Module 1: Rational Numbers, Expressions, Equations, and Inequalities

MODULE OVERVIEW

CONTENT STANDARDS

ENDURING UNDERSTANDINGS AND ESSENTIAL QUESTIONS

STARTING POINTS

UNIT IN CONTEXT

COMPONENTS OF RIGOR

STRATEGIES FOR DIVERSE LEARNERS

COMMON MISCONCEPTIONS

MODULE OVERVIEW

In this module, students will learn to simplify algebraic expressions, solve up to two-step equations & inequalities, and apply these skills to solving geometric and real world problems.

Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems. (www.corestandards.org)

Note: Items preceeded by ** are for the accelerated Grade 7 curriculum only.

CONTENT STANDARDS

7.EE.A.1 - Apply properties of operations as strategies to add, subtract, factor, and expand expressions with rational coefficients. (Major)

7.EE.A.2 - Understand that rewriting an expression in different forms in a problem can shed light on the problem of how the quantities in it are related. Note: Include applying formulas to problem-solving situations. (Major)

7.EE.B.3 - Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. (Major)

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. (Major)

4a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
4b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.

7.G.B.5 - Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (Major)

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. (Major)

1a. Describe situations in which opposite quantities combine to make 0.
1b. 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.
1c. 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.
1d. 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 multiply and divide rational numbers. (Major)

2a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1)=1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
2b. 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 −(pq)=(−p)q=p(−q)−(pq)=(−p)q=p(−q). Interpret quotients of rational numbers by describing real-world contexts.
2c. Apply properties of operations such as strategies to multiply and divide rational numbers.
2d. 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.

7.NS.A.3 - Solve real-world and mathematical problems involving the four operations with rational numbers. Note: This standard should include the order of operations. (Major)

**8.EE.B.5 - Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. (Note: Build on student understanding of constant of proportionality (7.RP.A.2) to build understanding of slope.) (Major)

**8.EE.B.6 - Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b. (Major)

**8.EE.C.7 - Solve linear equations in one variable.

7a. 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). (Major)
7b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (Major) Note: Check understanding of solving equations involving the distributive property (7.EE.A.1 and 7.EE.B.4) and build on prior knowledge.

**8.EE.C.8 - Analyze and solve pairs of simultaneous linear equations. (Major)

**8.F.B.4 - Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (Major)

ENDURING UNDERSTANDINGS AND ESSENTIAL QUESTIONS

ENDURING UNDERSTANDINGS

ESSENTIAL QUESTIONS

Any rational number can be expressed as a fraction in an infinite number of ways.

How can rational number be expressed?
Which representation of a rational number is most appropriate for the given situation?

Number operations can also apply to integer and rational number operations.

How do operations with integers compare to operations with rational numbers?

Expressions are powerful tools for exploring, reasoning about, and representing situations.

Algebraic properties can generate equivalent expressions even when their symbolic forms differ.

How can an expression be rewritten in equivalent forms?

It is often important to find the value of a variable for which two expressions represent the same quantity this is known as solving an equation. Solving an equation requires series of symbolic transformations to the equation so that in one expression we are left with only the variable, all by itself.

How can you justify that a series of symbolic transformations makes sense to solve a 2-step equation?
How does a series of symbolic transformations help in checking for the reasonableness to a solution for an equation?

Inequalities indicate that the value of one expression is greater than (or greater or equal to) the value of the other expression.

In solving an inequality, multiplying or dividing both expressions by a negative number reverses the sign that indicates the relationships between the two expressions.

What does the solution to an inequality represent in the context of the situation?

Properties of angle relationships provide the content to create, solve, and verify solutions to equations to determine an unknown for an angle measure.

How do angle relationships relate to one another?
How can solving equations be applied to angle relationships?

**A rate of change describes how one variable quantity changes with respect to another, in other words, a rate of change describes the co-variation between two variables.

**Describe the proportionality that exists within linear functions. How can this confirmed using an equation, a table of values, and/or a graph?

**The intersection(s) of two or more linear functions can be found using a variety of representations and techniques such as graphing, linear combination, and substitution. The existence or non-exsistence of intersection(s) can reveal information about the linear functions.

**Describe the various techniques for solving systems of linear equations and when is each strategy most effective to use?
What does the solution to a system of equations means in the context of the problem?

**The equal sign can indicate that two expressions are equivalent. It is often important to find the value(s) of a variable for which two expressions represent the same quantity. Finding the value(s) of a variable for which two expressions represent the same quantity is know as solving an equation.

**What strategies can be used to solve multi-step equations?
What situations will produce equations with no solutions or infinite solutions?

STARTING POINTS

Each standard is represented by the number of days it is addressed in the module. This is not the same as the number of instructional days within the module as some lessons address multiple standards.

STANDARD(S)

DAYS

BIG IDEAS

7.EE.A.1

Review properties of addition and multiplication (commutative, associative, identity, multiplicative property of zero, distributive property, and additive inverse)
Use properties to generate equivalent expressions with rational coefficients

7.EE.A.2

Formulate expressions to represent real-world scenarios
Identify equivalent expressions
Rewrite expressions in equivalent forms

7.EE.B.3

Apply understanding of operations with rational numbers to solve multi-step problems
Convert between forms of positive and negative rational numbers
Justify conclusions using mathematical reasoning

7.EE.B.4a

Write equations to represent real-world scenarios
Use manipulatives, scales, and/or diagrams to solve equations of the form
px + q = r or p(x + q) = r
Solve equations of the form px + q = r or p(x + q) = r algebraically
Verify solutions and reason about the solution in the context of the problem

7.EE.B.4b

Review writing and graphing inequalities to represent real-world scenarios
Solve two-step inequalities involving rational numbers and negative coefficients and graph solutions on a number line
Verify solutions and reason about the solutions in the context of the problem

7.G.B.5

Identify angle relationships, including vertical, adjacent, complementary, and supplementary
Use angle relationships to create and solve equations to determine an unknown for an angle figure
Verify solution and reason about angle measures in the context of the problem

7.NS.A.1a

Describe situations in which opposite quantities combine to make zero

7.NS.A.1b

Understand concepts of zero pairs and additive inverse

7.NS.A.1c

Understand subtraction of rational numbers as adding the additive inverse

7.NS.A.1d

Apply properties of operations as strategies to add and subtract rational numbers

7.NS.A.2a

Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
Interpret products of rational numbers by describing real-world contexts.

7.NS.A.2b

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
Interpret quotients of rational numbers by describing real-world contexts

7.NS.A.2c

Apply properties of operations as strategies to multiply and divide rational numbers

7.NS.A.2d

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.

7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers (Computations with rational numbers extend the rules for manipulating fractions to complex fractions)

**8.EE.B.5

Build on understanding of constant of proportionality (7.RP.A.2) to develop understanding of slope
Graph and compare different proportional relationships when given scenario, equation, and/or table of values, and relate to similar triangles (review from Grade 7)
Identify direct variation situations from other linear situations and write the equation y = mx to represent the function (review from Grade 7)

**8.EE.B.6

Apply understanding of translations in the coordinate plane to apply a vertical translation of y = mx to generate the linear function y = mx + b
Make connection that for any linear function the rate of change is constant
Identify functions of the form y = mx + b as linear functions and graph the functions
Find the slope of a line given different representations of that line (graph, table of values, equation)

**8.EE.C.7a

Apply understanding of linear functions to solve real-world problems with linear equations
Write and solve equations from real-world scenarios

**8.EE.C.7b

Solve linear equations in one variable, including equations with one solution, infinitely many solutions, or no solutions
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms

**8.EE.C.8

Solve pairs of simultaneous linear equations

**8.F.B.4

Find the slope and the y-intercept of a line
Use the slope and the y-intercept to write an equation of a line
Write equations in slope-intercept form to solve real-life problems
Use the point of a line and the slope to write an equation of the line
Use any two points to write an equation of a line
Write equations in point-slope form to solve real-life problems

UNIT IN CONTEXT

How does this unit relate to the progression of learning? What prior learning do the standards in this unit build upon? How does this unit connect to essential understandings of later content in this course and in Algebra?The table below outlines key standards from previous and future courses that connect with this instructional unit of study.

REACH BACK STANDARDS

IN-COURSE CONNECTIONS

REACH AHEAD STANDARDS

7.EE.A.1

6.EE.A.3 - Apply the properties of operations to generate equivalent expressions.
6.EE.4 - Identify when two expressions are equivalent

7.EE.A.2

6.EE.A.3 - Apply the properties of operations to generate equivalent expressions.
6.EE.4 - Identify when two expressions are equivalent

7.EE.B.3

7.NS.A.3 - Solve real-world and mathematical problems involving the four operations with rational numbers. *Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

7.EE.B.4

6.EE.B.6 - Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.7 - Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

7.G.B.5

4.MD.C.7 - Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

7.NS.A.1

6.NS.C.6 - Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.7 - Understand ordering and absolute value of rational numbers.

7.NS.A.2

6.NS.A.1 - Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

7.NS.A.3

6.NS.B.3 - Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

8.EE.B.5 *Accelerated Material

7.RP.A.2 - Recognize and represent proportional relationships between quantities.
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.

8.EE.B.6 *Accelerated Material

7.G.A.1 - Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.RP.A.2 - Recognize and represent proportional relationships between quantities.
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.

8.EE.B.7 *Accelerated Material

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.
7.RP.A.2 - Recognize and represent proportional relationships between quantities.
7.EE.A.1 - Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.2 - Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

8.F.B.4 *Accelerated Material

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.
7.RP.A.2 - Recognize and represent proportional relationships between quantities.
8.F.A.3 - Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

7.EE.A.1

7.EE.A.2 - Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.

7.EE.A.2

7.EE.A.1 - Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.B.3

7.NS.A.3 - Solve real-world and mathematical problems involving the four operations with rational numbers.

7.EE.B.4

7.RP.A.2 - Recognize and represent proportional relationships between quantities.
7.NS.A.3 - Solve real-world and mathematical problems involving the four operations with rational numbers.

7.G.B.5

7.NS.A.1

7.NS.A.2 - Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

7.NS.A.2

7.NS.A.3 - Solve real-world and mathematical problems involving the four operations with rational numbers.

7.NS.A.3

7.EE.B.3 - Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
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.

8.EE.B.5 *Accelerated Material

8.EE.B.6 - Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

8.EE.B.6 *Accelerated Material

8.EE.B.5 - Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

8.EE.B.7 *Accelerated Material

8.SP.A.3 - Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

8.F.B.4 *Accelerated Material

8.F.B.5 - Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally
8.SP.A.2 - Know that straight - lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.A.3 - Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

7.EE.A.1

8.EE.C.7.a - Solve linear equations in one variable.
8.EE.C.7.b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

7.EE.A.2

8.EE.C.7 - Solve linear equations in one variable.

7.EE.B.3

8.EE.A.4 - Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

7.EE.B.4

8.EE.C.7 - Solve linear equations in one variable.

7.G.B.5

8.G.A.1 - Verify experimentally the properties of rotations, reflections, and translations:
Lines are taken to lines, and line segments to line segments of the same length.
Angles are taken to angles of the same measure.
Parallel lines are taken to parallel lines.

7.NS.A.1

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.

7.NS.A.2

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.

7.NS.A.3

8.EE.A.2 - Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

8.EE.B.5 *Accelerated Material

8.F.A.2- Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.EE.C.8 - Analyze and solve pairs of simultaneous linear equations.

8.EE.B.6 *Accelerated Material

8.EE.C.8 - Analyze and solve pairs of simultaneous linear equations.
8.F.A.2 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
8.F.A.3 - Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

8.EE.B.7 *Accelerated Material

8.F.B.4 *Accelerated Material

COMPONENTS OF RIGOR

Standard(s)

Conceptual Understanding

Procedural Fluency

Application

7.NS.A.1a

Using two-color counters, number lines, or other visual models to add and subtract integers
Showing that a number and its opposite have a sum of zero

Adding and subtracting integers
Identifying and using zero pairs
Modeling the use of zero pairs for addition of subtraction of rational numbers

Solving real-world problems that involve addition and subtraction of rational numbers
Solving real-world problems involving zero pairs

7.NS.A.1b

Demonstrating an understanding of additive inverse
Connecting distances on the number line to the appropriate subtraction statement

Modeling, using mats, number lines, counters, equations, and so on, different combinations of positive and negative integers and explain how they reason their solutions
Discovering formal rules for adding rational numbers with different signs

Solving and interpreting real-world and mathematical problems involving addition and subtraction of rational numbers
Interpreting sums of rational numbers by describing real-world contexts.

7.NS.A.1c

Discovering that subtraction and adding with an additive inverse provide the same results

Modeling real world context that involves subtraction of rational numbers using a number line

Clarifying their reasoning about subtraction and additive inverse through oral or written communication

7.NS.A.1d

Discovering that properties of operations apply to addition and subtraction of rational numbers by identifying examples and non-examples

Solving numerical addition and subtraction equations using the properties of the operations

Clarifying their understanding of the properties of operations as they apply to addition and subtraction of rational numbers

7.NS.A.2a

Discovering the rules for multiplying rational numbers by reasoning from real world examples

Using different interpretations for the (-) sign such as "negative" or "the opposite of" to make sense of real world context using rational numbers

Concluding that the properties of the operations for multiplication hold for rational number multiplication

7.NS.A.2b

Using reasoning to determine that division by zero is not defined
Discovering that division as the inverse of multiplication holds true with rational numbers

Generating an algorithm for multiplying rational numbers
Generating and using a rule or algorithm for dividing rational numbers
Developing fluency through practice with multiplication and division of rational numbers

Solving and interpreting real-world and mathematical problems involving the products of rational numbers
Solving and interpreting real-world and mathematical problems involving the quotients of rational numbers
Clarifying their own understanding of their relationship between multiplication and division of rational numbers through writing

7.NS.A.2c

Understanding that properties can be used as strategies for multiplying and dividing rational numbers

Using properties of the operations to explain the solution to real world problems

Clarifying their understanding of properties of operations by discussing and explaining their reasoning using mathematical vocabulary

7.NS.A.2d

Understanding that fractions can be written as decimals

Using long division to convert rational numbers in fraction form to decimal form
Sorting the decimal form of rational numbers into two types, ending in 0/terminating or repeating

Explaining why and how they know a long division quotient will repeat

7.NS.A.3

Extending the work with order of operations to all rational numbers

Using and applying the order of operations to rational numbers
Applying the properties of operations to solve problems
Converting rational numbers to decimals using long division
Computing with complex fractions

Solving and interpreting real-world and mathematical problems involving addition, subtraction, multiplication, and/or division of rational numbers
Solving and interpreting real-world and mathematical problems involving the properties of operations involving the multiplication and/or division of rational numbers

7.EE.A.1

Understanding the different properties and how these properties can be used to produce equivalent expressions
Understanding that factors in a product can be decomposed into two or more parts
Using algebra tiles to model equivalent expressions while making connections to the algebraic processes

Using properties to generate equivalent expressions
Changing an expression into an equivalent expression using properties of operations and combining like terms
Representing real world problems with equivalent expression using properties of operations, combining like terms and substitution, and solve them

Defending why two expressions are or are not equivalent

7.EE.A.2

Understanding that rewriting an expression in different forms can better show the relationship among the terms in the expression

Using properties to generate equivalent expressions
Modeling contextual problems with multiple variable expressions

Explaining, using precise mathematical vocabulary, how two equivalent expression relate the quantities

7.EE.B.3

Understanding how to apply mathematical properties to solve problems
Using variables and mathematical symbols correctly
Evaluating the reasonableness of solutions

Performing operations with positive and negative rational numbers
Converting between forms for expressions and equations
Simplifying expressions and solving equations

Creating expressions and equations to model real-world situations
Selecting a strategy to solve a real-world problem

7.EE.B.4a

Understanding how inverse operations are used to isolate the variable
Understanding why inverse operations may be used
Comparing algebraic solutions to algebraic solutions
Using visual models such as tape diagrams, algebra tiles, balance scales, etc. to represent and solve equations

Solving equations of the form px + 1 = r and p(x + q) = r
Performing operations with rational numbers
Applying an algorithm to solve an equation

Writing and solving an equation to represent a real-world situation
Applying a variety of strategies (including symbolic, graphical, and/or numerical methods) to solve equations

7.EE.B.4b

Understanding when an inequality should be used to represent a real-world situation
Understanding why the inequality symbol is reversed when multiplying or dividing inequalities by a negative value

Solving inequalities involving rational numbers
Graphing inequalities on a number line

Writing inequalities to represent a real-world situation
Interpreting solutions to an inequality in the context of a real-world or mathematical context

7.G.B.5

Reasoning about angle relationships
Representing angle relationships with equations

Identifying the angle relationship
Solving equations for an unknown angle measure

Interpreting tasks with context in order to write and solve for an unknown angle

8.EE.B.5

*Accelerated

Comparing proportional relationships
Representing proportional relationships as unit rate

Graphing proportional relationships
Interpreting unit rate as slope

Comparing a distance-time equation to a distance-time graph to determine which object is moving at a faster speed

8.EE.B.6

*Accelerated

Using similar triangles
Using slope to explain points on the coordinate plane

Deriving the equation y = mx + b for a line through the origin
Deriving the equation y = mx + b for a line intersecting the vertical axis at point b

Comparing slopes on the coordinate plane and determining if they are the same.
Using similar triangles to find slopes of points on the coordinate plane

8.EE.C.7a

*Accelerated

Understanding that linear equations can have one, zero, or infinitely many solutions

Solving one variable linear equations

Analyzing one variable linear equations
Explaining whether the solution has one, zero, or infinitely many solutions

8.EE.C.7b

*Accelerated

Understanding that linear equations can have one, zero, or infinitely many solutions

Solving linear equations with rational coefficients
Using the distributive property and combining like terms where appropriate

Explaining whether the solution has one, zero, or infinitely many solutions

8.EE.C.8

*Accelerated

Understanding that points of intersection are solutions to pairs of simultaneous linear equations

Solving pairs of simultaneous linear equations

Solving real-world problems that lead to pairs of simultaneous linear equations

8.F.B.4

*Accelerated

Determining the rate of change of a function
Determining the initial value of a function

Constructing a function to model a linear relationship between two quantities
Using a table to compare the quantities and create the function

Interpreting the rate of change and the initial value and apply the interpretation in context to the function constructed

STRATEGIES FOR DIVERSE LEARNERS

"Access and equity in mathematics at the school and classroom levels rest on beliefs and practices that empower all students to participate meaningfully in learning mathematics and to achieve outcomes in mathematics outcomes in mathematics that are not predicted by or correlated with student characteristics... Support for access and equity requires, but is not limited to, high expectations, access to high-quality mathematics curriculum and instruction, adequate time for students to learn, appropriate emphasis on differentiated processes that broaden students' productive engagement with mathematics, and human and material resources." - Principles to Actions (NCTM 20014, p. 60)

COMMITTMENT TO UDL PRINCIPLES AND DIVERSE LEARNERS

ACPS is a proponent of the Gradual Release of Responsibility (GRR) Instructional Framework, with full integration of UDL Principles. Through the work of Dr. George Brown from ACPS, our district has merged the GRR and UDL components into a format which enables teachers to implement both seamlessly into their lesson planning. In addition, ACPS utilizes the Principles for the Design of Mathematics Curricula: Promoting Language and Content Development (2017) from the Graduate School of Education at Stanford University. Guidance from this document is included with each ACPS mathematics lesson. Both of these documents may be found under the Resource tab for each module.

COMMON MISCONCEPTIONS

Students may make computation errors when simplifying numeric expressions with integers.
Students may think the absolute value symbol means to take the opposite of the value.
Students may incorrectly use the additive inverse when working with operations of integers.
Students may have confusion and misapplication of a complex fraction.
Students may think that a number divided by zero is a defined value.
Students may not follow the correct order of inverse operations when isolating the variable.
Students may rely too much on the key word strategy to represent word problems.
Students may use the same operation rather than the inverse operation to solve equations.
Students may have difficulty determining the inverse operation with fractional coefficients.
Students may forget to reverse the direction of the inequality symbol when multiplying or dividing by a negative number. Students may not understand why the inequality sign is reversed.
Students might confuse the symbolic components of the graph for an inequality, i.e. open vs. closed circle, directional arrows.
Students may not realize solutions to equations and inequalities may not just be integers.
Students may not check the reasonableness of their solution(s) in the context of the problem.

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