student create descriptions of situations that fit given graphs?
• Can the student study graphs and answer questions based on the information given in the graphs?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 47
• April 2006, grade 9, item 51
• Spring 2003, grade 11, item 37
• Spring 2004, grade 11, item 47

(A.2.D) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

Clarifying Activity

In groups, students investigate the relationship between two variables by collecting, organizing, representing, modeling, and interpreting two variable data sets in a problem situation.

An example of a problem situation and analysis:

Students investigate the relationship between shoe size and height.

Students plot the data on graph paper and draw a line of best fit.

Students input the data into a graphing calculator to determine the function that describes the line of best fit. Students predict the shoe size of a person whose height is 49 inches, 66 inches, and so on.

Students should realize that shoe sizes for men and women differ, as the following chart illustrates. A simple way to include data for both genders is to convert to one gender's shoe size, as was done in the previous table. According to the chart, women's shoe sizes are roughly two sizes larger than men's. Students, however, could convert men's and women's American sizes to the common metric sizes.

Assessment Connections

Ask . . .

Start with . . .

• Tell me about the relationship between a person’s shoe size and his/her height.

Probe further with . . .

• What do you consider to determine a reasonable interval of values and a scale for each axis to construct a scatterplot of the data?
• Based on your scatterplot, would you say that shoe size and height are positively correlated, negatively correlated, or not correlated?
• What might you expect the shoe size to be for a person who is 66 inches tall? How did you determine this? How tall would you expect a person who wears size-10 shoes to be? How did you determine this?
• Are shoe sizes for men and women consistent? How might you measure shoe size for men and women using a more standard unit of measure? (You might convert men’s sizes to women’s by adding two sizes to men’s or using metric measurements.)
• Are shoe size and height functionally related? How do you know?

Listen for . . .

• Does the student use appropriate vocabulary to describe the data including units?
• Can the student verbally describe the numerical relationship between shoe size and height? (For example, "As one increases, the other increases.")
• Can the student verbally describe the data? (For example, "These numbers are the shoe sizes recorded for students, and the other numbers are the heights of the students.")
• Can the student verbally describe the connections between two or more representations? (For example, "The column in this table is represented by the axis in this graph, or this ordered pair is located here on the graph.")

Look for . . .

• Can the student collect and organize the collected data (relate shoe size and height to the same person)?
• Can the student use a graphing calculator to create lists, tables, and scatterplots and to determine a line of best fit and so on?
• Can the student identify the correlation between the data sets as positive, negative, or no correlation (around 0) for data approximating linear situations?
• Can the student predict a person’s shoe size given his/her height?
• Can the student predict a person’s height given his/her shoe size?
• Can the student model the relationship between shoe size and height using a linear function and use that model to make predictions?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• April 2006, grade 9, item 33
• Spring 2004, grade 10, item 12
• Spring 2003, grade 11, item 60

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(A.3) Foundations for functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.

(A.3.A) Foundations for functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to use symbols to represent unknowns and variables.

Clarifying Activity

When given a situation, students use symbols to represent the quantities in the situation and indicate what the symbols represent.

An example situation:

A plumbing company charges a flat fee of $35 plus $42 per hour for a residential service call. How can you determine algebraically how much the company will charge for a service call if you know how much time the plumber spent at the residence?

Assessment Connections

Ask . . .

Start with . . .

• How did you use symbols to indicate how much the plumbing company will charge for a service call?

Probe further with . . .

• What are the unknown quantities in this situation?
• What values change? What remains constant in the situation?
• What variables could be used to represent the unknown quantities in this situation? Could you use different variables to represent the unknown quantities in the situation?
• Why is it important to describe what the symbols represent when you use them?

Listen for . . .

• Does the student use appropriate language when describing how he/she utilizes symbols to represent unknown values and variables?
• Does the student clearly explain what the symbols represent?

Look for . . .

• Can the student use mathematical symbols to represent variables to describe relationships?
• Does the student write the meaning of the symbols used to represent the variables?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 45
• April 2006, grade 9, item 6
• Spring 2004, grade 10, item 32
• April 2006, grade 10, item 36
• April 2006, grade 11, item 56

(A.3.B) Foundations for functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to look for patterns and represent generalizations algebraically.

Clarifying Activity

Students look for patterns and express generalizations in a situation.

An example situation:

A plumbing company charges a flat fee of $35 plus $42 per hour for a residential service call.

Students set up a table using the variables h and c to model the relationship between the hours worked and the total charge.

Assessment Connections

Ask . . .

Start with . . .

• What algebraic expression describes the relationship between the number of hours worked and the total charge of the plumbing company?

Probe further with . . .

• What patterns do you observe? Describe the pattern in words and then in symbols.
• What generalizations can you make?
• What is the function rule relating the number of hours worked and the charge?
• How does the cost change as the number of hours worked changes?
• If the charge is $497.00, how many hours did the plumber work? How do you know?

Listen for . . .

• Does the student discuss how the pattern continues?
• Can the student verbalize the rule and justify his/her thinking?
• Can the student correctly make predictions based on the rule?
• Does the student use appropriate vocabulary?

Look for . . .

• Does the student organize information? Can the student use a systematic process to develop the rule?
• Does the student use symbols to describe the pattern or rule?
• Does the student make a table?
• Does the student recognize flaws or errors in the rule and adjust it accordingly?
• Does the student use the calculator appropriately?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 43
• Spring 2003, grade 11, item 54

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(A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

(A.4.A) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations.

Clarifying Activity

Students solve problem situations by finding specific function values, simplifying polynomial expressions, transforming equations, factoring, and solving equations.

An example situation:

A bowling ball is dropped from the top of a 144-foot building.

The function that specifies the height of the bowling ball in terms of the time is h(t) = 144 - 16t², where h(t) is feet above the ground and t is seconds.

Students answer questions such as:

• What is the height of the bowling ball after 2 seconds? (Find a specific function value.)
• How much time has passed when the bowling ball is 48 feet above the ground? (Transform and solve the equation 144 - 16t² = 48.)
• When will the bowling ball hit the ground? [Students note that the height of the bowling ball when it hits the ground is 0. They set up and solve the equation 144 - 16t² = 0. That is, (12 + 4t)(12 - 4t) = 0.]

Assessment Connections

Ask . . .

Start with . . .

• How did you find the solution to each question?

Probe further with . . .

• What equations can be solved to determine the solutions?
• How can you transform and solve the equation 144 - 16t² = 48?
• What is the height of the object as it hits the ground? How can you use this to determine when it will hit the ground?
• Does this solution make sense?
• What is the parent function for this function? Is it linear? Is it quadratic?

Listen for . . .

• Can the student explain his/her thinking?
• Does the student use appropriate vocabulary when describing how he/she solved the problem?
• Does the student check for reasonableness?

Look for . . .

• Can the student find specific function values?
• Can the student simplify polynomial expressions?
• Can the student transform and solve equations?
• Can the student start with a function and determine an equation based on the function to solve a problem situation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 16
• April 2006, grade 9, item 16
• Spring 2003, grade 10, item 31
• Spring 2003, grade 11, item 22

(A.4.B) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to use the commutative, associative, and distributive properties to simplify algebraic expressions.

Clarifying Activity

Students use the commutative, associative, and distributive properties to determine equivalent expressions for the area of the following rectangle in terms of x.

[image to support this activity coming soon]

Assessment Connections

Ask . . .

Start with . . .

• Express the area of the rectangle of width x + 5 and height x + 3.

Probe further with . . .

• How do you determine the area of any rectangle?
• How do you expand and simplify this expression?
• How do you relate the concrete model for the area of the rectangle to each step in the multiplication process?

Listen for . . .

• Does the student use appropriate vocabulary when explaining how to simplify the algebraic expression?
• Can the student name properties that are used to simplify algebraic expressions?
• Can the student verbally describe the connections between the symbolic manipulation to simplify the expression and the concrete/visual model of the area of the rectangle?

Look for . . .

• Can the student apply the commutative, associative, and distributive properties to simplify algebraic expressions?
• Does the student simplify the expressions appropriately?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 20
• Spring 2004, grade 10, item 25
• April 2003, grade 10, item 16
• Spring 2003, grade 11, item 25

(A.4.C) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

Clarifying Activity

Students are given a problem situation and a table of input-output values from the functional relationship modeled by the situation. They determine an algebraic rule from the table. They discuss an alternate way to express the functional relationship using function notation. Students make connections between y = and f(x) = notation. The two notations mean the exact same thing. Function notation, however, is more flexible and provides more information. Students can relate function notation to a “function machine.” They then use the relationship expressed using function notation to find values for the specific input values used in the table. Students determine that the output values are identical; they compare the output values to the output column of the table.

An example situation:

A skateboarding park charges $10 a month for membership fees and $5 a day to skate. A table of input-output values for the functional relationship is:

Determine an algebraic rule relating the total charge for a month to the number of days skated. Use x for the number of days skated in one month and y for the total cost to skate for the month.

Another way to express the relationship found in the situation is to use function notation: f(x) = 10 + 5x, where x is the number of days skated in one month and f(x) is the total cost to skate for that month. Compare the relationship expressed in function notation to the algebraic rule that you determined from the table.

Function notation helps show that the total cost is a function of the number of days. Use the input values from the table to determine the total cost for the month using the relationship expressed in function notation. Compare these results to the output values in the table.

Use the input-output table and the relationship expressed in function notation to determine how many days per month you could skate for a total cost of $40.

Assessment Connections

Ask . . .

Start with . . .

• How can you determine an algebraic rule using the table of input-output values?

Probe further with . . .

• How does the notation f(x) help show that cost depends on the number of days in this situation?
• What information does function notation provide that equation notation does not?
• What are the similarities and differences between function and equation notation?
• Express the function in equation and function notation using variables other than y, x, and f(x). Explain what each of your variables represents.

Listen for . . .

• Can the student explain his/her thinking?
• Does the student use appropriate vocabulary when describing how he/she solved the problem?
• How effectively does the student describe the similarities and differences between equation and function notations?

Look for . . .

• Does the student connect equation notation with function notation, such as y = 10 + 5x and f(x) = 10 + 5x?
• Can the student express the function in equation and function notation using variables other than y, x, and f(x)?

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(A.5) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations.

(A.5.A) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to determine whether or not given situations can be represented by linear functions.

Clarifying Activity

Students work in pairs to determine whether or not the following data exhibit a constant rate of change and can be represented by a linear function.

A possible student response:

Assessment Connections

Ask . . .

Start with . . .

• Can this data be represented by a linear function? How do you know?

Probe further with . . .

• Represent the data in the table using a graph. What type of relationship do you find?
• What is the rate of change between (–1, –1) and (0, 0)? (0, 0) and (1, –1)? (1, –1) and (2, –4)? Is the rate of change constant? How do you know?
• What is the rate of change between (–2, 3) and (–1, 1)? (–1, 1) and (0, –1)? (0, –1) and (1, –3)? Is the rate of change constant? How do you know?
• What rule can be used to represent the linear function?

Listen for . . .

• Can the student determine whether or not a relationship is a linear function? How easily?
• Does the student clearly and accurately describe what it means for a relationship between two quantities to be a linear function?
• Does the student express an understanding that linear functions can be represented in a variety of ways?

Look for . . .

• Does the student demonstrate an understanding that for linear functions the rate of change between data points is constant?
• Can the student use data sets to determine whether or not a relationship is a linear function?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• April 2006, grade 9, item 15
• April 2006, grade 10, item 55

(A.5.B) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to determine the domain and range for linear functions in given situations.

Clarifying Activity

Students discuss problem situations for linear functions and determine reasonable domain and range values.

An example problem situation:

Desmond does yard work on the weekends to earn spending money. He charges $3.50 an hour. He wants to buy an MP3 player that costs $130. How many hours of yard work must he do to pay for the MP3 player?

Assessment Connections

Ask . . .

Start with . . .

• What are reasonable input and output values for the function described in this problem situation?

Probe further with . . .

• What would be reasonable number choices to use in the Number of Hours Worked column?
• What values would you expect in the Money Earned column?
• What functional relationship is described in this situation?
• What values for the domain do not make sense? For the range?
• What is the mathematical domain and range? The situational domain and range?
• How do the mathematical and situational domains compare?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Can the student describe the mathematical domain and range for the linear function?
• Can the student describe reasonable domain and range values for this situation?
• Can the student compare the mathematical and situational domains? Ranges?

Look for . . .

• Can the student identify the mathematical domains and ranges and determine reasonable domain and range values for given situations?
• Does the student self-monitor and self-correct?

(A.5.C) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

Clarifying Activity

Students are given a problem situation that can be represented by a linear function. They represent the function in a variety of ways, using one representation to make another. Students create a table, graph, algebraic rule, and verbal description from the problem situation and make connections among these representations.

An example situation:

Andrea paints houses during her summer vacations. This year, she charged $6.00 per hour to paint.

Assessment Connections

Ask . . .

Start with . . .

• How will you use words, symbols, a table, and a graph to represent the relationship between the amount of money Andrea earned and the number of hours she painted?

Probe further with . . .

• How much did Andrea earn if she painted 1 hour? 2 hours? 3 hours? 10 hours? n hours? How do you know?
• Describe in words the dependency of the number of hours Andrea painted on the amount of money she earned.
• Describe the dependency using symbols.
• What is the function rule relating the number of hours painted and the amount earned?
• Is the rate of change constant in the amount earned with respect to the number of hours painted? How do you know?
• How would each of your representations change if Andrea increased her hourly rate by $0.50?

Listen for . . .

• Can the student verbalize the pattern between the number of hours Andrea painted and the amount of money she earned?
• Does the student express an understanding that linear functions can be represented in a variety of ways?
• Can the student verbally describe the connections between two or more representations? (For example, "This column in the table is represented by this axis on the graph or this ordered pair is located here on the graph.")
• Can the student explain how various representations of the functional relationship (table, graph, description, rule) change when Andrea alters her hourly rate?

Look for . . .

• Can the student represent the function describing the relationship between the two quantities using a variety of ways, including a table, graph, description, picture, and rule?
• Can the student determine the effect a change in hourly rate has on each representation?
• Does the student check his/her work for reasonableness?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 41
• April 2006, grade 9, item 15
• Spring 2003, grade 10, item 49
• April 2006, grade 10, item 55
• Spring 2003, grade 11, item 16

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(A.6) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations.

(A.6.A) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to develop the concept of slope as rate of change and determine slopes from graphs, tables, and algebraic representations.

Clarifying Activity

Students walk up a portable ramp with a side view and mark where they make their first step, second step, and so forth.

They then measure and record the horizontal and vertical distances of each step and express the steepness of the ramp as a ratio of the measure of vertical change to the measure of horizontal change.

Students record their measurements in a table, use the measurements to create a graph, write a function to describe the linear relationship of their measurements, and determine the slope using each representation (table, graph, algebraic representation).

Assessment Connections

Ask . . .

Start with . . .

• How can you find the slope (steepness) of this ramp?

Probe further with . . .

• Are the total horizontal and vertical distances traveled linearly related? How do you know?
• What function rule represents the linear relationship between the total horizontal and vertical distance traveled? How did you determine this rule?
• If the steepness of the ramp is a ratio of the measure of vertical change to the measure of horizontal change, how does the steepness of your ramp relate to the slope of your graph?
• How did you find the slope using the table? The graph? The function rule? What was it?
• What did you notice about the slope for each representation?

Listen for . . .

• Does the student use appropriate language when describing the slope for the different representations?
• Does the student express an understanding that linear functions can be represented in a variety of ways?
• Does the student express an understanding that the rate of change (slope) for linear functions is constant?
• Does the student clearly explain that the determination of slope for different representations of the same linear function should yield the same result?

Look for . . .

• Does the student organize and record his/her findings in a table?
• Can the student find an algebraic representation for the linear function?
• Can the student graph the linear function? Does he/she plot points or use the function rule to graph the function?
• Did the student use an appropriate strategy to determine the slope for each representation?
• Can the student find the slope of a linear function from a table, graph, and algebraic representation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 39
• Spring 2004, grade 9, item 19
• April 2006, grade 9, item 17
• Spring 2003, grade 10, item 46
• April 2006, grade 10, item 40

(A.6.B) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.

Clarifying Activity

Students are given a problem situation, a table with data that models the situation, a graph of the data, and/or a function rule describing the linear function related to the situation. They interpret the meaning of the slope and intercepts.

An example situation:

The table, graph, and function below indicate the amount of money Pierce earned working.

Assessment Connections

Ask . . .

Start with . . .

• What do the different representations tell you about the amount of money Pierce earned? How do you know this?

Probe further with . . .

• Interpret the meaning of the slope in terms of this situation. How did you determine this? Is there another way you could have found the slope?
• Interpret the meaning of the y-intercept in terms of this situation. Justify your reasoning. Is there another way you could have found the y-intercept?
• Interpret the meaning of the x-intercept in terms of this situation. Justify your reasoning. Is there another way you could have found the x-intercept?

Listen for . . .

• Does the student accurately describe the problem situation?
• Does the student clearly explain the meanings of slope and intercepts?

Look for . . .

• Can the student identify the slope and intercepts using data, rules, or graphs?
• Can the student interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 34
• April 2006, grade 9, item 11
• April 2006, grade 10, item 12

(A.6.C) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b.

Clarifying Activity

Students use a graphing calculator to graph the line y = x. Then, for each function given in the table below, students predict what they think the new graph will look like. Next, students check their prediction with the graphing calculator, sketch the graph, and describe the change between y = x and the new graph. After investigating several functions, students write a general statement about the effects of changing . . .

1. m on the graph of y = mx + b and
2. b on the graph of y = mx + b.

Assessment Connections

Ask . . .

Start with . . .

• How does changing m affect the graph of y = mx + b?
• How does changing b affect the graph of y = mx + b?

Probe further with . . .

• What does m represent? What does b represent?
• What is the same in all graphs of y = mx + b? What is different?
• What is the slope for y = x? What is the y-intercept? How do you know?
• What happens when m increases to integer values greater than 1? What happens when m decreases to positive fractions less than 1?
• What happens when m is negative?
• What happens when b is a positive integer?
• What happens when b is negative?
• How do the lines corresponding to y = 2x and y = –2x compare?
• How do the lines corresponding to y = x and y = x – 2 compare?

Listen for . . .

• Does the student express in words patterns that he/she observes to generalize the effect of changing m and b?
• Can the student describe the effects of changes in m and b on the graph of y = mx + b?
• Does the student use appropriate vocabulary, such as slope and y-intercept, when describing the effects of changing m and b on the graph of y = mx + b?
• Does the student explain his/her reasoning about how changes in m and b affect the graph?

Look for . . .

• What strategy does the student use to investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b?
• Does the student use technology appropriately to investigate the effects of changes in m and b on the graph of y = mx + b?
• Can the student use a graphing calculator to graph y = mx + b?
• Is the student organizing and recording his/her findings in a way that supports pattern recognition?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 18
• April 2006, grade 9, item 12
• Spring 2003, grade 10, item 24
• Spring 2004, grade 10, item 8
• Spring 2004, grade 11, item 17

(A.6.D) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.

Clarifying Activity

When given characteristics such as two points, a point and slope, or a slope and y-intercept, students find the equation of a line in y = mx + b form and graph the line.

Example characteristics:

• through the points (2, 4) and (-1, -5)
• through the point (2, 4) and a slope of 4
• with a slope of -2 and a y-intercept of 5

Assessment Connections

Ask . . .

Start with . . .

• What lines have these characteristics? Use a graph and an algebraic rule to represent the lines.

Probe further with . . .

• What is the graph of the line through the points (2, 4) and
  (–1, –5)? How does the graph compare with the graph of y = x?
• What is the graph of the line that passes through the points
  (2, 4) and has a slope of 4? How does this graph compare with the graph of y = x? Is the point (0, 0) on this line? How do you know?
• How can you use the points (2, 4) and (–1, –5) to determine the equation of a line that contains these points? How do you know? What is the line’s slope? What is its y-intercept? How do you know?
• What is the equation of the line through the points (2, 4) and has a slope of 4? How do you know? What is its y-intercept? How do you know?
• What is the graph of the line with a slope of –2 and a y-intercept of 5?
• What is the equation of the line with a slope of –2 and a y-intercept of 5? How do you know? What is its x-intercept?
• How does the graph compare with the graph of y = x?

Listen for . . .

• Does the student use appropriate language when describing the slope, intercept, equation, and points on a line?

Look for . . .

• Does the student use appropriate strategies to determine the equation of a line given some of the line’s characteristics?
• Can the student identify points on a line?
• Can the student find the slope given two points?
• Can the student graph a line given two points, the slope and a point, and the slope and the y-intercept?
• Can the student find the equation of a line given two points, the slope and a point, and the slope and the y-intercept?
• Which characteristics are easiest for the student to use to determine the equation of a line?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 44
• Spring 2004, grade 9, item 10
• Spring 2004, grade 10, item 44
• April 2006, grade 11, item 4

(A.6.E) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations.

Clarifying Activity

Students find the x- and y-intercepts from the following graph.

Graph:

Students find the zeros using the following representations

Table:

Algebraic rule:

y = x + 3

Assessment Connections

Ask . . .

Start with . . .

• What are the x- and y-intercepts for the function represented in the graph?
• What are the zeros for the function represented in the table and the algebraic rule?

Probe further with . . .

• When given a graph, how do you find the x-intercept? y-intercept?
• How do you find the zeros of the function using a graph?
• How do you find the zeros of the function using the table?
• Compare the y-intercept of the graph to the zeros of the function found in the table and the algebraic rule.
• When given a function rule, how do you find the y-intercept?
• What is the y-intercept of the graph of y = x + 3? The zero?

Listen for . . .

• Can the student clearly explain how to determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic rules?
• Can the student accurately describe the connection between the x-intercepts of a graph and the zeros of the function represented by the graph?

Look for . . .

• Can the student determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 42
• April 2006, grade 9, item 19
• Spring 2003, grade 10, item 26
• Spring 2003, grade 11, item 53
• April 2006, grade 11, item 49

(A.6.F) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to interpret and predict the effects of changing slope and y-intercept in applied situations.

Clarifying Activity

Students graph a linear function represented by a problem situation. They determine the slope and the y-intercept and give their meanings in this situation. The students then connect changes in problem situations to changes in the graph, slope, and y-intercept.

An example situation:

A caterer charges a $25 set-up fee plus $4 per person to serve a lunch meal. The caterer wants to increase her prices. She is considering changing either the set-up fee to $30 or the price per person to $5.

Assessment Connections

Ask . . .

Start with . . .

• Which change should the caterer make to produce the biggest increase in her earnings?

Probe further with . . .

• If the caterer originally charges a $25 set-up fee plus $4 per person, what is the functional relationship between the charges and the number of people?
• What is the graph of the linear function representing the catering charges and the number of people if she charges a $25 set-up fee plus $4 per person?
• What are the slope and the y-intercept? What are their meanings in this situation? The graph?
• If the caterer changes the price per person to $5 but leaves the set-up fee at $25, how does this affect the functional relationship between the charges and the number of people? The graph? The slope? The y-intercept?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Does the student use appropriate vocabulary when describing how changes in the situation affect the corresponding linear function?
• Does the student explain his/her reasoning when commenting about the effect of the changes?

Look for . . .

• Can the student find a linear function that describes the situation?
• Can the student graph a linear function that describes the situation?
• Can the student connect changes in problem situations to changes in the graph, slope, and y-intercept?

(A.6.G) Linear functions. The student understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. The student is expected to relate direct variation to linear functions and solve problems involving proportional change.

Clarifying Activity

Students relate direct variation to linear functions in problem situations. Students graph the function and solve problems related to the situation.

An example situation:

The amount of chlorine needed in a pool increases proportionally (varies directly) as the amount of water increases. The amount of chlorine needed for 550 gallons of water is 5 units.

Assessment Connections

Ask . . .

Start with . . .

• If 5 units of chlorine are needed for 550 gallons of water, what function represents the amount of chlorine needed for x gallons of water?

Probe further with . . .

• What does it mean for the amount of chlorine needed to increase proportionally as the amount of water increases?
• How would you graph the function that describes the relationship between the amount of chlorine needed for a pool and the amount of water in the pool? How much chlorine is needed for a pool with no water? With 110 gallons of water? With 1 gallon of water? With 850 gallons of water? How do you know?
• How many gallons of water are required if you have 4 units of chlorine?
• Can you use your graph to find the amount of chlorine needed for 850 gallons of water?
• Find a linear equation that describes the amount of chlorine needed for x gallons of water, and use this equation to determine how much chlorine is needed for 850 gallons of water.

Listen for . . .

• Is the student using an effective strategy to predict the amount of chlorine needed for a certain amount of water?
• Is the student explaining his/her reasoning?
• Does the student express an understanding of the problem situation?

Look for . . .

• Can the student relate direct variation to linear functions?
• Can the student solve problems involving proportional change by using the linear function describing the situation?
• Can the student use a variety of representations and strategies to solve problems involving proportional relationships?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 2
• Spring 2004, grade 11, item 45
• April 2006, grade 11, item 28

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(A.7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

(A.7.A) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to analyze situations involving linear functions and formulate linear equations or inequalities to solve problems.

Clarifying Activity

Students are given a problem situation that can be described using a linear function. They answer questions arising from the situation by formulating and solving equations and inequalities.

An example situation:

Sara wants to rent a car for a week and pay no more than $150. How many miles can she drive if the car rental costs $49 a week plus 13 cents per mile?

Assessment Connections

Ask . . .

Start with . . .

• How can you develop an equation or inequality to solve this problem?

Probe further with . . .

• What function rule can you use to describe the relationship between the cost in dollars for the car rental and the number of miles driven? How did you determine the rule?
• How can you use your knowledge about the slope and the y-intercept to write a rule for the linear relationship?
• How can you use the function rule to find the total cost if Sara drives 100 miles?
• Write an inequality to identify how many miles can Sara drive for $150.
• Can Sara drive 100 miles? 200 miles? 500 miles? 1,000 miles? How do you know?
• If Sara drives 150 miles each day, will the total rental cost more than $150?
• What other methods can you use to determine how many miles Sara can drive for $150? How are the different methods connected?

Listen for . . .

• Does the student identify the relationship as a linear function?
• Can the student identify the slope and y-intercept of the linear function from the problem situation?
• Does the student clearly explain and justify his/her strategies for developing and using equalities and inequalities to answer questions arising from the situation?
• Does the student use appropriate vocabulary during explanations?

Look for . . .

• Can the student check and support the reasonableness of his/her answer?
• Does the student formulate equations or inequalities to answer questions arising from the situations?
• Can the student solve an equation or inequality to answer questions arising from the situation?
• Does the student correctly manipulate the equation or inequality to answer questions arising from the situations?
• Can the student write a function rule to describe the relationship between the quantities described in the situation?
• Does the student use technology to graph the function?
• Does the student use technology to answer questions related to the situation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 50
• Spring 2004, grade 9, item 7
• April 2006, grade 9, item 38
• Spring 2003, grade 10, item 4
• April 2006, grade 10, item 50

(A.7.B) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities.

Clarifying Activities

Students are given a problem situation that can be described using a linear function. They answer a question arising from the situation by formulating and solving an inequality using concrete models or graphs.

[One method students use to find the solution to the inequality f(x) < C is by graphing the functions y = f(x) and y = C, locating their point of intersection, and then identifying which values of x in y = f(x) give y < C.]

An example situation:

Sara wants to rent a car for a week and pay no more than $150. How many miles can she drive if the car rental costs $49 a week plus 13 cents per mile?

Assessment Connections

Ask . . .

Start with . . .

• How can you develop inequality to solve this problem?
• How can you use a graph to answer the question?

Probe further with . . .

• What function rule can you use to describe the relationship between the cost in dollars for the car rental and the number of miles driven? How did you figure this out?
• Write an inequality to identify how many miles can Sara drive for $150.
• How can you represent the inequality on a graph? How is the solution represented on the graph?
• Is your solution reasonable?
• What other methods can you use to determine how many miles Sara can drive for $150?
• How can you solve this using algebra tiles?

Listen for . . .

• Does the student clearly explain and justify his/her strategies for developing and using equalities and inequalities to answer questions arising from the situation?
• Does the student describe an accurate understanding of the problem situation?
• Can the student explain how he/she uses various representations of the functional relationship (table, graph, description, picture, function rule) to answer questions arising from the situation?

Look for . . .

• Does the student formulate and solve equations or inequalities to answer questions arising from the situations?
• Does the student investigate a variety of methods for solving linear equations and inequalities, including concrete models, graphs, and the properties of equality?
• Can the student represent the inequality using a graph?
• Does the student use technology to graph the function?
• Does the student interpret and determine the reasonableness of solutions to linear equations and inequalities?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 8
• Spring 2004, grade 9, item 36
• April 2006, grade 9, item 10
• April 2006, grade 10, item 39
• April 2006, grade 11, item 21

(A.7.C) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to interpret and determine the reasonableness of solutions to linear equations and inequalities.

Clarifying Activity

Students discuss when particular solutions for linear equations and inequalities are NOT reasonable.

Assessment Connections

Ask . . .

Start with . . .

• When are solutions to problem situations described by linear equations not reasonable?

Probe further with . . .

• When are fractional answers not reasonable? (Answers referring to the number of people, coins, or other such discrete objects do not make sense.)
• When are negative number answers not reasonable? (Negative numbers representing distance, hours worked, elapsed time, and so on are unreasonable.)
• When should you check for reasonableness of solutions? (This analysis should be a part of every problem situation.)

Listen for . . .

• Does the student clearly describe when answers to problems are not reasonable?

Look for . . .

• Does the student know what to look for to interpret and determine the reasonableness of solutions to linear equations and inequalities?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 5
• Spring 2004, grade 10, item 46
• April 2006, grade 10, item 20
• Spring 2003, grade 11, item 23
• April 2006, grade 11, item 55

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(A.8) Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

(A.8.A) Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to analyze situations and formulate systems of linear equations in two unknowns to solve problems.

Clarifying Activity

Students are given a problem situation that can be represented by a linear system of equations. They write a system of equations to represent problem situations.

An example situation:

At Marti's Mexican Restaurant, 2 tacos and 3 enchiladas cost $7.17. If you buy 4 tacos and 5 enchiladas, the cost is $12.69.

Assessment Connections

Ask . . .

Start with . . .

• What system of linear equations could you use to determine the cost of 1 taco?

Probe further with . . .

• If the cost of 1 taco is t and 1 enchilada is e, what expression can you use to denote the cost of 2 tacos and 3 enchiladas?
• If the cost of 1 taco is t and 1 enchilada is e, what expression can you use to denote the cost of 4 tacos and 5 enchiladas?
• What system of equations did you set up to find the cost of a taco?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Does the student accurately describe the connection between the taco/enchilada combination and the total price?
• Does the student clearly explain and justify his/her strategies for developing and using the linear system of equalities?

Look for . . .

• Can the student accurately determine each linear equation from the problem situation?
• Can the student analyze situations and formulate systems of linear equations in two unknowns to solve problems?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 39
• April 2006, grade 9, item 32
• Spring 2004, grade 10, item 54
• Spring 2004, grade 11, item 53
• April 2006, grade 11, item 1

(A.8.B) Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to solve systems of linear equations using concrete models, graphs, tables, and algebraic methods.

Clarifying Activity

Students write a system of equations to represent a problem situation (see A.8.A). They solve systems of linear equations using concrete models, graphs, tables, and algebraic methods.

An example situation:

At Marti's Mexican Restaurant, 2 tacos and 3 enchiladas cost $7.17. If you buy 4 tacos and 5 enchiladas, the cost is $12.69. What is the cost of 1 taco?

Assessment Connections

Ask . . .

Start with . . .

• How can you find the cost of 1 taco?

Probe further with . . .

• How can you formulate a system of linear equations to find the value of 1 taco if 2 tacos and 3 enchiladas cost $7.17 while 4 tacos and 5 enchiladas cost $12.69?
• What methods would you use to solve this system of equations?
• How can you model and solve the situation using algebra blocks?
• How can you transform the system of equations to express the equations into slope-intercept form?
• How can you use a graph to solve the system? Tables?
• How can you use the tracing features of a graphing calculator to solve the system of equations?
• How can you solve the system of equations algebraically?

Listen for . . .

• Can the student analyze problem situations involving linear systems?
• Does the student clearly explain and justify his/her strategies for developing and using the linear system of equalities to answer questions arising from the situation?
• Does the student use appropriate vocabulary during explanations?
• Can the student explain how he/she uses various methods to solve the linear system of equations?
• Does the student self-monitor and self-correct, noticing that the different solution methods should result in the same answers to questions?
• Can the student verbally describe the connections between two or more solution methods?

Look for . . .

• Can the student formulate systems of linear equations from problem situations?
• Does the student investigate a variety of methods for solving linear systems of equations, including concrete models, graphs, and the properties of equality?
• Can the student solve systems of linear equations using concrete models, graphs, tables, and algebraic methods?
• Does the student organize and record the pattern arising from the situation in a table?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 10, item 36
• Spring 2004, grade 10, item 24
• April 2006, grade 10, item 47
• Spring 2004, grade 11, item 29
• April 2006, grade 11, item 32

(A.8.C) Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to interpret and determine the reasonableness of solutions to systems of linear equations.

Clarifying Activity

Students interpret and discuss the solutions to this system of equations and determine if their solutions are reasonable. Students use their solutions from A.8.A and A.8.B.

An example situation:

At Marti's Mexican Restaurant, 2 tacos and 3 enchiladas cost $7.17. If you buy 4 tacos and 5 enchiladas, the cost is $12.69. What is the cost of 1 taco?

Assessment Connections

Ask . . .

Start with . . .

• What are the solutions to the system of equations that describe this problem situation?

Probe further with . . .

• What do the solutions (the numerical values) mean in this context of the situation? Explain using one representation utilized to find the solutions.
• What is the cost of 1 enchilada? 1 taco? Do these costs seem right? How do you know?
• How can you use the problem situation or the system of equations to check the reasonableness of your answers?
• How do you know when it is important to check your answers for reasonableness?

Listen for . . .

• Does the student accurately interpret the meaning of the solutions within the problem situation?
• Does the student accurately describe instances when solutions should be checked for reasonableness?

Look for . . .

• Does the student know what to look for to interpret and determine the reasonableness of solutions to linear equations and inequalities?
• Can the student accurately connect the cost of 1 enchilada and 1 taco back to the system of equations?

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(A.9) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions.

(A.9.A) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to determine the domain and range for quadratic functions in given situations.

Clarifying Activity

Students discuss problem situations that can be described by quadratic functions. They identify the domain and the range.

An example situation:

Dale is raising a calf for the local livestock show. The calf now weighs 200 pounds. The current price for calves is $1.50 per pound. The calf is gaining 5 pounds a week, but the price per pound is dropping 3 cents a week.

Assessment Connections

Ask . . .

Start with . . .

• What is the domain and range for the functional relationship described in this problem situation?

Probe further with . . .

• What values for the domain do not make sense? The range?
• What would be reasonable numerical values for the Number of Weeks column (domain)?
• What values would you expect in the Selling Price column (range)?
• Write the mathematical domain and range for this situation.
• What are reasonable domain and range values for this situation?
• How do the mathematical and situation domains compare?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Can the student describe the mathematical domain and range for the quadratic function?
• Can the student describe reasonable domain and range values for this situation?
• Does the student explain his/her reasoning?
• Can the student compare the mathematical and situation domains?

Look for . . .

• Does the student use an efficient strategy to determine the range and domain of the function? What strategy does he/she use?
• Does the student build a table?
• Can the student write the mathematical domain and range for this situation?
• Can the student identify the mathematical domains and ranges and determine reasonable domain and range values for given situations?

(A.9.B) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to investigate, describe, and predict the effects of changes in a on the graph of y = ax² + c.

Clarifying Activity

Students use a graphing calculator to graph the quadratic y = x². Then, for several functions in the y = ax² form, students predict what the new graph will look like, check their prediction with the graphing calculator, sketch the graph, and describe the change. After investigating several functions in y = ax² form, students write a general statement about the effects of changing a: “The effects of changing a on the graph of y = ax² + c are . . .”

Assessment Connections

Ask . . .

Start with . . .

• What are the effects of changing a on the graph of y = ax²?

Probe further with . . .

• What is the value of a in y = ax² + c if we graph variations of y = ax²?
• What is similiar and different about the various graphs of y = ax²?
• What is the x-intercept for y = x²? How do you know?
• What happens when a increases to values greater than 1? What happens when a decreases to positive fractions less than 1? What happens when a is negative?
• How do the graphs corresponding to y = 4x² and y = -4x² compare?
• How do the graphs corresponding to y = x² and y = 3x² compare?
• What general statements can you make about the effects of changing a?

Listen for . . .

• Can the student describe the effects of changes in the parameters of quadratic functions?
• Does the student verbalize patterns that he/she observes to generalize the effect of changing a?
• Can the student clearly describe the effects of changes in a on the graph of y = ax + c?
• Does the student explain his/her reasoning about how changes in a affect the graph?

Look for . . .

• What strategy does the student use to investigate, describe, and predict the effects of changes in a on the graph of y = ax² + c?
• Can the student predict the effects of changes in a on the graph of y = ax² + c?
• Does the student use technology appropriately to investigate the effects of changes in a on the graph of y = ax² + c?
• Does the student organize and record his/her findings in a way that supports pattern recognition?
• Does the student indicate an understanding that the graphs of quadratic functions are affected by the parameters of the function?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 10, item 17
• April 2006, grade 10, item 37
• April 2006, grade 11, item 46

(A.9.C) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to investigate, describe, and predict the effects of changes in c on the graph of y = ax² + c.

Clarifying Activity

Students use a graphing calculator to graph the curve y = x². Then, for several functions in the y = ax² + c form, students predict what they think the new graph will look like, check their prediction with the graphing calculator, sketch the graph, and describe the change. After they investigate several functions in y = ax² + c form, students write a general statement about the effects of changing c: "The effects of changing c on the graph of y = ax² + c are . . ."

Assessment Connections

Ask . . .

Start with . . .

• What are the effects of changing c on the graph of y = ax² + c?

Probe further with . . .

• What does c represent on the graph?
• What is similar and different in all graphs of y = ax² + c?
• What is the y-intercept for y = x²? How do you know?
• What happens when c is greater than 0? Less than 0?
• How do the graphs corresponding to y = x² and y = x² + 2 compare?
• How do the graphs corresponding to y = 2x² - 2 and y = 2x² + 2 compare?
• What general statements can you make about the effects of changing c?
• What would you expect the graph of y = -x² + 2 to look like? Check it. How accurate was your prediction? Predict and check y = -x² - 2, y = -(x² - 6), and y = x² + 1.

Listen for . . .

• Does the student verbalize patterns that he/she observes to generalize the effect of changing c on the graph?
• Can the student describe the effects of changes in c on the graph of y = ax² + c?
• Does the student use appropriate vocabulary such as y-intercept when describing the effects of changing c on the graph of y = ax² + c?
• Does the student explain his/her reasoning about how changes in c affect the graph?

Look for . . .

• What strategy does the student use to investigate, describe, and predict the effects of changes in c on the graph of y = ax² + c?
• Can the student predict the effects of changes in c on the graph of y = ax² + c?
• Does the student use technology appropriately to investigate the effects of changes in c on the graph of y = ax² + c?
• Does the student organize and record his/her findings in a way that supports pattern recognition?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 9, item 52
• Spring 2004, grade 9, item 2
• April 2006, grade 9, item 46
• April 2006, grade 10, item 18
• April 2006, grade 11, item 7

(A.9.D) Quadratic and other nonlinear functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. The student is expected to analyze graphs of quadratic functions and draw conclusions.

Clarifying Activity

Students are given a graph of a quadratic function that describes a situation. They analyze the graph to answer questions arising from the situation.

An example situation:

The function that specifies the height [h(t)] of an object in terms of the time (t) after it is shot into the air is h(t) = 96t – 16t², where h is feet above ground level and t is seconds.

Assessment Connections

Ask . . .

Start with . . .

• What do you know about the height of the object from your graph?

Probe further with . . .

• What is the initial height of the object?
• Is the change in height constant?
• When will the object hit the ground? How do you know? What is the height of the object when it hits the ground?
• After how many seconds will the object be at a height of 128 feet? How do you know?
• When is the object at its highest and what is that height?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Is the student clearly explaining his/her reasoning?
• Does the student accurately describe the scale of the graph?

Look for . . .

• Does the student analyze graphs to draw conclusions for situations that can be described by quadratic functions?
• Does the student self-monitor and self-correct?
• Does the student use technology to graph the quadratic function?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 10, item 1
• Spring 2004, grade 10, item 55
• Spring 2004, grade 11, item 4

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(A.10) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.

(A.10.A) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student is expected to solve quadratic equations using concrete models, tables, graphs, and algebraic methods.

Clarifying Activity

Students use a variety of ways to solve a quadratic equation. They use concrete models (such as algebra tiles), make a table of data, and plot the data on a graph to determine the zeros of the function (graphs). They factor and find the zeros as well as use the quadratic formula (algebraic methods).

A sample equation:

x² + 5x + 6 = 0

1. Concrete Models: Students use algebra tiles to determine the factors of x² + 5x + 6 that correspond to the sides of the rectangle in the area model.
   
2. Graphs and Tables: Using a graphing calculator, students input the function Y1 = x² + 5x + 6 and graph it. They trace on the graph to find the roots of the equation x² + 5x + 6 = 0. Students use the table in the graphing calculator to confirm the roots.
3. Algebraic Approach: Students solve the equation x² + 5x + 6 = 0 by factoring or using the quadratic formula.

Assessment Connections

Ask . . .

Start with . . .

• How did you solve x² + 5x + 6 = 0?

Probe further with . . .

• How can you solve x² + 5x + 6 = 0 using algebra blocks? A graph? A table? Algebraic manipulation?
• How many solutions will you have? How do you know?
• What is the function corresponding to the equation?
• What is the relationship between the solutions of quadratic equation and the roots (zeros) of the corresponding functions?
• How would you use a graph to solve x² + 5x + 6 = 12? A table? Algebra tiles? Algebraic manipulation?

Listen for . . .

• Does the student use appropriate vocabulary when describing his/her strategies for solving the quadratic equation?
• Does the student effectively explain the strategies employed to solve the quadratic equation?
• Does the student relate the different solution strategies?
• Does the student self-monitor and self-correct?

Look for . . .

• Does the student understand that there is more than one way to solve a quadratic equation?
• Can the student solve quadratic equations using appropriate methods?
• Can the student solve quadratic equations using a variety of methods, including concrete models, tables, graphs, and algebraic methods?
• Can the student factor the quadratic equation?
• Is there appropriate use of technology to solve the quadratic equation?
• Does the student relate the solutions for an equation with the roots (zeros) of the corresponding function?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2003, grade 10, item 27
• Spring 2004, grade 11, item 37

(A.10.B) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student is expected to make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.

Clarifying Activity

Students investigate the relationship between the solutions of quadratic equations and the roots (zeros) of the corresponding functions through problem situations.

An example situation:

Students determine when a football hits the ground if the height [h(t)] of a football with respect to time (t) when kicked into the air is h(t) = 80t – 16t², where h is feet above ground level and t is seconds.

Assessment Connections

Ask . . .

Start with . . .

• How can you use the graph of the quadratic function to determine when the football hits the ground?

Probe further with . . .

• What equation can you use to find the time when the ball hits the ground?
• What are the solutions to the equation? When does the ball hit the ground? How do you know?
• What methods might you use to solve the quadratic equation?
• Graph the height of the ball with respect to time described by the function.
• How can you use a graphing calculator to graph the function?
• What are the x-intercepts of the graph?
• How do the zeros of the function compare with the roots of the equation?
• What is the relationship between the solutions of quadratic equations and the zeros of the corresponding functions?
• How long is the football in the air? Use the graph and the equation to support your answer.

Listen for . . .

• Does the student express an understanding of the problem situation?
• Does the student explain his/her reasoning?
• Does the student use appropriate vocabulary when explaining his/her thinking?
• Does the student accurately describe the connection between the x-intercepts of the graph, the roots of the quadratic equation, and the zeros of the related function?

Look for . . .

• Does the student make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the functions?
• Can the student graph the quadratic function? Without prompting, which strategy did the student use to find when the ball hits the ground?
• Does the student understand that there is more than one way to solve a quadratic equation?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 10, item 40
• April 2006, grade 10, item 42
• April 2006, grade 11, item 18

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(A.11) Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations.

(A.11.A) Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to use patterns to generate the laws of exponents and apply them in problem-solving situations.

Clarifying Activity

Students use a calculator to explore all of the laws of exponents. Given a family of problems designed to reveal a law, they answer and organize problems in tables, look for patterns, and generate the appropriate law of exponents.

After developing the laws of exponents, students apply them.

An example family of problems designed to explore multiplication of exponential expressions with like bases:

Based on their findings, students apply the law of exponents for multiplication to complete the following table.

Students develop the other laws of exponents in a similar manner.

Assessment Connections

Ask . . .

Start with . . .

• What patterns do you notice in the table?

Probe further with . . .

• What do you notice about your answers to each problem?
• What do you notice about the base of each exponential expression?
• What pattern in the exponents do you notice?
• If we consider 2¹⁰ to be the simplification of exponential expression in this family, how would you simplify 2² · 2⁸?
• Based on your observations, how would you simplify 2² · 2¹²?
• Can you describe how you would simplify the product of two exponential expressions with base 2? base 3?
• Can you describe how you would simplify the product of two exponential expressions with the same base?
• How would you write the law symbolically a^m · aⁿ = a^m+n?

Listen for . . .

• Does the student use appropriate vocabulary when explaining his/her reasoning?
• Can the student verbalize patterns?

Look for . . .

• Can the student use a calculator to evaluate exponential expressions?
• Is the student organizing and recording his/her finding in a table?
• Can the student use patterns to generate the laws of exponents?
• Can the student apply the laws of exponents?

TAKS Connections

Released TAKS items related to this activity can be found at the Texas Education Agency website:

• Spring 2004, grade 9, item 44
• April 2006, grade 9, item 34
• Spring 2004, grade 11, item 54
• April 2006, grade 11, item 16

(A.11.B) Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.

Clarifying Activity

Students work in groups and discuss situations that involve inverse variation. They use concrete models, tables, graphs, and algebraic methods to analyze and represent the situation.

An example situation:

Three students share the costs of a limousine for prom. It will cost them $50 each for this luxury. Being short on cash, they decide to include others. How will including other students affect the cost per student?

1. Play Money: Students represent the cost per student by dividing the money into piles. They form two equal piles of money to represent 2 students sharing a limousine, three equal piles to represent 3 students, four equal piles to represent 4 students, and so on. They note that as the number of piles increases, the cost per student (amount of money in each pile) decreases.
2. Table:

   

3. Graph: Students use the information from the table to arrive at the cost per student expressed as a function of the number of students. Let y = 150/x. They graph the function and use the graph to determine how many students are needed to make the cost per student less than $20.

Assessment Connections

Ask . . .

Start with . . .

• What patterns do you notice in the table?

Probe further with . . .

• What is the cost of the limousine if only 1 student rides in it? What is the cost if 2 students share? 3 students? 4 students? n students? What patterns do you observe?
• Can you describe in words the dependency of cost of the limousine and the number of students sharing it?
• What is the function rule relating the cost of the limousine and the number of students sharing it?
• Is the relationship between the cost of the limousine and the number of students sharing it a linear function? How do you know?
• What does it mean to say that the cost of the limousine and the number of students sharing it vary inversely?
• How many students must share the cost of the limousine if they do not spend more than $25 each? How did you determine this?
• Does the ordered pair (6, 20) belong to the graph of the function representing the relationship between the cost of the limousine and the number of students sharing it? How do you know? Is there another way you could determine this?
• How would the function change if the limousine company increased its rate by $50?

Listen for . . .

• Can the student verbalize the pattern of the number of students sharing the limousine and its cost?
• Can the student verbally describe the function for the number of students sharing the limousine and its cost?
• Can the student explain how he/she uses various representations of the functional relationship (table, graph, description, picture, rule) to answer questions arising from the situation?
• Can the student verbally describe the connections between two or more representations? (For example, “The column in this table is represented by this axis on the graph or this ordered pair is located here on the graph.”)

Look for . . .

• Does the student understand that there are situations modeled by functions that are neither linear nor quadratic?
• Can the student represent the function that describes inverse variations using a variety of ways, including a table, graph, description, picture, and rule?
• Can the student use various representations of the functional relationship (table, graph, verbal description, diagram, rule) to answer questions arising from the situation?
• Can the student write a function rule to describe the relationship between the quantities described in the situation?

(A.11.C) Quadratic and other nonlinear functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. The student is expected to analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.

Clarifying Activities

Students investigate a situation that can be described by exponential growth and represent the exponential growth function using a variety of ways, including concrete models, tables, graphs, and algebraic descriptions.

An example situation:

Students fold a sheet of square paper in half repeatedly. After each fold, they open the paper and count the number of sections found in the paper (created by the folds). They express the number of sections with respect to the number of folds made. Students predict how many sections there will be after 8 folds.

Alternate method:

Students relate the area of each folded section to the number of folds made. They determine the area of each folded section after 8 folds.

Example investigation:

1. Concrete Models: Students fold a piece of paper and generate data for the table.
2. Table:

   

3. Graph: Students plot the information on a graph and use the graph to predict the number of sections in the paper after 8 folds.
4. Algebraic Symbols: Students recognize the pattern in the table as the function y = 2^x, where x is the number of folds.

Example investigation (area):

1. Concrete Models: Students fold a piece of paper and generate area data for the table.
2. Table:

   

3. Graph: Students plot the information on a graph and use the graph to predict the area after 8 folds.
4. Algebraic Symbols: Students recognize the pattern in the table as the function y = ()x, where x is the number of folds and y is the area of each section.

Assessment Connections

Ask . . .

Start with . . .

• How can you relate the number of sections in the folded paper to the number of folds made?

Probe further with . . .

• How many sections are there after 1 fold? 2 folds? 3 folds? 4 folds? n folds? How do you know?
• What patterns do you observe?
• Can you describe in words the dependency of the number of sections in the paper to the number of folds made?
• What is the function rule relating the number of sections in the paper and the number of folds made?
• Is the relationship between the number of sections in the paper and the number of folds made a linear function? Is it quadratic? How do you know?
• What does it mean to say that the number of layers of paper grows exponentially with the number of folds made?
• What does it mean to say that the area of each folded section decays exponentially with the number of folds made?

Listen for . . .

• Does the student express an understanding of the problem situation?
• Does the student express an understanding that there are situations modeled by functions that are neither linear nor quadratic?
• Can the student verbalize the patterns?
• Can the student verbally describe the connections between two or more representations? (For example, “The column in this table is represented by this axis on the graph or this ordered pair is located here on the graph.”)

Look for . . .

• Does the student organize and record his/her findings in a table?
• Can the student represent the function that describes inverse variations using a variety of ways, including a table, graph, description, picture, and rule?
• Can the student use various representations of the functional relationship (table, graph, verbal description, diagram, rule) to answer questions arising from the situation?
• Can the student write a function rule to describe the relationship between the quantities in the situation?
• Does the student appropriately use technology to graphically represent the function and answer questions arising from the situation?

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