Download Algebra I v.2014

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
Content: Foundation Algebra
1|Page
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?
Skills:
Assessment:




Compare and order real numbers
Apply the order of operations
Apply the distributive property
Add/Subtract/Multiply/Divide real numbers




Add, subtract, multiply, divide integers
Simplify: 3(x+8)
What property is shown here: (3 ∙ 8) ∙ 5 = 3 ∙ (8 ∙ 5)
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 - (3 ∙ 8) ∙ 5 = 3 ∙ (8 ∙ 5); Commutative property - 16 + 3 = 3 + 16;
Vocabulary: Distributive property - −2(4x + 9) = −8x − 18; Identity property - 7 + 0 = 7
Comments: Emphasis on pgs. 47-66
Grade Level/ Subject: Algebra I
Content: Solving Equations
2|Page
Duration: September (3 weeks)
Essential
How do we set up and solve problems to find unknown pieces of information?
Question:


Skills:
Assessment:




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
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:
2x + 3 = 17
2(x + 4) − 6 = 3(x − 1) + 2x
px − mx = h 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 variable.
Standards: 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
Vocabulary: equation – an equation that relates multiple variables and has 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
Content: Exponents and Exponential Functions
3|Page
Duration: October (4 weeks)
Essential
How can we write expressions more simply?
Question:
Skills:
Assessment:




Apply the properties of exponents to simplify rational expressions.
Convert between radical and rational exponent forms
Write exponential equations
Graph exponential equations
x0 (2x3 )5
(x2 )−2
3
√(3x)5 =

Simplify expressions using exponent properties such as:


Convert expression between rational and radical form:
(3x)5/3
Write, graph, and model exponential equations given tables and real-world problems
(y = abx )
Resources: Pearson Algebra I Common Core: Chapter 7: pgs. 418-472
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 change that is increasing at a
Vocabulary:
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
4|Page
Content: Radical Expression and
Equation
Duration:
October/November (3
week)
Essential Question: How do we represent solutions exactly?
Skills:




Assessment:



Simplify square roots
Simplify/evaluate expressions involving properties/laws of expo
problems.
Apply the Pythagorean theorem to determine unknown measure
Add/Subtract/Multiply radical expressions.
Given two sides of a right triangle, use the Pythagorean Theorem
(including examples with irrational sides that need to be reduced
Reduce radicals with and without variable expressions
Add/Subtract and Multiply Radicals
Resources: Pearson Algebra I Common Core: Chapter 10: pgs. 615-631
Standards: CC.2.1.HS.F.2 - Apply properties of rational and irrational numbers to s
mathematical problems.
CC.2.1.HS.F.1 - Apply and extend the properties of exponents to solve
exponents.
Vocabulary: Pythagorean theorem – a2 + b2 = c 2 is a relationship between the sid
hypotenuse of a right triangle.
Comments:
Content: Solving Inequalities
Duration: November/December (2
Grade Level/ Subject: Algebra I
5|Page
weeks)
Essential
How do we represent a range of solutions to an inequality relationship?
Question:
Skills:
Assessment:





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.
Solve multi-step inequalities
Solve compound inequalities
Solve equations and inequalities with absolute values






Solve and graph solutions on a number line:
3x + 4 < 13
30 > −(5z + 15) + 10z
−3 < m − 4 < −1
|3x + 4| − 6 < 13
|3x + 4| − 6 = 13
Resources: Pearson Algebra I Common Core: Chapter 3: pgs. 165-220
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; Inequality – a relationship where one thing is
greater than or less than another; Intersection – looks for values that are found in multiple sets;
Vocabulary: 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
Content: Introduction to Functions
6|Page
Duration: December(2 weeks)
Essential
How do we map an input to an output?
Question:
Skills:
Assessment:






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.
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.
1
Find the domain and range of a table, graph, ordered pair (some equations eg. y = x ).


Define a function.
Use function notation F(x) = 3x + 2, Find F(3)
Resources: Pearson Algebra I Common Core: Chapter 4: pgs. 234-280
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 many different inputs have the
Vocabulary: 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
Content: Linear Functions
7|Page
Duration: Jan (5 weeks)
Essential
How do we model data that changes at a constant rate?
Question:




Skills:






Assessment:





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
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 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.
Resources: Pearson Algebra I Common Core: Chapter 5: pgs. 293-351
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.
C.2.1.HS.F.3 - Apply quantitative reasoning to choose and interpret units and scales in formulas,
Standards: 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.
Linear function – a function with a constant rate of change; Parallel lines – two lines with the
Vocabulary: same slope; Perpendicular lines – two lines with opposite reciprocal slopes; Slope – the rate of
change of a graph, table or equation; x-intercept – the place where a graph crosses the x-axis; yintercept – the place where a graph crosses the y-axis
Comments:
Grade Level/ Subject: Algebra I
Content: Systems of Equations and Inequalities
8|Page
Duration: Feb. (3 weeks)
Essential
How do we find and what is the meaning of the intersection of two relationships?
Question:
Skills:





Assessment:



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.
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.
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.
Resources: Pearson Algebra I Common Core: Chapter 6: pgs. 364-406
CC.2.2.HS.D.10 - Represent, solve, and interpret equations/inequalities and systems of
equations/inequalities algebraically and graphically.
Standards: CC.2.2.HS.C.3 - Write functions or sequences that model relationships between two 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 parallel lines with different yVocabulary: 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
Content: Exponents and Exponential Functions
9|Page
Duration: March (1 week)
Essential
How can we write expressions more simply?
Question:
Skills:
Assessment:




Apply the properties of exponents to simplify rational expressions.
Convert between radical and rational exponent forms
Write exponential equations
Graph exponential equations
x0 (2x3 )5
(x2 )−2
3
√(3x)5 =

Simplify expressions using exponent properties such as:


Convert expression between rational and radical form:
(3x)5/3
Write, graph, and model exponential equations given tables and real-world problems
(y = abx )
Resources: Pearson Algebra I Common Core: Chapter 7: pgs. 418-472
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.
– relationship with a rate of change that is decreasing at a constantly decreasing rate; –
relationship with a rate of change that is increasing at a constantly increasing rate; – the
Vocabulary:
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
10 | P a g e
Essential
How do we represent solutions exactly?
Question:


Skills:



Assessment:


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.
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
Resources: Pearson Algebra I Common Core: Chapter 10: pgs. 615-631
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 – a2 + b2 = c 2 is a relationship between the side lengths and
Vocabulary: hypotenuse of a right triangle.
Comments:
Content: Data Analysis
Duration: March/April/May ( 1week)
Grade Level/ Subject: Algebra I
11 | P a g e
Essential
How do we quantify and interpret a data filled world?
Question:


Skills:


Assessment:



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.
Analyze data, make predictions, and/or answer questions based on displayed data (boxand-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 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; Interquartile range –
Vocabulary: the middle 50 percent of a data set; Probability – the likelihood of an event occurring
Comments:
Content: Polynomials and Factoring
Duration: March/April (5 weeks)
Grade Level/ Subject: Algebra I
12 | P a g e
Essential
How do we model and simplify linear and non-linear relationships?
Question:

Skills:





Assessment:


Add/Subtract/Multiply polynomial expressions up to binomials times trinomials (FOIL
method)
Apply the distributive property to polynomials
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 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.
Resources: Pearson Algebra I Common Core: Chapter 8: pgs. 486-533
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; Polynomial –
Vocabulary: 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
13 | P a g e
Content: Rational Expressions & Equations
Duration: May (1 weeks)
Essential
How do create a model for a relationship defined by the ratio of polynomials?
Question:
Skills:
Assessment:



Factor polynomials
Reduce/simplify rational expressions by cancelling like factors from numerators and
denominators
Multiply/Divide rational expressions

Simplify rational expressions by factoring problems like:

Multiply and Divide rational expressions like:
2𝑥 2 −5𝑥−12
2𝑥 2 −9𝑥+4
4𝑥+12
𝑥
∙
𝑥 2 −2𝑥 6𝑥+18
Resources: Pearson Algebra I Common Core: Chapter 11: pgs. 664-669
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; Rational
Vocabulary: Equation/Function – an equation that contains a rational expression; Complex Fraction – a
fraction that contains a fraction within the numerator or denominator;
Comments:
Grade Level/ Subject: Algebra I
14 | P a g e