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RESOURCE UNDER REVISION. The activities on this page have been aligned to the revised Algebra I TEKS. However, these activities are currently UNDER REVISION to better meet the revised Algebra I TEKS. Please check back often for the updated activities. The revised TEKS were adopted by the Texas State Board of Education in 2005, with full implementation scheduled for 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 write a sentence describing a dependent relationship and identify the independent and dependent quantities.

(For example: My grade in this course depends on the number of hours I study; independent quantity—number of hours I study, dependent quantity—grade in the course.)

(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

Students use data sets to determine whether a relationship is and is not a function.

Example of a relationship that is not a function: (age of student, month of birth). Example of a relationship that is a function: (height of ramp, distance car rolls).

Additional Clarifying Activity

Students roll toy cars down a ramp and record the height of the ramp and the distance the car rolls. Students change the height of the ramp several times and record the distance the car rolls each time. Information should be recorded in a table (height of ramp, distance car rolls), and students should determine the relationship between height and distance (the distance the car rolls depends upon the height of the ramp).

(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 discuss a problem similar to the following:

Scott has a friend in another town. A phone call to this town is long distance. The charge is $.75 cents for access to long distance plus $0.10 for each minute.

The information given in the problem situation specifies the cost in dollars for a phone call in terms of the length of the call in minutes. Students describe this as a functional relationship (the cost is a function of the length), set up a table to find costs for different lengths of calls, and graph the function using technology, and they represent the function algebraically in a formula such as y = 0.10x + 0.75.

Students answer such questions as "How much will a 15 minute call cost?" by evaluating the function at x = 15: the cost is 0.10(15) + 0.75 = $2.25.

They answer such questions as "How many minutes can I talk for $3.00?" by setting up an equation 3.00 = 0.10x + 0.75 based on the function y = 0.10x + 0.75, and solving the equation for x: the number of minutes is x = (3.00 - 0.75) / 0.10.

x = 22.5 minutes.

(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 Activities

Students use cubes to construct buildings with towers. (representation - concrete models)

Students make a table relating the number of cubes in a tower to the total number of cubes used. (representation - table)

Students translate the data in the table to a graph. (representation - graph)

Students describe the pattern. (representation - verbal description): The total number of cubes is two more than the number of cubes in the tower.

Students determine a rule that describes the relationship between the number of cubes in the tower (x) and the total number of cubes in the building (y). (representation - formula): y = x + 2

Students use the formula to set up an equation to answer questions such as "How many blocks are in the tower of a 40-block building?" (40 = x + 2)

Additional Clarifying Activity

Student discuss a problem similar to the following: An empty box weighs 20 grams and a pamphlet weighs 8 grams. Often, several pamphlets are put into a box, and the box is mailed off. Represent the relationship between the number of pamphlets and the mailing weight of the box in at least three different ways.

Responding to this problem, students might:

Build a table:

Draw a graph:

Click here for a larger version of the graph.

Write an algebraic formula:

Let n = number of pamphlets and w = mailing weight. Then

The formula w = 8n + 20 expresses the mailing weight of the box in grams as a function of the number of pamphlets in the box. The fomula n = (1/8)(w -20) expresses the number of pamphlets as a function of the weight in grams.

Give a verbal description:

To find the mailing weight in grams multiply the number of pamphlets by 8 and add 20.

(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 use the table, graph, or formula constructed in A.1.D to predict the total number of cubes in a building that has a 10 cube tower.

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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

Students graph y = x and y = x², describe the similarities and the differences between the two functions, and identify each as linear or quadratic.

(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

Students describe a domain and range for the following example: Chris rents roller blades for $12 a day. His total cost is a function of the number of days he rents the roller blades.

Students state the domain and range and describe values which would not make sense.

(verbal description)

• Domain: the number of days he rents the roller blades
• Range: the total cost of renting the roller blades

(mathematical description)

• Domain: {x: x is zero or a positive integer}
• Range: {y : y is zero or a positive multiple of 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

Students work in groups, and each group is given a set of graphs. For each graph in that set, the groups create a situation in which the value of one variable depends on the value of the other.

Sample situation: Mary began to walk to school. She stopped at the stoplight, then ran the rest of the way.

(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

Students gather data in two-variable format (such as height vs. shoe size).

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 size of shoe a person would wear whose height is 49 inches, 66 inches, etc.

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

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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

Students discuss a problem similar to the following: A plumbing company charges $42 per hour plus $35 for the service call.

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

Students answer questions such as:

• What values remain constant? $42 and $35
• What values vary? hours worked and total charge
• What variables could be used to represent the quantities in this situation? Sample answers: h—hours and c—total charge

(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 discuss a problem similar to the following: A plumbing company charges $42 per hour plus $35 for the service call.

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

Students look for a pattern.

• Sample answer: The total charge is increasing by $42.
• Sample answer: In the process column, the 42 and the 35 remain constant; the only quantity changing is the number of hours worked.

Students define a function to describe the relationship between the number of hours worked and the total charge. Answer: 42(h) + 35 = c

Using this function, students set up an equation to answer questions such as, "How many hours were worked if the charge is $497.00?"

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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 a problem such as the following:

An object is dropped from the top of a 144 foot building. The function that specifies the height of the object in terms of the time is h = 144 -16t², where h is in feet above the ground and t is in seconds.

Students answer questions such as:

• What is the height after 2 seconds? (find a specific function value)
• How much time has passed when the object is 48 ft above the ground? (transform and solve the equation 144 - 16t² = 48)
• When will the object hit the ground? (Students should note that the height of the object when it hits the ground is 0. They should set up and solve the equation 144 - 16t² = 0.)

Additional Clarifying Activity

Students simplify the polynomial expression (x + 5)(x + 3) to express the area of the following rectangle in terms of x.

(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 express the area of the following rectangle in terms of x.

Additional Clarifying Activity

Have students use the commutative property of multiplication to investigate the following problem:

If you are buying something on sale, would you rather the store clerk give you the discount first, then figure the sales tax on the discounted amount, or figure the sales tax first on the original amount, then apply the discount to the total?

(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 recognize that functions can be denoted in many forms: y = , d = , y₁ = , f(x) = . Only the form f(x) allows us to see both input and output values. Students choose what form they need to fit the problem: for example, at a rate of 25 miles per hour, distance is a function of time; d = 25t. A student could enter this in a graphing calculator as y₁ = 25x. Or if a student wants to focus on several specific times, f(t) = 25t might be a better choice.

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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 determine whether or not the following data have a constant rate of change and can be represented by a linear function.

(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 the following problem and determine the reasonable domain and range values.

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

Students answer questions such as:

• What would be reasonable number choices to use in the number of hours worked column (domain)?
• What values would you expect to get in the money earned column (range)?

(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

Given the following situation:

Russell does yard work on the weekends to earn spending money. He charges $3.50 an hour. (verbal description)

Students make a table showing the relationship between the number of hours worked and money earned. (tabular description)

Students plot the data on a graph and determine the line of best fit. (graphical description)

Students determine the function that describes the line of best fit. (algebraic description)

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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 ramp and mark where they make their first step, their second step, and so forth. Then they measure and record the horizontal and vertical distances between each of their steps 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 for each representation (table, graph, algebraic representation).

Additional Clarifying Activity

Here is a picture of a storefront. The wall above and below the windows is covered with square tiles. What is the slope of the sidewalk in front of the store?

(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

For the table, graph, and function below, students determine the amount a person earns working at a $3.00 hourly rate. Students find the slope of the line shown in the graph and tell what the slope represents in this situation. (For every hour worked, the person earns $3.) Students explain what the y-intercept represents. (no work, no money)

(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, students predict what they think the new graph will look like, use the graphing calculator to check their prediction, sketch the graph, and describe the change. After investigating several functions, students write a general statement about . . .

• the effects of changing "m";
• the effects of changing "b".

(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

Students find the equation of a line in y = mx + b form.

• Students write the equation of the line below.

  

• Students write the equation of a line through the points (2,4) and (-3, -5).
• Students write the equation of a line through the point (2,4) and a slope of 4.
• Students write the equation of a line with a slope of -2 and a y-intercept of 5.

(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 intercepts from the following representations.

Graph:

Table:

Algebraic representation:

y = x + 3

(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 connect changes in problem situations to changes in graphs by considering problems such as the following:

A caterer charges a $25 fee plus $4 per person. Graph the function represented by this problem. State the slope and y-intercept and give their meanings in this situation.

Students answer questions such as:

• If the caterer changes the fee to $30, how will this affect the graph? (change in y-intercept)
• If the caterer changes the price per person to $5, how will this affect the graph? (change in slope)

(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 such as the following:

The amount of chlorine needed in a pool increases proportionally (varies directly) as the amount of water increases. If 5 units of chlorine is the amount needed for 550 gallons of water, what function represents the amount of chlorine needed for x gallons of water?

Students graph the function and predict the amount of chlorine needed for 850 gallons of water.

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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 write an inequality for the following problem:

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

(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 solve an equation such as 4x + 2 = 3x - 5 using concrete models, graphs, or the properties of equality.

Students use algebra blocks to build the following equation and use the tiles to simplify and solve.

(concrete models)

build the equation:

Students graph y = 4x + 2 and y = 3x - 5 and find the intersection point in order to determine the value of x. (graphs)

Students use symbolic manipulation and the properties of equality to solve the equation 4x + 2 = 3x - 5.

Additional Clarifying Activity

Students graph the solution to the following problem:

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

To graph the solution to the inequality $150 < $49 + 0.13 m when m is in miles, students first graph the functions y = $49 + 0.13 m and y = $150, locate their point of intersection, and identify which values of m in y = $49 + 0.13m give y < $150.

(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 would not be reasonable. (For example, fractional answers for problems that refer to the number of people, coins, or other such discrete objects would not make sense. Negative numbers would not be used to represent distance, hours worked, elapsed time, etc. This analysis should be a part of every problem solving situation.)

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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 write a system of equations to represent problem situations such as the following:

At Marti's Mexican Restaurant, 2 tacos and 3 enchiladas cost $2.39. If you buy 4 tacos and 5 enchiladas, the cost is $4.23. What is the cost for one taco?

Set-up:

2t + 3e = 2.39

4t + 5e = 4.23

(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 problem situations such as the following:

At Marti's Mexican Restaurant, 2 tacos and 3 enchiladas cost $2.39. If you buy 4 tacos and 5 enchiladas, the cost is $4.23. What is the cost for one taco?

Set-up:

2t + 3e = 2.39

4t + 5e = 4.23

Students model and solve the situation using algebra blocks.

Students represent the system of equations by putting the equations into slope-intercept form and using graphing technology, then solve the system using table, graph, intersection, and tracing features.

Students solve the system algebraically by a method of their choice: substitution or linear combination.

(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 discuss when answers to linear systems would not be reasonable. (For example, fractional answers for problems that refer to the number of people, coins, or other such discrete objects would not make sense. Negative numbers would not be used to represent distance, hours worked, elapsed time, etc. This analysis should be a part of every problem-solving situation.)

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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 the following problem:

Dale is raising a calf for the local livestock show. It now weighs 200 pounds. She could sell it now for $1.50 per pound. The calf is gaining 5 pounds a week, but the price per pound is dropping 3 cents a week. Analyze graphically the value of the calf over time.

Quadratic Function:

value = (weight in pounds)(price per pound)

v = (200 + 5t)(1.50 - .03t)

where t = number of weeks

Students identify the values where the function makes sense (0 ≤ t ≤ 50 to prevent "negative weeks" and negative values for the calf) and what the best time would be to sell the calf (at the value of t where v is a maximum on the graph).

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

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

(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 analyze graphs of quadratic functions in problem situations such as the following:

The function that specifies the height of an object in terms of the time after it is shot into the air is h = 96t - 16t² where h is in feet above ground level and t is in seconds, 96 ft/sec is the initial velocity of the ball, and 16t² is the change in height attributed to gravity.

Students use a graphing calculator to graph the height of the object with respect to time and determine after how many seconds the object will be at a height of 128 feet.

(Note: Students will need to analyze the graph to see that it will hit 128 feet twice, once at 2 seconds and again at 4 seconds.)

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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:

Example: x² + 5x + 6 = 0

• (use concrete models): Students use algebra blocks to determine the factors of x² + 5x + 6.
  
• (use graphing and tables): Students input the function as Y₁ = x² + 5x + 6 and graph it. Students trace to find the roots of the equation x² + 5x + 6 = 0. (Some calculators allow students to calculate the roots or use a table feature to locate the roots.)
• (use algebraic approach): Students solve the equation x² + 5x + 6 = 0 by

  Factoring:

  

  Using the Quadratic Formula:

  

  x = -2 or -3

(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 in problem situations such as the following:

The function that specifies the height of a football with respect to time when kicked into the air is h = 80t - 16t² where h is in feet above ground level and t is in seconds.

Students determine when the football will hit the ground. (Students should note that the height of the ball when it hits the ground is zero. After substituting 0 for h, they may then use the method of their choice to solve the quadratic equation. Students should then graph the height of the ball with respect to time described by the function and note that the roots (x-intercepts or zeros) have the same values as the solutions to the equation.)

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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 complete the table below to generate the laws of exponents.

Students answer questions such as:

• What did you notice about your answers to each problem? They were all 1024.
• What did you notice about the base of each exponential expression? They were all 2.
• What pattern in the exponents could you use as a shortcut? The sum of the exponents was always 10.

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

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

Students apply the law of exponents in a problem-solving situation such as: Light travels at about 3*10⁵ kilometers per second. There are 6.048*10⁵ seconds in one week. How far does light travel in one week?

Answer:

(3*10⁵)(6.048*10⁵) = 18.144*10¹⁰. Therefore, light will travel 1.8144*10¹¹ kilometers in one week.

(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 the following problem that involves inverse variation.

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?

• (using play money): Students represent the cost per student by dividing the money into piles. Students form two equal piles of money to represent two students sharing a limousine, three equal piles of money to represent three students, four equal piles of money to represent four students sharing a limousine, etc. Students note that as the number of piles increases, the cost per student (amount of money in each pile) decreases.
• (using a table):

  

• (using a graph): Students use the information from the table to arrive at cost per student expressed as a function of the number of students.

  Let Y₁ = 150 / x,

  Students graph the function and use the graph to determine how many students are needed to make the cost per student less than $ 20.00.

(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 the following exponential growth situation:

A sheet of paper is folded in half repeatedly. Express the number of layers of paper obtained with respect to the number of folds made.

• (using concrete models): Students actually fold a piece of paper to begin generating data for the table.
• (using a table):

  

• (using a graph): Students plot the information on a graph and use the graph to predict the number of layers after 7 folds.
• (using algebraic symbols): Students recognize the pattern in the table as the function y = 2^x where x is the number of folds.

Additional Clarifying Activity

Students investigate the following exponential decay situation:

A sheet of paper whose area is 1 square unit is folded in half repeatedly. Express the area of each of the folded sections with respect to the number of folds made.

• (using concrete models): Students actually fold a piece of paper to begin generating data for the table.
• (using a table):

  

• (using a graph): Students plot the information on a graph and use the graph to predict the area after 7 folds.
• (using algebraic symbols): Students recognize the pattern in the table as the function y = (1/2)^x where x is the number of folds.

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