ALGEBRA 1
Texas Essential Knowledge and Skills
Annotated by TEA for Pre-AP rigor
Introduction
As the committee began its examination of the Texas Essential Knowledge and Skills (TEKS), we were often surprised by what was included or left
out of courses that preceded or followed those that we normally teach.
"Do they really expect eighth graders to be able to do that?"
"Where are the sequences and series that we used to do in Algebra II?"
Ultimately, we agreed that all of the concepts and skills necessary to prepare students for success in AP* Statistics and AP Calculus would be
covered if the TEKS were interpreted in a particular way. Due to time constraints, we were reluctant to add any additional topics to the TEKS, though
a teacher might choose to do so.
The problem is particularly acute at the middle school level when all of the TEKS for grades 6-8 are often covered in only two years in order for
students to take Algebra I in grade 8. Having students just skip over a year of elementary or middle school mathematics is a dangerous proposition
that can have serious repercussions in subsequent courses. A well-planned and instructed Pre-AP* middle school program combines, streamlines, and
collapses the material in such a way that all of the TEKS are addressed at a deeper and more complex level.
At one point, someone on the committee said, "The problem is not that the TEKS are incomplete; it is that all of these things are treated equally.
Some of these TEKS are three-minute topics, and some of them are three-week topics." That gave us our idea for the structure of the charts in this
section. We went through the TEKS and sorted them into three groups.

The TEKS in regular font are topics with which students already have some familiarity due to previous instruction and which are being
revisited through the spiraling curriculum or are topics that can be covered in minimal time. These topics might provide foundational
knowledge (such as definitions) that will be used for future topics throughout the course.

The TEKS typed in italics are topics that might be addressed throughout the course on multiple occasions or might be addressed to greater
depth than the previous topics.

The TEKS in a bold, slightly larger, font are those that merit greater time commitment and greater depth of understanding for the
Pre-AP student. These topics should be taught with a particular emphasis toward preparing students for AP Calculus or AP
Statistics.
After categorizing the TEKS, we looked for problems or activities that would exemplify those TEKS in the third group and included them in the
second column as examples of what we felt were good Pre-AP mathematics problems and activities. Remember that these are only examples;
students will have to do many more than the few problems that we were able to include here in order to be well-prepared for AP Statistics and AP
Calculus. These are meant to give you ideas and get you started in understanding what makes a good Pre-AP mathematics problem. You will also
find in the second column additional comments about the TEKS or sample problems that we felt might be important.
TEKS: Algebra I
TEKS
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.
Examples
Commentary
(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.
1. Given the formula for the slope of a line, solve for y:
(A) The student finds specific function
m = (y - y1) / (x - x1)
values, simplifies polynomial expressions,
transforms and solves equations, and factors
as necessary in problem situations.
(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
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.
(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?
(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.
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) The student develops the concept of slope Denise is walking to her friend’s house. Her distance from home at any given
as rate of change and determines slopes from time is shown on the graph below.
graphs, tables, and algebraic
representations.
(B) The student interprets the meaning of
slope and intercepts in situations using data,
symbolic representations, or graphs.
(C) The student investigates, describes, and
predicts the effects of changes in m and b on
the graph of y = mx + b.
dista nce
(D) The student graphs and writes equations a. In this situation, explain the meaning of:
time
of lines given characteristics such as two
points, a point and a slope, or a slope and yb. During what time interval is Denise walking the slowest? Explain your
intercept.
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 dista nce
time
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.
(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, PreAP 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 green hot air balloon is 20 feet above the ground and is rising at a rate of 5
(A) The student analyzes situations and
formulates systems of linear equations in two 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.
unknowns to solve problems.
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.
(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
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.
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 (xintercepts) 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 Find the missing exponent (n) in each problem.
laws of exponents and applies them in
2
5
7
n
18
problem solving situations.
a. x  x  x  x  x
b.
(a n b 3 ) 4  a16b12
c.
x 4 n  x 3n  x
x 2n
d.
28  2 5
2n
x4 1
e. n  2
x
x
f. 3  1
n
g.
(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.
(b 4 ) n  b 24
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, PreAP students can use properties of exponents to
solve these equations.