1
Algebra I
Expanded Test Blueprint
Reporting Category: Expressions and Operations
Number of Items: 12
Standards of Learning:
A.1
The student will represent verbal quantitative situations algebraically and
evaluate these expressions for given replacement values of the variables.
TI-84: Use the STO key.
Evaluate x 2  x  6 when x   1 .
TI-84: Use the graph and table features of the calculator.
Evaluate x 2  x  6 when x   1 .
When x   1; y  8 .
Therefore, evaluating
x 2  x  6 when x   1
yields 8.
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A.2
The student will perform operations on polynomials, including
a) applying the laws of exponents to perform operations on expressions;
b) adding, subtracting, multiplying, and dividing polynomials; and
c) factoring completely first- and second-degree binomials and trinomials in
one or two variables. Graphing calculators will be used as a tool for
factoring and for confirming algebraic factorizations.
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Summary of Exponent Properties:
4
When adding/subtracting polynomials  add/subtract LIKE terms.
When multiplying polynomials  apply the Distributive Property.
Factoring reverses polynomial multiplication.
1. Find the greatest common factor (GCF) of all terms of the polynomial and then
apply the distributive property.
Steps to factoring:
1. LIST all possible factor pairs of the constant term
c.
(Combinations of two integers whose product is equal to the constant term
c ).
2. DECIDE the signs for the parentheses based on the constant term (c).
If
If
c
c
is positive
is negative
use the same signs.
use different signs.
(2 positive signs OR 2 negative signs)
3. CHOOSE the factor combination that SUMS to the coefficient of the linear term (b).
4. WRITE the answer as the product of two binomials.
5. CHECK your answer (FOIL).
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3. A SPECIAL TRINOMIAL - Perfect Square Trinomials:
a 2  2ab  b2   a  b  a  b    a  b 
2
a 2  2ab  b2   a  b  a  b    a  b 
2
4. A SPECIAL BINOMIAL - Difference of two squares:
a2  b2   a  b  a  b 
5.
Steps for factoring trinomials by grouping:
1. MULTIPLY A C .
2. LIST all possible factor combinations whose product is equal to 𝐴 ∙ 𝐶 .
3. DECIDE the signs for the parentheses based on the constant term
If c is positive
If c is negative
use the same signs.
use different signs.
c ).
(2 positive signs OR 2 negative signs)
4. CHOOSE the factor combination that SUMS to the coefficient of the linear term 𝐵.
5. REWRITE 𝐵𝑥 as the sum of the two factors. There will be 4 terms.
6. FACTOR by grouping:
a. Group the first two terms and the last two terms
b. Factor the GCF out of each group {the parentheses should match}
c. Use the distributive property to write the answer as the product of two binomials
7. CHECK your answer (FOIL).
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Using the Polynomial Root Finder and Simultaneous Equation Solver v2.0  PlySmlt2
The polynomial root finder can solve simple linear equations, quadratic equations, or
any polynomial up to a degree of 10 with fractions or decimals and real or imaginary answers.
To use the Poly Solver App:
1. Press the APPS button on your calculator.
2. Choose the “Plysmlt2” option.
3. Press any key to continue.
4. Choose Option 1: “Polynomial Root Finder”.
5. Make the selections so that your screen matches the one below:
TI-84 Plus
TI-84 Plus CE
6. Push the “GRAPH” key to continue to the NEXT step.
7. Enter the values of a, b, and c. Find the roots of y  2 x 2  x  3 .
8. Push the “GRAPH” key to SOLVE.
9. x1 and x2 are the solutions to the quadratic equation.
3
and 1 .
2
The factors of y  2 x 2  x  3 are  2 x  3 x  1 .
The roots, x-intercepts, solutions, zeros are x 
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A.3
The student will express the square roots and cube root of whole numbers and
the square root of a monomial algebraic expression in simplest radical form.
A square root in simplest form is one in which the radicand (argument)
has no perfect square factors other than one.
In the real number system, the argument of a square root must be
nonnegative while the argument of a cube root may be any real number.
A cube root in simplest form is one in which the argument has no perfect
cube factors other than one.
Use
on the TI-84 and compare decimals.
Simplify 75 .
Algebraically: Factor out the perfect root.
75 
25 3 
5 5 3  5 3
TI:84: Y1 = 75 / x 2  Table of Values:
75  5 3
Reporting Category: Equations and Inequalities
Number of Items: 18
Standards of Learning:
A.4
The student will solve multistep linear and quadratic equations in two
variables, including
a) solving literal equations (formulas) for a given variable;
b) justifying steps used in simplifying expressions and solving equations,
using field properties and axioms of equality that are valid for the set of
real numbers and its subsets;
c) solving quadratic equations algebraically and graphically;
d) solving multistep linear equations algebraically and graphically;
e) solving systems of two linear equations in two variables algebraically and
graphically; and
f) solving real-world problems involving equations and systems of equations.
Graphing calculators will be used both as a primary tool in solving
problems and to verify algebraic solutions.
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Properties:
Addition
Multiplication
Identity
a  0  0  a  a
Inverse
a  (a )  (a)  a  0
a 1  1 a  a
1
1
a
 a  1
a
a
ab  ba
a  0
Commutative a  b  b  a
 a  b   c  a  b  c   ab c  a bc 
Associative
Distributive Property
a  b  c   ab  ac
Multiplicative Property of Zero
a  0  0 or 0  a  0
Substitution Property
If a  b , then b can replace a
in a given equation or inequality.
Reflexive Property
aa
Symmetric Property
If a  b , then b  a .
Transitive Property of Equality
If a  b and b  c , then a  c .
A solution to an equation is the value or set of values that can be
substituted to make the equation true.
The solution of an equation in one variable can be found by graphing the
expression on each side of the equation separately and finding the
x-coordinate of the point of intersection.
Solve Quadratic Equations Algebraically:
**Set trinomial equal to 0.
Factor.
Set each factor = 0 and SOLVE.
x  6 x  27
2
x  6 x  27  0
( x  9)( x  3)  0
( x  9)  0 or ( x  3)  0
x  9
or
x  3
2
x  9,  3
9,0
are called zeros or solutions.
and  3,0 are x-intercepts of the graph.
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**Apply the Quadratic Formula: x2  6 x  27  0
x 
x 
x 
x 
(6) 
(6) 2  4(1)(27)
2(1)
6 
36  108
2
6 
144
2

6  12
2
18
6
 9 or x 
 3
2
2
The following statements are equivalent:
h is a zero of the polynomial function f ;
x
 h  is a factor of f ( x) ;
h is a solution of the polynomial equation f ( x)  0 ; and
h is an x-intercept for the graph of y  f ( x) .
Solve Quadratic Equations Graphically:
**Graph the function  answers are where the graph crosses the x-axes.
x  9,  3 are called zeros, solutions, x-intercepts of the function.
**PlySmlt2 APP  answers are given.
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A solution to a system of equations is
an ordered pair that satisfies both equations and
the point of intersection on the graph.
Parallel Lines
The lines coincide.
Set builder notation may be used to represent solution sets of equations
and inequalities.
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Use the TI-84 Plus calculator to find accurate points of intersection for two graphs.
Adapted from Jeff McCalla and C. C. Edwards from Ti-84 Plus Graphing Calculator For Dummies, 2nd Edition
Using the arrow keys in a graph activates a free-moving trace. However, using a freemoving trace rarely locates the point of intersection of two graphs but instead gives you
an approximation of that point.
To accurately find the coordinates of the point where two functions intersect, perform the
following steps:
1. Graph the functions in a viewing window that contains the point of intersection
of the functions.
2. Press [2nd][TRACE] to access the Calculate menu.
3. Press [5] to select the intersect option.
4. Select the first function.
If the name of one of the intersecting functions does not appear in the border at the top
of the screen, repeatedly press the up- and down-arrow keys until it does. This is
illustrated in the first screen. When the cursor is on one of the intersecting functions,
press [ENTER] to select it.
5. Select the second function.
If the calculator does not automatically display the name of the second intersecting
function in the border at the top of the screen, repeatedly press the up- and downarrow keys until it does. This is illustrated in the second screen. When the cursor is on
the second intersecting function, press [ENTER] to select it.
6. Use the right- and left-arrow keys to move the cursor as close to the point of
intersection as possible.
This is illustrated in the third screen.
7. Press [ENTER] to display the coordinates of the point of intersection.
SHORTCUT TIP:
 If there are only two functions in the Y= editor, you can save time by pressing
[2nd][TRACE][ENTER][ENTER] to choose your functions.
 If there is only one point of intersection of the two functions, then press [ENTER] again to
calculate the point of intersection. It is only necessary to make a guess when there is more than one
point of intersection.
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A.5
The student will solve multistep linear inequalities in two variables, including
a) solving multistep linear inequalities algebraically and graphically;
b) justifying steps used in solving inequalities, using axioms of inequality and
properties of order that are valid for the set of real numbers and its subsets;
c) solving real-world problems involving inequalities; and
d) solving systems of inequalities.
A solution to an inequality is the value or set of values that can be
substituted to make the inequality true.
When you divide or multiply BOTH sides of an inequality by a NEGATIVE #
 switch the symbol.
2 x  14
2 x
14

2
2
x  7
A.6
The student will graph linear equations and linear inequalities in two variables,
including
a) determining the slope of a line when given an equation of the line, the
graph of the line, or two points on the line. Slope will be described as rate
of change and will be positive, negative, zero, or undefined; and
b) writing the equation of a line when given the graph of the line, two points
on the line, or the slope and a point on the line.
The graph of a line represents the set of points that satisfies the equation
of a line.
A line can be represented by its graph or by an equation.
Slope-intercept Form of a line: y  mx  b
m 
slope, b  y-intercept
Point-Slope Form of a line: y  y1  m  x  x1  m  slope,  x1 , y1  is the pt
Standard Form of a line: Ax  By  C
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Slope is the ratio of vertical change to horizontal change.
Slope 
y  y1
rise
 2
;
run
x2  x1
x2  x1
HOY : Horizontal Line  Zero Slope  y  a number
VUX : Vertical Line  Undefined Slope  x  a number
Positive Slope  increases/rises from left to right
Negative Slope  decreases/falls from left to right
Parallel lines have equal slopes
The product of the slopes of perpendicular lines is 1 unless one of the
lines has an undefined slope.
The graph of the solutions of a linear inequality is a half-plane bounded
by the graph of its related linear equation. Points on the boundary
are included unless it is a strict inequality.
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How to Start and Quit the “Inequalz” APP on the TI-84 Plus
Adapted from Ti-84 Plus Graphing Calculator For Dummies, 2nd Edition By Jeff McCalla and C. C. Edwards
To start the Inequality app, press [APPS].
Then, press [ALPHA] [x2]
if necessary,
use the down-arrow key to move the cursor to the Inequalz app, and press [ENTER]
to select the app. The official name of the app is the Inequality Graphing app.
After choosing the app, you are confronted with one of the last two screens shown here.
If no other apps are running, you see the second screen.
Press any key to enter the Inequality Graphing app.
If Inequality Graphing is already running, you see the third screen.
Press [1] to re-enter the app.
After you enter Inequality Graphing, you are placed in the Y= editor so that you can enter
functions and inequalities.
The functions previously housed in this editor appear on the screen along with the
inequality symbols at the bottom of the screen, as illustrated in the first screen.
Cursor is on the equal sign.
Cursor is NOT on the equal sign.
If you move the cursor so that it is not on an equal sign, the inequality symbols at the
bottom of the screen vanish, as in the second image in this figure.
On the TI-84 Plus C, Context Help in the border at the top of the screen reads,
"Select Relation – Press Alpha f1–f5."
If you forget a step while using this app, use the Context Help to guide you!
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Most of the time, you don't even know the Inequality Graphing app is running unless you
are actively using the app or unless you place the cursor on an equal sign in the
Y= editor to display the inequality symbols, as in the first screen.
Example: To graph the system of inequalities:
Y1  2 x  1
Y2   3 x  5
.
Move the cursor so that it is on the equal sign, press Alpha f2 for <.
Key 2 x  1 in as your normally would.
Move the cursor so that it is on the equal sign, press Alpha f5 for  .
Key 3x  5 in as your normally would.
Press Graph.
To quit (exit) this app on the TI-84 Plus C,
On the TI-84 Plus, press [APPS] choose Inequalz and press [2] to quit the app.
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Reporting Category: Functions and Statistics
Number of Items: 20
Standards of Learning:
A.7
The student will investigate and analyze function (linear and quadratic)
families and their characteristics both algebraically and graphically, including
a) determining whether a relation is a function;
b) domain and range;
c) zeros of a function;
d) x- and y-intercepts;
e) finding the values of a function for elements in its domain; and
f) making connections between and among multiple representations of
functions including concrete, verbal, numeric, graphic, and algebraic.
Each element in the domain of a relation is the abscissa of a point of the
graph of the relation.
Each element in the range of a relation is the ordinate of a point of the
graph of the relation.
The values of f ( x) are the ordinates of the points of the graph of f.
Relation  a set of ordered pairs
Function  a relation where the
 Domain (x) does not repeat
 Graph passes the Vertical Line Test (vertical line intersects
graph at exactly one point)
An object x in the domain of f is an x-intercept or a zero of a function f
if and only if f ( x)  0 .
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A.8
The student, given a situation in a real-world context, will analyze a relation to
determine whether a direct or inverse variation exists, and represent a direct
variation algebraically and graphically and an inverse variation algebraically.
Direct Variation: If y varies directly with x, y  kx and k 
y
; k  0.
x
As x increases, y increases.
As x decreases, y decreases.
Graph is a line through (0, 0) with positive slope  y  mx  0 .
Inverse Variation: If y varies inversely with x, y 
k
x
and k  xy ; k  0 .
As x increases, y decreases.
As x decreases, y increases.
Graph contains 2 curves that are reflections of each other.
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A.9
The student, given a set of data, will interpret variation in real-world contexts
and calculate and interpret mean absolute deviation, standard deviation, and
z-scores.
Descriptive statistics may include measures of center and dispersion.
Variance, standard deviation, and mean absolute deviation measure the
dispersion of the data.
The sum of the deviations of data points from the mean of a data set is 0.
Standard deviation
is expressed in the original units of measurement of the data.
addresses the dispersion of data about the mean.
is calculated by taking the square root of the variance.
The greater the value of the standard deviation, the further the data tend
to be dispersed from the mean.
For a data distribution with outliers, the mean absolute deviation may be
a better measure of dispersion than the standard deviation or variance.
A z-score (standard score) is a measure of position derived from the
mean and standard deviation of data.
A z-score derived from a particular data value tells how many standard
deviations that data value is above or below the mean of the data
set. It is positive if the data value lies above the mean and negative
if the data value lies below the mean.
a measure of the spread of a data set
a measure of the spread of a data set
The number of standard deviations
an element is away from the mean.
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Using the TI-84 to calculate the Standard Deviation of data.
Example: Find the Standard Deviation of the following set of data:
{3, 5, 7, 10, 11, 10, 5, 25, 5, 12}
1. Enter the data into L1.
From the home screen, press
To enter the data into L1,
.
on “1:Edit”.
In L1 enter each element of the data set.
Enter the first element and press ENTER to move the next line.
Continue until all of the elements have been entered into L1.
2. Calculate standard deviation.
Press
.
Press ENTER on option “1:1 –Var Stats”.
Press ENTER to compute the 1-variable statistics.
The calculator defaults to L1.
Enter the list name after 1-Var Stats if
data are in another list.)
Enter on “Calculate”.
Standard Deviation  5.98
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A.10 The student will compare and contrast multiple univariate data sets, using
box-and-whisker plots.
Measures of Central Tendency:
Mean  average of data
(add up all of the data and divide by the # of objects)
Median  middle # when data are in order of least to greatest
Mode  most frequently occurring data
Range of Data  maximum data – minimum data
Box-and-Whisker Plot  read directions & look at the graphs
IQR = Inner Quartile Range = Q3 – Q1
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A.11 The student will collect and analyze data, determine the equation of the curve
of best fit in order to make predictions, and solve real-world problems, using
mathematical models. Mathematical models will include linear and quadratic
functions.
Curve of Best Fit  graph that goes through the most points
Regression on the TI-84
To find the Equation of BEST fit on the TI-84…
1.
Enter your values:
Press STAT choose 1: EDIT
2.
Type x – values into L1
Type y – values into L2
Display your data: Turn the STATPLOT on.
Press 2ND Y= Choose 1: Plot1
ENTER on “On”.
Choose Scatterplot… Xlist:L1… Ylist:L2
Press ZOOM9: ZoomStat to graph the data in the correct window.
3.
Choose your function type:
Press STAT CALC 4: LinReg
Your Calculator should read “LinReg (ax+b)”
Press STAT CALC 5: QuadReg
Your Calculator should read “QuadReg”
Enter your lists: L1 and L2 are chosen by default.
Have the calculator store the equation in Y1.
Type , VARS  Y-VARS ENTER on 1: Function ENTER on 1: Y1
Press ENTER to have TI-84 calculate the curve of BEST fit.
4. Write your equation. Round according to the directions.