Written by:
Leslie Gentile and Debra Wahle
Reviewed by:
Matthew A. Mingle
Director of Curriculum and Instruction
Chris Kenny
Supervisor of Special Services
Kathryn Lemerich
Supervisor of Mathematics and Business
Approval date:
August 26, 2014
Members of the Board of Education:
Lisa Ellis, President
Kevin Blair, Vice President
Shade Grahling, Curriculum Committee Chairperson
David Arthur
Johanna Habib
Thomas Haralampoudis
Leslie Lajewski
James Novotny

Replacement Pre-Algebra addresses previously introduced topics from 7th and 8th grades, along with reinforcement of elementary Algebraic skills. The Course will incorporate 8 units of study as follows: Language of Algebra, Number Sense, Logic of Algebra, Coordinate Geometry (Points,
Slopes and Lines), Word Problems, Modeling Polynomial Operations, Exponents, and Rational
Numbers and Expressions. Each unit will allow students to establish a greater depth of understanding of algebraic ideas with an emphasis on developing Algebraic “Habits of Mind”
(Puzzling and Persevering, Seeking and Using Structure, Using Tools Strategically, Describing
Repeated Reasoning, Communicating with Precision) . The purpose of the course is to adequately prepare students to move seamlessly into a typical Algebra 1 curriculum. Students will be provided with a variety of instructional opportunities to develop their procedural as well a perceptual understanding of concepts and ideas. Opportunities for exploration using manipulatives, technological tools, and other course resources will be provided consistent with the Common Core
State Standards for Math 8 and Algebra 1.
This course aims to:
● enable students to make sense of various types of problems and the reasonableness of their answers
● build students’ confidence with the various approaches to solving a problem so they persevere in solving them
● encourage students to become abstract thinkers who make sense of quantities and their relationships in problem situations
● develop students’ ability to cooperatively discuss, make conjectures and critique ideas of one another
● use, apply, and model mathematics to solve problems arising in everyday life, society, and the workplace
● consider the variety of available tools when solving a mathematical problem
● communicate mathematical ideas precisely and effectively to others
● determine a pattern or analyze structure within mathematical content to apply to related ideas
● use repeated reasoning to follow a multi-step process through to completion
Suggested activities and resources page

Unit Title: Introduction to Algebraic Language
Unit Summary:
This unit provides an introduction to algebraic notation. Students will work with variables, algebraic expressions and the commutative and distributive properties. They will be able to solve algebraic equations using inverse operations by establishing a solid understanding of why and how to use inverse operations to balance equations and arrive at a solution.
Suggested Pacing: 10 lessons
Unit Essential Questions:
● How can one use a model to represent the process for solving linear equations?
● How does one interpret the number of solutions of linear equations in one variable?
Unit Enduring Understandings:
● A linear equation is a balance of equivalent quantities.
● A variable represents an unknown quantity that can be found in equations by balancing the equation using inverse operations.
Unit Benchmark Assessment Information:
Unit 1 Benchmark Assessment Project

Objectives
(Students will be able to…)
Students will be able to use verbal, pictorial, algebraic and symbolic notations, think logically about equivalence, and use known information to determine unknown information.
Essential
Content/Skills
Content/Skills:
Explore number tricks and balancing mobile puzzles.
Use mental mathematics to apply the distributive property to numbers.
Suggested
Assessments
Partner activity:
Create a physical mobile puzzle to represent an algebraic equation, write the key and the equation on an index card.
(Mobiles can be hung in the classroom)
Give students the backwards steps to complete a number puzzle then have students write the steps forward. Have students test out their steps of their puzzle on classmates to see if they work.
Standards
(NJCCCS CPIs, CCSS, NGSS)
8.EE.C.7
Solve linear equations in one variable.
8.EE.C.7.A
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no
solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results
(where a and b are different numbers).
Pacing
10 lessons

Unit Title: Number Sense
Unit Summary:
This unit provides a full review of working within the number system. Students will perform operations with integers, rational and irrational numbers, fractions and decimals. The number line will be used as a tool for reasoning and making sense of concepts. Students will be able to approximate the value of irrational numbers and determine which rational numbers they fall between and use absolute value to find the distance between two numbers.
Suggested Pacing: 13 lessons
Unit Essential Questions:
● Why does one need to distinguish between rational and irrational numbers?
● How does one locate irrational numbers on a number line?
● How are division and multiplication of a fraction by a fraction related?
● In what context would I need to add or subtract fractions?
Unit Enduring Understandings:
● Division by a rational number may result in a quotient whose value is bigger than, equal to, or smaller that the value of the dividend.
● Absolute Value is a tool that can be used to find distance.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to use a number line to order numbers
(including decimals and fractions), find distance between two numbers, reason abstractly regarding placement of numbers
(including irrational), and represent inequalities.
Students will be able to perform four operations on rational numbers.
Essential
Content/Skills
Content:
Whole Numbers,
Integers, Rational vs.
Irrational Numbers,
Decimals, Fractions
Skills:
Compare numbers on a number line, estimate the value of irrational numbers.
Evaluate and estimate sums and differences of decimals and fractions.
Suggested
Assessments
Explain what happens to a whole number when it is divided by a proper fraction and by an improper fraction.
Geogebra “Absolute
Value Assessment”
IXL.com
“Inequalities”
Standards
(NJCCCS CPIs, CCSS, NGSS)
8.NS.A.1
Know that numbers that are not rational are called irrational.
Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.A.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions
(e.g., π2). For example, by truncating the decimal expansion of
√2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get
better approximations.
7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.1.B
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0
(are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
CC 7.NS.A.1.C
Understand subtraction of rational numbers as adding the additive inverse, p - q = p+ (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7 .NS.A.1.D
Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to
Pacing
13 lessons

multiply and divide rational numbers.
7.NS.A.2.B
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q =
p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.A.2.C
Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.D
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in
0s or eventually repeats.

Unit Title: Modeling the Logic of Algebra
Unit Summary:
This unit explores the use of models to reason numerical and polynomial multiplication. The method of multiplication seeks to visually organize an abstract process to create a structural model of area.
Mobile models and puzzles explored in Unit 1 are reintroduced to build students’ facility with manipulating expressions and equations in a thoughtful, logical, and accurate manner.
Suggested Pacing: 12 lessons
Unit Essential Questions:
● How can multiplication with numbers and polynomials be modeled using area?
● How can mobiles be used to model the solving of algebraic equations?
Unit Enduring Understandings:
● The quantities/shapes inside a figure can be calculated in parts then added together as parts to represent the product.
● To solve an algebraic equation is to maintain balance.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to extend the behavior of multiplication of numbers to multiplication of polynomials using an area model.
Students will be able to apply the distributive property.
Students will equate the steps to solving mobile puzzles to solving algebraic equations.
Essential
Content/Skills
Content:
Distributive Property,
Multiplication of
Polynomials, Solve
Linear Equations in
One Variable.
Skills:
Use Area models for multiplication, translate between symbolic expressions and area models, use mobile models to extend thinking to the solving of equations.
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Students are paired to write six equations
(in one variable) with an answer key.
Groups swap papers and solve each other’s equations. Two pairs of students group together to discuss accuracy of their work and help fix mistakes.
Students go to
IXL.com link: http://www.ixl.com/ math/grade-8 to complete Z.11 assessment “Multiply
Polynomials Using
Algebra Tiles”
8.EE.7 Solve Linear Equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x - a, a=a, or a=b results (where a and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Pacing
12 lessons

Unit Title: Coordinate Geometry
Unit Summary:
This unit will build students’ familiarity with the coordinate plane, graphs, and their associated components. Basic transformations will be explored as well as the process for determining whether or not an ordered pair represents a solution to a linear equation.
Suggested Pacing: 15 lessons
Unit Essential Questions:
● What do ordered pairs represent on a line?
● How can you determine whether or not an ordered pair is a solution to an equation?
● What happens to a shape when its x-coordinates, y-coordinates or both are transformed?
Unit Enduring Understandings:
● Students will come to understand graphs as collections of solution points.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to plot points on a coordinate grid.
Students will be able to transform points of a graph.
Students will be able to determine if an ordered pair is a solution to a linear equation.
Essential
Content/Skills
Content:
Coordinate plane as perpendicular number lines, ordered pairs, solutions to a linear equations
Skills: plot points as ordered pairs, transform points, substitute coordinates of a given ordered pair into an equation to determine if it is a solution.
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Students create a coordinate plane foldable.
See the link: http://tothesquareinc h.wordpress.com/201
2/01/19/coordinate-g rid-foldable/
Given pre-made graphs, students draw the requested transformed graph on the same coordinate plane and label vertex points on the new graph.
A.REI.10: 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
(which could be a line).
8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Pacing
12 lessons

Unit Title: Problem Solving
Unit Summary:
This unit will help students gain confidence with determining the appropriate procedure to use when solving a word problem. Students will determine known vs. unknown information, reason the initial steps to take, and create algebraic equations representing the given facts.
The focus will be placed on the organization of information, considering the variety of methods applicable to solve the problem, and the accuracy of their answers.
Suggested Pacing: 15 lessons
Unit Essential Questions:
● What steps do I take to start working the problem?
● What do I know for sure and what do I need to figure out?
● How can I organize the given information into an algebraic equation?
Unit Enduring Understandings:
● Students will dissect a word problem into comprehendible pieces.
● Students will build stamina to interpret word problems and persevere to its conclusion.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will determine what they know and don’t know in a given word problem.
Students will be able to organize information from a word problem.
Students will be able to assign variables to unknowns and determine an algebraic equation using the given information.
Essential
Content/Skills
Content:
Word Problems,
Tables, Charts, Given information, Writing
Algebraic Equations
Skills: sort information given a word problem, determine known facts in a problem, choose variables to represent unknown information, use an algebraic equation to solve the word problem.
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Students will create a graphic organizer of steps to follow when solving a word problem.
Use the logic puzzle at: http://www.mathsisf
un.com/puzzles/fourpeople-travel.html
(search the site for more number puzzles to explore)
7.EE.B.4
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
HSA.CED.A.1
Cr Create equations and inequalities in one variable and use them to solve problems.Include equations arising from linear and quadratic functions, and simple rational and
exponential functions.
9.1.12.A.1
Apply Critical thinking and problem-solving strategies during structured learning experiences.
Pacing
15 lessons

Unit Title: Modeling Polynomial Operations
Unit Summary: This unit will focus on the use of area models to represent multiplication and division processes. Students will use area puzzles to begin to establish an understanding of factoring of polynomial expressions. The area model tool will be expanded to include factoring of trinomials into two binomials. Students will also use mental math activities to determine product/sum factors leading them to application of the zero-product property.
Suggested Pacing: 14 lessons
Unit Essential Questions:
● How can an area model be used to represent processes of multiplication, division, and factoring?
● What is the Zero product property and how does it get applied to solve quadratic equations?
Unit Enduring Understandings:
● Division and factoring a ways of “undoing” multiplication.
● Quadratic equations can be solved by factoring.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to translate area models into algebraic equations showing multiplication and division.
Students will be able to use area models to find a missing factor of a polynomial or to factor a trinomial into two binomials.
Students will be able to solve a quadratic equation.
Essential
Content/Skills
Content:
Area Models,
Factoring,
Zero-Product Property
Skills: use area models to represent multiplication and division processes, complete are model puzzles with various pieces of information missing, use factoring to simplify rational expressions, solve a quadratic equation by factoring and using the zero-product property .
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Partner Activity:
Each student creates
2 area models representing multiplication of polynomials. (see page 6 of Unit 10 in
TA). Pieces of the model will be missing and filled in by the partner.
Model the practice multiplication problem on top of page and # 1 of
“SmartPal” binder p.
77 while students copy on clearboard.
Students remove paper and complete #
2 and 3 as a quiz grade.
A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
a. I Interpret parts of an expression, such as terms, factors, and coefficients.
b. I Interpret complicated expressions by viewing one or more of their parts as a single entity.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it.
A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
proa. Factor a quadratic expression to reveal the zeros of the function it defines.
F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Pacing
14 lessons

Unit Title: Working With Exponents
Unit Summary:
This unit will extend students’ working knowledge of positive integer exponents to include zero, negative integers and rational exponents. Students will make sense of properties of exponents. By experiencing how repeated multiplication differs from repeated addition, students will develop an an understanding of exponential growth compared to that of arithmetic growth. Graphs of linear and exponential functions will be explored. Previously used area models will be utilized for polynomial multiplication.
Suggested Pacing: 14 lessons
Unit Essential Questions:
● What does a zero, negative or rational number exponent mean?
● How do you multiply or divide expressions with a common base that involve exponents?
● What is the difference between linear vs. exponential growth?
● How do linear and exponential function graphs differ?
Unit Enduring Understandings:
● Exponential growth is a repeated multiplication while arithmetic growth is a repeated addition.
● Any quantity raised to a zero exponent is equivalent to one.
● Negative exponents involve reciprocals.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to describe how growth by multiplication differs from growth by addition.
Students will be able to extend the logic of positive exponents to negative, zero and rational exponents.
Students will be able to multiply and divide exponential expressions with a common base.
Essential
Content/Skills
Content: negative, zero and rational exponents, area models, products and quotients of powers.
Skills: use area models to multiply polynomials, simplify products and quotients of exponential expressions, recognize linear vs. exponential graphs.
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Students research on the internet and provide real-life examples of growth by addition and growth by multiplication.
Quiz: Students explain in words what it means for patter of numbers to grow by addition vs. growing by multiplication.
Give numerical examples of each.
Make a foldable of exponent rules...see the link: https://docs.google.co
m/a/madisonnjps.org
/file/d/0B7ciNQ8KxMI
DQk9yUENnMmZMen
M/edit
(titles can be altered to suit your needs) or the link: http://www.teachers
payteachers.com/Pro duct/Laws-of-Exponen ts-Foldable-337553
8.EE.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions.
For example, 3
1/27.
2 × 3 -5 = 3 -3 = 1/3 3 =
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
Pacing
14 lessons

Unit Title: The Logic of Rational Numbers and Expressions
Unit Summary:
This unit will explore rational relationships focusing on the four basic operations with fractions.
Instead of memorizing rules, students will seek to establish an understanding of patterns and proportional reasoning. Tools such as Cuisenaire rods, number lines, and area models will be used to develop a deeper understanding of addition, subtraction and multiplication of fractions and algebraic expressions involving fractions. Rates of change of linear graphs will be introduced with the purpose of application to further studies regarding the concept of slope.
Students will establish an understanding of equivalent operations (such as multiplication of ½ vs. division by 2) and use such operations to solve algebraic equations.
Suggested Pacing: 14 lessons
Unit Essential Questions:
● How can tools such as a number line, mobile puzzles and cuisenaire rods be used to make sense of fractions?
● What is the process for adding and subtracting fractions?
● What is the process for multiplying and dividing fractions?
● When are fractions used to solve algebraic equations?
Unit Enduring Understandings:
● Fractions are just numbers found on a number line.
● Operations with fractions can be reasoned without memorizing a rule.
● Multiplication and division are inverse operations.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to add fractions with like and unlike denominators.
Students will be able to identify and write equivalent expressions for numerical and rational expressions.
Students will be able to multiply and divide fractions.
Students will be able to solve basic algebraic equations involving fractions.
Essential
Content/Skills
Content: proportional relationships, fractions, expressions involving fractions, inverse operations.
Skills: use cuisenaire rods and other tools
(number lines, area model) to make sense of fraction operations, apply 4 operations to fractions, solve algebraic equations involving fractions.
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Make a foldable explaining how to add, subtract, multiply and divide numeric fractions and algebraic fractions.
Create three equations which involve the use of fractions to solve.
Have a classmate solve your equations explaining each step to you.
A.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
8.E E.7 Solve linear equations in one variable. b b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it.
7.NS.A.1
: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.2
: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Pacing
14 lessons

Unit Title: Linear Equations
Unit Summary:
This unit will explore the characteristics of lines. Students will find the distance between two points on a line and explore the pythagorean theorem to establish understanding of the distance formula.
The concept of slope will be a major focus. Students will discover the slope formula as a ratio of vertical to horizontal distance. Equations of lines will be written given two points on the line or one point and the slope of the line. Students will also use the concept of slope to test for collinearity of points. Important connections between Algebra and Geometry will be emphasized.
Suggested Pacing: 13 lessons
Unit Essential Questions:
● How do you find the distance between two points on a coordinate plane?
● How does the Pythagorean theorem relate to the distance formula?
● How do you find the slope of a line given two points?
● How can you write the equation of a line given two points on the line?
● How can you write the equation of a line given a point on the line and the slope?
● How do I know if points are collinear?
Unit Enduring Understandings:
● To be able to write the equation of a line, two things are needed; slope and a point.
● Algebra and Geometry are inter-related.
Unit Benchmark Assessment Information:

Objectives
(Students will be able to…)
Students will be able to find the distance between two points on a line.
Students will be able to find the slope of a line.
Students will be able to determine if three or more points are collinear using slope.
Students will be able to write the equation of a line.
Essential
Content/Skills
Content: points, lines, distance, slope, collinearity,
Pythagorean theorem.
Skills: find distance between points, find slope, understand the connection between the distance formula and the Pythagorean
Theorem, write equations of lines.
Suggested
Assessments
Standards
(NJCCCS CPIs, CCSS, NGSS)
Have students stand at given points on a taped grid on the floor.
Have them find the distance between given students, and write the equation of the line through those points (students can hold a string to represent the line through the points).
Also, determine the collinearity of specific students.
Use foldable on page
18-19 from “Dinah
Zike’s Notebook
Foldables Algebra 1”.
Have students glue down all four foldables, under each foldable, student is to write an appropriate situation that models the line. (teacher pre-writes labels on x and y axes).
S.ID.7: Interpret the slope (rate of change) and the intercept
(constant term) of a linear model in context of the data.
8.G.8 Apply the Pythagorean
Theorem to find the distance between two points in a coordinate system.
A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Pacing
13 lessons