Download Level Distinctions - South Orange Maplewood School District

School District of South Orange and Maplewood
Mathematics Department
Appendix C: Level Distinctions in Algebra 1
Leveling is not a function of intelligence or mathematical talent or the ability to learn. Leveling in math begins
with a consideration of the mathematical content that has to be developed, takes a measure of students’ prior
learning, and enacts a plan to maximize learning across a spectrum of student achievement.
This appendix addresses the first of two categorical distinctions regarding the variation in students’ preparation to do mathematics in the various levels. This
appendix describes the parameters of content by level for each learning objective based on the content outline. At each level, students will address every
learning objective prescribed by State Standards. Variations and modifications in the content outline are based on evidence of foundational knowledge, the
instructional time required for students to obtain mastery of essential aspects of content, and the opportunity to make mathematical decisions as the content
is developed. The mathematics identified here establishes the expectations for mastery in each level.
In a broader sense, the second distinction addresses the Mathematical Practices, that is, the processes and strategic competencies students need to use in
order to think mathematically and develop mathematical concepts and skills. The curriculum document which will be submitted for adoption in July will
describe those processes and instructional approaches and generally describe differences within and across levels. The Mathematical Practices are defined
and required by national and state standards.
Mathematics Curriculum: Algebra
Objective
1. Approach solving
equations as a
process of reasoning
and explaining the
reasoning.
Solve single-variable
linear equations with
rational coefficients.
Construct viable
arguments to justify
solution methods.
Recognize, express
and solve problems
that can be modeled
using single-variable
linear equation
NJCCSS: N-Q-1,2,3;
A-SSE-1,2;A-CED1,2,3; A-REI-1,3
2. Recognize and
solve real world
problems involving
proportional reasoning
and rational numbers,
including percent
problems.
NJCCSS: N-Q 1
May 17, 2012 am
Level 2
Level 3
Level 4
Content Outline
*1-2 weeks spent
generating the knowledge
of “doing-undoing” via
tasks with increasingly
higher levels of demand.
*1-2 weeks spent
working with equivalence
and balance to
conceptual fair-moves in
algebraic manipulation.
*1-2 weeks time
practicing the skill of
solving such equations,
with connections to above
concept attainment.
*1 week on word
problem, initially with
repeated problems with
similar to contexts as an
access point to the skill.
Assess prior knowledge
of each topic listed in the
content outline through
problems, examples and
discussion of concepts
and skills.
For example, as a
student solves linear
equations, their
justification for actions
refers to maintaining the
balance of equality on
both sides of the
equation.
Instructional time spent
only where gaps or
weaknesses are
identified.
Prior mastery of topic
expected.
1. Represent real world situations with
linear equations.
2. Explain solutions in the context of the
problem.
3. Solve simple linear equations.
4. Compare and contrast solution
methods; make decisions as to which is
preferable for given types of problems.
6. Solve equations using combining like
terms and the distributive property.
7. Solve equations with rational
coefficients.
8. Solve equations with variables on both
sides, including those with no and
infinite solutions.
* 1 week on real world
applications and other
word problems with
emphasis on setting up
the proportions using
language that evidences
proportional reasoning.
Repeated practice for
problems using similar
contexts. This provides
access for some
students who need to
see more examples of
abstraction.
Recognize / discuss
proportionality in real
situations and
determine appropriate
contexts for applying
percent/proportionality in
the real world.
There is no
instruction planned
for this objective.
Prior mastery of topic
expected.
There is no
instruction planned
for this objective.
1. Represent, apply and explain methods
for solve problems involving rates and
ratios, proportions and percents.
2. Apply proportions to similar figures
Translate from situations
or other representations
to symbolic form,
formulating an
appropriate algebraic
representation for the
identified contexts.
Page 2
Mathematics Curriculum: Algebra
Objective
Level 2
3. Reason
quantitatively
and use units to
solve problems.
Solve equations
involving several
variables for one
variable in terms
of the others.
*1-2 weeks As with all word
problems, emphasis is
mathematical structure. At first,
no extra computation is required
to provide the expected unit
type. Several examples towards
the end of the week include the
problems for which extra
computation/ conversion is
required. Conceptual
understanding of proportion, as
developed Obj 2, and unit
conversion go together.
*1 week – Transforming
formulas is always grounded in
context. Before transforming,
students become familiar with
formulas, and they build
relational thinking and flexibility
through practice. E.g. the
relationship between radius and
circumference must be well
understood before transforming.
Then, while making connections
to conceptual understandings of
‘fair-moves’, the transformed
formula is grounded in context.
E.g., “Now, this is the formula
that describes circumference in
terms of the radius.”
As before, practice ensues.
NJCCSS:
N-Q-1,2,3
A-CED-4
May 17, 2012 am
Level 3
Use previously
discussed, accepted
means of algebraic
manipulation
regardless of the
manipulated terms
being number or
variable.
(Typically a big
obstacle for students)
Use formulas students
have previously used
to give purpose to the
variable manipulation
(What if I want to know
the base of my
rectangle? How can I
write a formula to give
me that knowing area
and height?)
Level 4
Content Outline
Transform a variety of
formulas and other
equations to solve for
one variable in terms
of the others.
1. Transform equations
a. define appropriate quantities for the
purpose of descriptive modeling
2. Work with formulas
a. Use units as a way to understand
problems and to guide the solution of
multi-step problems
b. interpret units consistently in a
formula
c. Choose a level of accuracy
appropriate to the limitations on
measurement when reporting
quantities.
Example:
1
Given 𝑉 = 𝜋𝑟 2 ℎ,
3
solve for r.
Page 3
Mathematics Curriculum: Algebra
Objective
4. (A) Model and
explain the concept
of a function using
situations, graphs,
tables, and
functional notation.
(B) Interpret
functions that arise
in applications in
terms of the
context.
(C) Analyze
functions using
different
representations.
Graph linear
functions given an
equation or table
and interpret the
graph in the context
of the problem.
Recognize slope as
rate of change and
use the slope and
intercepts to
answer questions
about a problem
with or without
context.
NJCCSS:
F-IF 1 - 6, 9
A-REI-10
May 17, 2012 am
Level 2
*1-2 weeks Develop flexibility
among representations with
repeated use of contextualized
prompts that students can do
using more than one model.
E.g., “Use the table to show
how much it costs when you
get in a taxi before it starts to
move.”
Deliver these prompts in
context for sense making of
the process. The repeated
questions then elicit
understanding that slope and
intercepts are represented in
table, equation and graph (and
again made sense of with
context/situation)
-Review coordinate plane and
plotting points.
-Limited exposure to f(x),
mostly use y=
- Limited content and
application to straightforward
wording. Many times give
students the equations “You
are helping to plan an awards
banquet for your school, and
you need to rent tables to seat
180 people. Tables come in
two sizes. Small tables seat 4
people, and large tables seat 6
people. This situation can be
modeled by the equation
Level 3
Determine domain
and range from a
table of values and
a graph.
Distinguish
between examples
of and nonexamples of
functions based on
relationships
depicted in tables
and graphs.
Determine rate of
change and yintercepts from
tables and graphs.
(Varied intervals for
x and y).
Graphing lines
given in any form
using rational
slopes and rational
y-intercepts.
Translate between
the forms of linear
equations and
recognize their
equivalence.
Interpret x- and yintercepts in the
context of a
Level 4
Find 𝑓(𝑐) for
a given 𝑓(𝑥)
ℎ(𝑡) =
−16𝑡 2 +
14𝑡 + 7
means height
is a function
of time
Given a
standard form
word
problem, write
and equation,
graph using
correct
domain and
range, find
and interpret
intercepts in
context.
Interpret
domain and
range in
context.
Given a linear
function in
any form,
rewrite in any
form.
Content Outline
A. 1. Demonstrate understanding/apply and explain
the concept of a function (If f is a function and x is
an element of its domain, then f(x) denotes the
output of f corresponding to the input x.)
2. Use function notation to…
a. evaluate functions for inputs in their domains
b. interpret statements that use function notation in
terms of a context
c. Recognize that sequences are functions,
sometimes defined recursively, whose domain is a
subset of the integers.
B 1. Interpret, connect and analyze different
representations for functions.
a. Graph linear functions using a table.
b. Write and graph functions for horizontal and
vertical lines.
Interpret key features of graphs and tables in terms
of the quantities, and sketch graphs showing key
features given a verbal description of the
relationship it describes. Key features include:a.
intercepts; b. intervals where the function is
increasing/decreasing, positive/ negative; c. slope
2. Relate the domain of a function to its graph and,
where possible, to the quantitative relationship it
describes.
Represent situations using standard form or slopeintercept form of an equation. Find pairs of values
for the linear relationship 4. Interpret x and y
intercept in the context of the problem. Use these
values to graph equations in standard form.
1. Find and interpret constant rate of change
(slope) with or without context.
2. Graph equations w/ the slope and y-intercept.
3. Understand that the graph of an equation in two
variables is the set of all its solutions plotted in the
coordinate plane, often forming a curve.
Page 4
Mathematics Curriculum: Algebra
Objective 4 cont’d
4x+6y=180. Where x is the
number of small tables and y is
the number of large tables.”
-Graphing linear relationships
using a table, standard form
and, Slope intercept form
-Describe, recognize, and be
able to distinguish characterize
of slope with or without
context.
Develop a flexibility among
representations through
repeated prompting of context
questions that students answer
with the use of more than one
representation. E.g., “Use the
table to tell me how much it
costs when you get in a taxi,
before it moves.”
Deliver these prompts in
context for sense making of
the process. The repeated
contextual questions then elicit
understanding that slope and
intercepts are represented in
table, equation and graph
May 17, 2012 am
problem. What
happens in the
context of the
function (what
happens when the
input is zero, what
happens when the
output is zero?)
Describe how and
why application or
contexts can limit a
domain of their
functional
representations.
Page 5
Mathematics Curriculum: Algebra
Objective
Level 2
5. Recognize,
describe and
represent linear
relationships
using words,
tables, numerical
patterns, graphs,
and equations.
Translate among
these
representations.
2-3 weeks – Consistent practice moving
among the four representations of function.
Recognizing the variety of problems, for
example linear equations in slope-intercept
versus standard form, requires a concerted
effort, either through days designated for
each type of problem, or doing at least one of
each type of problem a day. Encouragement
of appropriate table use.
Describe,
analyze and use
key
characteristics of
linear functions
and their graphs
- Exposure to converting from slope-intercept
to standard form (limit intercepts to integers)
-Interpret and analyze representations of
data on a scatter plot and line of best fit.
(Students are given data and asked to
predict a trend leading to line of best fit)
Interpret and
compare linear
models for data
that exhibit a
linear trend
including
contextual
problems.
NJCCSS : S-ID6, 8
G-GEPE-5
May 17, 2012 am
-Point-slope form
-Write for parallel lines
-Consistent practice moving among the four
representations of functions. Recognizing the
variety of problems, for example linear
equations in y-intercept form versus standard
form, requires a concerted effort, either
through days designated for each type of
problem, or doing at least one of each type of
problem a day.
- Recognizing the variety of problems, for
example linear equations in y-intercept form
versus standard form, requires a concerted
effort, either through days designated for
each type of problem, or doing at least one of
each type of problem a day. Encouragement
of appropriate table use, such as a y=0, 1, 2
table vs a x= 0, y=0 table.
Level 3
Write the equation of a line given
the graph of the line: lines with
integer and rational slopes yintercepts, graphs where the yintercept is not shown.
Write equation of a line given two
points, a point and the slope, a
point and a parallel line, a point
and a horizontal or vertical line,
both intercepts, or a point and
an intercept, find an equation for
the line in slope-intercept form,
point-slope form or standard form
Given a linear equation or graph
of a line, identify the slope of a
perpendicular line.
Application Problems – Identify
the most appropriate from of a
line to represent a given
situation. Identify the meaning of
the slope and y-intercept in the
context of a problem. For
example: Given the cost of a gym
membership after two months
and after four months, identify
the slope and what it represents,
identify the intercept and what it
represents. Identify the most
appropriate from of a line to
represent a given situation.
Create scatterplots, analyze
correlation of data, find, write an
equation for, and use trend lines
for prediction.
Level 4
Content Outline
Given a
graph, two
points, a
point and the
slope, a point
and a
perpendicular
or parallel
line, a point
and a
horizontal or
vertical line,
both
intercepts, or
a point and
an intercept,
find an
equation for
the line in
slopeintercept
form, pointslope form or
standard
form.
A. Write linear equations
in slope-intercept form,
point-slope form (level 4
only) and standard form,
and use these equations
to solve problems.
B. Write equations for
parallel and perpendicular
lines, interpret in context.
C. Prove the slope criteria
for parallel and
perpendicular lines and
use them to solve
geometric problems (e.g.,
find the equation of a line
parallel or perpendicular
to a given line that passes
through a given point)
D. Scatterplots
1. Plot bivariate data with
a scatterplot and describe
the nature of any possible
linear trends: positive or
negative, weak or strong.
2. Approximate a trend
line by drawing on the
Create
scatterplot through the
scatterplots,
cloud of points. Give the
analyze
correlation of equation for this line.
3. Use these trend lines to
data, find,
predict values.
write an
equation for, 4. Compute (using
technology) and interpret
and use
trend lines for the correlation coefficient
of a linear fit.
prediction.
Page 6
Mathematics Curriculum: Algebra
Objective
6. Solve
inequalities and
graph solutions
on a number
line.
Model real world
situations with
inequalities and
explain solutions
in the context of
the problem.
NJCCSS ACED-1,
3 A-REI-3
May 17, 2012 am
Level 2
*2-3 Weeks. Use this unit as an opportunity to
build on previous knowledge. That is, every
day a student learns about solving and
modeling linear inequalities, the student will
also review the concepts and skills of linear
equations.
Significant work in building the concept of an
algebraic inequality also requires bolstering up
a conceptual appreciation for inequality.
Possible activities include comparing quantities
and describing the possibilities for x when x > 5.
---Logically interpret and identify equivalent
inequalities
(i.e. x > 2 is the same as 2 < x )
-Limit solutions to common fractions and
integers.
-Explain solutions in context to the problem.
-Exploring why the inequality changes when
you divide or multiply by a negative.
-Students should realize that there are an
infinite number of solutions to statements
such as x > 5
-Also focus on the difference between a
greater than and a greater than or equal to.
-If students are asked to write inequalities
that use common phrases like “more than”,
“product”, “is”, and “difference”
Level 3
Reinforce logical
interpretation of equivalent
inequalities.
Solve multistep inequalities,
including those with infinite
solutions or no solution.
Graph solutions on a
number line where possible.
Interpret solutions in context.
Use any rational number in
the expression of the
inequality or its solution.
Level 4
Solve multistep
inequalities,
including those
with infinite
solutions or no
solution.
Graph solutions
on a number line
where possible.
Interpret
solutions in
context.
Level 4 Only
Represent real
world problems
with compound
inequalities.
Solve compound
inequalities.
Interpret
solutions in
context.
Content Outline
1. Represent real world
situations with linear
inequalities
2. Explain solutions in the
context of the problem.
3. Solve simple linear
inequalities
6. Solve inequalities using
combining like terms and
distributive law.
7. Solve inequalities with
rational coefficients.
8. Solve inequalities with
variables on both sides,
including those with no
and infinite solutions.
Solve compound
inequalities (Level 4 only)
Page 7
Mathematics Curriculum: Algebra
Objective
7. Solve
equations
involving the
absolute
value of a
linear
expression.
NJCCSS
A-REI-1;
F-IF-1
Level 2
*2-3 Weeks. Use this unit as an
opportunity to build on previous
knowledge. That is, every day a
student learns about solving
absolute value equations, the
student will also review the
concepts and skills of linear
equations.
Significant work in building the
concept of absolute value may
be required.
---Re-explain meaning of absolute
value (distance away form
zero).
-Emphasize that absolute value
cannot be negative. (Bring up
distance, “Even if you walk
backwards, you are still moving
a distance”)
-All word problems are limited to
either giving student the
equation or in the form
Level 3
Reinforce general
descriptions or student
definitions for absolute
value
Solve multi-step equations
with absolute value
expressions.
Distinguish when absolute
value equations have no
solution
Level 4
Solve multi-step equations with
absolute value expressions, including
expressions with no solution.
Review the objective through error
analysis, and pattern comparisons.
Content Outline
1. Solve equations with
absolute value
expressions.
x  center  range
May 17, 2012 am
Page 8
Mathematics Curriculum: Algebra
Objective
Level 2
8. Graph and
analyze the graph
of the solution set
of a two-variable
linear inequality.
*1-2 weeks – Meaning should drive
the graphing of two-variable
inequalities. For many students,
this may require consistent use of
the table for graphing the line as
well as using the test-point for
NJCCSS F-IF-1,5 shading as follows: Plug in the
point into the inequality; Determine
if the inequality is satisfied (“Yes”
or “No”); Write “Yes” or “No” on the
graph where the test-point
appears; Write the opposite word
“Yes” or “No” on the other sides of
the line; Shade on the side of the
line where you wrote “Yes.” Such a
scripted procedure should be
developed via meaning making of
the inequality
Level 3
Level 4
Content Outline
Distinguish between notation
and visual representation
(<,dashed; &solid)
Emphasis is on writing
and graphing linear
inequalities.
Use a test point to determine
direction to shade for solution.
Analyze and solve real
world problems that
can be represented
with a two-variable
linear inequality, and
interpreting solutions in
context.
1. Explain how any point in
the shaded region satisfies
the equation.
2. Model a real world
situation with a two-variable
linear inequality and explain
the solution set in the context
of the problem.
3. Graph a linear equality with
correct shading and
dotted/dashed lines.
Contrast the differences
between the solution to a
linear inequality and a linear
equation.
Write the linear inequality
given its graph.
Write the linear
inequality given its
graph. Interpret the
graph in context.
-Review how to graph a line
-Relate how > looks on a number
line verse how it looks in a linear
equations (open circle is going to
represent dashed line).
-Limited exposure to word
problems and real world situations.
May 17, 2012 am
Page 9
Mathematics Curriculum: Algebra
9. Solve systems
of linear
equations in two
variables using
algebraic and
graphic
procedures.
Recognize,
express and
solve problems
that can be
modeled using
one or twovariable
inequalities; or
two variable
systems of linear
equations.
Interpret their
solutions in terms
of the context of
the problem.
NJCCSS A-REI-6
May 17, 2012 am
1 week – This topic is less essential than
others in the curriculum. In order to spend
more time on the developmental pieces in
Algebra 1, here the topic is reduced to
problems that only require the knowledges
from above topics. I.e., students use their
skills to graph two-variable inequalities
(see previous) to draw two of these on one
coordinate plane. There is little time for
modeling with this technique.
- Review how to graph including a
coordinate plane review.
- Solve by graphing, substitution and
elimination.
Graphing:
-Equations are mostly are in slope intercept
form.
-All solutions are integers
Substitution:
- One or both of the equations are x=, y=
- If both are in standard form one of the
variables have a coefficient of 1.
-All coefficient are integers
-To facilitate checking of solutions, All
solutions are common fractions or integers
Elimination:
- All coefficient are integers
-To facilitate checking of solutions, All
solutions are common fractions or integers.
Applications/Word Problems:
-Make problems with situations students
can relate to (money problems).
-Stress to students they need two linear
equations with two variables.
Solving Graphically:
Solutions limited to
integer coordinates.
Lines given in any form.
Substitution and
Elimination
Equations are written in
any form.
Solutions are rational
numbers to facilitate
checking of solutions.
Determine if the system
of equations has no
solution, one solution,
or infinite solutions from
solving by all three
methods.
Given real world
situations, write and
solve the system.
Interpret the solutions
in context.
Determine the solution
of a system of linear
inequalities
Given a system,
choose and justify
an approach for
solving. Solve
systems in any form
including those with
no solution or
infinite solutions.
Given an equation,
write another
equation to make a
system with a given
characteristic
(example: given
3𝑥 + 2𝑦 = 7 write
another equation so
that the system has
no solution. Justify
your answer.)
Write a system with
a given solution
(eg:(4,7)).
1. Explain that the solution to
a system of linear equations
is the ordered pair that
satisfies both equations.
2. Use tables to confirm the
solution to a system.
3. Solve systems graphically
by finding the point of
intersection, both with
technology and by hand.
4. Confirm that a solution
satisfies both equations
algebraically.
5. Solve systems using
algebraic techniques.
6. Decide which method is
best to solve a given system.
7. Given an algebraic
representation, represent the
solution set to a system of
linear inequalities graphically.
8. Model real world situations
using systems of linear
equations and inequalities.
Given real world
situations, write and
solve the system.
Interpret solutions
in context.
…..
Page 10
Mathematics Curriculum: Algebra
Objective
10. Summarize, represent,
and interpret data on a
single count or
measurement variable
(1-4), or on 2categorical
and quantitative
variables(5-6).
Interpret linear models
(7-9)
May 17, 2012 am
Level 2
2-3 days. This topic is
less essential to
development of algebraic
thinking and therefore
should be satisfied with
experiential activities that
require students to
represent data using
dotplots, histograms and
scatterplots. They should
be instructed in finding
means, medians and
modes. Students may be
assigned a small, in-class
project as an
assessment, rather than a
test.
Level 3
Level 4
Content Outline
1. Represent data with plots on the
real number line (dot plots,
histograms, and box plots).
2. Use statistics appropriate to the
shape of the data distribution to
compare center (median, mean)
and spread (interquartile range,
standard deviation) of two or
more different data sets.
3. Interpret differences in shape,
center, and spread in the context
of the data sets, accounting for
possible effects of extreme data
points (outliers).
4. Use the mean and standard
deviation of a data set to fit it to a
normal distribution and to
estimate population percentages.
Recognize that there are data
sets for which such a procedure
is not appropriate. Use
calculators, spreadsheets, and
tables to estimate areas under
the normal curve.
5. Summarize categorical data for
two categories in two-way
frequency tables. Interpret
relative frequencies in the
context of the data (including
joint, marginal, and conditional
relative frequencies). Recognize
possible associations and trends
in the data.
9. Distinguish between correlation
and causation.
Page 11
Mathematics Curriculum: Algebra
Objective
Level 2
Level 3
Level 4
Content Outline
11. Determine and
evaluate random
processes underlying
statistical experiments
(1-2).
2-3 days: Again, keeping this
limited due to time
constraints, students learn
about sample surveys
through doing an in-class
project assignment. They are
coached regarding effective
questioning and data
tabulation and presentation.
Instruction limited to
randomization, a few
simulations, and a
sampling survey to
estimate population mean
or proportion
A brief review and
extensions, as
required, for each
aspect of content
outline.
1. Understand statistics as a
process for making inferences
about population parameters based
on a random sample from that
population.
Make inferences and
justify conclusions from
sample surveys,
experiments and
observational studies
(3-6).
NJCCSS S-IC-1-6
2. Decide if a specified model is
consistent with results from a given
data-generating process, e.g., using
simulation.
3. Recognize the purposes of and
differences among sample surveys,
experiments, and observational
studies; explain how randomization
relates to each.
4. Use data from a sample survey to
estimate a population mean or
proportion; develop a margin of
error through the use of simulation
models for random sampling.
5. Use data from a randomized
experiment to compare two
treatments; use simulations to
decide if differences between
parameters are significant.
6. Evaluate reports based on data.
May 17, 2012 am
Page 12
Mathematics Curriculum: Algebra
Objective
Level 2
Level 3
Level 4
12. Apply probability
concepts to determine the
likelihood an event will
occur in practical
situations.
NJCCSS S-CP-1-7
Construct sample
space.
Define a sample space
Describe events as subsets of a
sample space (the set of
outcomes) using characteristics
(or categories) of the outcomes,
or as unions, intersections, or
complements of other events
(“or,” “and,” “not”).
Content Outline
3. Understand the conditional
probability of A given B as P(A
and B)/P(B), and interpret
independence of A and B as
saying that the conditional
probability of A given B is the
same as the probability of A, and
the conditional probability of B
Find the probability of a
Understand that two events A
given A is the same as the
single event or compound
Distinguish between Distinguish between
and B are independent if the
probability of B.
event occurring with or
independent and
independent and
probability of A and B occurring 4. Construct and interpret twowithout replacement.
dependent events.
dependent events.
together is the product of their
way frequency tables of data
Content outline
probabilities, and use this
when two categories are
1. Describe events as
characterization to determine if associated with each object
subsets of a sample space
they are independent.
being classified. Use the two-way
(the set of outcomes) using
table as a sample space to
characteristics (or
Distinguish between Distinguish between
Determine the probability of
decide if events are independent
categories) of the
overlapping and
mutually exclusive and
mutually exclusive or overlapping and to approximate conditional
outcomes, or as unions,
mutually exclusive
overlapping events in
events
probabilities.
intersections, or
events
order to determine the
5. Recognize and explain the
complements of other
-For example:
probability of an
concepts of conditional
events (“or,” “and,” “not”).
addition rule
occurrence
probability and independence in
2. Demonstrate or explain
Find the probability of a single
everyday language and everyday
that two events A and B
event or compound event
situations.
are independent if the
Differentiate between Differentiate between and occurring with or without
6. Find the conditional probability
probability of A and B
of A given B as the fraction of B’s
experimental and
calculate experimental
replacement.
outcomes that also belong to A,
occurring together is the
theoretical probability. and theoretical
product of their
probabilities
and interpret the answer in terms
probabilities, and use this
of the model.
characterization to
7. Apply the Addition Rule, P(A
determine if they are
or B) = P(A) + P(B) – P(A and B),
independent.
and interpret the answer in terms
of the model.
(continued in last column)
May 17, 2012 am
Page 13
Mathematics Curriculum: Algebra
Objective
Level 2
13. Use counting
principles to determine
the number of ways an
event can occur. Interpret
and justify solutions.
1. Determine the number
1. Find the number of
or outcomes by applying
possible outcomes for
the counting principle.
single or compound
events- selected
situations
NJCCSS S-CP-9
Level 3
2. Apply counting principle
Level 4
Content Outline
1. Find the number of
possible outcomes for
single or compound
events- challenging
situations
1. Find the number of
possible outcomes for
single or compound
events.
2. Distinguish between a
combination and
permutation and use these
to solve real world
problems.
3. Use permutations and
combinations to compute
probabilities of compound
events and solve problems.
14. Apply the laws of
exponents to numerical
and algebraic expressions
with integral exponents to
rewrite them in different
but equivalent forms or to
solve problems.
NJCCSS N-RN-2
May 17, 2012 am
2 weeks: Laws of
exponents are developed
with consistent reference to
the definition of exponent
(as repeated
multiplication). E.g.,
student outcome is to
describe the product of
powers rule with an
example.
Automaticity with exponent
rules should also be the
focus, and several days are
devoted to practice with
these rules.
Investigate exponential
relationships to determine
the properties of integer
exponents
Review properties of
integer exponents.
Use properties of
exponents to
simplify/evaluate
expressions
Use these properties to
simplify complex
expressions.
Apply appropriate
properties of exponents to
solve equations
Scientific notation is
reviewed as an application
of exponents
Introduce rational
exponents.
2. Distinguish between a
combination and
permutation and use these
to solve problems
3. Use permutations and
combinations to compute
probabilities of compound
events and solve problems.
1. Use product and
quotient properties to
rewrite exponential
expressions
2. Rewrite expressions so
that all exponents are
positive (Equivalent forms)
3. Solve problems using
exponents
4. Convert between
standard form and
scientific notation
Review scientific notation.
Page 14
Mathematics Curriculum: Algebra
Objective
Level 2
Level 3
Level 4
Content Outline
15. Model and solve
problems involving
exponential growth and
decay.
1 week:
Repeated use of simple
contexts that are modeled
with exponential growth
and decay.
Determine the
characteristic differences
between linear and
exponential functions
(represented in
tables/graphs/equations)
Review recursive routines
for exponential functions.
Connect the recursive
routine to an explicit
formula.
1. Model exponential
function form y = abx
NJCCSS A-SSE-1,2,3
Work simultaneously with
contexts that are NOT
modeled with exponential
growth/decay (like linear
and quadratic).
Consistent use of the table,
context and graph as with
other modeling units as
above and below.
Analyze graphs or tables to
determine the equation for
an exponential function
Apply exponential functions
to real world scenarios
(Interest on a bank
account/half-life of
substance/etc.)
Determine the
behavior/characteristic
differences between
exponential growth and
decay
Find an equation for a
exponential function given
points or a graph or a table
of values..
Predict a function to
exponential vs linear given
a table of values.
2. Analyze y-intercept and
constant multiple to write a
function rule
3. Graph exponential
functions as continuous
and positive
4. Model and/or solve real
world problems
Use exponential functions
to model real world
situations involving growth
and decay given growth or
decay factors or rates of
increase or decrease.
Analyze the graph of an
exponential function for
domain and range, and key
features such as
increasing, decreasing,
and y-intercept.
May 17, 2012 am
Page 15
Mathematics Curriculum: Algebra
Objective
Level 2
Level 3
Level 4
16. Add, subtract
and multiply
polynomial
expressions with
or without a
context.
1 week –
With multiplying, make
connections to the
distributive property.
Review the concepts of
combining like terms and the
distributive property
Identify like terms for powers of variables
greater than 2.
NJCCSS
A-APR-1,2
Identify like terms in
polynomials based on their
For some students, the use integer power
of a multiplication matrix
may be helpful, especially Apply distributive property to
polynomial multiplication
for multiplying a trinomial
(requires extension to
against a binomial.
multiplying more than just
monomial times binomial)
Add, subtract and multiply polynomial
expressions with or without a context.
Extended distribution i.e. A binomial and a
trinomial.
Simplifying multi-term polynomial
expressions
Apply newly learned concepts
to simplify polynomial
expressions
Investigate common patterns
in polynomial multiplication
17. Factor
simple
polynomial
expressions with
or without
context.
Solve factored
polynomial
equations with or
without context.
NJCCSS
A-APR-3
May 17, 2012 am
2 weeks:
The majority of problems
require factoring out a
common monomial
and/or a quadratic when
a=1. Consistent
emphasis on the
connection to
multiplication (as the
inverse). Also, direct
guidance, such as
requiring students to write
out "Find the pair of
factors of c that add to b,"
before finding the pair
and writing the factored
polynomial.
Review what it means to be a
“factor”
Apply concept of a factor to
identify a greatest common
monomial factor of a
polynomial expression
Solve equations using the
“zero-product property”( in
factored form or by algebraic
manipulation to appear in
factored form)
Determine the factors of a
trinomial through various
strategies (i.e. recognition of
special patterns)
1. Increase knowledge of greatest monomial
factor by using multi-variable/power
polynomials.
2. Factor polynomials by understanding and
recognizing the patterns of special factoring as
well i.e. difference of two squares. perfect
square trinomials, factor by grouping.
3. Solving real world problems in context
which could be representative of the product
of binomials for example area, vertical motion
models, and other polynomial applications.
4. Interpret and apply the zero product
property to the solution of a factored quadratic
trinomial with or without context.
5.Recognize and interpret extraneous/nonextraneous solutions after solving a quadratic
equation.
Content Outline
1Classify, add,
subtract, & multiply
polynomials
2Use polynomials to
represent real world
situations; solve
problems using
polynomials
3 Understand that
polynomials form a
system analogous
to the integers,
namely, they are
closed under the
operations of
addition,
subtraction, and
multiplication.
1. Identify and
factor out the GCF.
2. Factor
trinomials. Use
products of
binomials to
represent area.
3. Identify and
factor the
difference of
squares.
4. Use the zeroproduct property to
solve quadratic
equations.
Page 16
Mathematics Curriculum: Algebra
Objective
18. Recognize,
describe,
represent and
analyze a
quadratic function
using words,
tables, graphs and
equations.
Analyze a table,
numerical pattern,
graph, equation or
context to
determine whether
a linear or
quadratic
relationship could
be represented.
Recognize and
solve problems
that can be
modeled using a
quadratic function.
Interpret the
solution in terms of
the context of the
original problem.
NJCCSS A-SSE-3,
A-REI-4,7
May 17, 2012 am
Level 2
1 week:
Repeated use
of simple
contexts that
are modeled
with quadratic
functions.
Work
simultaneously
with contexts
that are NOT
modeled with
quadratic
functions (like
linear and
exponential).
Consistent use
of the table,
context and
graph as with
other
modeling units
as above.
Level 3
Recognize a parabola
as the graphic
representation of a
quadratic function
Determine the
symmetry of a
quadratic function
Identify the
characteristics of a
parabola and their
algebraic means for
calculation based on
the equation
Investigate the
changes in the graph
of a quadratic function
based on alterations
made to the
coefficients of its
equation
Level 4
1. Identify the attributes of a parabola: the
shape, the location of the axis of
symmetry and its vertex.
2. Referencing the quadratic function of the
parabola, recognize the significance of
the coefficients of a, b, and for example if
a is positive or negative based on the
shape of the parabola, and the value of c
from the graph and that it represents the
y intercept.
3. Analyze a quadratic trinomial for
graphing attributes: opening up/down, x
and y intercepts, axis of symmetry,
location of vertex (above or below x
axis),
4. Given an equation, describe
transformations to the parent function y =
x2 . Compare and contrast different
graphs of quadratic functions, noting
differences in width and how this will
relate to their relative equations.
5. Calculate the axis of symmetry and
vertex of the parabola given the
equation.
6. Identify equations whose graphs will be
symmetric to the y-axis (b=0).
7. Model real world situations that provide a
given context to address pertinent
questions relative to quadratic equations
including vertical motion model and area.
8 . Determine the number of solutions of a
quadratic equation by graphing.
Content Outline
1. Find the value of the vertex
from a table (noting the
symmetry).
2. Determine positive or
negative value of a based on
the graph. Determine the
value of c from the graph.
3. Compare and contrast
different graphs of quadratic
functions, noting differences
in width and how this will
relate to their relative
equations.
4. Identify equations whose
graphs will be symmetric to
the y-axis (b=0).
5. Model real world situations
with a quadratic equation,
including vertical motion and
area.
6. Find the vertex from the
equation of a parabola.
7. Use the quadratic model of a
given context to answer
contextual questions.
8. Distinguish between functions
with a maximum and
minimum value, and this may
relate to a given context.
Page 17
Mathematics Curriculum: Algebra
Objective
Level 2
19. Solve
quadratic
equations.
1 week:
Introduce
through
quadratic
contexts.
NJCCSS
A-SSE-3,
A-REI-4,7
Tables and
graphs are
juxtaposed
with linear
and
exponential
so as to
highlight
quadratic
features.
Level 3
Determine the
significance of a
solution to a
quadratic equation
(numerically and
graphical
implications)
Use the quadratic
formula
Level 4
1. Solve quadratic equations using appropriate methods to
determine the number and nature of the solutions for
example: by factoring, inverse operations, quadratic
formula and completing the square.
2. Use and understand multiple terms pertaining to solving
quadratic equations, for example finding roots, finding
zeros and x intercepts of a quadratic function.
3. Use the discriminant to determine number of solutions.
Understand the discriminant’s predictive ability as the
square root piece of the quadratic formula.
Content Outline
1. Solve using inverse
operations when b=0.
2. Solve using the quadratic
formula for all quadratic
equations.
3. Solve a binomial square by
taking the square root of both
sides. (level 4 only)
4. Solve graphically, by hand
(approximation) and with
technology.
5. Interpret solution in the
context of the problem.
4. Determine the number of the solutions from the
discriminant.
5. .Anticipate the location of the vertex of the parabola,
above or below the x axis, based on analyzing the equation
for a minimum or maximum vertex in conjunction with the
discriminant and analyzing the number of solutions.
6 Solve graphically, by hand (approximation) and with
technology.
7. Model real world problems.
8. Interpret solutions in the context of the problem.
May 17, 2012 am
Page 18
Mathematics Curriculum: Algebra
Objective
20. Analyze a table,
numerical pattern, graph,
equation or context to
determine whether a
linear, quadratic or
exponential relationship
could be represented.
Or, given the type of
relationship, determine
the elements of the table,
numerical pattern or
graph.
NJCCSS
A-REI-11
F-IF-4-7,9
Level 2
2 weeks:
This topic is worked on
throughout the year. When
students learn linear functions,
they are given quadratic and
exponential problems as well so
as to distinguish the liinear
features. Again during the
quadratic unit, quadratic
contexts/tables/graphs are
juxtaposed with linear and
exponential so as to highlight
quadratic features. Similar
approach with exponential.
Level 3
Analyze the characteristics of a
given representation to indicate
which category of functions the
representation can be applied.
21. Use the properties
of radicals to rewrite
numerical and algebraic
expressions containing
square roots in different
but equivalent forms or to
solve problems.
1 week: This topic is less
connected to others in the
Algebra 1 curriculum (because of
a lack of attention to square root
functions, contexts, graphs and
tables) and is also heavily
emphasized in the Algebra 2
curriculum. Only an introduction
to the topic is included,
specifically introducing that the
square root is the inverse of
squaring, and simplifying only
stand-alone square roots (no
addition, subtraction,
multiplication of radicals).
As it is in Level 2, this objective is Perform operations
introductory versus a broader
with radicals which
ranged development. All
includes simplifying,
descriptive described in Level 2
rationalizing the
are appropriate for Level 3
denominator,
content development for this
adding, subtracting,
objective.
multiplying and
dividing and solving
real world
problems.
NJCCSS
N-RN-1,2;
A-APR-6; A-REI-2
May 17, 2012 am
Level 4
Content outline
1. Use the shape of 1. Use the shape of the
the graph and/or a
graph to distinguish
table of values to
between functions.
distinguish between 2. Analyze the change in
linear, quadratic,
the y-values to distinguish
exponential and
between functions.
radical functions.
3. Complete a table given
2. Given a function the type of function.
select the
4. Find other points given
appropriate domain the type of function.
to construct a
5. Solve real world
graph as it relates
problems.
to solving real
world problems.
1. Write square root
radicals in simplest
form
2. Find products and
quotients of radicals
3. Rationalize the
denominator
4. Add, subtract and
multiply radical
expressions
5. Solve real world
problems
Page 19