• # TEKS: Algebra I

TEKS

Examples

Commentary

Foundations for functions: knowledge and skills and performance descriptions.

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

(A) The student describes independent and dependent quantities in functional relationships.

(B) The student gathers and records data, or uses data sets, to determine functional relationships between quantities.

(C) The student describes functional relationships for given problem situations and writes equations or inequalities to answer questions arising from the situations.

(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

(E) The student interprets and makes decisions, predictions, and critical judgements from functional relationships.

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

(A) The student identifies and sketches the general forms of linear (y = x) and quadratic (y = x2) parent functions.

(B) The student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations, both continuous and discrete.

(C) The student interprets situations in terms of given graphs or creates situations that fit given graphs

(D) The student collects and organizes data, makes and interprets scatter plots (including recognizing positive, negative, or no correlation for data approximating linear situations), and models, predicts, and makes decisions and critical judgments in problem situations.

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

(A) The student uses symbols to represent unknowns and variables.

(B) The student looks for patterns and represents generalizations algebraically.

(A.4) 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) The student finds specific function values, simplifies polynomial expressions, transforms and solves equations, and factors as necessary in problem situations.

1. Given the formula for the slope of a line, solve for y:

m = (y - y1) / (x - x1)

This standard is developed throughout the Algebra I curriculum. Symbolic manipulation is a major component of Algebra I, and Pre-AP* students should have many meaningful opportunities to practice these skills.

(B) The student uses the commutative, associative, and distributive properties to simplify algebraic expressions.

(C) The student is expected to connect equation notation with function notation, such as y = x + 1 and f(x) = x +1.

2. Solve for m:

y + mx1 = mx + y1

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

(A) The student determines whether or not given situations can be represented by linear functions.

(B) The student determines the domain and range for linear functions in given situations.

(C) The student is expected to use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

Collect data to determine the effect that distance has on the width of what you can see through an empty paper towel roll. You and your partner will measure the length you are from the wall and the diameter of the circle you can see through the roll. You are to collect six pairs of data using different distances and then complete the following:

1. List your data points and tell which variable you assigned as your independent variable and which variable as your dependent variable.

2. Draw a scatter plot of the data. Be sure to label the independent and dependent variables.

3. Write the equation you think best fits the data. Explain why you think the equation you wrote is a good fit and/or any problems you think there might be with the equation.

4. Give the value of the slope of the line and the y-intercept. In terms of the problem, explain their meanings.

5. What source of error do you think you had in collecting your data? How could you have reduced that error?

6. Based on your equation in part 3:

a. If you were 8.7 feet from the wall, what would you expect to be the width of the circle that you could see? (Show your substitution step.)

b. If you were 35 feet from the wall, what would you expect to be the width of the circle you would see? (Show your substitution step.)

c. Of the two answers you got in parts (a) and (b), which one are you more confident about and why?

This introduces the AP* Statistics concepts of analyzing patterns in scatter plots and the least squares regression line. The concepts of interpolation and extrapolation are important in the AP Statistics curriculum.

(A.6) 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) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations.

(B) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs.

Denise is walking to her friend’s house. Her distance from home at any given time is shown on the graph below. a. In this situation, explain the meaning of: b. During what time interval is Denise walking the slowest? Explain your answer. What is her speed in feet per second?

c. During what time interval is Denise walking the fastest? Explain. What is her speed in feet per second during that interval?

d. During what time interval is Denise not moving? What is the during this time interval?

e. During what time interval is Denise walking 1 foot per second?

f. Find Denise's average walking rate during the time interval 0-6 seconds.

g. Write a story that describes Denise's walk.

h. Write a function for Denise's distance d(t) in terms of the time t during the time interval 3 < t < 6

i. Using the function d(t) you wrote in part h, find the exact value for:

d(5) - d(3.5)
5 - 3.5

What does this value represent?

j. The last three seconds of Denise's walk are not shown on the graph. If she walks at a constant rate during the last three seconds, write a function f(t) that models the last three seconds of her walk, given f(10) = 6 and f(13) = 15.

k. Use the function you wrote in part j to find f(12). What does this value represent?

Understanding the relationship between rate of change and the slope of a line is a foundation for AP Calculus that is found in the Algebra I curriculum. This problem is based on problem #21 on the 2000 Algebra I EOC Test and problem #2 on the 2000 AB Calculus AP Exam.

(C) The student investigates, describes, and predicts the effects of changes in m and b on the graph of y = mx + b.

(D) The student graphs and writes equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.

(E) The student determines the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations.

(F) The student interprets and predicts the effects of changing slope and y-intercept in applied situations.

(G) The student relates direct variation to linear functions and solves problems involving proportional change.

(A.7) 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) The student analyzes situations involving linear functions and formulates linear equations or inequalities to solve problems.

(B) The student investigates methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, selects a method, and solves the equations and inequalities.

The linear function C defined C(h) = 32h + 121 gives the cost (in dollars) to hire a disc jockey for a school dance for h hours.

a. Explain the meaning of the numbers 32 and 121 in the cost function in the context of this problem.

b. Graph C.

c. Your school has budgeted \$350 to spend for a disc jockey. Find the maximum number of hours your school can hire the disc jockey for the dance.

d. If the principal of your school decides that the school will spend between \$225 and \$350 on the disc jockey, what is the time interval that your school can hire the disc jockey?

e. When you call the disc jockey, you discover that he rounds each fraction of an hour that he works to the next hour. For example, if he works 2.2 hours, he will charge you for three hours. Complete the given table of values for disc jockey costs then use the table to help you graph the costs. How does this graph compare with the graph in part b?

 hours 1.25 1.5 1.75 2 2.25 2.5 2.75 3 cost

Part e illustrates how a question can be extended to provide students with a glimpse into an advanced topic. Although step functions will not be fully developed until later math courses, Pre-AP Algebra I students can gain insight into linear functions by comparing them to functions that are not linear.

(C) For given contexts, the student interprets and determines the reasonableness of solutions to linear equations and inequalities.

(A.8) 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) The student analyzes situations and formulates systems of linear equations in two unknowns to solve problems.

A green hot air balloon is 20 feet above the ground and is rising at a rate of 5 feet per minute. A red hot air balloon is 150 feet above the ground and is descending at a rate of 20 feet per minute.

a. Write a function for the height (in feet) of each balloon in terms of the time (in minutes).

b. Make a table of values for each function that you wrote in part a. Use your table of values to estimate the time the balloons will be the same distance from the ground. Approximate the height of the balloons at this time.

c. On graph paper, sketch the graph of this situation. Use your graph to estimate at what time the balloons will be the same distance from the ground. Approximate the height of the balloons at this time.

d. Use algebraic methods to determine the time that the balloons will be the same distance from the ground. Use algebraic methods to determine the height of the balloons at this time.

e. During what interval of time is the red balloon's height greater than that of the green balloon?

f. If the red balloon continues to descend at this rate, when will it reach the ground?

(B) The student solves systems of linear equations using concrete models, graphs, tables, and algebraic methods.

Let R be the region enclosed by the graphs of y=2x, y=x and x=4.

a. Graph and shade the region R on a coordinate plane.

b. Find the area of R.

c. Explain how you found the area of region R.

It is important to discuss the different approaches to finding the area of R. The variety of methods will help students feel more comfortable taking risks and trying new methods with more difficult problems.

(C) The student interprets and determines the reasonableness of solutions to systems of linear equations.

Quadratic and other nonlinear functions: knowledge and skills and performance descriptions.

(A.9) 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 equations.

(A) The student determines the domain and range values for which quadratic functions in given situations.

(B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax2 + c.

(C) The student investigates, describes, and predicts the effects of changes in c on the graph of y = ax2 + c.

(D) The student analyzes graphs of quadratic functions and draws conclusions.

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

(A) The student solves quadratic equations using concrete models, tables, graphs, and algebraic methods.

An object is hurled upward from the ground at an initial velocity of 128 ft/s. The height (h) in feet of the object at any given time (t) in seconds is:

h(t) = 128t – 16t2

a. When will the object reach a height of 192 feet?

b. When will the object reach the ground?

c. When will the object reach its maximum height?

d. What is the maximum height of the object?

e. Find the exact time that the object is 150 feet high.

f. For what values of time (t) is the object higher than 150 feet?

g. Graph this situation using the values for time and height found in parts a-e.

h. State the reasonable domain and the range of the function in the context of this problem.

Quadratic functions in Pre-AP Algebra 1 can be used to introduce optimization problems found in AP Calculus.

(B) 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.

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

(A) The student uses patterns to generate the laws of exponents and applies them in problem solving situations.

Find the missing exponent (n) in each problem.

a. b. c. d. e. f. g. Students should be allowed to investigate these problems on their own or in groups of two to three. Although this set of problems looks like exponential equations found in Algebra II, Pre-AP students can use properties of exponents to solve these equations.

(B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.

(C) The student analyzes data and represents situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.