Mathematics Algebra I – Year in Detail (SAMPLE)

Transcription

Mathematics
Algebra I – Year in Detail (SAMPLE)
Univariate
Statistics
Unit 12
Bivariate
Statistics: Linear,
Quadratic, and
Exponential
Unit 11
Piecewise
Functions
Unit 10
Radical Functions
Unit 9
Quadratic
Functions
Unit 8
Solving Quadratic
Equations
Unit 7
Arithmetic with
Polynomials
Unit 6
Linear and
Exponential
Functions
Unit 5
Evaluating
Functions
Unit 4
Systems of Linear
Equations and
Inequalities
Unit 3
Linear Equations
and Inequalities
Unit 2
Solving SingleVariable
Equations and
Inequalities
Unit 1
12 days
10 days
15 days
15 days
10 days
15 days
20 days
20 days
14 days
15 days
12 days
12 days
A-CED.A.1
A-CED.A.4
A-REI.B.3
N-Q.A.1
N-RN.B.3
8.NS.A.1
8.NS.A.2
A-REI.D.10
A-REI.D.12
N-Q.A.1
F-IF.C.7a
8.EE.B.5
A-CED.A.3
A-REI.C.5
A-REI.C.6
A-REI.D.12
8.EE.C.8
F-IF.A.1
F-IF.A.2
F-IF.A.3
F-IF.B.5
F-IF.C.9
F-BF.B.3
8.F.A.3
A-SSE.B.3c
A-CED.A.1
A-CED.A.2
F-LE.A.1
F-LE.A.2
F-LE.A.3
F-LE.B.5
8.EE.A.1
8.F.A.3
8.F.B.4
8.F.B.5
A-SSE.A.1
A-APR.A.1
A-SSE.A.2
A-REI.A.1
A-REI.B.4
A-SSE.B.3a
A-SSE.B.3b
8.EE.A.1
8.EE.A.2
F-IF.B.4
F-IF.B.5
F-IF.B.6
A-APR.B.3
F-IF.C.7a
F-IF.C.8a
F-IF.C.9
F-BF.B.3
F-IF.B.4
F-IF.B.5
F-IF.B.6
F-IF.C.7b
F-IF.C.9
A-REI.11
F-IF.B.4
F-IF.C.7b
S-ID.C.7
S-ID.C.8
S-ID.C.9
F-BF.A.1a
N-Q.A.1
N-Q.A.2
S-ID.B.5
S-ID.B.6
8.EE.B.5
8.SP.A.1
8.SP.A.3
8.SP.A.4
N-Q.A.3
S-ID.A.1
S-ID.A.2
S-ID.A.3
Major Clusters
A-SSE Seeing Structure in Expressions (1, 2)
A-APR Arithmetic with Polynomials and Rational
Expressions (1)
A-CED Creating Equations (1, 2, 3, 4)
A-REI Reasoning with Equations and
Inequalities (1, 3, 4, 5, 6, 10, 11, 12)
F-IF Interpreting Functions (1, 2, 3, 4, 5, 6)
S-ID Interpreting Categorical and Quantitative
Data (7, 8, 9)
Supporting Clusters
N-Q Quantities (1, 2, 3)
A-SSE Seeing Structure in Expressions (3)
A-APR Arithmetic with Polynomials and
Rational Expressions (3)
F-IF Interpreting Functions (7, 8, 9)
F-BF Building Functions (1)
F-LE Linear, Quadratic, and Exponential
Models (1, 2, 3, 5)
S-ID Interpreting Categorical & Quantitative
Data (5, 6)
Additional Clusters
N-RN The Real Number System (3)
F-BF Building Functions (3)
S-ID Interpreting Categorical &
Quantitative Data (1, 2, 3)
Other
MP – Standards for
Mathematical Practice
Potential Gaps in Student
Pre-Requisite Knowledge
(click to see description)
8.NS – 1, 2
8.EE – 1, 2, 5, 8
8.F. – 2, 3, 4, 5
8.SP – 1, 3, 4
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Mathematics
This plan is meant to support districts creating their own curriculum or pacing guides. The scope and sequence of curricular resources such as Eureka Math
and others will likely not match this sample plan exactly. The standards do not require only one order to achieve mastery. Thus, many curricular tools will
suggest different scope and sequences for the standards. Districts should follow the guidance they feel is most appropriate for their students.
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Summary of Year for Algebra I
Algebra I will begin where 8 grade ended: linear equations in one and two variables and linear functions. In Algebra I students will increase their procedural skill and fluency in
solving linear equations and inequalities in one variable. Additionally, students will deepen their understanding of linear equations and inequalities in two variables. Also, the
course will emphasize modeling with linear equations and inequalities, culminating with solving systems of both linear equations and inequalities. From there, the course shifts
to developing a deeper understanding of functions. Students will focus on linear and exponential functions by exploring situations that could be modeled by either a linear
function or an exponential function. Then students will move to studying quadratic equations and functions, including identifying key elements of graphs, transformations with
functions, and identifying domain and range. Students will apply these newly developed understandings and skills with functions to simple radical functions and then to
piecewise-defined functions. The course concludes with a study of bivariate and univariate statistics to develop necessary understandings and skills the students will need for
their study of statistics in Algebra II.
Standards Clarification for Algebra I
•
•
Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Comments
are included throughout the document to clarify and provide additional background for each unit.
Some standards are addressed in both Algebra I and Algebra II. The relevance to Algebra I is discussed in the Assessment Limits and Clarifications column.
Mathematical Practices Recommendations for Algebra I
Two overarching practices relevant to Algebra I are:
• Make sense of problems and persevere in solving them (MP.1).
• Model with mathematics (MP.4). Specific modeling standards appear throughout the high school standards indicated by a star (★) symbol.
Indeed, other mathematical practices in Algebra I might be seen as contributing specific elements of these two. The intent of the following set is not to decompose the above
mathematical practices into component parts but rather to show how the mathematical practices work together.
• Reason abstractly and quantitatively (MP.2). This practice standard refers to one of the hallmarks of algebraic reasoning, the process of decontextualization and
contextualization. Much of elementary algebra involves creating abstract algebraic models of problems (A-CED, F-BF) and then transforming the models via algebraic
calculations (A-SSE, A-APR, F-IF) to reveal properties of the problems.
• Use appropriate tools strategically (MP.5). Spreadsheets, a function modeling language, graphing tools, and many other technologies can be used strategically to gain
understanding of the ideas expressed by individual content standards and to model with mathematics.
• Attend to precision (MP.6). In algebra, the habit of using precise language is not only a mechanism for effective communication but also a tool for understanding and
solving problems. Describing an idea precisely (A-CED, A-REI) helps students understand the idea in new ways.
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• Look for and make use of structure (MP.7). For example, writing 49x + 35x + 6 as (7x) + 5(7x) + 6, a practice many teachers refer to as “chunking,’’ highlights the structural
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similarity between this expression and z + 5z + 6, leading to a factorization of the original: ( (7x) + 3) ( (7x) + 2) (A-SSE, A-APR).
• Look for and express regularity in repeated reasoning (MP.8). Creating equations or functions to model situations is harder for many students than working with the
resulting expressions. An effective way to help students develop the skill of describing general relationships is to work through several specific examples and then express
what they are doing with algebraic symbolism (A-CED). For example, when comparing two different text messaging plans, many students who can compute the cost for a
given number of minutes have a hard time writing general formulas that express the cost of each plan for any number of minutes. Constructing these formulas can be
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Mathematics
facilitated by methodically calculating the cost for several different input values and then expressing the steps in the calculation, first in words and then in algebraic
symbols. Once such expressions are obtained, students can find the break-even point for the two plans, graph the total cost against the number of messages sent, and make
a complete analysis of the two plans.
Fluency Recommendations for Algebra I
A/G: Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a
slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using
systems of linear inequalities in two variables).
A-APR.1: Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in algebra, as well as in their symbolic work with functions.
Manipulation can be more mindful when it is fluent.
A-SSE.1b: Fluency in transforming expressions and chunking (parts of an expression as a single object) is essential in factoring, completing the square, and other algebraic
calculations.
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Mathematics
Possible time frame:
12 days
Unit 1: Solving Single-Variable Equations and Inequalities
In this unit students extend the work of the last three years and use linear equations and inequalities in one variable to model and solve real-world and mathematical problems.
The ultimate goal of this unit is for students to be able to rearrange a formula to highlight a quantity of interest and understand that this process as uses the same reasoning as
solving equations. Understanding how the process of solving an equation relates to using a formula will allow for students to use units as a way to understand problems and to
guide their solution to multi-step problems by choosing and interpreting units consistently in formulas. Students will continue to use their conceptual understanding as well as
their procedural skill and fluency in solving equations throughout the course to solve systems of linear equations and quadratic equations.
Assessment Limits
and Clarifications
Major Cluster Standards
Create equations that describe numbers or relationships
HSA-CED.A.1 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. ★
HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★
Modeling standard
HAS-CED-A.1
i) Assessment tasks are limited to linear, quadratic, or
exponential equations with integer exponents.
★
Solve equations and inequalities in one variable
HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
Assessment Limits
and Clarifications
Supporting Cluster Standards
Reason quantitatively and use units to solve problems.
HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs
and data displays. ★
Modeling standard
HSN-Q.A.1 The skills associated with this standard can and
should be used in nearly every unit of this course. This standard
will culminate in Unit 11 when students study bivariate statistics.
★
HSN-Q.A.1 Does not include choosing and interpreting the scale
and the origin in graphs and data displays.
Additional Cluster Standards
Use properties of rational and irrational numbers.
8.NS.A.1 8.NS.A.2 Click to see wording of potential gaps whose content is a prerequisite for HSN-RN.B.3.
HSN–RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational
number and an irrational number is irrational; and that the product of a nonzero rational number and an
irrational number is irrational.
Assessment Limits
and Clarifications
N-RN.B.3 This standard can be taught through working with
formulas involving rational and irrational numbers.
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Mathematics
10 days
Unit 2: Linear Equations and Inequalities
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In 8 grade students developed a deep conceptual understanding of the equation for a line and slope as a rate of change. Additionally students will leave the 8 grade able to
fluently graph linear equations in the form y=mx+b. In this unit students will extend their work in graphing linear equations to graphing linear inequalities. Students will also
understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. This understanding can then be extended to the graph of
a linear inequality in two variables.
Assessment Limits
and Clarifications
Represent and solve equations and inequalities graphically
8.EE.B.5 Click to see wording of a potential gap whose content is a prerequisite for concepts in this unit.
HSA-REI.D.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).
HSA-REI.D.12 Limited to a
single linear inequality.
Students will graph systems of
linear inequalities in Unit 3.
HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict
inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Assessment Limits
and Clarifications
HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. ★
Analyze functions using different representations
HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Modeling standard
★
HSN-Q.A.1 The focus in this
unit is choosing and
interpreting the scale and the
origin in graphs.
HSF-IF.C.7a Limited to linear
equations. Students will graph
piece-wise defined functions in
Unit 10.
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Mathematics
15 days
Unit 3: Systems of Linear Equations and Inequalities
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In 8 grade students were introduced to systems of linear equations and developed a conceptual understanding of the solution to a system of linear equations as the point of
intersection of their graphs which is the point that satisfies both equations simultaneously. Students also solved systems of linear equations algebraically as well as estimated
the solution to a system by graphing. This unit will extend the students work with systems of linear equations to include a focus on modeling with systems of linear equations
and systems of linear inequalities. In this unit students will increase their procedural skill and fluency in solving systems of linear equations exactly and approximately (e.g., with
graphs) and develop the ability to select a solution method based on the specific system they are trying to solve.
Assessment Limits
and Clarifications
Modeling standard
★
8.EE.C.8 Click to see wording of a potential gap whose content is a prerequisite for concepts in this unit.
HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as
viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods. ★
Solve systems of equations
HSA-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple
of the other produces a system with the same solutions.
HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two
variables.
HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict
inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Students will use their
procedural skill and fluency in
graphing a linear inequality to
graph a system of linear
inequalities. (A-REI.D.12 from
Unit 2)
HSA-REI.C.6 This unit should
include a heavy emphasis on
using systems of equations
(and inequalities) to model and
solve real-world problems.
i) Assessment tasks have a realworld context. ii) Assessment
tasks have hallmarks of
modeling as a mathematical
practice (less defined tasks,
more of the modeling cycle,
etc.).
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Mathematics
15 days
Unit 4: Evaluating Functions
In this unit students extend their work with functions in grade 8 to develop a more sophisticated understanding of a function including function notation. After function notation
is established, students will evaluate functions. As students evaluate functions using their graphs they will begin developing an understanding of function transformations which
will be discussed more thoroughly in Unit 9. Also in this unit students will graph functions by creating a table of values. Function types are not limited in this unit but should
focus on linear, exponential, quadratic, and radical functions. Through graphing their own functions, students will deepen their understanding of function transformations.
Graphing will also provide students the opportunity to apply their conceptual understanding of domain and range as it relates to the function rule and its corresponding graph.
8.F.A.3 Click to see wording of a potential gap whose content is a prerequisite for this unit.
Understand the concept of a function and use function notation
HSF-IF.A.1 Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element of the range. 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. The graph of f is the graph of the equation y = f(x).
HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.
HSF-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose
domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively
by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context
HSF-IF.B.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.★
HSF-IF.C.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 quadratic function and an algebraic expression for another, say which
has the larger maximum.
Build new functions from existing functions
HSF-BF.B.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 k given 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.
Assessment Limits and Clarifications
Modeling standard
★
HSF-IF.B.5 As students explore specific function types in Units 8, 9, and 10, they will
continue to relate the domain of a function to its graph and, in context, to the
quantitative relationship it describes.
HSF-IF.C.9 i) Assessment tasks are limited to linear functions, quadratic functions, square
root functions, cube root functions, piecewise-defined functions (including step functions
and absolute value functions), and exponential functions with domains in the integers.
HSF-BF.B.3 As students explore quadratic functions in Unit 8, they will formalize their
understanding of function transformations and then use this understanding in Units 9
and 10.
HSF-BF.B.3
i) Identifying 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 are limited to linear and quadratic functions.
ii) Experimenting and illustrating an explanation of the effects on the graph using
technology is limited to the functions in HSF-IF.B.4, HSF-IF.B.6, and HSF-IF.C.9.
iii) Assessment tasks do not involve recognizing even and odd functions.
Page 7
Mathematics
10 days
Unit 5: Linear and Exponential Functions
After a broad study of functions in Unit 4 that focused on the concept of a function and the relationship between inputs and outputs, students will take a closer look at linear
and exponential functions. Students will be very familiar with relationships which grow linearly, but this will be their first exposure to relationships which grow exponentially.
Students will recognize real-world situations can be modeled with linear and with exponential functions. Additionally, students will write linear and exponential functions,
including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or two input-output pairs. Students will use the functions they create to
solve problems and will interpret the parameters of both linear and exponential functions in terms of a context.
Write expressions in equivalent forms to solve problems
8.EE.A.1 Click to see wording of a potential gap whose content is a prerequisite for HSA-SSE.B.3.
HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity
represented by the expression.★
c. Use the properties of exponents to transform expressions for exponential functions. For example the
expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest
rate if the annual rate is 15%.
8.F.A.3 Click to see wording of a potential gap whose content is a prerequisite for HSA-CED.A.1.
HSA-CED.A.1 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. ★
HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations
on coordinate axes with labels and scales. ★
Modeling standard
HSA-SSE.B.3c Students will only evaluate and use exponential
functions with domains in the integers.
i) Assessments have a real-world context. There is interplay
between the mathematical structure of the expression and
the structure of the situation such that choosing and
producing an equivalent form of the expression reveals
something about the situation. ii) Tasks are limited to
exponential expressions with integer exponents.
HAS-CED.A.1 i) Assessments limited to linear, quadratic, or
exponential equations with integer exponents.
HSA-CED.A.2 Although students will be presented with graphs
of exponential functions, students will not be expected to
graph exponential functions.
Construct and compare linear, quadratic, and exponential models and solve problems
HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. ★
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by
equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to
another.
8.F.B.4 Click to see wording of a potential gap whose content is a prerequisite for HSF-LE.A.2.
HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs (include reading these from a table). ★
HSF-LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more generally) as a polynomial function. ★
Interpret expressions for functions in terms of the situation they model
8.F.B.5 Click to see wording of a potential gap whose content is a prerequisite for HSF-LE.B.5.
HSF-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. ★
★
Modeling standard
HSF-LE.A.2 Students will fit exponential and linear functions to
data sets represented by scatter plots in Unit 11.
i) Assessment tasks are limited to constructing linear and
exponential functions in simple context (not multi-step).
★
HSF-LE.B.5 i) Tasks have a real-world context. ii) Exponential
functions are limited to those with domains in the integers.
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Mathematics
15 days
Unit 6: Arithmetic with Polynomials
Units 6 and 7 will focus on developing the algebraic skills and fluencies that are essential to the work with quadratic functions beginning in Unit 8. This unit will focus on
performing arithmetic operations with polynomials. Students will apply their understanding and skill in generating equivalent expressions involving combining like terms, the
distributive property, and the properties of integer exponents to begin multiplying polynomials. Through their work with polynomial expressions with and without context,
students will interpret parts of expressions including terms, factors, and coefficients; moreover, students will interpret complicated expressions in context by viewing one or
more of their parts as a single entity.
Assessment Limits
and Clarifications
Interpret the structure of expressions
HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.★
a. Interpret parts of an expression, such as terms, factors, and coefficients.
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b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) as the product
of P and a factor not depending on P.
Perform arithmetic operations on polynomials
HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition,
subtraction, and multiplication; add, subtract, and multiply polynomials.
Modeling standard
★
HSA-APR.A.1 Multiplication of
polynomials should extend
beyond the product of two
binomials.
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Mathematics
Unit 7: Solving Quadratic Equations
20 days
After learning in Unit 6 how to multiply polynomials, students will learn in this unit how to undo the process of multiplying polynomials to factor polynomials. Students
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began factoring work in the 6 grade where they applied the distributive property to simplify an expression involving a variable as well as to factor out a greatest common
factor of an expression involving a variable. This unit will build on this work by increasing the complexity of the expressions and the complexity of applying the distributive
property. This unit will begin by teaching students the necessary conceptual understanding and procedural skill and fluency in factoring quadratics to enable them to finish
the unit by solving quadratic equations. Students will also solve quadratic equations that cannot be solved by factoring requiring the introduction of completing the square
and then further develop into the quadratic formula. Once these alternate methods have been established, students will reengage in solving quadratic equations and
extend their work to include quadratics with irrational solutions. By the end of this unit, students will be able to select the most appropriate solution method based on the
initial form of the equation.
Interpret the structure of expressions
4
4
2 2
2 2
HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x – y as (x ) – (y ) ,
2
2
2
2
thus recognizing it as a difference of squares that can be factored as (x – y )(x + y ).
Students will solve single variable equations to find the
solutions for a quadratic equation. (A-REI.B.3 from Unit 1)
Understand solving equations as a process of reasoning and explain the reasoning
HSA-REI.A.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.
Solve equations and inequalities in one variable
8.EE.A.2 Click to see wording of a potential gap whose content is a prerequisite for HSA-REI.B.4.
HSA-REI.B.4 Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the
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form (x – p) = q that has the same solutions. Derive the quadratic formula from this form.
2
b. Solve quadratic equations by inspection (e.g., for x = 49), taking square roots, completing the square, the
quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Write expressions in equivalent forms to solve problems
8.EE.A.1 Click to see wording of a potential gap whose content is a prerequisite for HSA-SSE.B.3.
HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression.★
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it
defines.
HAS-SSE.A.2
i) Assessment tasks are limited to numerical expressions
and polynomial expressions in one variable. ii) Examples:
2
2
Recognize 53 - 47 as a difference of squares and see an
opportunity to rewrite it in the easier-to-evaluate form
2
(53-47)(53+47). See an opportunity to rewrite a + 9a + 14
as (a+7)(a+2).
HAS-REI.A.1 i) Assessment tasks are limited to quadratic
equations.
HSA-REI.B.4 Students will use the skills associated with
this standard in Unit 8 to graph quadratic functions.
HSA-SSE.B.3 Quadratic functions are not introduced until
Unit 8; therefore the solutions for a quadratic function
should not be referred to zeros until then.
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Mathematics
20 days
Unit 8: Quadratic Functions
2
Students will begin this unit revisiting the function f(x)=x that they worked with in Unit 4 and graph the function by using by creating a table of values. From there students will
2
reengage with function transformations to graph quadratic functions and to generate a general form for a quadratic function (f(x)=a(x-h) +k). After developing their conceptual
understanding of quadratic functions and procedural skill and fluency in graphing them, students will begin working with quadratic functions in different forms that will lead
them to using other methods for graphing. Quadratic functions that model real-world relationships between quantities will be presented to students in this unit, and students
will compare quadratic functions presented in different forms As students analyze and graph quadratic functions, the topic of domain should be revisited in this new context.
Students will also identify key features of quadratic graphs and will interpret the meaning of these features in the context of the real-world relationship the graph models.
HSF-IF.B.4 For a function that models a relationship between two quantities, 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. Key features include: intercepts; intervals where the function is increasing,
decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and
periodicity.★
HSF-IF.B.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.★
HSF-IF.B.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.★
Modeling standard
HSF-IF.B.4 i) Assessment tasks have a real-world context. ii)
Assessment tasks are limited to linear functions, quadratic
functions, square root functions, cube root functions, piecewisedefined functions (including step functions and absolute value
functions), and exponential functions with domains in the integers.
Understand the relationship between zeros and factors of polynomials
HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
HSF-IF.C.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.
HSF-IF.C.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 quadratic
function and an algebraic expression for another, say which has the larger maximum.
Modeling standard
HSA-APR.B.3 This standard should be addressed early in this unit
when students start connecting solutions for the equation to zeros
on the graph. Although cubic functions should be addressed as part
of this standard, the main focus should still be quadratic functions.
i) Assessment tasks are limited to quadratic and cubic polynomials
in which linear and quadratic factors are available. For example,
2
find the zeros of (x - 2)(x - 9).
HSF-IF.C.7a Limited to only quadratic functions.
HSF-IF.C.9 i) Assessment tasks are limited to linear functions,
quadratic functions, square root functions, cube root functions,
piecewise-defined functions (including step functions and absolute
value functions), and exponential functions with domains in the
integers.
★
★
Page 11
Mathematics
Build new functions from existing functions
HSF-BF.B.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 k given 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.
HSF-BF.B.3
i) Identifying the effect on the graph of replacing f(x) by f(x) + k,
kf(x), f(kx), and f(x+k) for specific values of k (both positive and
negative) is limited to linear and quadratic functions. ii)
Experimenting with cases and illustrating an explanation of the
effects on the graph using technology is limited to linear functions,
piecewise-defined functions (including step functions and absolute
value functions), and exponential functions with domains in the
integers. iii) Tasks do not involve recognizing even and odd
functions.
Page 12
Mathematics
14 days
Unit 9: Radical Functions
Students will be coming in to this unit with a good understanding of how to use function transformations, how to evaluate functions, and how to graph linear and quadratic
functions. Much of this conceptual understanding and procedural skill and fluency will be the foundation for this unit on radical functions. In Unit 4 students had limited
exposure to radical functions as they explored the concepts of functions. In this unit students will be responsible for graphing square root and cube root functions. Students will
graph the parent functions using a table of values and will again experiment with transformations to show that the familiar function transformation rules extend to radical
functions. Radical functions that model real-world relationships between quantities will be presented. Students will compare radical functions presented in different forms. The
topic of domain should be revisited in this new context. Students will also identify key features of radical graphs and will interpret the meaning of these features in the context
of the real-world relationship the graph models.
periodicity.★
HSF-IF.B.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.
★
HSF-IF.B.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.★
Modeling standard
functions), and exponential functions with domains in the
integers.
functions), and exponential functions with domains in the
integers.
★
b. Graph square root, cube root, and piecewise-defined functions, including step functions and
absolute value functions.
HSF-IF.C.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 quadratic
function and an algebraic expression for another, say which has the larger maximum.
Modeling standard
HSF-IF.C.7b Limited to square root and cube root functions.
Students do not solve square root or cube root equations in this
course.
HSF-IF.C.9 i) Assessment tasks are limited to linear functions,
piecewise-defined functions (including step functions and
absolute value functions), and exponential functions with
domains in the integers.
★
Page 13
Mathematics
15 days
Unit 10: Piecewise Functions
This unit will continue to build on students’ conceptual understanding and procedural skill and fluency in working with functions. In this unit students will be responsible for
graphing various piecewise-defined functions with an initial focus on linear functions leading to the concepts of step functions and absolute value functions. To begin this unit,
students will explore real-world situations that require multiple function rules to model a given situation. Students will begin by graphing piecewise-linear functions and will
extend this to include all previously graphed function types. Absolute value and step functions are among the various piecewise-defined functions students will graph in this
unit. Students will again use their conceptual understanding of function transformations to graph various absolute value functions and step functions. Students will also identify
key features of a graph and interpret the meaning of these features in the context of the real-world relationship the graph models.
HSA-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y
= g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find successive approximations. Include cases
where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic
functions.★
periodicity.★
Modeling standard
★
HAS-REI.D.11 i) Tasks that assess conceptual understanding of the
indicated concept may involve any of the function types mentioned
in the standard except exponential and logarithmic functions. ii)
Finding the solutions approximately is limited to cases where f(x)
and g(x) are polynomial functions.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and
absolute value functions.
Modeling standard
★
HSF-IF.C.7b Limited to piecewise-defined functions including, but
not limited to, step functions and absolute value functions. The
piece-wise functions that students will graph in this unit should
include linear, quadratic, and radical functions as pieces. (F-IF.C.7
from Units 2, 8, and 9)
Page 14
Mathematics
15 days
Unit 11: Bivariate Statistics: Linear, Quadratic, and Exponential
This unit is designed to provide students the opportunity to apply their conceptual understanding of functions (emphasizing linear, quadratic, and exponential) in a statistics
context. This unit will support and enhance the students’ conceptual understanding of functions and their ability to model real-world situations. The students will begin this unit
by summarizing categorical data for two categories in two-way frequencies tables. From there students will represent data on two quantitative variables on a scatter plot and
describe how the variables are related. They will extend their work to begin fitting a function to the data and using those functions to solve problems in the context of the data.
Beyond informally assessing the fit of a function to a data set, students will compute (using technology) and interpret the correlation coefficient of a linear fit. Finally, students
will establish and distinguish between correlation and causation in the context of the data.
Interpret linear models
8.EE.B.5 8.SP.A.3 Click to see wording of potential gaps whose content is a prerequisite for HSS-ID.C.7.
HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the
context of the data.
HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
HSS-ID.C.9 Distinguish between correlation and causation.
Build a function that models a relationship between two quantities
HSF-BF.A.1 Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
HSN-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the origin in
graphs and data displays. ★
HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. ★
Summarize, represent, and interpret data on two categorical and quantitative variables
8.SP.A.4 Click to see wording of a potential gap whose content is a prerequisite for HSS-ID.B.5.
HSS-ID.B.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.
8.SP.A.1 Click to see wording of a potential gap whose content is a prerequisite for HSS-ID.B.6.
HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the
variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the
data. Use given functions or choose a function suggested by the context. Emphasize linear,
quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Modeling standard
Students will write linear and exponential functions to model the
relationship between two quantitative variables in a data set. (ACED.A.2 from Unit 5)
HSF-BF.A.1 i) Assessment tasks have a real-world context. ii)
Assessment tasks are limited to linear functions, quadratic functions,
and exponential functions with domains in the integers.
HSN-Q.A.1 This standard culminates in this unit by having students
choose and interpret the scale and the origin in data displays.
HSN-Q.A.2 This standard will be assessed in Algebra I by ensuring
that some modeling tasks (involving Algebra I content or securely
held content from grades 6-8) require the student to create a
quantity of interest in the situation being described (i.e., a quantity
of interest is not selected for the student by the task). For example,
in a situation involving data, the student might autonomously decide
that a measure of center is a key variable in a situation, and then
choose to work with the mean.
HSS-ID.B.6a i) Assessment tasks have a real-world context. ii)
Exponential functions are limited to those with domains in the
integers.
★
Page 15
Mathematics
12 days
Unit 12: Univariate Statistics
In this unit students will use univariate statistics to compare and make inferences about two data sets. This unit will utilize statistical concepts and skills the students learned in
th
th
the 6 and 7 grades and use them to develop more sophisticated statistical concepts and skills. Students will represent data sets graphically by plotting on the real number line
including dot plots, histograms, and box plots and use their graphical displays to make observations about the shape and distribution of the data. Students will then use
measures of center (median, mean) and spread (interquartile range, standard deviation) appropriate to the shape of the data distribution to compare two or more data sets.
Additionally students will explore more closely the differences in shape, center, and spread accounting for possible effects of extreme data points (outliers) and interpret these
differences within the context of the data sets. As students work with statistical measures in context they should choose levels of accuracy (and precision) appropriate to the
limitations on measurement.
Assessment Limits
and Clarifications
HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ★
Summarize, represent, and interpret data on a single count or measurement variable
HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
HSS-ID.A.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.
Modeling standard
★
Assessment Limits
and Clarifications
HSS-ID.A.2 Standard deviation
is the only statistical measure
the students have not been
exposed to prior to this unit.
HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data
points (outliers).
Page 16
Algebra I Potential Gaps in Student Pre-Requisite Knowledge
This document indicates pre-requisite knowledge gaps that may exist for Algebra I students based on what the Grade 8 common core math standards expect.
Column four indicates the Algebra I common core standard which could be affected if the Grade 8 gap exists. Other gaps may exist for other reasons;
therefore, it important that teachers diagnose their students’ needs as part of the planning process.
Domain
The Number
System (NS)
Grade 8
CCSS
8.NS.A.1
Go to Unit 1
8.NS.A.2
Equations and
Expressions (EE)
Go to Unit 2
8.EE.A.1
8.EE.A.2
Go to Unit 3
Go to Unit 5
8.EE.B.5
Go to Unit 7
Go to Unit 11
.
8.EE.C.8
Wording of Grade 8 CCSS Potential Gap
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.
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.
Know and apply the properties of integer exponents to generate equivalent numerical
expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
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 √2is irrational.
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.
Analyze and solve pairs of simultaneous linear equations.
Algebra I
CCSS
HSN-RN.B.3
HSN-RN.B.3
HSA-SSE.B.3
HSA-REI.B.4
Pre-requisite
for Unit 2 on
Linear
Equations
HSA-CED.A.3
HSA-REI.C.5
HSA-REI.C.6
Page 17
Algebra I Potential Gaps in Student Pre-Requisite Knowledge
Functions (F)
Go to Unit 4
8.F.A.3
Go to Unit 5
8.F.A.4
8.F.B.5
Statistics and
Probability (S)
8.SP.A.1
Go to Unit 11
8.SP.A.3
8.SP.A.4
.
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. For example, the function A = s2 giving the area
of a square as a function of its side length is not linear because its graph contains the points
(1,1), (2,4) and (3,9), which are not on a straight line.
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.
Construct and interpret scatter plots for bivariate measurement data to investigate patterns
of association between two quantities. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
Construct and interpret scatter plots for bivariate measurement data to investigate patterns
of association between two quantities. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
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.
Understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table. Construct and interpret a
two-way table summarizing data on two categorical variables collected from the same
subjects. Use relative frequencies calculated for rows or columns to describe possible
association between the two variables. For example, collect data from students in your class
on whether or not they have a curfew on school nights and whether or not they have assigned
chores at home. Is there evidence that those who have a curfew also tend to have chores?
Pre-requisite
for Units 4
and 5
HSF-LE.A.2
HSS-LE.B.5
HSS-ID.B.6
HSS-ID.C.7
HSS-ID.B.5
Page 18

Mathematics Algebra I – Year in Detail (SAMPLE)

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