Curriculum Framework-Syllabus
_____________ALGEBRA II______________
(course or subject title)
PART I
DESIRED RESULTS:
SAS - Curriculum Framework (Algebra II 8-12 )
Big Idea
Numbers,
measures,
expressions,
equations, and
inequalities can
represent
mathematical
situations and
structures in
many
equivalent
forms.
Essential Questions
How can you extend
algebraic properties and
processes to quadratic,
exponential and polynomial
expressions and equations
and then apply them to
solve real world problems?
Concepts
Competencies
Algebraic
properties,
processes and
representations
Extend algebraic
properties and
processes to
quadratic,
exponential, and
polynomial
expressions and
equations and to
matrices, and apply
them to solve real
world problems.
Exponential
functions and
equations
Extend algebraic
properties and
processes to
quadratic,
exponential, and
polynomial
What are the
advantages/disadvantages
of the various methods to
represent exponential
functions (table, graph,
Patterns exhibit equation) and how do we
relationships
choose the most
that can be
appropriate
extended,
representation?
described, and
How do quadratic
generalized.
equations and their graphs
Relations and
and/or tables help us
functions are
interpret events that occur
mathematical in the world around us?
relationships
How do you explain the
that can be
benefits of multiple
represented
methods of representing
and analyzed
polynomial functions
using words,
tables, graphs, (tables, graphs, equations,
and equations. and contextual situations)?
There are some
mathematical
relationships
that are always
true and these
relationships
Standards /
Eligible
Content
2.1.A2.A,
2.1.A2.B,
2.1.A2.D,
2.1.A2.F,
2.2.A2.C,
2.3.A2.C,
2.3.A2.E,
2.5.A2.A,
2.8.A2.B,
2.8.A2.C,
2.8.A2.E,
A2.1.1.1.2,
A2.1.1.2.1,
A2.1.1.2.2,
A2.1.2.1.1,
A2.1.2.1.2,
A2.1.2.1.3,
A2.1.2.1.4,
A2.1.2.2.1,
A2.1.2.2.2,
A2.2.1.1.1,
A2.2.1.1.2,
A2.1.3.1.1,
A2.1.3.1.2,
A2.1.3.1.3,
A2.1.3.2.1,
A2.1.3.2.2
2.1.A2.A,
2.1.A2.B,
2.1.A2.D,
2.1.A2.F,
2.2.A2.C,
2.3.A2.C,
are used as the
rules of
arithmetic and
algebra and are
useful for
writing
equivalent
forms of
expressions and
solving
equations and
inequalities.
Mathematical
functions are
relationships
that assign each
member of one
set (domain) to
a unique
member of
another set
(range), and the
relationship is
recognizable
across
representations.
Families of
functions
exhibit
properties and
behaviors that
can be
recognized
across
representations.
Functions can
be transformed,
combined, and
composed to
create new
functions in
mathematical
and real world
Quadratic
functions and
equations
expressions and
equations and to
matrices, and apply
them to solve real
world problems.
2.3.A2.E,
2.5.A2.A,
2.8.A2.B,
2.8.A2.C,
2.8.A2.E,
A2.1.1.1.2,
A2.1.1.2.1,
A2.1.1.2.2,
A2.1.2.1.1,
A2.1.2.1.2,
A2.1.2.1.3,
A2.1.2.1.4,
A2.1.2.2.1,
A2.1.2.2.2,
A2.2.1.1.1,
A2.2.1.1.2,
A2.1.3.1.1,
A2.1.3.1.2,
A2.1.3.1.3,
A2.1.3.2.1,
A2.1.3.2.2
Extend algebraic
properties and
processes to
quadratic,
exponential, and
polynomial
expressions and
equations and to
matrices, and apply
them to solve real
world problems.
2.1.A2.A,
2.1.A2.B,
2.1.A2.D,
2.1.A2.F,
2.2.A2.C,
2.3.A2.C,
2.3.A2.E,
2.5.A2.A,
2.8.A2.B,
2.8.A2.C,
2.8.A2.E,
A2.1.1.1.2,
A2.1.1.2.1,
A2.1.1.2.2,
A2.1.2.1.1,
A2.1.2.1.2,
A2.1.2.1.3,
A2.1.2.1.4,
A2.1.2.2.1,
A2.1.2.2.2,
A2.2.1.1.1,
A2.2.1.1.2,
A2.1.3.1.1,
situations.
Bivariate data
can be modeled
with
mathematical
functions that
approximate
the data well
and help us
make
predictions
based on the
data.
A2.1.3.1.2,
A2.1.3.1.3,
A2.1.3.2.1,
A2.1.3.2.2
Polynomial
functions and
equations
Extend algebraic
properties and
processes to
quadratic,
exponential, and
polynomial
expressions and
equations and to
matrices, and apply
them to solve real
world problems.
2.1.A2.A,
2.1.A2.B,
2.1.A2.D,
2.1.A2.F,
2.2.A2.C,
2.3.A2.C,
2.3.A2.E,
2.5.A2.A,
2.8.A2.B,
2.8.A2.C,
2.8.A2.E,
A1.2.1.1.1,
A2.1.1.1.2,
A2.1.1.2.1,
A2.1.1.2.2,
A2.1.2.1.1,
A2.1.2.1.2,
A2.1.2.1.3,
A2.1.2.1.4,
A2.1.2.2.1,
A2.1.2.2.2,
A2.2.1.1.2,
A2.1.3.1.1,
A2.1.3.1.2,
A2.1.3.1.3,
A2.1.3.2.2
Algebraic
properties,
processes and
representations
Represent
exponential
functions in
multiple ways,
including tab les ,
graphs, equations,
and contextual
situations, and
make connections
among
representations;
relate the
2.1.A2.F,
2.3.A2.E,
2.6.A2.C,
2.8.A2.B,
2.8.A2.D,
2.8.A2.E,
2.8.A2.F,
2.11.A2.A,
2.11.A2.B,
A2.1.2.1.4,
A2.2.1.1.3,
A2.2.1.1.4,
Degree and
direction of
linear
association
between two
variables is
measurable.
growth/decay rate
of the associated
exponential
equation to each
representation.
A2.2.2.1.2,
A2.2.2.1.3,
A2.2.2.1.4,
A2.2.2.2.1,
A2.1.3.1.3,
A2.1.3.1.4,
A2.2.3.1.1
Exponential
functions and
equations
Represent
exponential
functions in
multiple ways,
including tab les ,
graphs, equations,
and contextual
situations, and
make connections
among
representations;
relate the
growth/decay rate
of the associated
exponential
equation to each
representation.
2.1.A2.F,
2.3.A2.E,
2.6.A2.C,
2.8.A2.B,
2.8.A2.D,
2.8.A2.E,
2.8.A2.F,
2.11.A2.A,
2.11.A2.B,
A2.1.2.1.4,
A2.2.1.1.3,
A2.2.1.1.4,
A2.2.2.1.2,
A2.2.2.1.3,
A2.2.2.1.4,
A2.2.2.2.1,
A2.1.3.1.3,
A2.1.3.1.4,
A2.2.3.1.1
Algebraic
properties,
processes and
representations
Represent a
quadratic function
in multiple ways,
including tab les ,
graphs, equations,
and contextual
situations, and
make connections
among
representations;
relate the solution
of the associated
quadratic equation
to each
representation.
2.3.A2.E,
2.6.A2.C,
2.8.A2.B,
2.8.A2.D,
2.8.A2.E,
2.8.A2.F,
2.11.A2.A,
A2.2.1.1.3,
A2.2.1.1.4,
A2.2.2.1.1,
A2.2.2.1.3,
A2.2.2.1.4,
A2.2.2.2.1,
A2.1.3.1.1,
A2.1.3.1.2,
A2.2.3.1.1
Quadratic
functions and
equations
Represent a
quadratic function
in multiple ways,
including tab les ,
graphs, equations,
and contextual
situations, and
make connections
among
representations;
relate the solution
of the associated
quadratic equation
to each
representation.
2.3.A2.E,
2.6.A2.C,
2.8.A2.B,
2.8.A2.D,
2.8.A2.E,
2.8.A2.F,
2.11.A2.A,
A2.2.1.1.3,
A2.2.1.1.4,
A2.2.2.1.1,
A2.2.2.1.3,
A2.2.2.1.4,
A2.2.2.2.1,
A2.1.3.1.1,
A2.1.3.1.2,
A2.2.3.1.1
Algebraic
properties,
processes and
representations
Represent a
polynomial function
in multiple ways,
including tab les ,
graphs, equations,
and contextual
situations, and
make connections
among
representations;
relate the solution
of the associated
polynomial
equation to each
representation.
2.1.A2.B,
2.3.A2.E,
2.6.A2.C,
2.8.A2.B,
2.8.A2.D,
2.8.A2.E,
2.8.A2.F,
2.11.A2.A,
A2.1.2.2.1,
A2.1.2.2.2,
A2.2.1.1.3,
A2.2.1.1.4,
A2.2.2.1.3,
A2.2.2.1.4,
A2.2.2.2.1,
A2.1.3.1.1,
A2.1.3.1.2,
A2.2.3.1.1
Polynomial
functions and
equations
Represent a
polynomial function
in multiple ways,
including tab les ,
graphs, equations,
and contextual
2.1.A2.B,
2.3.A2.E,
2.6.A2.C,
2.8.A2.B,
2.8.A2.D,
2.8.A2.E,
situations, and
make connections
among
representations;
relate the solution
of the associated
polynomial
equation to each
representation.
2.8.A2.F,
2.11.A2.A,
A2.1.2.2.1,
A2.1.2.2.2,
A2.2.1.1.3,
A2.2.1.1.4,
A2.2.2.1.3,
A2.2.2.1.4,
A2.2.2.2.1,
A2.1.3.1.1,
A2.1.3.1.2,
A2.2.3.1.1
PART II
FAIR ASSESSMENTS, EVIDENCE OF LEARNING:
(The following is an example of a mid-term (semester) and final exam
assessment that is not expected to be listed in the syllabus because it is the
exam itself. It is provided as an example on two ways to measure student
learning. The project summary report format listed below would be
included in the on-line syllabus because it is important information
regarding how the assessment will be graded).
Project Summary Report Format
All summary reports must use the format described below.
I. Title
II. Name/Class/Period
III. Materials (10 points)
List of all the items you used to complete the project.
IV. Problem (20 points)
Several sentences explaining the purpose of the project. The purpose
may be a connection or an example of Algebra II in the real world or just
a practice exercise of skills learned.
V. Procedure (20 points)
List all of the steps involved in completing the project in your own words
and using appropriate math terminology.
VI. Data (20 points)
Answer ALL of the questions in the project. Make sure these are
sequential with the project. Include any attachments such as data charts
and graphs.
VII. Conclusion (30 points)
Type two pages (12 pt. font; double spaced) on your personal
observations of the project. Was the project difficult or easy and why?
What did you learn? What would you like to see different for other
students? Any other questions or concerns?
Semester 1 Mini-Project
Problem:
We have been asked to advise the Calcutta Cutting Corners Corporation to stop cutting too many corners! They are
having financial difficulties because of the design of their packaging and the pending recession.
When deciding on the packaging of consumer items, important factors such as costs, maximum volume, appearance,
and uniqueness are considered. Many of these factors are mathematically driven, so you need to influence the
company on the proper packaging procedures. You also need to influence the unions to change how they work.
Equipment
- Graphing Tool (computer/calculator tool)
- 8 pieces of 8.5 x 11 paper
- Scissors
- Ruler
- Tape
- Pencil
Procedure
1. Cut each of the 8 pieces of paper into squares measuring 8 inches by 8 inches.
2. Using a pencil and ruler, mark each side off in 1/2 inch increments so that you have created a piece of
graph paper that has 16 squares by 16 squares. Repeat this procedure for all 8 pieces of paper.
3. Take one sheet of graph paper and cut out one square out of each of the 4 corners. Fold up the 4 sides to
create a box (without a lid) that has 14 squares across the bottom and has a height of 1 square up. You may
want to tape the sides in place. This is your first of 8 boxes.
4. Repeat step #3 cutting 1 inch squares from each of the 4 corners and folding up the sides. Continue with
the remaining 6 pieces of paper increasing the size of the squares that you will cut out by 1/2 inch. Can the
last piece of paper have 4 inch squares cut from each corner? Why or why not?
5. Create a data table like the example data table below.
Note: The measurements are in inches, not squares. Each square is 1/2 inch. Volume should be given in
cubic inches.
6. Find the length, width, height, and volume (V = lwh) of each of the 7 boxes, to complete the data table.
Questions
1. Which box has the maximum volume?
2. With the Graphing Tool, graph the data by placing the height on the horizontal axis and the volume on the
vertical axis. Use a range of x from -5 to 10 and y from -5 to 40. Title and label the axes of your graph. To
graph the data click on the button below.
I would recommend pasting the graph into a Word document. Then you can view it throughout the rest of the
activity. Choose File, Copy from the menu and then just paste it into a document that you can continuously
view.
Very Important: Make sure you use the "cubic" option for the line that best fits your graph. If the line fits, then
your data fits a cubic function.
3. What type of function is your graph? Why?
4. What is the equation of your function? Copy the equation and record the equation on a piece of paper or in
the Word document.
5. Use the function plotting of your Graphing Tool to graph the function. You can either paste the equation
into the tool or you can just type in the equation that you recorded in part 4. Use a range of x from -5 to 10
and y from -5 to 40. Again, I would recommend pasting the graph into a Word document. It will be easier to
compare the two graphs and easier to save and submit them.
6. Where does your function graph intersect the x-axis? Get the coordinates. Round your answers to 2
decimal places and remember, y=0 or close to it. These points are called the zeros of the function. Notice
that you are able to see the zeros in the graph of the function but not in the graph of your actual data. What is
the reason for that? Think about the differences between real data and the graph of an equation.
7. Locate the highest point on your graph using the tracker icon. What does this point represent?
8. What type of product could be packaged in the container that has the maximum volume? Why?
Algebra II Culminating Project
Project Summary Report Format
All summary reports must use the format described below.
I. Title
II. Name/Class/Period
III. Materials (10 points)
List of all the items you used to complete the project.
IV. Problem (20 points)
Several sentences explaining the purpose of the project. The purpose may be a connection or an example of
Algebra II in the world or just a practice exercise of skills learned in a module.
V. Procedure (20 points)
List all the steps involved in completing the project in your own words and using appropriate math
terminology.
VI. Data (20 points)
Answer ALL of the questions in the project. Make sure these are sequential with the project. Include any
attachments like data charts and graphs.
VII. Conclusion (30 points)
Type two pages (12 pt. font; double spaced) on your personal observations of the project. Was the project
difficult or easy and why? What did you learn? What would you like to see different for other students? Any
other questions or concerns?
EXERCISE I:
We might need to find housing for some of our student teachers, and we thought that the Tuckahoe Heights
Apartments might meet our immediate needs. You are to check the facilities and the community to see if our
employees could reside at Tuckahoe Heights temporarily.
Equipment: Calculator
Purpose: The purpose of this lab is to practice multiplying, dividing, adding, and subtracting polynomials with a purpose
in mind. This exercise does review all of these concepts. For the school district, the purpose here is to preview some
possible living space for its student teachers.
The formulas that you will need are perimeter equals the sum of twice the length and twice the width (P=2L+2W), and
area equals the product of the length and the width (A=LW).
Procedure:
Part 1: Using your knowledge of polynomial addition and subtraction (combining like terms), answer the following
questions. Write your work and answers in an Assignment Area labeled Part 1.
1. Find entire length of this two bedroom apartment in terms of x.
2. Find the entire width of this two-bedroom apartment a (not counting the patio) in terms of x. (Be careful, the width
of bath 1 is not given, but the width of the entry is the same as the width of the pantry. With that information, you
can find the width of the left hand side of the house.)
Part 2: Using your knowledge of the distributive property and polynomial addition and subtraction, answer the following
questions. (Remember, P=2L+2W.) Write your answers in the same assignment area labeled Part 2.
3. Find the perimeter of the entire 2-bedroom apartment (excluding the patio) in terms of x.
4. Find the perimeter of the living room and dining room in terms of x.
5. Find the perimeter of the master bedroom, excluding the sitting area and the closets.
Part 3: Using your knowledge of solving linear equations, answer the following questions. Write your work and answers in
the same assignment area labeled Part 3.
6. If the perimeter of the living room and dining room is 60 feet, what is the value of x?
7. If the perimeter of the master bedroom excluding the sitting area and the closets is 42 feet, what is the value of x?
8. If the length of the entire apartment is 57 feet, what is the value of x?
Part 4: You have reviewed polynomial multiplication and division in this module, practice now. (Remember, A=LW.)
9. Find the area of the entire living room/dining room.
10. Find the area of the patio.
11. Find the area of the entire apartment (excluding the patio). Note: don't forget to subtract for the cutout area in the
upper left corner.
12. If the area of the kitchen was 2x - 4, what would be the width? Write the expression or number for the width.
13. If the area of the dining room and living room is 300 sq. ft., what is the equation that would be used to solve for x?
Write the equation with zero on the right, ready to solve.
14. If the area of the entire apartment is 800 sq. ft., what is the equation that would be used to solve for x? Use the
length and width from questions 1 and 2. Do not worry about the area of the patio or the corner that is cut out of the
living room/dining room. Solve.
Part 5:
15. Find the area of your bedroom, Label it A. Using the dimensions of the Master Bedroom in Figure 1, Write an equation using
(x + 8) as length- L and (2x – 3) as the width-W and the actual area you found for your room-A. Manipulate the equation so
that it is set equal to zero, in the form: ax2+ bx + c = 0... DO NOT SOLVE!!!
EXERCISE II
Purpose: Application of conic sections.
Equipment: A graphing tool/program.
Problem
The orbit of a planet around the sun can be described as an ellipse with the sun as one of the foci. The point
in its orbit where a planet is closest to the sun is called the perihelion. The point at which the planet is
farthest from the sun is called the aphelion. There are several situations that may occur in a planet's orbit.
This is where we need your help.
Procedure and Questions
1. Make a graph of an ellipse that shows the location of the sun, a planet, the perihelion point and the
aphelion point. (Use the graphing tool and a paint program to label the points.)
2. Suppose a planet's distance from the sun at aphelion and at perihelion are the same. What is the
shape of the planet's orbit? Make a graph and label the points.
3. Suppose that the aphelion and perihelion both lie on the x-axis, and the perihelion is located on the
positive x-axis. Where is the aphelion located? Make a graph and label the points.
4. If the distance from a planet to the sun at aphelion is A, and the perihelion distance is P, what is the
distance from aphelion to the center?
5. What is the distance from the sun to the center in terms of A and P?
6. Why must the distances from D to the sun be greater than P and less than A?
7. Write a problem using coordinates for the planet D, Perihelion, Aphelion, the center of the ellipse, and
the Sun (focus). Write the equation of the ellipse that would support your coordinates. Graph the
ellipse.
EXERCISE III:
Purpose: Apply Parabolic Concepts
We are going to do some estimation. This is a
lesson to connect parabolic concepts to a
calculus concept of summation and area under
a curve. That concept develops into the concept
of the integral. You could say we are looking
into your future. You will be taking Calculus,
right?
Equipment:
15 rectangular paper strips 1/2 inch by 8 1/2 inches and the graph of the parabola
above.
Problem: I have drawn a curve (see above). You are to use it, and estimate the area
in three different ways.
Procedure:
Part 1:
a. Expand the graph above to fit on an 8 ½” x 11” piece of paper. Using your
paper strips, cut them to fit outside the curved area. They should all be 1/2 inch in
width, and you cut the length to fit. Line them up touching sides, with one end on the
horizontal line at the bottom, and you cut the top end to fit just outside the curve.
These paper strips must be rectangles. Don't make any sloped cuts. Calculate the
area of each rectangle and add them to find the estimated area of the parabolic
figure. This area will be larger than the actual area.
b. Using your paper strips, cut them to fit inside the curved area. They should all be
1/2 inch in width, and you cut the length to fit. Line them up with one end on the
horizontal line at the bottom, and you cut the top end, again, straight across.
Calculate the area of each inscribed rectangle and add them to find the
estimated area of the parabolic figure. This area will be smaller than the actual
area.
c. Find the average of the two areas.
d. Thinking: The average area will be a closer approximation of the actual area of the
parabolic figure. Let's think about the new area. It's the average of the two areas.
Formulas: A=bh or base times height is the formula for the area of each original
rectangle. That's the same as length times width, right? Now if we take the base of
the rectangles in the first estimate as b1 and the base of the rectangles in the second
estimate as b2, the average is then multiplied by h (height) we have A = ??? That
formula should ring a bell. If not, get out a math book, and look for geometric figures,
or area. This formula represents the area of what geometric figure?
e. Thinking: If the height (that's the half inch measure) gets smaller and smaller, will
the estimate get closer to the actual area?? That's calculus. The height approaches
zero and the number of figures used to estimate area approaches infinity.
Now lets use our Algebra II skills a bit.
Part 2:
Let's put an equation with that curve. Use y = x2 - 8x +7
a. What is the value of the discriminant?
b. What types of roots will this equation have? Ex: two complex roots, one repeated
root, two real rational roots or two real irrational roots?
c. Find the roots by factoring and solving.
d. Find the roots by using the quadratic equation.
e. Find the roots by graphing and using a graphing tool.
f. Find the y-intercept.
g. Complete the square, and write that form of the equation.
h. Find the vertex of the parabola.
i. Can you think of a reason why this equation could not possibly be the equation of
the curve you printed?
Answer all questions that are in bold print, and write a one-two page conclusion.
PART III
LEARNING EXPERIENCES, INSTRUCTION:
First-Degree Equations/Inequalities, Relations and Functions
Unit Length: 10 weeks
Content
 Expressions, formulas, order of
operations, real number
properties
Objectives/Activities
Instructional Strategies
(Students Will …)
(Teachers Will…)
Use a variety of literacy strategies
designed to engage students in
reading, writing, talking, listening,
and thinking. Suggested strategies
will include but are not limited to the
following:









 Equations/inequalities in one
variable, including absolute
value
Think/Pair/Share (daily)
Do Now and Ticket Out the
Door (daily)
Word Wall Activities
Sorter Activities
Comparison Alley
Collins Writing Type 1
and/or Type 2 (daily)
Text-rendering
Predicting/Summarizing
Anticipation guides
Assessments
Includes a minimum of 10 grades
per marking period from any of
the following:
homework
class participation
reflective journals
quizzes
tests
open-ended problems
group projects/presentations
quarterly benchmark
assessments
Semester Project - in the form of
a performance based assessment
 Relations and functions
Final Year-End Project - in the
form of a performance based
assessment
First-Degree Equations/Inequalities, Relations and Functions (cont.)
Content
Objectives/Activities
Instructional Strategies
(Students Will …)
(Teachers Will…)
 Perform operations with functions and
represent functions algebraically,
graphically, numerically, and verbally
 Use functions to interpret trends and
predict outcomes.
 D.1.1.2
Identify functional patterns based on the
mathematical definition of a function and
describe why it is a function; take the
patterns and describe in a threedimensional way in both graphic model and
real-life situations.
Use a variety of literacy strategies
designed to engage students in
reading, writing, talking, listening,
and thinking. Suggested
strategies will include but are not
limited to the following:
Assessments
Includes a minimum of 10 grades
per marking period from any of the
following:
homework
class participation
 Linear equations/inequalities
 Know and use vocabulary and symbols
related to linear equations/inequalities
 Graph the solution set of a linear
equation/inequality in the coordinate
plane using different methods.
 D.2.1.3
Write linear equations to represent
mathematical models of real-world
problems, solve the equations, and then
use the solutions to determine possible
solutions for the real-world problem.
 D.3.2.2
Compare/contrast the three different forms
of a linear equation using a comparison
alley, given equations in a variety of realworld situations.
 D.2.1.2
Complete tasks that enable them to graph
 Think/Pair/Share (daily)
 Do Now and Ticket Out the
Door (daily)
 Word Wall Activities
 Sorter Activities
 Comparison Alley
 Collins Writing Type 1 and/or
Type 2 (daily)
 Text-rendering
 Predicting/Summarizing
 Anticipation guides
reflective journals
quizzes
tests
open-ended problems
group projects/presentations
quarterly benchmark assessments
Semester Project - in the form of a
performance based assessment
solution sets of linear inequalities on a
coordinate plane, which also demonstrates
their proficiency in solving equations.
D.3.2.1
Discuss the concept of slope by relating it
to as many classroom objects that can be
generated; then apply slope to linear
algebraic expressions as well as real-world
applications.
 Slope and writing linear
equations
Final Year-End Project - in the
form of a performance based
assessment
First-Degree Equations/Inequalities, Relations and Functions (cont.)
Content
 Scatter plots and linear
regression
Objectives/Activities
Instructional Strategies
(Students Will …)
(Teachers Will…)
 E.4.2.1/E.4.2.2
Create a scatter plot given a real-world
problem or situations; then use a line of
best fit to make predictions.
Use a variety of literacy strategies
designed to engage students in
reading, writing, talking, listening,
and thinking. Suggested
strategies will include but are not
limited to the following:
 Systems of
equations/inequalities
 Linear Programming
Assessments
Includes a minimum of 10 grades
per marking period from any of the
following:
homework
class participation
 Write a system of linear
equations/inequalities that can be used
to solve a real-world problem.
 D.2.1.4
Compare/Contrast the three methods that
can be used to solve systems of equations
containing two equations in two variables.
 D.3.2.3/D.4.1.1
Extrapolate the slope and y-intercept given
the graph of a linear equation and generate
a table of coordinates that are contained on
the function.
 Think/Pair/Share (daily)
 Do Now and Ticket Out the
Door (daily)
 Word Wall Activities
 Sorter Activities
 Comparison Alley
 Collins Writing Type 1 and/or
Type 2 (daily)
 Text-rendering
 Predicting/Summarizing
 Anticipation guides
reflective journals
quizzes
tests
open-ended problems
group projects/presentations
quarterly benchmark assessments
Semester Project - in the form of a
performance based assessment
Final Year-End Project - in the
form of a performance based
assessment.
Polynomials, Rationals, Radicals, and Quadratic Equations/Inequalities
Unit Length: 10 weeks
Content
 Polynomials – adding,
subtracting, multiplying, dividing,
factoring
Objectives/Activities
Instructional Strategies
(Students Will…)
(Teachers Will…)
 Know and use vocabulary and symbols
related to working with polynomial
expressions/equations.
 D.2.2.1
Add, subtract, multiply, divide, and factor
binomials and trinomials to reduce to
simplest form.
Use a variety of literacy strategies
designed to engage students in
reading, writing, talking, listening,
and thinking. Suggested
strategies will include but are not
limited to the following:
Assessments
Includes a minimum of 10 grades
per marking period from any of the
following:
homework
 Polynomial equations
 Rational expressions/equations
 Write polynomial equations and solve by
factoring.
 Write rational expressions/equations and
solve.
 D.2.2.3
Apply the number properties to algebraic
rational expressions and will
compare/contrast the use of the properties
with real numbers versus polynomial
expressions.
 A.2.1.2
Solve real-world problems using
direct/joint/inverse variations and
proportions.
 Direct/Joint/Inverse variation
 Think/Pair/Share (daily)
 Do Now and Ticket Out the
Door (daily)
 Word Wall Activities
 Sorter Activities
 Comparison Alley
 Collins Writing Type 1 and/or
Type 2 (daily)
 Text-rendering
 Predicting/Summarizing
 Anticipation guides
class participation
reflective journals
quizzes
tests
open-ended problems
group projects/presentations
quarterly benchmark assessments
Semester Project - in the form of a
performance based assessment
 Radical expressions
A.1.1.1/A.1.1.3/A.1.3.1
Simplify and use numerical expressions
involving square roots and locate on a
number line
 Rational exponents
 Discuss the connections between radical
expressions and rational exponents.
Rewrite/simplify radical expressions by
using rational exponents and vice versa.
Final Year-End Project - in the
form of a performance based
assessment
Polynomials, Rationals, Radicals, and Quadratic Equations/Inequalities (cont.)
Content
 Radical equations/inequalities
 Complex numbers
Objectives/Activities
Instructional Strategies
(Students Will…)
(Teachers Will…)
 Write and solve radical
equations/inequalities and graph solution
sets in the coordinate plane.
Use a variety of literacy strategies
designed to engage students in
reading, writing, talking, listening,
and thinking. Suggested
strategies will include but are not
limited to the following:
 Know and use the sets of imaginary and
complex numbers in addition to the set of
real numbers used in unit 1 and reason
why each set is needed.
 Make connections between the sets of
numbers.
 Know and use vocabulary, phrases, and
symbols related to the complex number
system.
 D.2.1.5/D.2.2.2
Review, compare, and apply the methods
of factoring in order to solve quadratic
equations that are factorable, including
differences of squares and trinomials
 Quadratic functions/inequalities
 Solve quadratic equations using
factoring, the quadratic formula,
completing the square, or technology.
 Write quadratic equations/inequalities
that could model real-world problems
and then solve.
 Compare/contrast the different methods
for solving quadratic equations using a
comparison alley.
Assessments
Includes a minimum of 10 grades
per marking period from any of the
following:
homework
 Think/Pair/Share (daily)
 Do Now and Ticket Out the
Door (daily)
 Word Wall Activities
 Sorter Activities
 Comparison Alley
 Collins Writing Type 1 and/or
Type 2 (daily)
 Text-rendering
 Predicting/Summarizing
 Anticipation guides
class participation
reflective journals
quizzes
tests
open-ended problems
group projects/presentations
quarterly benchmark assessments
Semester Project - in the form of a
performance based assessment
Final Year-End Project - in the
form of a performance based
assessment
Exponential/Logarithmic Relations and Advanced Functions/Relations
Unit Length: 6 weeks
Content
 Exponential relations
 Logarithms
Objectives/Activities
Instructional Strategies
(Students Will…)
(Teachers Will…)
 Know and use the vocabulary and
symbols related to exponentials and
logarithms.
 Compare/contrast exponential relations
and logarithmic relations using a
comparison alley.
 Simplify expressions involving
exponentials and logarithms.
 Write exponential/logarithmic equations
to model real-world problems.
Use a variety of literacy strategies
designed to engage students in
reading, writing, talking, listening,
and thinking. Suggested
strategies will include but are not
limited to the following:
Assessments
Includes a minimum of 10 grades
per marking period from any of the
following:
homework
 Solve and graph the solution sets of
exponential and logarithmic
equations/inequalities.
 Exponential and logarithmic
equations/inequalities
 Compare/contrast common logarithms
and natural logarithms using a
comparison alley.
 Think/Pair/Share (daily)
 Do Now and Ticket Out the
Door (daily)
 Word Wall Activities
 Sorter Activities
 Comparison Alley
 Collins Writing Type 1 and/or
Type 2 (daily)
 Text-rendering
 Predicting/Summarizing
 Anticipation guides
class participation
reflective journals
quizzes
tests
open-ended problems
group projects/presentations
quarterly benchmark assessments
 Base e and natural logarithms
 Evaluate expressions and solve
equations/inequalities using natural
logarithms.
 Use logarithms to solve real-world
problems involving growth and decay.
Semester Project - in the form of a
performance based assessment
 Exponential growth and decay
 Special Functions (step,
constant, identity, piecewise,
absolute value, etc.)
 Compare/contrast the various special
functions and graph/sketch on a
coordinate plane with/without technology.
 Analyze the graphs of special functions
in order to interpret trends and predict
outcomes.
Final Year-End Project - in the
form of a performance based
assessment
Conic Sections
Unit Length: 4 weeks
Content

Parabolas 
 Hyperbolas
 Circles
 Ellipses
Objectives/Activities
Instructional Strategies
(Students Will…)
(Teachers Will…)
C.3.1.1
Calculate the distance and/or
midpoint between 2 points on a
coordinate plane.
 Write and graph equations of
parabolas, hyperbolas, circles,
and ellipses.
 Identify conic sections and
compare/contrast each type.
 Apply conic sections to realworld problems.
Use a variety of literacy
strategies designed to
engage students in reading,
writing, talking, listening,
and thinking. Suggested
strategies will include but
are not limited to the
following:
Assessments
Includes a minimum of
10 grades per marking
period from any of the
following:
homework
 Think/Pair/Share (daily)
 Do Now and Ticket Out
the Door (daily)
 Word Wall Activities
 Sorter Activities
 Comparison Alley
 Collins Writing Type 1
and/or Type 2 (daily)
 Text-rendering
 Predicting/Summarizing
 Anticipation guides
class participation
reflective journals
quizzes
tests
open-ended problems
group
projects/presentations
quarterly benchmark
assessments
Semester Project - in the
form of a performance
based assessment
Final Year-End Project in the form of a
performance based
assessment.
Part IV
VOCABULARY:
Distributive Properties
The Symmetric Property of
Equality
Vertical Line Rule
The Commutative Property of
Addition
The Reflexive Property of
Equality
Slope-Intercept form of a
Linear Equation
The Commutative Property of
Multiplication
Order of Operations
Point-Slope form of a
Linear Equation
The Associative Property of
Addition
<=, >=, <, >
y-coordinate
The Associative Property of
Multiplication
Combined Inequality
x-coordinate
The Identity Property of Addition
Relation
x-intercept
The Identity Property of
Multiplication
Function
y-intercept
The Inverse Property of Addition
(Additive Inverse)
Direct variation
Parallel Slope
The Inverse Property of
Multiplication (Multiplicative
Inverse)
Inverse Variation
Perpendicular Slope
The Transitive Property of
Equality
Joint Variation
Cartesian Coordinate
System
Variables
Factoring
Term
Greatest Common Factor
Like Terms
Difference of Squares
Constant
Sum of Squares
Degree of a Polynomial
Perfect Square Trinomial
Polynomial
Difference of Cubes
Monomial
Sum of Cubes
Binomial
Factor by Grouping
Trinomial
Quadratic Equation
Synthetic Division
Pure Quadratic Equation
Axis of Symmetry
Perfect Square Root
Discriminate
Perfect Cube Root
Complete the Square
Prime Number
Rational Numbers
Simplify a Radical
Irrational Numbers
Rationalize a Denominator
Complex Numbers
Quadratic Formula
Imaginary Number
Conjugate
Value of i2
Parabola
Vertex of a Parabola
Matrix
Systems of Equations
Element of a Matrix
Solving Systems Graphically
Dimensions of a Matrix
Solving Systems Algebraically
Column Matrix
Solving Systems by Substitution
Row Matrix
Solving Systems by Addition
Square Matrix
Least Common Multiple
Scalar Multiplication
Systems of Inequalities
Identity Matrix
Slope of a Line (m)
Inverse Matrix
y-intercept of a line (b)
Solving Systems by the Inverse Matrix
Method
Minor Axis of an Ellipse
Conic Section
Foci of an Ellipse
Circle
Equation of a Parabola (General Form)
Equation of a circle (General Form)
Focus of a Parabola
Radius of a Circle
Directrix of a Parabola
Center of a Circle
Hyperbola
SOHCAHTOA
Transverse Axis of a Hyperbola
Pythagorean Formula
Conjugate Axis of a Hyperbola
Ellipse
Foci of a Hyperbola
Equation of an Ellipse (General Form)
Aphelion
Major Axis of an Ellipse
Perihelion
Extraneous Root
Rational Expression
Remainder
Restrictions on the Domain of a Rational
Expression
Intermediate Value Theorem
Multiplying Rational Expressions
Fundamental Theorem of Algebra
Dividing Rational Expressions
Descartes' Rule of Signs
Reciprocal of a Rational Expression
Even Function
Complex Fraction
Odd Function
Least Common Denominator
Increasing Interval
Adding Rational Expressions
Decreasing Interval
Vertical Asymptote
Domain
Hole
Range
Horizontal Asymptote
Upper & Lower Bounds
Antilog
Radical Form of an Expression
Antiln (ex)
Logarithmic Form of an Expression
Product Property of Logarithms
Exponential Notation
Quotient Property of Logarithms
Natural Log (ln)
Power Property of Logarithms
Base of a Logarithm
Permutations
Median
Sequence
Factorial
Mode
Arithmetic Sequence &
Formula
Combinations & Formula Quartile
Series
Probability
Deviation
Arithmetic Series &
Formula
Experimental Probability
Standard Deviation
Summation & Notation
Theoretical Probability
Variance
Geometric Sequence &
Formula
Random Sample
Measures of Central
Tendency
Common Ratio
Stratified Sample
Box & Whisker Plot
Geometric Series &
Formula
Statistics
Histogram
Fundamental Counting
Principle
Mean
Stem & Leaf Plot
Part V
MATERIALS:
Textbook title, supplemental names of resources, websites, etc.
Part VI
INTERVENTIONS:
All teachers will access, utilize and be afforded the opportunity to contribute to the
Commonwealth’s intervention website at:
http://www.pdesas.org/main/fileview/Sec-RtII-Tier1.pdf
Secondary Response to Instruction and Intervention (RtII)
Tier 1 Core Instruction
Introduction and Purpose of the Document
Response to Instruction and Intervention (RtII) is an assessment and instruction framework for
conceptualizing, organizing, and implementing Pennsylvania’s Standards Aligned System (SAS). The
overarching goal of RtII is to improve student achievement using research-based curriculum, instructional
practices, and tiered interventions matched to the assessed needs of students.
A robust core curriculum (Tier 1) is central to Pennsylvania’s Secondary Response to Instruction and
Intervention framework. The purpose of this document is to provide supplementary information and
guidance to Pennsylvania secondary educators on the definition of the development and implementation
of Tier 1 core instruction.
About the Secondary RtII “Core” (Tier 1)
First, secondary schools must identify the core curriculum that serves as the foundation in each secondary
classroom. The curriculum in all subject areas must be aligned with and address Pennsylvania standards
for reading, writing, speaking and listening (i.e., Reading/Language Arts courses), as well as other content
area standards including science, math, and social studies. RtII implementation efforts must consider the
course content and focus for Language Arts/Communication Arts/English courses, as well as the
integration of a literacy curriculum across all subject areas. Additionally, the secondary foundational core
includes the adoption and use of universal instructional design principles and high leverage instructional
strategies across all subject areas. These practices provide instructional continuity across the content areas
and promote high levels of achievement for all students.
While strong Tier 1 “core” curriculum and instructional practice represent the starting point and
foundation of Secondary RtII, an infrastructure, consistent with effective middle and high school
practices, must be in place. Team processes, schedules, and course requirements, and other secondary
contextual variables are important to the successful implementation of the Tier 1 core and the overall RtII
framework. Formative and benchmark assessment procedures must be in place to monitor student
learning in core instruction to ensure the fidelity and effectiveness of instructional practice.
Document Focus
This document highlights universal instructional design principles and several high impact strategies from
the research literature. It is not intended as an exhaustive list but rather serves as a starting point for
schools/districts seeking to initiate Secondary RtII. Since RtII represents systems change and must be
implemented strategically and over time, we encourage schools to select and implement a “critical few”
of these strategies (at a time) deeply and with fidelity.