GSE Algebra I
GSE Algebra I Pacing Guide Part A
Unit 1: Relationships between Quantities and Expressions
In this unit students will:
• Use units of measure (linear, area, capacity, rates, and time)as a way to understand
problems.
• Interpret units in the context of the problem.
• Convert units and rates using dimensional analysis (English–to–English and
Metric–to– Metric without conversion factor provided and between English and Metric
with conversion factor);
• Identify the different parts of the expression or formula and explain their meaning.
• Decompose expressions and make sense of the multiple factors and terms by
explaining the meaning of the individual parts.
• Understand similarities between the system of polynomials and the system of integers.
• Understand that the basic properties of numbers continue to hold with polynomials.
• Draw on analogies between polynomial arithmetic and base–ten computation,
focusing on properties of operations, particularly the distributive property.
• Operate with polynomials with an emphasis on expressions that simplify to linear or
quadratic forms.
• Rewrite (simplify) expressions involving radicals.
• Use and explain properties of rational and irrational numbers.
• Explain why the sum or product of rational numbers is rational; why the sum of a
rational number and an irrational number is irrational; and why the product of a
nonzero rational number and an irrational number is irrational.
Standards for Unit 1
Use properties of rational and irrational numbers.
MGSE9–12.N.RN.2: Rewrite expressions involving radicals and rational exponents using the
properties of exponents.(i.e., simplify and/or use the operations of addition, subtraction, and
multiplication, with radicals within expressions limited to square roots).
MGSE9–12.N.RN.3: Explain why the sum or product of rational numbers is rational; why the
sum of a rational number and an irrational number is irrational; and why the product of a
nonzero rational number and an irrational number is irrational.
Reason quantitatively and use units to solve problems.
MGSE9–12.N.Q.1: Use units of measure (linear, area, capacity, rates, and time) as a way to
understand problems:
a.Identify, use, and record appropriate units of measure within context, within data
displays, and on graphs;
b.Convert units and rates using dimensional analysis (English–to–English and Metric–
to-Metric without conversion factor provided and between English and Metric with
conversion factor);
c. Use units within multi–step problems and formulas; interpret units of input and
resulting units of output.
MGSE9–12.N.Q.2: Define appropriate quantities for the purpose of descriptive modeling.
Given a situation, context, or problem, students will determine, identify, and use appropriate
quantities for representing the situation.
MGSE9–12.N.Q.3: Choose a level of accuracy appropriate to limitations on measurement
when reporting quantities. For example, money situations are generally reported to the nearest cent (hundredth).
MGSE9–12.A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
MGSE9–12.A.SSE.1a: Interpret parts of an expression, such as terms, factors, and coefficients, in context.
MGSE9–12.A.SSE.1b: Given situations which utilize formulas or expressions with multiple terms and/or factors, interpret the meaning (in context) of individual terms or factors.
Perform arithmetic operations on polynomials
MGSE9–12.A.APR.1: Add, subtract, and multiply polynomials; understand that polynomials
form a system analogous to the integers in that they are closed under these operations.
Unit 1 should be complete in 2 ½ weeks.
A unit test will be given at the completion of the Odysseyware unit and Notebook concerning the unit.
Unit 2: Reasoning with Linear Equations and inequalities
In this unit studentswill:
• Solve linear equations in one variable.
• Justify the process of solving an equation.
• Rearrange formulas to highlight a quantity of interest.
• Solve linear inequalities in one variable.
• Solve a system of two equations in two variables by using multiplication and addition.
• Justify the process of solving a system of equations.
• Solve a system of two equations in two variables graphically.
• Graph a linear inequality in two variables.
• Analyze linear functions using different representations.
• Interpret linear functions in context.
• Investigate key features of linear graphs.
• Recognize arithmetic sequences as linear functions.
Standards for Unit 2
Create equations that describe numbers or relationships
MGSE9-12.A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential Functions (integer inputs only).
MGSE9-12.A.CED.2: Create linear, quadratic, and exponential equations in two or more
variables to represent relationships between quantities; graph equations on coordinate axes
with labels and scales. (The phrase “in two or more variables” refers to formulas like the
compound interest formula, in which A = P(1 + r/n) nt has multiple variables.)
MGSE9-12.A.CED.3: Represent constraints by equations or inequalities, and by systems of
Equations and/or inequalities, and interpret data points as possible (i.e. a solution) or not
possible (i.e. a non-solution) under the established constraints.
MGSE9-12.A.CED.4: Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. Examples: Rearrange Ohm’s law V = IR to highlight
resistance R; Rearrange area of a circle formula A = π r2to highlight the radius r.
Understand solving equations as a process of reasoning and explain the reasoning
MGSE9-12.A.REI.1: Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.
Solve equations and inequalities in one variable
MGSE9-12.A.REI.3: Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.
Solve systems of equations
MGSE9-12.A.REI.5: Show and explain why the elimination method works to solve a system
of two-variable equations.
MGSE9-12.A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with
graphs), focusing on pairs of linear equations in two variables.
Represent and solve equations and inequalities graphically
MGSE9-12.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.
MGSE9-12.A.REI.11: Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.
MGSE9-12.A.REI.12: Graph the solution set to a linear inequality in two variables.
Build a function that models a relationship between two quantities
MGSE9-12.F.BF.1: Write a function that describes a relationship between two quantities.
MGSE9-12.F.BF.1a: Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with $15 and earns $2 a day, the explicit expression “2x+15” can be described recursively (either in writing or verbally) as “to find out how much money Jimmy will have tomorrow, you add $2 to his total today.”
MGSE9-12.F.BF.2: Write arithmetic and geometric sequences recursively and explicitly, use them to model situations, and translate between the two forms. Connect arithmetic Sequences to linear functions and geometric sequences to exponential functions.
Understand the concept of a function and use function notation
MGSE9-12.F.IF.1: Understand that a function from one set (the input, called the domain) to
another set (the output, called the range) assigns to each element of the domain exactly one
element of the range, i.e. each input value maps to exactly one output value. If f is a
function, x is the input (an element of the domain), and f(x) is the output (an element of the
range). Graphically, the graph is y = f(x).
MGSE9-12.F.IF.2: Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
MGSE9-12.F.IF.3: Recognize that sequences are functions, sometimes defined recursively,
whose domainis a subset of the integers. (Generally, the scope of high school math defines this subset as the set of natural numbers 1,2,3,4...) By graphing or calculating terms, students should be able to show how the recursive sequence a1=7, an=an-1+2; the sequences n= 2(n-1)+ 7; and the function f(x) = 2x + 5 (when x is a natural number) all define the same sequence.
Interpret functions that arise in applications in terms of the context
MGSE9-12.F.IF.4: Using tables, graphs, and verbal descriptions, interpret the key
characteristics of a function which models the relationship between two quantities. Sketch
a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
MGSE9-12.F.IF.5: Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
MGSE9-12.F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Analyze functions using different representations
MGSE9-12.F.IF.7: Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.
MGSE9-12.F.IF.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context).
MGSE9-12.F.IF.9: Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For example,
given a graph of one function and an algebraic expression for another, say which has the larger maximum.
Unit 2 must be complete by the end of Week 5 of the minimester.
A unit test will be given at the completion of the Odysseyware Unit and Notebook.
Unit 3: Modeling and Analyzing Quadratic Functions
In this unit students will:
• focus on quadratic functions, equations, and applications
• explore variable rate of change
• learn to factor general quadratic expressions completely over the integers and to
solve general quadratic equations by factoring by working with quadratic functions
that model the behavior of objects that are thrown in the air and allowed to fall subject
to the force of gravity
• learn to find the vertex of the graph of any polynomial function and to convert the
formula for a quadratic function from standard to vertex form
• apply the vertex form of a quadratic function to find real solutions of quadratic
equations that cannot be solved by factoring
• explore only real solutions to quadratic equations
• explain why the graph of every quadratic function is a translation of the graph of the
basic function f(x) =x2
• apply the quadratic formula
• justify the quadratic formula
Standards for Unit 3
Interpret structure of expressions
MGSE9‐12.A.SSE.2: Use the structure of an expression to rewrite it in different equivalent forms. For example, see x4–y4 as (x2) 2–(y2)2, thus recognizing it as a difference of squares that can be factored as (x2–y2) (x2+y2).
Write expressions in equivalent forms to solve problems
MGSE9–12.A.SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
MGSE9–12.A.SSE.3a: Factor any quadratic expression to reveal the zeros of the function definedby the expression.
MGSE9–12.A.SSE.3b: Complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression.
Create equations that describe numbers or relationships
MGSE9–12.A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, simple rational, and exponential functions (integer inputs only).
MGSE9-12.A.CED.2: Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (The phrase “in two or more variables” refers to formulas like the compound interest formula, in which A = P(1 + r/n)nt has multiple variables.)
MGSE9– 12.A.CED.4: Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. Examples: Rearrange Ohm’s law V = IR to highlight resistance R; Rearrange area of a circle formula A =π r2 to highlight the radius r.
Solve equations and inequalities in one variable
MGSE9‐12.A.REI.4: Solve quadratic equations in one variable.
MGSE9–12.A.REI.4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x–p)2=q that has the same solutions. Derive the quadratic formula from ax2+bx+c= 0.
MGSE9–12.A.REI.4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions).
Build a function that models a relationship between two quantities.
MGSE9–12.F.BF.1: Write a function that describes a relationship between two quantities.
Build new functions from existing functions.
MGSE9–12.F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of kgiven the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Understand the concept of a function and use function notation.
MGSE9–12.F.IF.1: Understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range, i.e. each input value maps to exactly one output value. If f is a function, x is the input (an element of the domain), and f(x) is the output (an element of the range). Graphically, the graph is y = f(x).
MGSE9–12.F.IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Interpret functions that arise in applications in terms of the context.
MGSE9–12.F.IF.4: Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
MGSE9–12.F.IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person–hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
MGSE9–12.F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Analyze functions using different representations.
MGSE9–12.F.IF.7: Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.
MGSE9–12.F.IF.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context).
MGSE9–12.F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
MGSE9–12.F.IF.8a: 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 termsof a context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms.
MGSE9–12.F.IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one function and an algebraic expression for another, say which has the larger maximum.
Unit 3 must be complete at the end of week 8 of the minimester.
Unit 3 Test will be given at the completion of Unit 3 Odysseyware and notebook on Unit 3.
A midterm exam will be given covering the material in Units 1,2, and 3.
