2014 - 2015 algebra 1 a mathematics curriculum...

Algebra 1 A MATHEMATICS Curriculum Map

2014 - 2015

Common Core State Standards

Mathematics Department Algebra 1A Curriculum Map Volusia County Schools June 2014

Common Core State Standards Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity.

2. Reason abstractly and quantitatively. (MAFS.K12.MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols.

3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence.

4. Model with mathematics. (MAFS.K12.MP.4) Many everyday problems can be solved by modeling the situation with mathematics.

5. Use appropriate tools strategically. (MAFS.K12.MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding.

6. Attend to precision. (MAFS.K12.MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations.

7. Look for and make use of structure. (MAFS.K12.MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea.

8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently.


Algebra 1A: Common Core State Standards

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, called units, deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Relationships Between Quantities and Reasoning with Equations: By the end of eighth grade students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. All of this work is grounded on understanding quantities and on relationships between them. SKILLS TO MAINTAIN: Reinforce understanding of the properties of integer exponents. The initial experience with exponential expressions, equations, and functions involves integer exponents and builds on this understanding. Linear and Exponential Relationships: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.

Algebra 1A resources for all Units:

Illustrative mathematics: provides the standards with example problems that cover the standard. http://www.illustrativemathematics.org/standards/hs The Math Dude: http://www.montgomeryschoolsmd.org/departments/itv/mathdude/ Algebra Nation- http://www.algebranation.com/

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Algebra 1A: Common Core State Standards At A Glance

N-RN: Real Number System A-CED: Create Equations that Describe Relationships A-REI: Reasoning with Equations and Inequalities A-APR: Arithmetic with Polynomials & Rational Expressions F-BF: Building Functions F-IF: Interpreting Functions F-LE: Linear, Quadratic and Exponential Models A-SSE: Seeing Structure in Expressions S-ID: Interpreting Data N-Q: Quantities

First Nine Weeks Second Nine Weeks Third Nine Weeks Fourth Nine Weeks

Unit 1- Real Number System MAFS.912.N-RN.2.3 Unit 2- Solving Equations and Inequalities MAFS.912.A-CED.1.1 MAFS.912.A.-REI.1.1 MAFS.912.A-CED.1.3 MAFS.912.A-CED.1.4 MAFS.912.A-REI.2.3

Unit 3- Linear Equations and Their Graphs MAFS.912.F-IF.1.1 MAFS.912.F-IF.1.2 MAFS.912.F-IF.2.5 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.6 MAFS.912.A-CED.1.2 MAFS.912.A-REI.4.10 MAFS.912.F-IF.3.7a MAFS.912.F-LE.2.5

Unit 4- Systems of Linear Equations and Inequalities MAFS.912.A.-REI.3.5 MAFS.912.A.-REI.3.6 MAFS.912.A-REI.4.11 MAFS.912.A.-REI.4.12

Unit 5-Properties of Exponents MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 Unit 6- Non-Linear Functions MAFS.912.A-SSE.1.1 MAFS.912.F-IF.3.7a MAFS.912.F-IF.3.8a

Unit 6- Non-Linear Functions (con’t) MAFS.912.F-IF.3.7 b-e MAFS.912.F-IF.3.9 MAFS.912.F-BF.2.3 MAFS.912.F-IF.1.3 MAFS.912.F-LE.1.2 MAFS.912.F-BF.1.1 MAFS.912.F-LE.1.3 MAFS.912.A-CED.1.1 MAFS.912.F-LE.1.1

MAFS.912.N-Q.1.1, MAFS.912.N-Q.1.2, MAFS.912.N-Q.1.3: these standards should be applied and referenced throughout the curriculum.


Fluency Recommendations 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 (seeing parts of an expression as a single object) is essential in factoring, completing the square, and other mindful algebraic calculations. The following Mathematics and English Language Arts CCSS should be taught throughout the course: MAFS.912.N-Q.1.1: 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. MAFS.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling. MAFS.912.N-Q.1.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. LACC.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing tasks, attending to special cases or exceptions defined in the text. LACC.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in context and topics. LACC.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information expressed visually or mathematically into words. LACC.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners. LACC.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of each source. LACC.910.SL.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. LACC.910.SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning. LACC.910.WHST.1.1: Write arguments focused on discipline-specific content. LACC.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LACC.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.


Course: Algebra 1A

Unit 1: The Real Number System

Essential Question(s): How does knowledge of integers help when working with rational and irrational numbers?

Standard The students will:

Learning Goals I can:

Remarks

Resources

MAFS.912.N-RN.2.3. 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. SMP #2, #3

• classify real numbers as rational or irrational according to their definitions.

• add, subtract, multiply and divide real numbers.

• explain why the sum of two rational numbers is rational.

• explain why the product of two rational numbers is rational.

• explain why the sum of a rational and irrational is irrational.

This unit should be treated as a review and be completed within 7 days.


Course: Algebra 1A Unit 2: Solving Equations and Inequalities

Essential Question(s):

How can algebra describe the relationship between sets of numbers? In what ways can the problem be solved, and why should one method be chosen over another?



Remarks

Resources

MAFS.912A-CED.1.1. Create equations and inequalities in one variable and use them to solve problems. SMP#4

• identify the variables and quantities represented in a real world problem.

• write the equation or inequality that best models the problem.

• solve linear equations and inequalities.

• interpret the solution in the context of the problem.

Students may believe that solving an equation such as 3x + 1 = 7 involves “only removing the 1,” failing to realize that the equation 1 = 1 is being subtracted to produce the next step.

*AlgebraNation.com Mini-Projects/Tasks= http://insidemathematics.org/problems-of-the-month/pom-onbalance.pdf http://insidemathematics.org/common-core-math-tasks/high-school/HS-A-2003%20Number%20Towers.pdf

MAFS.912A.REI.1.1 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. SMP#4

• explain a process to solve equations. • apply the distributive property when

necessary to solve equations. • construct a viable argument to justify

a solution method.

When using Distributive Property, students often multiply the number (or variable) outside the parentheses by the first term in the parentheses, but neglect to multiply that same number by the other term(s) in the parentheses. Regarding variables on both sides, students often will try to combine the terms as if they are on the same side of the equation rather than eliminating one of the variables.

http://insidemathematics.org/problems-of-the-month/pom-onbalance.pdf�




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Course: Algebra 1A Unit 2: Solving Equations and Inequalities (cont)


How can algebra describe the relationship between sets of numbers? In what ways can the problem be solved, and why should one method be chosen over another?



Remarks

Resources

MAFS.912A.CED.1.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. SMP #4

• identify the variable and quantities represented in a real-world problem.

• determine the best models for a real-world problem.

• write inequalities that best models a problem.

Students may confuse the rule of reversing the inequality when multiplying or dividing by a negative number, with the need to reverse the inequality anytime a negative sign shows up in solving the last step of the inequality. Example: 3x > -15 or x < - 5 (Rather than correctly using the rule: -3x >15 or x< -5)

Mars Tasks: http://insidemathematics.org/common-core-math-tasks/high-school/HS-A-2003%20Number%20Towers.pdf http://insidemathematics.org/common-core-math-tasks/high-school/HS-F-2008%20Functions.pdf

MAFS.912A.CED.1.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. SMP #4

• solve a formula for a given variable.

• solve problems involving literal equations.

Students may struggle to solve literal equations/ formulas due to not containing any numbers, so reiterating that the same steps (inverse operations) are used whether dealing with eliminating a variable or number may be helpful.

MAFS.912A.REI.2.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. SMP #5, #7

• solve linear equations and inequalities in one variable.








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Course: Algebra 1A Unit 3: Linear Equations and Their Graphs


In what ways can the problem be solved, and why should one method be chosen over another?



Remarks

Resources

MAFS. 912.F-IF.1.1 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, the 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). SMP #6, #7

• define relation, domain and range. • define a function as a relation in which

each input (domain) has exactly one output (range).

• determine if a graph, table or set of ordered pairs represent a function.

• determine if stated rules (both numeric and non-numeric) produce ordered pairs that form a function.

• explain that when ‘x’ is an element of the input of a function f(x) represents the corresponding output.

• explain that the graph of ‘f’ is the graph of the equation y=f(x).

Students may believe that all relationships having an input and an output are functions, and therefore, misuse the function terminology. Students may also believe that the notation f(x) means to multiply some value f times another value x. The notation alone can be confusing and needs careful development. For example, f(2) means the output value of the function f when the input value is 2. Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domains. Other letters can be used for functional notation e.g. g(x), p(x), etc…

Tasks & Mini-Projects http://insidemathematics.org/common-core-math-tasks/high-school/HS-F-2008%20Functions.pdf http://insidemathematics.org/common-core-math-tasks/high-school/HS-F-2004%20Graphs2004.pdf http://insidemathematics.org/common-core-math-tasks/high-school/HS-F-2006%20Printing%20Tickets.pdf






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Course: Algebra 1A Unit 3: Linear Equations and Their Graphs (cont)





Remarks

Resources

MAFS. 912.F-IF.1.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. SMP #7 MAFS. 912.F-IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. SMP #4

• decode function notation and explain how the output of a function is matched to its input.

• convert a table, graph, set of ordered pairs or description into function notation by identifying the rule used to turn inputs into outputs and writing the rule.

• use order of operations to evaluate a function for a given domain value.

• identify the numbers that are not in the domain of a function. • choose and analyze inputs (and outputs) that make sense based

on the problem. • explain how the domain of a function is represented in its graph. • state, defend and explain the appropriate domain of a function that

represents a problem situation.

f(x)= 2x2 + 4….squares the input, doubles the square and adds four to produce the output Given h(x) = √𝑥 the domain has to be positive numbers

MAFS. 912.F-IF.2.4 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. SMP #1, #7, #8

• locate the information that explains what each quantity represents. • interpret the meaning of an ordered pair. • determine if negative inputs and/or outputs make sense in the

problem. • identify and explain the x and y intercept • define intervals of increasing and decreasing of a table or graph. • identify and explain relative maximums and minimums. • identify reflective and rotational symmetries in a table or graph. • explain why the function has symmetry in the context of the

problem. • identify and explain positive and negative end behavior of a

function. • define and identify a periodic function from a table or graph. • explain why a function is periodic.


Course: Algebra 1A

Unit 3 - Linear Equations and Their Graphs (cont)

Essential Question(s): In what ways can the problem be solved, and why should one method be chosen over another?

Standard

The students will: Learning Goals

I can:

Remarks Resources

MAFS. 912.F-IF.2.6 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. SMP #4, #5

• define and explain interval, rate of change and average rate of change.

• calculate the average rate of change of a function, represented either by function notation, a graph or a table over a specific interval.

• compare the rates of change of two or more functions

• interpret the meaning of the average rate of change in the context of the problem.

Students may also believe that the slope of a linear function is merely a number used to sketch the graph of the line. In reality, slopes have real-world meaning, and the idea of a rate of change is fundamental to understanding major concepts from geometry to calculus.

Get the Math - http://www.thirteen.org/get-the-math/files/2011/10/vidgamesfulllesson.pdf

MAFS.912.A-CED.1.2 Create equation in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales SMP #4

• set up coordinate axes using an appropriate scale and label the axes.

• write equations of lines in various forms. • graph equations on coordinate axes with

appropriate labels and scales.

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Course: Algebra 1A Unit 3: Linear Equations and Their Graphs (cont)





Remarks

Resources

MAFS.912.A-REI.4.10. 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). SMP #2

• explain that every ordered pair on the graph of an equation represents values that make the equation true.

• verify that any point on a graph will result in a true equation when their coordinates are substituted into the equation.

MAFS.912.F-IF.3.7a Graph linear functions by hand and show intercepts, maxima, and minima. SMP #7,#8

• identify that the parent function for lines is the line f(x) = x. • identify and graph a line in the point-slope form: y-y1=m(x-x1). • identify and graph a line in slope-intercept form: f(x)=mx+b. • identify the standard form of a linear function as Ax + By = C. • use the definitions of x and y intercepts to find the intercepts of a

line in standard form and then graph the line. • relate the constants A, B, and C to the values of the x and y

intercepts and slope.

MAFS.912.F-LE.2.5 Interpret the parameters in a linear function in terms of a context. SMP #2, #4

• identify the names and definitions of the parameters ‘m’ and ‘b’ in a linear function f(x)=mx+b.

• explain the meaning (using appropriate units) of the slope, y-intercept and other points on the line when then line models a real-world relationship.

• compose an original problem situations and construct a linear function to model it.


Course: Algebra 1A Unit 4: Systems of Linear Equations and Inequalities


In what ways can the problem be solved, and why should one method be chosen over another? Standard


I can:

Remarks Resources

MAFS.912A-REI.3.5 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. SMP # 3

• solve a system of two linear equations by graphing and determining the point of intersection.

• solve a system of two linear equations algebraically using substitution.

• solve a system of two linear equations algebraically using elimination.

Most mistakes that students make are careless rather than conceptual. Teachers should encourage students to learn a certain format for solving systems of equations and check the answers by substituting into all equations in the system.

Mars Tasks: http://insidemathematics.org/common-core-math-tasks/high-school/HS-A-2006%20Graphs2006.pdf Illuminations – Supply and Demand - http://illuminations.nctm.org/LessonDetail.aspx?ID=L382

MAFS.912.A-REI.3.6 Solve systems of linear equations exactly and approximately (with graphs), focusing on pairs of linear equations in two variables. SMP # 7

• explain why some linear systems have no solutions or infinitely many solutions.

• solve a system of linear equations algebraically to find an exact solution.

• graph a system of linear equations and determine the approximate solution to the system of linear equations by estimating the point of intersection.

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Course: Algebra 1A Unit 4: Systems of Linear Equations and Inequalities


In what ways can the problem be solved, and why should one method be chosen over another? Standard


I can:

Remarks Resources

MAFS.912.A-REI.4.11 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 equations f(x)=g(x) using technology to graph the functions, make tables of values, or find successive approximations.

• explain that a point of intersection on the graph of a system of equations represents a solution to both equations.

• infer that the x-coordinate of the points of intersections are solutions for f(x) = g(x).

• use a graphing calculator to determine the approximate solutions to a system of equations.

Students will want to give an ordered pair answer instead of just the x-coordinate. Linear functions only in this unit.

MAFS.912.A-REI.4.12 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 corresponding half-planes. SMP #5

• solve and graph linear inequalities with two variables. • solve and graph system of linear inequalities. • explain that the solution set for a system of linear

inequalities is the intersection of the shaded regions of both inequalities and check points in the shaded region to verify the solution.


Course: Algebra 1A

Unit 5: Properties of Exponents

Clarification: Radicals are now included in this sections. This will not only include the previous NGSSS standard for radicals (adding, subtracting, multiplying, and dividing radicals) but it will also include converting between radical and rational exponents.

Essential Question(s): How does knowledge of integers help when working with rational and irrational numbers?

How can the relationship between quantities best be represented? Standard


I can:

Remarks Resources

MAFS.912.N-RN.1.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. SMP #3, #7, #8 MAFS.912.N-RN.1.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. SMP #7

• evaluate and simplify expressions containing zero and integer exponents.

• multiply monomials. • apply multiplication properties of

exponents to evaluate and simplify expressions.

• divide monomials. • apply division properties of exponents

to evaluate and simplify expressions. • apply properties of rational exponents

to simplify expressions. • convert between radicals and rational

exponents.

Students sometimes misunderstand the meaning of exponential operations, the way powers and roots relate to one another, and the order in which they should be performed. Attention to the base is very important. Consider examples: (−81

34 ) and (−81)

34 . The

position of a negative sign of a term with a rational exponent can mean that the rational exponent should be either applied first to the base, 81, and then the opposite of the result is taken, (−81

34 ), or the rational exponent should

be applied to a negative term (−81)34. The

answer of √−814 will be not real if the denominator of the exponent is even. If the root is odd, the answer will be a negative number. Students should be able to make use of estimation when incorrectly using multiplication instead of exponentiation. Students may believe that the fractional exponent in the expression 36

13 means the same

as a factor of 1/3 in multiplication expression, 36 ● 1/3 and multiple the base by the exponent.

Tasks & Mini-Projects http://insidemathematics.org/common-core-math-tasks/high-school/HS-A-2009%20Quadratic2009.pdf Manipulating Radicals - http://api.ning.com/files/oteyilu*j6qVWKtAVizsGFTd*nP1b3gsbfNN3t7yx-PqAFE9x*X2Vo50qq2gDApK8pn75oIJuNVQovyELA6yBVO9Vt*yL7Ye/2manipulating_radicals_complete.pdf

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Course: Algebra 1A Unit 6 – Non-Linear Functions

Essential Question(s): How can the relationship between quantities best be represented?

In what ways can functions be built? Can you determine which function best models a situation?



Remarks

Resources

MAFS.912.A-SSE.1.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. SMP #7

• factor polynomials by using the greatest common factor. • factor polynomials using the grouping method. • factor quadratic trinomials when a=1. • factor quadratic trinomials when a>1. • factor perfect square trinomials. • factor the difference of two squares. • form a perfect-square trinomial from a given quadratic

binomial.

Some students may believe that factoring and completing the square are isolated techniques within a unit of quadratic equations. Teachers should help students to see the value of these skills in the context of solving higher degree equations and examining different families of functions.

MAFS.912.F-IF.3.7a Graph quadratic functions and show intercepts, maxima and minima. SMP #7, #8 MAFS.912.F-IF.3.8a 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.

SMP #2, #7

• explain that the parent function for quadratic functions is a parabola f(x) = x2 .

• explain that the minimum and maximum of a quadratic is called the vertex.

• identify whether the vertex of a quadratic will be a minimum or maximum by looking at the equation.

• find the y-intercept of a quadratic by substituting 0 for ‘x’ and evaluating.

• estimate the vertex of a quadratic by evaluating different values of ‘x’.

• decide if the quadratic has x-intercepts and if so estimate their value(s).

• graph a quadratic using evaluated points • use technology to graph a quadratic and to find precise

values for the x-intercept(s) and maximum or minimum.

The expectation is for F.IF.3.7a &3.7e to focus on linear and exponential functions in Algebra I. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two exponential functions. In Algebra I for F.IF.3.7b, compare and contrast absolute value, step and piecewise- defined functions with linear, quadratic, and exponential functions.

Forming Quadratics – http://map.mathshell.org/materials/download.php?fileid=700

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MAFS.912.F-IF.3.7b,c,d,e Graph function expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated functions. SMP #7, #8

• graph using evaluated points and technology and explain the parent function of the following: square root, cube root, piecewise, absolute value, step, and exponential functions.

• find intercepts, maximums and minimums, end behavior, and horizontal asymptotes (by observation) when present.

• explain how the domain of a function is represented in its graph.

Highlight issues of domain, range, and usefulness when examining piecewise- defined functions.

MAFS.912.F-IF.3.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

• state the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain.

• identify the degree of a polynomial. • classify exponential functions as growth or decay. • approximate the factored equation of a polynomial

function when looking at a graph of a function. • determine the multiplicity of the x-intercepts of a

polynomial when looking at a graph of the function.

Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

http://www.math.hmc.edu/calculus/tutorials/transformations/

MAFS.912.F-BF.2.3 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 SMP #5, #7

• explain how ‘k’ translates the original graph • determine the value of ‘k’ when given the graph of a

transformed function.

‘k’ translations include: left, right, up, down, vertical stretch and shrink, horizontal stretch and shrink Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities among all functions and their graphs. Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4 units to the left (due to the subtraction symbol). Examples should be explored by

http://www.math.hmc.edu/calculus/tutorials/transformations/�





hand and on a graphing calculator to overcome this misconception. Students often confuse the shift of a function with the stretch of a function.


Course: Algebra 1A Unit 6: Non-Linear Functions (cont)





Remarks

Resources

MAFS.912.F-IF.1.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. SMP #2, #7, #8 MAFS.912.F-LE.1.2 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). SMP #2, #7, #8

• convert a sequence into a function. • explain that a recursive formula tells

me how a sequence starts and tells me how to use the previous value(s) to generate the next element of the sequence.

• explain that an explicit formula allows me to find any element of a sequence.

• distinguish between explicit and recursive formulas for sequences.

• relate arithmetic sequences to linear functions.

• relate geometric sequences to exponential functions.

• determine if a function is linear or exponential given a sequence, a graph, a verbal description or a table.

• describe the algebraic process used to construct the linear function and exponential function that passes through two points.

In F.IF.1.3 draw a connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. 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. Students should be able to explain that a recursive formula tells how a sequence starts and how to use the previous value(s) to generate the next element of the sequence. Students should be able to explain that an explicit formula allows them to find any element of a sequence without knowing the element before it (e.g., If I want to know the 11th number on the list, I substitute the number 11 into the explicit formula). Algebra 1 Assessment Limits and Clarifications i) this standard is part of the Major work in Algebra I and will be assessed accordingly. ii) Tasks are limited to constructing linear and exponential functions in simple context (not multi- step).

http://www.kenston.k12.oh.us/khs/academics/math/AA_11-2B_arithmetic_sequences_recursive.pdf Prentice Hall Textbook Section 4.7 (arithmetic sequences only)

http://www.kenston.k12.oh.us/khs/academics/math/AA_11-2B_arithmetic_sequences_recursive.pdf�













Remarks

Resources

MAFS.912.F-BF.1.1 Write a function that describes the relationship between two quantities. SMP #4, #7 MAFS.912A-CED.1.1. 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. SMP#4

• identify the quantities being compared in a real-world problem.

• write a function to describe a real-world problem.

• compose two or more functions.

Students may believe that the process of rewriting equations into various forms is simply an algebra symbol manipulation exercise, rather than serving a purpose of allowing different features of the function to be exhibited.

http://www.montgomeryschoolsmd.org/departments/itv/mathdude/MD_Algebra1_7-3.shtm Prentice Hall Textbook Section 4.5

MAFS.912.F-LE.1.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. SMP #2, #8

• understand exponential functions and how they are used.

• recognize differences between graphs of exponential functions with different bases.

• apply exponential functions to model applications that include growth and decay in different contexts.

http://www.montgomeryschoolsmd.org/departments/itv/mathdude/MD_Algebra1_7-3.shtm�










Remarks

Resources

MAFS.912.F-LE.1.1 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.

SMP #3, #4, #8

• define a linear function and exponential function.

• demonstrate that an exponential function has a constant multiplier or equal intervals.

• identify situations that display equal ratios of change over equal intervals and can be modeled with exponential functions.

• distinguish between situations modeled with linear functions and with exponential functions when presented with a real-world problem.

Compare tabular representations of a variety of functions to show that linear functions have a first common difference (i.e., equal differences over equal intervals), while exponential functions do not (instead function values grow by equal factors over equal x-intervals). Apply linear and exponential functions to real-world situations. For example, a person earning $10 per hour experiences a constant rate of change in salary given the number of hours worked, while the number of bacteria on a dish that doubles every hour will have equal factors over equal intervals. Provide examples of arithmetic in graphic, verbal, or tabular forms, and have students generate formulas and equations that describe the patterns. Examine multiple real-world examples of exponential functions so that students recognize that a base between 0 and 1 (such as an equation describing depreciation of an automobile [ f(x) = 15,000(0.8)x represents the value of a $15,000 automobile that depreciates 20% per year over the course of x years]) results in an exponential decay, while a base greater than 1 (such as the value of an investment over time [ f(x) = 5,000(1.07)x represents the value of an investment of $5,000 when increasing in value by 7% per year for x years]) illustrates growth.

http://spot.pcc.edu/~kkling/Mth_111c/SectionII_Exponential_and_Logarithmic_Functions/Module3_Comparing_Linear_and_Exponential/Module3_Comparing_Linear_and_Exponential.pdf

http://spot.pcc.edu/~kkling/Mth_111c/SectionII_Exponential_and_Logarithmic_Functions/Module3_Comparing_Linear_and_Exponential/Module3_Comparing_Linear_and_Exponential.pdf�
















Remarks

Resources

MAFS.912.N-Q.1.1 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.

• label units through multiple steps of a problem

• choose appropriate units for real world problems involving formulas

• use and interpret units when solving formulas

• choose an appropriate scale and origin for graphs and data displays

• interpret the scale and origin for graphs and data displays

MAFS.912.N-Q.1.2 Define appropriate quantities for the purpose of descriptive modeling.

• Identify the variable or quantities of significance from the data provided

• Identify or choose the appropriate unit of measure for each variable or quantity

MAFS.912.N-Q1.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

• report measured quantities in a way that is reasonable for the tool used to make the measurement (e.g., 5.1 cm is a reasonable way to report a length measured on a typical ruler, but 5.1476 cm is not. If time is measured on a stopwatch, 1:23:45.78 is a reasonable way to report a time; it is not appropriate if time is measured on a wall clock)

• report calculated quantities using the same level of accuracy as used in the problem statement

2014 - 2015 algebra 1 a mathematics curriculum...

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