Transition Mathematics Project:
College Readiness Standards
STANDARD 7
The student accurately describes and applies concepts and procedures from
algebra.
NOTE: This standard assumes the student is already proficient with the relevant concepts and procedures described in Component 1.5
of the Washington State Grade Level Expectations through grades 9/10.
Components
7.1 Use properties of equality to solve equations
through a series of equivalent equations, appropriate
properties to simplify expressions, resulting in an
equivalent expression, and factors and terms
appropriately to simplify algebraic expressions.

7.2 Combine and simplify algebraic expressions
which contain polynomials, rational
expressions, radicals, or rational exponents.
7.3 Solve the following types of equations and inequalities
numerically, graphically, and algebraically. Interpret
solutions algebraically and in the context of the
problem. Distinguish between exact and approximate
answers. [see GLE 1.5.6, grade 9/10]
Evidence of Learning
Explain the distinction between factor and
term
EX:
Is the following a true statement? Explain why or why
not.

Find the sum, difference, or product of two
polynomials, then simplify the result.
EX:
32x  4  5x  2
form a  b by its conjugate will always
result in an expression with no radical sign.
Give a numerical example where neither
 a
nor b are perfect squares.
EX:
Solve the following equation in more than one way:
2
EX:
Explain why or why not the following represents a
true statement:
x3
3
x
EX:
If possible, order the following numbers
from
smallest to largest. Explain your reasoning.
3

EX:
Solve the following equation and explain your solution:
Explain why multiplying an expression of the
a b  a b
2
 Solve linear equations in one variable.
200
 3 , 3
200
 ,3
200 2
200
3
200
21 v  9  31 v
EX:
Solve the equation 7x = 8x.
EX:
Solve the equation

x2
x
 4   and verify your solution.
3
2

Explain the distinction between expression
and equation
EX:
As written,
x 1
x
is best described as

x 2 x  3
a) a rational function
b) a sum of rational expressions
c) a product of rational expressions
d) a rational equation.

Explain your choice.


EX:
Perform the indicated multiplication and express
the answer using positive exponents. Assume all
variables are real. Assume all expressions are
defined. Show your work.
1
5  5
5  1
 1

4 3 x 2 4 3  x 2 ;z 2  3z 2  3




EX:
Write in the form ax2+bx+c: (x+2)2-4(x3)(2x+1)+8x2+2

Explain the distinction between simplify and
solve
Know what it means to have a solution to an
equation.
EX:
Simplify:
x
2
x
2





Which of the following cannot be solved? Explain
your 
choices.

1. 4 x  7
2. 11c 1  0

2
3. x  1  0
4. 3x  4 x

5. f (x)  2x 13
1x 2  5x  4
Note: The 3/11 workgroup decided the spirit of the
two bullets below would be better served if they were
rewritten in a positive fashion. My suggestions are in
red below them.

 When working with an expression, student
does not attempt to isolate a variable or use
techniques that require action on both sides

8a bc 
2

3
2
 Solve systems of linear equations in two variables.
EX:
Solve the system of equations: 9x+8y=12 and 5x-3y=11
EX:
Determine the value of b so that the following system of
equations is dependent:
 3x  4 2x 2  4x  7
x 2  7x  4
 x  1
EX:
Complete the following equation so the solution of the
1
4

 2
equation is 4: 8x  7  7x  ____
x  1 x  3x  4
a2b  3a b
EX:

 Solve linear inequalities in one variable, including those
involving “and” and “or.”
EX:
3-4x<2+x

4x 16y  4
2x  by  2
EX:
Construct a system of two linear equations with a solution
of (2, 6) so that one line is horizontal and the other line is
vertical.
 Solve linear inequalities in two variables (graphically
only).
EX:
Solve the system of inequalities by graphing: 5x+4y>10
and 2x-3y<5
3
Factor out the greatest common factor from
polynomials of any degree, including and
from expressions involving rational
exponents.
EX:
Explain how the two terms in the expression
2x  3 42x  3 can be added by factoring out the
greatest common factor.
EX:
Maggie claimed the following expression was factored.
Explain why Maggie is correct or incorrect.
x2x 1  52x 1

 Solve absolute value equations of the form |ax + b| = c.
EX:
Solve |3x-8|=9
 Solve quadratic equations (including those with
irrational or complex number solutions; students are not
expected to check complex solutions algebraically).
EX:
Mark wants to solve the quadratic equation
x  5x  2  1. He didn’t want to go to all the bother
of an equation.
Recognize and use appropriate techniques in
simplifying expressions.

x 
1
2
Factor the expression x

x
Todd claimed the expression
x  3
x3
2
was equivalent to the
expression x  3 . Is Todd correct? Explain.



2
 25x
1
2

EX:

3
EX:
Factor the expression 4ax 3  20ax 2  2ax
EX:
Give an example of a real number for which
2
EX:
When working with an equation, if an action
leads toone side of the equation becoming
zero, the student writes the zero and retains
an equation rather than omitting the zero and
equals sign, thereby creating an expression.
Recognize and use appropriate techniques in
solving equations.
EX:
Jan solved the following equation in this way: 
3x  4 x


3x
4x

3x
3x
4
x
3
Explain how you know whether Jan is right or wrong.
If Jan is wrong, solve the equation correctly.
Factor quadratic polynomials with integer
coefficients into a product of linear terms, if

possible.
EX:
Can the quadratic function graphed below be factored
into a product of linear terms? Explain your answer.
(insert graph here of ½(x-1)^2 – 1)
EX:
Factor each of the following into a product of linear
terms, if possible:
x 2  34 x  35
x 2  7xy  6y 2
x 2 10 x  90


EX:
Does the equation x2-2x+6=0 have a real solution? Explain.
 Solve equations in one variable containing a single
radical.
EX:
x  5  x 1
2x  5 = 8
 Solve exponential equations in one variable
(numerically and graphically).
EX:
Below is a graph of the equation y  3x . Use the graph to solve
x
the equation 3  27 .

Simplify quotients of polynomials given in
factored form, or in a form which can be
factored.
EX:
Explain why
ab
 1 for all values of a and b
ba
except when a = b.
EX:

Simplify each of the following expressions, if possible:

to multiply the binomials on the left side and subtract 1 to
get zero on one side (so he could use the zero-factor
principle). Since 1 times 1 is 1, he thinks he can set each
factor to 1. What mathematical help can you give him? (A
complete answer will say whether Mark is right or wrong
and show why.)
xy
x
a(2x  1)  b
a 2  b2
6x 2  2x  20 x 3
2x
EX: Solve 3
x 2
= 81
EX:

The following table represents ordered pairs on the graph of
x
1 
f (x)    . x -2 -1 0 1
9 
2
Y 81 9 1 1/9 1/81
a) Use the table to find f(0).
b) Use the table to find x so that f(x) = 9.
c) Use the table to make a supported conjecture about the value of
f(-3).
 Solve rational equations in one variable that can be
transformed into an equivalent linear or quadratic
equation. (limited to monomial or binomial
denominators)

EX:
Solve the following two equations:
Add, subtract, multiply, and divide two
a
rational expressions of the form
,
bxc
1
3

p 1 p 1
4
1
11
b)


k  2 3k  6 9
a)
where a, b, and c are real numbers such that
bx  c  0 .

EX:
Given the two expressions
15
3
and
,
4x 1
4x

Solve
x2
=x+1
2x  3
Solve
1
3
 2
4
x 1 x 1
a) Add the expressions.
15
3
from
.
4x 1
4x


c) Multiply the expressions.
15
3
d) Find the quotient of
and
4x 1
4x


b) Subtract
EX:
The sum of arational expression
and

1
is
x5
2x
. Find this rational expression.
x  25
 Simplify products and quotients of single-term

expressions with rational exponents
(rationalizing denominators not necessary).
EX:
Simplify (a12b14c22)1/2
2

EX:
Which of the following are true for all values of
x and y, except possibly at 0?
x  y 
2


 x2  y2
1 1
1
4 x  3  
4x 3
Recognize the equivalence between
expressions with rational exponents and
a

radicals ( x b
 b x a , a, b integers, b0).
 Solve literal equations (formulas) for a particular
variable.
Solve PV=nRT for R
Solve A 
Solve
1
1
hb1  hb2 for h
2
2
xz
 5e m for v
2
vm
(Needs context, not necessarily a word
problem)
Transition Mathematics Project:
College Readiness Standards
STANDARD 8
The student accurately describes and applies function concepts and procedures
to understand mathematical relationships.
NOTE: This standard assumes the student is already proficient with the relevant concepts and procedures described in Component 1.5
of the Washington State Grade Level Expectations through grades 9/10.
Components
8.1 Recognize functional relationships
presented in words, tables, graphs, and
symbols (for any function). [ see GLE
1.5.3]

[Recognize functional relationships.]
EX: Do the following tables represent
functions? Explain why or why not.
X
q(x)
X
f(x)
1
½
2
¼
1
½
2
¼
3
1/2
2
1/8
4
1/16
EX:
State the type of function given in the
sketch: (next page—4 graphs)

(There are multiple types of function
8.2 Represent functions (linear,
quadratic, exponential, piecewise, and
reciprocal) using and translating among
words, tables, graphs, and symbols.
[see GLEs 1.5.2, 1.5.4]
8.3 Analyze and interpret features of
a function
8.4 Model situations and relationships using a
variety of functions (linear, quadratic,
exponential, piecewise, and reciprocal). [see
GLE 1.5.4]
Evidence of Learning

Evaluate functions to generate a graph.

EX:
Determine 3 ordered pairs on the graph of
1
the function f (x)  5x  , then graph the
2
function.
EX:
 the function
Given
3t
D(t)  
when 0  t  1
3  6(t  1) when t > 1
complete the following table and sketch a
graph of the function:
T 0
D
1
2.25
3
12
5
21.6

Describe patterns in the function's
rate of change, identifying intervals
of increase, decrease, and
constancy. If possible, relate the
patterns to the function's
description in words or its
graphical representation.
Identify y-intercepts and zeros using
symbols, graphs and tables.
f ( x)  2 x 2  9 x  5
EX: For
. What is the
y-intercept? How do you know? Factor
the expression and find the x-intercepts.
Plot the points and graph the function.
What x-value does the vertex have? How
can you find the corresponding y-value?

Choose a function suitable for modeling a
real-world situation presented using words or
data.

Determine and interpret the meaning of rates
of change, intercepts, zeros, extrema, and
trends.

Identify and justify whether a result obtained
from a function model has real-world
significance.
here…) [w/linear choice]

Determine the domain of a function.
EX: Describe the domain of the functions
f(x) =
f(x) =
2
and
4x  5
4x  5
b) f (x)  x 2  2x  8


Understand and interpret function
notation, particularly as it relates to

graphic displays of data.
EX: Understand the difference between

f  x   2, f  2  x .
EX:
A function N maps a person to the first
letter of their first name. (I.e., Don
maps to D). What is the domain and
range of N?
EX:
Describe the domain of the function
2x
f (x) 
4x  5
(no 2x just 2)
EX:

EX:
The two ordered pairs (-4, 0) and (2, 0) are
on the graphs of which of the following
functions?
a) f (x)  x 2  2x  8
Describe the domain of the function
f(x) = (2x)/(4x-5); f(x) = 4x-5
c) f (x)  x 3  2x 2  8x
d) f (x)  4x  2
Graph the functions you’ve selected by
finding additional ordered pairs for each.
 Describe relationships between the
algebraic features of a function and the
features of its graph and/or its tabular
representation.
EX:
Explain why the domain of the rational
3
2
expression x  1 has no excluded
values.
 Use simple transformations (horizontal
and vertical shifts, reflections about
axes) to create the graphs of new
functions using linear, quadratic and/or
absolute value functions.
 Algebraically construct new functions
using addition and subtraction (e.g.,
profit function).
EX:Mike has a job that pays $8.52 per hour.
a) Write a function, f, describing his salary per
hours worked, x.
b) If Mike saves 20% of his total salary, write a
function, g, describing the amount Mike saves per
hours worked.
c) Write a function, h, describing the amount of
money Mike has left after saving his 20%.

EX:
Line a has intercepts (0, 2) and (1, 0),
line b has intercepts (0, 4) and (2, 0) and
line c has intercepts (0, 6) and (3, 0).
Will these lines ever intersect? Explain.
EX:
What is the y-intercept of the graph
of y=3x2 – 5x + 7 ?
EX:
Find the zeros, y-intercept, and
domain of f(x) = (2x-3 ) - 5
EX:
What does ‘zero of a function’ mean?
EX:
x
Graph the function
1 
f (x)    .
2 
Find its zeros, y-intercepts, domain, and
discuss its end behavior.
EX:


Identify extrema and trends using
graphs and tables.
EX:
Find the vertex of the function
h(x)  x 2  8x  3
Additional Examples




CRS 7.1 (bullet 1)
EX:
Which of the following can be simplified by removing the common factor (2x – 3)? Justify your answer:
2x  3
x(2x  3)
x(2x  3)
2
x  (2x  3)
x(2x  3)
2x  3
x(2x  3)
x 2 (2x  3)
CRS 7.1 (bullet 5)
Carlos simplified this expression as follows. Is he right or wrong? Explain why.
2
3

x
x
2
3 
x  
x
x 
2x
3x


x
x
2 35

CRS 7.2 (bullet 5)
Simplify 1/(x-1) – 2/(x-1)
Simplify 3/(2x-6) ÷ -5/(-x+3)
CRS 7.2 (bullet 6)
Simplify (2x3/2y2)(3xy1/3)
Simplify (6x-3/4y3)/(12x2y-1/2)
CRS 7.2 (bullet 7)
EX: Assuming all expressions are defined, which of the following statements are true? Which are false? Explain.
1
1
x 3 y 2  2  x 3 y 2
1
x y 
3

2
2
 x 3 y 2
EX: If f(x) = x2/3, find f (8)

CRS 8.1 (bullet 1)
1. The USGS keeps track of the temperature inside a volcano on the island of Hawaii. Would these data represent a function?
2. Recognize examples of functions of each of the following types. (Assume x is the input variable and y is the output variable.)
i) Given by a table:
ii) Given by a graph:
iii) Given by a formula:
Recognize examples of i), ii), and iii) that are NOT functions, and explain why they are not.
CRS 8.1 (bullet 2)
3
1. Explain why the domain of the rational expression x  1 has no excluded values.
2
2. A function N maps a person to the first letter of their first name. (I.e., Don maps to D). What is the domain and range of N?
CRS 8.2 (bullet 2)
1. The table below is from y  h(x ) .
What are the zeros of f?
What are two factors of f?
x
y1
-7
-6
-5
-4
-3
0
-2
-2
0
4
2. The x-intercepts of the graph of y  g (x) are (-5, 0) and (0,0). What are two factors of g (x ) ? What are the zeros of
g (x ) ?
3. Given f (1)  0 , f (4)  0 and f (2)  0 :
What are the factors of f?
y  f (x) ?
What are the x-intercepts of the graph
EX:
The expression b  4ac found in the quadratic formula is so important that it gets its own name – the “discriminant.”
“Discriminant” shares the same root as the word “discriminate,” which means “to distinguish from another object.” (cf. Mirriam-Webster Online) What does the discriminant
distinguish between, and how do you use it to do so?
The values of the discriminant fall into three major categories. For each category, sketch the graph of a quadratic that would have a discriminant like the one you describe. (You
do not need to provide any quadratic equations as part of your answer.)
2
EX:
y  h(x ) .
The table below is from
What are the zeros of f?
What are two factors of f?
x
y1
-7
-6
0
-2
-5
-4
-3
-2
0
4
EX:
The x-intercepts of the graph of
EX:
y  g (x) are (-5, 0) and (0,0). What are two factors of g (x ) ? What are the zeros of g (x ) ?
f (1)  0 f (4)  0 and f (2)  0 :
Given
,
What are the factors of f?
What are the x-intercepts of the graph
y  f (x) ?
EX:
Given the graph of a polynomial h: (may be above level)
 Find the factors of h.
 Find the zeros of h.
 Find the x-intercepts of h.
EX:
Use the given graph of y  f (x) to answer the questions.
a. Find the domain of f.
b. Find the range of f.
c. Solve f ( x )  0
d. Find the output if the input is 3.
e. Find the input if the output is -3.
f. Find
f (1)
g. Find x such that
f ( x)  2
CRS 8.2 (bullet 3)
1. If f (2)  3 :
Write an ordered pair that must be on the graph of f ( x )  5
Write an ordered pair that must be on the graph of f ( x)  5
Write an ordered pair that must be on the graph of f ( x  2)
Write an ordered pair that must be on the graph of f ( x  3)
Write an ordered pair that must be on the graph of f ( x  1)  5
Describe the strategy you used to answer the above items.
2. Use the given table of values for y = f(x) to complete the following table.
x
0
1
2
3
4
y
7
1
–4
–4
–7
x
3
y = f(x - 3)
7
1
–4
–4
–7
5
8
8
8