Consensus Map Grade Level s1

Algebra I

Algebra is a study of how the world can be modeled and interpreted by first quantifying it mathematically. To be able to communicate meaning effectively and form connections between concepts there is a need for proficient variable manipulation. Algebra I will develop skills in students to help them solve real world problems with a focus on: (1) Write, solve, and interpret equations based on real-world situations and problems. (2) Identify the rate at which data is changing by so that they may be able to predict future events. (3) Create graphical representations of data to help communicate meaning and to identify solutions. Students will compare their graphs of data that changes at a constant rate to data that changes at other rates. (4) Simplify algebraic expressions by extending the properties of numbers. (5) Find where two relationships are both true in order to find the values of the unknowns. (6) Write inequalities to model real-world situations, find their solutions, and interpret their meanings. An algebraic inequality is a relationship where the solution for an unknown is a range of values. Grade Level/ Subject: Algebra I 2 | P a g e

Content: Foundation Algebra Duration: August/September (2 weeks)

How can we show that algebraic properties and processes are extensions of arithmetic Essential properties and processes, and how can we use algebraic properties and processes to solve Question: problems?

 Compare and order real numbers  Apply the order of operations Skills:  Apply the distributive property  Add/Subtract/Multiply/Divide real numbers

 Add, subtract, multiply, divide integers  Simplify: 3(x+8) Assessment:  What property is shown here:  Write an expression given a phrase: “Three more than b” = b+3

Resources: Pearson Algebra 1 Common Core: Chapter 1: pgs. 1-66

CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve problems with rational exponents. Standards: CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to solve real world or mathematical problems.

Associative property - ; Commutative property - ; Distributive property - Vocabulary: Identity property -

Comments: Emphasis on pgs. 47-66 Grade Level/ Subject: Algebra I 3 | P a g e

Content: Solving Equations Duration: September (3 weeks)

Essential How do we set up and solve problems to find unknown pieces of information? Question:

 Write, solve, and/or apply a linear equation.  Use and/or identify any algebraic property to justify any step in an equation- solving process.  Combining like terms in algebraic expressions/equations Skills:  Interpret solutions to a problem in the context of the problem-solving situation.  Apply the distributive property to algebraic equations.  Set up a proportion with similar figures

 Solve equation such as:  Assessment:   Solve for x.

Resources: Pearson Algebra I Common Core: Chapter 2: pgs. 81-114

CC.2.2.HS.D.8 - Apply inverse operations to solve equations or formulas for a given Standards: variable. CC.2.2.HS.D.9 - Use reasoning to solve equations and justify the solution method.

Coefficient – the number multiplied by a variable; Constant – a number not multiplied by a variable; Inverse operations – the opposite operation which solves an equation; Literal equation – an equation that relates multiple variables and has Vocabulary: an expression as a solution; Order of operations – an agreed upon order for performing arithmetic operations; Reciprocal – the multiplicative inverse of a number i.e. flip the fraction

Comments: pgs.115 – 147 not Algebra I Keystone Standards Grade Level/ Subject: Algebra I 4 | P a g e

Content: Exponents and Exponential Functions Duration: October (4 weeks)

Essential How can we write expressions more simply? Question:

 Apply the properties of exponents to simplify rational expressions.  Convert between radical and rational exponent forms Skills:  Write exponential equations  Graph exponential equations

 Simplify expressions using exponent properties such as:  Convert expression between rational and radical form: Assessment:  Write, graph, and model exponential equations given tables and real-world problems ()

CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve problems with rational exponents. Standards: CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to solve real world or mathematical problems. CC.2.2.HS.D.2 - Write expressions in equivalent forms to solve problems.

Exponential decay – relationship with a rate of change that is decreasing at a constantly decreasing rate; Exponential growth – relationship with a rate of Vocabulary: change that is increasing at a constantly increasing rate; Growth/decay factor – the multiplicative; Order of a root – the number times a factor must exist under a radical to reduced to a rational number

Comments: pgs. 453-472. Less emphasis, one day per section. Grade Level/ Subject: Algebra I 5 | P a g e

Content: Radical Expression and Equation Duration: October/November Essential Question: How do we represent solutions exactly?

Skills:  Simplify square roots  Simplify/evaluate expressions involving properties/laws of exponents, roots to solve problems.  Apply the Pythagorean theorem to determine unknown measurements of right triangles.  Add/Subtract/Multiply radical expressions.

Assessment:  Given two sides of a right triangle, use the Pythagorean Theorem to find the 3rd side (including examples with irrational sides that need to be reduced).  Reduce radicals with and without variable expressions  Add/Subtract and Multiply Radicals

Standards: CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to solve real world or mathematical problems. CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve problems with rational exponents.

Vocabulary: Pythagorean theorem – is a relationship between the side lengths and hypotenuse of a right triangle.

Comments:

Content: Solving Inequalities Duration: November/December Grade Level/ Subject: Algebra I 6 | P a g e

Essential Question: How do we represent a range of solutions to an inequality relationship?

 Solve one variable inequalities and represent solution set on a number line.  Identify or graph the solutions set to a linear inequality on the coordinate plane. Skills:  Solve multi-step inequalities  Solve compound inequalities  Solve equations and inequalities with absolute values

 Solve and graph solutions on a number line:   Assessment:   

CC.2.2.HS.D.7 - Create and graph equations or inequalities to describe numbers or relationships. Standards: CC.2.2.HS.D.10 - Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.

Absolute value – the distance away from zero; relationship where one thing is greater than or less than another; Intersection – looks for values that are found in Vocabulary: multiple sets; Solution set – the possible answers to an inequality; Unions – look for the total number of unique value in multiple sets Comments: Minimal emphasis on pgs. 194-199 and pgs. 214-220. If constrained by time, skip.

Grade Level/ Subject: Algebra I 7 | P a g e

Content: Introduction to Functions Duration: NovemberDecember(32 weeks)

Essential How do we map an input to an output? Question:

 Determine if a relation is a function in a graph, table, or equation.  Identify domain and range of a function in a graph, table, equation, set of coordinates. Skills:  Describe the characteristics of a function  Distinguishing linear equations versus non-linear equations  Writing equations to represent function, real-world application, patterns  Graphing functions on the coordinate plane

 Tell if a relation is a function in table, graph, ordered pair, equation.  Find the domain and range of a table, graph, ordered pair (some equations Assessment: eg. ).  Define a function.  Use function notation , Find )

CC.2.2.HS.C.1 - Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.2.HS.C.2 - Graph and analyze functions and use their properties to make connections between the different representations. Standards: CC.2.2.HS.C.3 - Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.4 - Interpret the effects transformations have on functions and find the inverses of functions. CC.2.2.HS.C.6 - Interpret functions in terms of the situations they model

Arithmetic sequence – a pattern of numbers with a common difference between them; Domain – the set of all possible inputs; Function – a relationship where every input has an output and only one output; Many-to-one – a function where Vocabulary: many different inputs have the same output; One-to-one – a function where only one input maps to one output; Range – the set of all possible outputs; Vertical line test – a test where a vertical line drawn on a graph may only cross a relationship once in order to be called a function Comments: Less emphasis on pgs. 274-281 Less emphasis on recursive formulas. Grade Level/ Subject: Algebra I 8 | P a g e

Content: Linear Functions Duration: n: DecemberJan (45 weeks)

Essential How do we model data that changes at a constant rate? Question:

 Identify rate of change in a table, graph, or equation  Write linear equations in point-slope, slope-intercept, and standard forms.  Model linear equations from real-world problems  Graphing linear equations given point-slope, slope-intercept, and standard form equations Skills:  Solving for x and y-intercepts; graphing linear equations using intercepts  Identify slope and write equations for parallel or perpendicular lines  Characterize the trend of a scatterplot  Write the equation for a trend line  Graph absolute value functions

 Given two points, a point and a slope, or a point and the y-intercept write the equation for the line in slope-intercept, point-slope, and standard form.  Graph a linear equation on a coordinate plane.  Write the equation for a trend line given a scatterplot. Student pick point Assessment: that follow the trend of the data.  Find x- and y-intercepts by substituting zero in for the other variable.  Write equations for linear relationships to model real-world (word) problems.  Graph absolute Value equations.

CC.2.4.HS.B.3 - Analyze linear models to make interpretations based on the data. CC.2.2.HS.C.3 - Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.5 - Construct and compare linear, quadratic, and exponential models to solve problems. Standards: C.2.1.HS.F.3 - Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.4.HS.B.2 - Summarize, represent, and interpret data on two categorical and quantitative variables. CC.2.1.HS.F.4 - Use units as a way to understand problems and to guide the solution of multi‐step problems. Grade Level/ Subject: Algebra I 9 | P a g e

Linear function – a function with a constant rate of change; Parallel lines – two lines with the same slope; Perpendicular lines – two lines with opposite reciprocal Vocabulary: slopes; Slope – the rate of change of a graph, table or equation; x-intercept – the place where a graph crosses the x-axis; y-intercept – the place where a graph crosses the y-axis Comments: Grade Level/ Subject: Algebra I 10 | P a g e

Content: Systems of Equations and Inequalities Duration: Jan./Feb. (3 weeks)

Essential How do we find and what is the meaning of the intersection of two relationships? Question:

 Write and/or solve a system of linear inequalities using graphing.  Write and/or solve a system of linear equations using graphing, substitution, or elimination. Skills:  Identify the best solution strategy given the system setup  Use strategies for solving systems of equations to solve real-world problem situations.

 Solve systems of equations and inequalities with 2 variables using graphing, substitution and elimination. Substitution should have problems where variables are not solved for already. Elimination should have problems where both equations need to be multiplied by a constant first. Assessment:  Write a system of equation and inequalities given a real-world problem situation.  Interpret the meaning of the point of intersection or the region of intersection.  Solve systems of equations with infinitely many solutions or no solutions.

CC.2.2.HS.D.10 - Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically. CC.2.2.HS.C.3 - Write functions or sequences that model relationships between two Standards: quantities. CC.2.1.HS.F.5 - Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Infinitely many solutions – the results of two lines that overlap will have infinitely many points where both equations are true; No solutions – the result of two Vocabulary: parallel lines with different y-intercepts will have no points where both equations are true; System of equations – two or more equations to be solved simultaneously

Comments: Grade Level/ Subject: Algebra I 11 | P a g e

Content: Exponents and Exponential Functions Duration: Feb/March (13 weeks)

Essential How can we write expressions more simply? Question:

 Apply the properties of exponents to simplify rational expressions.  Convert between radical and rational exponent forms Skills:  Write exponential equations  Graph exponential equations

 Simplify expressions using exponent properties such as:  Convert expression between rational and radical form: Assessment:  Write, graph, and model exponential equations given tables and real-world problems ()

CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve problems with rational exponents. Standards: CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to solve real world or mathematical problems. CC.2.2.HS.D.2 - Write expressions in equivalent forms to solve problems.

Exponential decay – relationship with a rate of change that is decreasing at a constantly decreasing rate; Exponential growth – relationship with a rate of Vocabulary: change that is increasing at a constantly increasing rate; Growth/decay factor – the multiplicative; Order of a root – the number times a factor must exist under a radical to reduced to a rational number

Comments: pgs. 453-472. Less emphasis, one day per section.

Content: Radical Expression and Equation Duration: March (1 week) Grade Level/ Subject: Algebra I 12 | P a g e

Essential How do we represent solutions exactly? Question:

 Simplify square roots  Simplify/evaluate expressions involving properties/laws of exponents, roots to solve problems. Skills:  Apply the Pythagorean theorem to determine unknown measurements of right triangles.  Add/Subtract/Multiply radical expressions.

 Given two sides of a right triangle, use the Pythagorean Theorem to find the 3rd side (including examples with irrational sides that need to be reduced). Assessment:  Reduce radicals with and without variable expressions  Add/Subtract and Multiply Radicals

CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to solve real world or mathematical problems. Standards: CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve problems with rational exponents.

Pythagorean theorem – is a relationship between the side lengths and Vocabulary: hypotenuse of a right triangle.

Comments:

Content: Data Analysis Duration: March/April/May (2 1weeks) Grade Level/ Subject: Algebra I 13 | P a g e

Essential How do we quantify and interpret a data filled world? Question:

 Calculate and/or interpret the range, quartiles, and interquartile range of data.  Estimate or calculate to make predictions based on a circle, line, bar graph, measure of central tendency, or other representation. Skills:  Analyze data, make predictions, and/or answer questions based on displayed data (box-and-whisker plots, stem and leaf plots, scatter plots, measure of central tendency, or other representation).  Find probabilities for compound events and represent as a fraction, decimal or percent.

 Make a box and whisker plot given a set of data points or a frequency table.  Interpret a box-and-whisker plot to identify the interquartile range and Assessment: quartiles.  Find the probability of multiple independent and dependent events happening (e.g. P(King and a Queen) with or without replacement).

Resources: Pearson Algebra I Common Core: pgs. 738-744; 746-751; 769-781

CC.2.4.HS.B.1 - Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.4 - Recognize and evaluate random processes underlying statistical experiments. Standards: CC.2.4.HS.B.5 - Make inferences and justify conclusions based on sample surveys, experiments, and observational studies. CC.2.4.HS.B.7 - Apply the rules of probability to compute probabilities of compound events in a uniform probability model.

Compound probability – the likelihood of multiple events occurring; Vocabulary: Interquartile range – the middle 50 percent of a data set; Probability – the likelihood of an event occurring

Content: Polynomials and Factoring Duration: March/April (45 weeks)

Essential How do we model and simplify linear and non-linear relationships? Question:

 Add/Subtract/Multiply polynomial expressions up to binomials times trinomials (FOIL method)  Apply the distributive property to polynomials Skills:  Factor algebraic expressions including differences of squares and trinomials.  Reduce/Simplify algebraic expressions in fractions by factoring  Simplify rational functions by canceling common factors.

 Classify polynomials by number of terms (monomials, binomials, etc.) and by degree (linear, quadratic, etc.)  Add/Subtract and multiply polynomials. (Largest polynomial to multiply is Assessment: binomial times trinomial).  Factor polynomials where a=1, that are perfect square trinomials, difference of two squares, have a common factor, and have a not equal to 1.

CC.2.2.HS.D.1 - Interpret the structure of expressions to represent a quantity in terms of its context. Standards: CC.2.2.HS.D.3 - Extend the knowledge of arithmetic operations and apply to polynomials. CC.2.2.HS.D.5 - Use polynomial identities to solve problems. CC.2.2.HS.D.6 - Extend the knowledge of rational functions to rewrite in equivalent forms.

Degree of a polynomial – the highest exponent of a variable of a polynomial; Vocabulary: Polynomial – One or more algebraic terms added or subtracted together; Term – A piece of a polynomial separated by addition or subtracted

Comments: Less emphasis on pgs.529-533

Grade Level/ Subject: Algebra I 15 | P a g e

Content: Quadratic Equations Duration: April/ May (3 weeks)

Essential How do we model and display data that varies quadratically? Question:

 Factor algebraic expressions including differences of squares and trinomials.  Identify line of symmetry, y-intercepts, x-intercepts, vertex and orientation of quadratic equations. Skills:  Apply the quadratic formula to solve quadratic equations.  Interpret results of the discriminant to determine the number of real solutions to a quadratic equation.

Given a quadratic equation:  Find vertex, line of symmetry, orientation, y-intercept, x-intercept  Solve quadratic equations by square roots, factoring, and quadratic formula.  Determine the number of solutions to a quadratic equation using the Assessment: discriminant  Reduce irrational solutions to the quadratic formula.  Compare Rate of Change in Linear, Quadratic and Exponential Models from a table, graph or equation.

Resources: Pearson Algebra I Common Core: Chapter 9: pgs. 544-572, 582-594

CC.2.2.HS.C.5 - Construct and compare linear, quadratic, and exponential models to solve problems. Standards: CC.2.2.HS.D.2 - Write expressions in equivalent forms to solve problems.

Quadratic Equation – a second degree Polynomial whose graph is a parabola; Discriminant – determines the number of real solutions to a quadratic equation; Vertex – The point where a quadratic equation changes from increasing to Vocabulary: decreasing or vice versa; Parabola – the shape of a quadratic equation; Zeros/Roots – Another name for the solutions and x-intercepts of a quadratic equation. Comments: Grade Level/ Subject: Algebra I 16 | P a g e Grade Level/ Subject: Algebra I 17 | P a g e

Content: Radical Expression and Equation Duration: May (2 weeks)

Essential How do we represent solutions exactly? Question:

 Simplify square roots  Simplify/evaluate expressions involving properties/laws of exponents, roots to solve problems. Skills:  Apply the Pythagorean theorem to determine unknown measurements of right triangles.  Add/Subtract/Multiply radical expressions.

 Given two sides of a right triangle, use the Pythagorean Theorem to find the 3rd side (including examples with irrational sides that need to be reduced). Assessment:  Reduce radicals with and without variable expressions  Add/Subtract and Multiply Radicals

CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to solve real world or mathematical problems. Standards: CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve problems with rational exponents.

Pythagorean theorem – is a relationship between the side lengths and Vocabulary: hypotenuse of a right triangle.

Content: Rational Expressions & Equations Duration: May May (12 weeks)

Essential How do create a model for a relationship defined by the ratio of polynomials? Question:

 Factor polynomials  Reduce/simplify rational expressions by cancelling like factors from Skills: numerators and denominators  Multiply/Divide rational expressions

 Simplify rational expressions by factoring problems like: Assessment:  Multiply and Divide rational expressions like:

Standards: CC.2.2.HS.D.6 – Extend the knowledge of rational functions to rewrite in equivalent forms.

Rational Expression – an expression with a polynomial divided by a polynomial; Excluded Value – Values that will make the function or expression undefined; Vocabulary: Rational Equation/Function – an equation that contains a rational expression; Complex Fraction – a fraction that contains a fraction within the numerator or denominator;

Comments: