Download Algebra II 2015

Course Title: Algebra 2
Content Area: Mathematics
Grade Level(s): 9 - 12
Course Description: This course extends the skills developed in an introductory algebra course. New topics are often introduced through application.
Topics to be studied in-depth include: the quadratic formula, advanced factoring, nth order roots, radical equations, synthetic division, complex numbers,
and sequences and series. Graphing calculators are an integral component of this course. PREREQUISITE: Algebra I. Geometry can be taken concurrently.
Credits: 5.0
Curriculum Writer(s): Andrea Cocco and Kim Partenfelder
Date Created: Spring 2015
Date Approved by Board of Education: October 2015
Pacing Guide
Unit 1: Review
Unit 2: Quadratics
Unit 3: Polynomials
Unit 4: Exponential and Logarithmic Equations
Unit 5: Trigonometry
Unit 6: Probability and Statistics
Unit 7: Sequences and Series
Marking Period 1
Marking Period 1 - 2
Marking Period 2
Marking Period 3
Marking Period 3
Marking Period 4
Marking Period 4
1
Unit 1: Algebra 1 Review
Unit Summary: Students will review essential Algebra 1 skills necessary to move on to true Algebra 2 topics.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.H
SA.SSE.A.1.A
Interpret parts of an expression, such as terms, factors, and coefficients.
CCSS.MATH.CONTENT.H
SA.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.
CCSS.MATH.CONTENT.H
SA.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.
CCSS.MATH.CONTENT.H
SA.CED.A.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
CCSS.MATH.CONTENT.H
SA.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.
CCSS.MATH.CONTENT.H
SA.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.
CCSS.MATH.CONTENT.H
SA.REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
2
CCSS.MATH.CONTENT.H
SA.REI.B.4
Solve quadratic equations in one variable.
CCSS.MATH.CONTENT.H
SA.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.
CCSS.MATH.CONTENT.H
SA.REI.C.6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two
variables.
CCSS.MATH.CONTENT.H
SF.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.*
CCSS.MATH.CONTENT.H
SF.IF.C.7.A
Graph linear and quadratic functions and show intercepts, maxima, and minima.
CCSS.MATH.CONTENT.H
SF.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).
CCSS.MATH.CONTENT.H
SG.GPE.B.5
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the
equation of a line parallel or perpendicular to a given line that passes through a given point).
CCSS.MATH.CONTENT.H
SS.ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Summative Assessments:
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●
Quizzes
Chapter Test
3
Formative Assessments:
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check homework
students will do problems on the board
exit tickets
mini white boards
use of technology
Pre-Assessments
do nows
oral questioning
closure
projects
technology based assessments
reflect and correct
writing
Enduring Understandings:
● The problem in front of you is a member of a larger class of problems.
● Number sense develops through experience.
● Operations create relationships between numbers.
● The relationships among the operations and their properties promote computational fluency.
● Real world situations can be represented symbolically and graphically.
● Algebraic expressions and equations generalize relationships from specific cases.
● There can be different strategies to solve a problem, but some are more effective and efficient than others are.
● The context of a problem determines the reasonableness of a solution.
● Functional relationships can be expressed in real contexts, graphs, algebraic equations, tables, and words; each representation of a given
function is simply a different way of expressing the same idea.
● Understand slope is a constant rate of change.
● Recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b).
● Understand linear equations describe the association between two quantities in bivariate data.
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Understand the connections between proportional relationships, lines, and linear equations.
Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns of change.
You can combine like radicals using properties of real numbers.
Essential Questions:
● What are the similarity and differences between solving a linear equation compared to a linear inequality?
● How do you know when units are appropriate for real-world situations?
● Why are units important in the problem-solving process?
● How do you determine the appropriate level of accuracy for a quantitative solution?
● How do you determine if a solution is viable?
● What is the purpose of function notation?
● Why are functions represented by different models?
● How do you decide which is the appropriate method of solution to a given system of linear equations?
● What is the purpose(s) of graphing systems of equations on coordinate plane?
Instructional Outcomes:
Students will know
● how to solve linear equations and inequalities.
● how to solve absolute value equations and inequalities.
● how to find slope.
● how to graph (slope intercept form, standard form, intercepts, absolute value equations).
● how to solve systems using substitution, elimination and graphing (2 or 3 variables).
● how to write equations of lines.
● how to simplify expressions involving exponents.
● how to add and subtract polynomials.
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how to identify variation.
how to write and graph direct variation equations.
how to use inverse and joint variation models.
how to simplify radicals.
how to add and subtract radicals.
how to multiply and divide radicals.
Suggested Learning Activities:
1. Noodle Tag 1: Each card has steps to solving an equation. Students collect all their colored cards and put them in order to solve the problem.
Can be used to review linear equations or to help teach radical equations, log equations, exponential equations. (Will need to get noodles and
then make color coded index cards with steps to solving equations)
2. Peer Edit Hop: Students will be given a few minutes to solve a lengthier equation, showing all steps. When completed, students will spread
their problems face down randomly on the floor. While music is playing, students will circulate around the room by hopping. When the music
stops, students will pick up the closet paper. They will go back to their seats and begin to check the student’s work. During the 1st round of
editing, the editor will only look at the 1st line of work. They will either put a check mark next to it if it’s correct or they will actually correct the
mistake. When time is called the students will put the papers back and the floor and hop again. During the 2nd round of editing, the student
will check the 1st and 2nd lines of work. Continue this until all lines are reviewed and corrected. Work should be given back to the original
student.
3. Review Plotting Ordered Pairs: Have students physically plot, name point, and name origin/axis where it is located. Students need to pick a
point that will keep them connected to a classmate via touching elbows. Teacher can take out or move students as needed.
4. Graphing Linear Equations: Have the students work in groups of 5 to graph a linear equation. The first student will plot the ordered pair for the
intercept on the y-axis. The next student will start on the same spot as the first student and then move forward or backward (the rise) and then
right or left (the run). The next student will start where the second student is standing and move again. The remaining students can fill in
between the 3 students that plotted themselves. They will form a straight line. The 1st and 3rd students need to remember to hold out their
outside arm (to function as the arrow that shows the line continues forever).
5. Graphing using Intercepts: Have the students wear signs to represent the different parts of an equation in standard form. To show the students
how to solve for intercepts using the cover up method, have the x-term walk away and have the students solve what is left. Then have one
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student plot that point. Then have the x-term join the problem again. Then have the y-term walk away and have the student solve what is left.
Have one student plot that point. Then have the rest of the class fill in to form a straight line between the 2 plotted points.
6. Graphing Absolute Value Equations: Have the students identify the vertex and have one student physically plot it. Then have students identify
the slope. Have 2 student start at the vertex and “rise” (forward or backward) together and then “run” in opposite directions the same number
of points (one goes left and one goes right). Have 2 more students start at those points and continue the pattern until all 5 points are plotted.
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
Notes/Comments:
Unit 2: Quadratics
Unit Summary: Students will understand the relationship between factoring and foiling. They will also understand there are multiple ways to solve
quadratics.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
7
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.H
SN.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.
CCSS.MATH.CONTENT.H
SA.SSE.B.3.A
Factor a quadratic expression to reveal the zeros of the function it defines.
CCSS.MATH.CONTENT.H
SA.SSE.B.3.B
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
CCSS.MATH.CONTENT.H
SA.APR.D.7
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition,
subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational
expressions.
CCSS.MATH.CONTENT.H
SA.REI.B.4.A
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q
that has the same solutions. Derive the quadratic formula from this form.
CCSS.MATH.CONTENT.H
SA.REI.B.4.B
Solve quadratic equations by inspection (e.g., for x2 = 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.
CCSS.MATH.CONTENT.H
SA.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).
CCSS.MATH.CONTENT.H
SF.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.*
8
CCSS.MATH.CONTENT.H
SF.IF.C.7.A
Graph linear and quadratic functions and show intercepts, maxima, and minima.
CCSS.MATH.CONTENT.H
SF.IF.C.8.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.
Summative Assessments:
●
●
Quizzes
Chapter Test
Formative Assessments:
●
●
●
●
●
●
●
●
●
●
●
●
check homework
students will do problems on the board
exit tickets
mini white boards
use of technology
Pre-Assessments
do nows
oral questioning
closure
projects
technology based assessments
reflect and correct
Enduring Understandings:
● You can factor many quadratic trinomials into products of two binomials.
● To find the zeros of a quadratic function, you must set the equation equal to zero.
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Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial.
You can solve a quadratic equation in standard form in more than one way. In general, you can find a formula that gives values of x in terms of a,
b, and c.
A basis for the complex numbers is a number whose square is -1.
Essential Questions:
● How do you determine the maximum and minimum value of a quadratic function from the equation?
● How does the geometric description of a parabola relate to the equation?
● What real-life situations can be graphed with parabolas?
● How do you derive the equation of a parabola?
● How do you find components of a parabola, including axis of symmetry, and direction of opening?
● How are the concepts of factoring quadratic trinomials and FOIL method related to each other mathematically?
● How do you use the concept of factoring by grouping to factor a quadratic trinomial?
● Why does the discriminant determine the number and nature of the roots to a quadratic equation and how does it relate to the Quadratic
Formula?
● When would you use the quadratic formula to find the roots of a quadratic function?
● How are the powers of ί derived, and how are they cyclic?
● Why is it when simplifying imaginary and complex numbers that the highest power of ί is one?
● How is multiplying any two polynomials just an expansion of the distributive property?
● How do you know when a rational expression can be simplified?
● What do complex numbers represent?
● When are complex numbers important in your mathematical studies?
● What are the similarities and differences of using the quadratic formula with real numbers and complex numbers?
● How do the patterns of i help simplify complex numbers?
● What are the similarities and differences between the real number system and the complex number system?
● Why is there more than one way to solve a quadratic equation?
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What is the meaning of complex solutions?
What are the similarities and differences of solving equations and rational equations?
Instructional Outcomes:
Students will know:
● how to identify key characteristics of a parabola by looking at the equation.
● how to graph parabolas.
● how multiplying and factoring polynomials are related.
● how to factor an expression by factoring a common factor from each term.
● how to factor quadratic trinomials with the quadratic coefficient of one.
● how to factor quadratic trinomials with the quadratic coefficient not equal to one and prime.
● how to factor quadratic trinomials with the quadratic coefficient not equal to one by grouping.
● how to factor the difference of two squares and perfect square trinomials.
● how to factor higher powered polynomials.
● how to factor sums and differences of cubes.
● how to solve quadratic equations using the square root method.
● how to solve quadratic equations by factoring and using the zero product property.
● how to plot and identify complex numbers on the complex plane.
● how to perform operations with complex numbers (add, subtract, multiply).
● how to evaluate powers of ί.
● how to solve quadratic equations with complex solutions.
● how to solve quadratic equations by completing the square.
● how to solve quadratic equations using the Quadratic Formula.
● how to simplify rational expressions.
● how to multiply and divide rational expressions.
● how to add and subtract rational expressions.
● how to solve rational equations.
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how to simplify complex fractions.
Suggested Learning Activities:
1. Plotting Complex Numbers: Have each student take a piece of paper to simulate their origin. Take steps to the right or left for the real part and
take steps forwards and backwards for the complex part. Jump to plot the point.
2. Graphing Quadratic Equations: Have students work in groups of 6 to graph a parabola (5 students will be points and the 6th student will help
with the counting). Students will need to determine the location of the vertex and they will need to determine the coordinates of 2 points on
either the left or the right side of the vertex. Once the math part is done, the student will begin to physically plot the graph. The first student
will plot the intercept. The next student will plot his/her point. The 6th person (the non-plot person) will count how far the student would need
to rise and run from the vertex to get to the 2nd point. The third student will start at the vertex and rise the same amount as the 2nd student
and run the same amount in the opposite direction. The 4th and 5th people find their plots the same way. Once they are plotted they need to
extend their arms too (to show that the graph continues forever).
3. Quadratic Formula: Help students memorize the quadratic formula by teaching it to the tune of Pop Goes the Weasel.
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
Notes/Comments:
12
Unit 3: Polynomials
Unit Summary: Students will apply prior knowledge of operations and graphing to polynomials.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.H
SN.RN.A.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. For example, we define 51/3 to be the
cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
CCSS.MATH.CONTENT.H
SN.RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
CCSS.MATH.CONTENT.H
SN.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
CCSS.MATH.CONTENT.H
SN.CN.A.2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex
numbers.
CCSS.MATH.CONTENT.H
SN.CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
CCSS.MATH.CONTENT.H
SN.CN.B.4
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers),
and explain why the rectangular and polar forms of a given complex number represent the same number.
CCSS.MATH.CONTENT.H
SN.CN.C.7
Solve quadratic equations with real coefficients that have complex solutions.
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CCSS.MATH.CONTENT.H
SN.CN.C.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
CCSS.MATH.CONTENT.H
SA.SSE.A.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it
as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
CCSS.MATH.CONTENT.H
SA.APR.B.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so
p(a) = 0 if and only if (x - a) is a factor of p(x).
CCSS.MATH.CONTENT.H
SA.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.
CCSS.MATH.CONTENT.H
SA.APR.D.6
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +r(x)/b(x), where a(x), b(x), q(x), and
r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more
complicated examples, a computer algebra system.
CCSS.MATH.CONTENT.H
SA.REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
CCSS.MATH.CONTENT.H
SA.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).
CCSS.MATH.CONTENT.H
SF.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).
CCSS.MATH.CONTENT.H
SF.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.
CCSS.MATH.CONTENT.H
SF.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:
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intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.*
CCSS.MATH.CONTENT.H
SF.IF.C.7.B
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
CCSS.MATH.CONTENT.H
SF.IF.C.7.C
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
CCSS.MATH.CONTENT.H
SF.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.
CCSS.MATH.CONTENT.H
SF.BF.B.4.A
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For
CCSS.MATH.CONTENT.H
SF.BF.B.4.B
(+) Verify by composition that one function is the inverse of another.
example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
Summative Assessments:
●
●
Quizzes
Chapter Test
Formative Assessments:
●
●
●
●
check homework
students will do problems on the board
exit tickets
mini white boards
15
●
●
●
●
●
●
●
●
use of technology
Pre-Assessments
do nows
oral questioning
closure
projects
technology based assessments
reflect and correct
Enduring Understandings:
● The properties of integers apply to polynomials.
● Solving polynomials involves the reversal of operations, the distributive property and rules of exponents.
● A polynomial function has distinguishing “behaviors”. You can look at its algebraic form and know something about its graph. You can look at its
graph and know something about its algebraic form.
● Knowing the zeros of a polynomial functions can help you understand the behavior of its graph.
● If (x-a) is a factor of a polynomials, then the polynomial has value 0 when x=a. If a is a real number, then the graph of the polynomial has (a,0) as
an x-intercept.
● You can divide polynomials using steps that are similar to the long division steps that you use to divide whole numbers.
● The degree of a polynomial equation tells you how many roots the equation has.
● You can write a radical expression in an equivalent form using a fractional (rational) exponent instead of a radical sign.
● Solving a square root equation may require that you square each side of the equation. This process can introduce extraneous solutions.
Essential Questions:
● What are the similarities and differences between long division and synthetic division?
● Why are exponents represented as fractions?
● What does the degree of a polynomial tell you about its related polynomial function?
● For a polynomial function, how are factors, zeros and x-intercepts related?
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For a polynomial function, how are factors and roots related?
How do the concepts of nth roots relate with rational exponents?
Instructional Outcomes:
Students will know:
● how to simplify expressions involving powers.
● how to evaluate nth roots and study rational exponents.
● how to simplify expressions involving rational exponents.
● how to identify and classify polynomials.
● how to perform direction and synthetic substitution.
● how to divide a polynomial using long division.
● how to divide a polynomial using synthetic division.
● how to find all zeros of a polynomial function
● how to analyze graphs using end behavior.
● how to determine if a function is odd or even.
● how to classify all the zeros of polynomial functions.
● how to perform operations with functions.
● how to find the inverse of functions.
● how to graph square root and cube root functions.
● how to solve radical equations.
Suggested Learning Activities:
1. Graphing Polynomials: Have students work together to physical act out the shapes of polynomials (degrees 0 to 4) with the correct end
behavior and placed in the general spot.
2. Families of Graphs: Break students into 4 teams and give students a set of equations on index cards and blank index cards for them to write the
headings (constant, linear, quadratic, cubic, exponential, logarithmic). They will need to organize them by type. They will complete a relay race
to get them from one side of the room to the other. Only one person can be standing at a time and each person can only carry one index card
at a time.
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3. Exponents: Multiplication: Have students hold a piece of paper that contains a variable. Have students stand next to each other to represent
expressions. Students will count up (add) the like terms.
4. Exponents: Division: Place tape on the floor to represent the division bar. Have students hold a piece of paper that contains a variable. Have
students stand in the numerator and the denominator. Students will see that one in the top cancels out one in the bottom and they will leave
the problem. Students will count what is left in the numerator and what is left in the denominator (using the exponent multiplication rule from
above).
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
Notes/Comments:
Unit 4: Exponential and Logarithmic Equations
Unit Summary: Students will be able to apply growth and decay models.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
18
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.H
SF.IF.C.7.E
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing
period, midline, and amplitude.
CCSS.MATH.CONTENT.H
SF.BF.B.5
(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems
involving logarithms and exponents.
CCSS.MATH.CONTENT.H
SF.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).
CCSS.MATH.CONTENT.H
SF.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.
CCSS.MATH.CONTENT.H
SF.LE.A.4
For exponential models, express as a logarithm the solution to abct = d where a, c, and dare numbers and the base b is 2, 10,
or e; evaluate the logarithm using technology.
Summative Assessments:
●
●
Quizzes
Chapter Test
Formative Assessments:
●
●
●
●
●
●
check homework
students will do problems on the board
exit tickets
mini white boards
use of technology
Pre-Assessments
19
●
●
●
●
●
●
do nows
oral questioning
closure
projects
technology based assessments
reflect and correct
Enduring Understandings:
● Real-life situations involving growth and decay can be modeled using exponential relationships.
● The exponential function y =𝑏𝑏 is one-to-one, so its inverse x = 𝑏𝑏 is a function.
● To express “y as a function of x” for the inverse you write y = 𝑏𝑏𝑏𝑏 𝑏.
● Logarithms and exponents have corresponding properties.
● You can use logarithms to solve exponential equations; and conversely, you can use exponents to solve logarithmic equations.
● The function y =𝑏𝑏 and y = ln x are inverse functions.
Essential Questions:
● How do you use the concept of exponential growth to work with the principle of compound interest?
● How do you convert between exponential and logarithmic form?
● How do you use the inverse relationship between exponential and logarithmic functions to solve equations?
● Why are exponential and logarithm functions inverses?
Instructional Outcomes:
Students will know:
● how to write and evaluate exponential expressions to model growth and decay situations.
● how to graph exponential growth and decay functions.
● how to write equivalent forms for exponential and logarithmic functions.
● how to write, evaluate, and graph logarithmic functions.
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●
●
●
●
●
●
how to use properties to simplify logarithmic expressions.
how to expand and condense logarithmic expressions.
how to translate between logarithms in any base.
how to solve exponential and logarithmic equations.
how to use the number e to write and graph exponential functions representing real world situations.
how to solve equations and problems involving e or natural logarithms.
Suggested Learning Activities:
1. Families of Graphs: Break students into 4 teams and give students a set of equations on index cards and blank index cards for them to write the
headings (constant, linear, quadratic, cubic, exponential, logarithmic). They will need to organize them by type. They will complete a relay race
to get them from one side of the room to the other. Only one person can be standing at a time and each person can only carry one index card
at a time.
2. Ball Toss for Evaluating Logs: The teacher will put a problem on the board and students will be paired together. They will do the problem in
their head and then throw the ball back and forth the correct number of times (as many as the solution).
3. Logarithm War: Like the card game war, but students must calculate the value of each of their cards to decide who wins the round.
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
21
Notes/Comments:
Unit 5: Trigonometry
Unit Summary: Students will see how trigonometry applies to triangles and circles.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.H
SF.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.
CCSS.MATH.CONTENT.H
SF.IF.C.7.E
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing
period, midline, and amplitude.
CCSS.MATH.CONTENT.H
SF.TF.A.1
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
CCSS.MATH.CONTENT.H
SF.TF.A.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed counterclockwise around the unit circle.
CCSS.MATH.CONTENT.H
SF.TF.A.3
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit
circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any
real number.
22
CCSS.MATH.CONTENT.H
SF.TF.A.4
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
CCSS.MATH.CONTENT.H
SF.TF.B.5
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*
CCSS.MATH.CONTENT.H
SG.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions
of trigonometric ratios for acute angles.
CCSS.MATH.CONTENT.H
SG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*
Summative Assessments:
●
●
Quizzes
Chapter Test
Formative Assessments:
●
●
●
●
●
●
●
●
●
●
●
check homework
students will do problems on the board
exit tickets
mini white boards
use of technology
Pre-Assessments
do nows
oral questioning
closure
projects
technology based assessments
23
●
reflect and correct
Enduring Understandings:
● There are three trigonometric functions and three trigonometric inverse functions.
● The characteristics of trigonometric and circular functions and their representations are useful in solving real-world problems.
● Certain right triangles have properties that allow their side lengths to be determined without using the Pythagorean Theorem.
● Ratios can be used to find side lengths and angle measures of a right triangle when certain combinations of side lengths and angles measures
are known.
● Angles can be measured in different units.
● The trigonometric functions describe periodic phenomena.
● The graphs of sine and cosine model real-world phenomena.
Essential Questions:
● How are trigonometric used in right triangles?
● What is radian measure?
● How can you evaluate trigonometric functions of any angle?
● How do the graphs of y = sin(x) and y = cos(x), compare to the graph of y = tan (x)?
Instructional Outcomes:
● how to use trig functions to find side lengths of right triangles.
● how to use trig functions to find angles of right triangles.
● how to draw angles in standard position.
● how to find co-terminal angles.
● how to convert between degrees and radians.
● how to find arc length and areas of sectors.
● how to operate with a unit circle.
24
●
how to graph sine, cosine, and tangent.
Suggested Learning Activities:
1. Make the Unit Circle: Students will create the unit circle on a paper plate.
2. Trig Foldable: Use a foldable to review right triangle trig.
3. Video to learn trig: https://www.youtube.com/watch?t=72&v=t2uPYYLH4Zo
4. Radians: http://mathequalslove.blogspot.com/2014/12/trigonometry-unit-2-interactive.html. Have students create a circle and find the center.
Use a pipe cleaner to represent the radius. Have students hold it outside of the circle and mark where it stops. It should create 6 equal spots
with a tiny bit left over. Each spot would be equal to 1 radian. ("Twinkle, Twinkle Little Star. Circumference equals 2 pi r.")(Never fear
when the word radian appears.)
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
Notes/Comments:
25
Unit 6: Data Analysis
Unit Summary: Students will be able to understand how data can be predicted and analyzed.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
CCSS/NJCCCS Number
CCSS/NJCCCS Content
CCSS.MATH.CONTENT.
HSS.ID.A.1
Represent data with plots on the real number line (dot plots, histograms, and boxplots).
CCSS.MATH.CONTENT.
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.
CCSS.MATH.CONTENT.
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).
CCSS.MATH.CONTENT.
HSS.ID.A.4
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population
percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
CCSS.MATH.CONTENT.
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.
CCSS.MATH.CONTENT.
HSS.IC.A.1
Understand statistics as a process for making inferences about population parameters based on a random sample
from that population.
26
CCSS.MATH.CONTENT.
HSS.IC.A.2
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For
example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you
to question the model?
CCSS.MATH.CONTENT.
HSS.IC.B.3
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain
how randomization relates to each.
CCSS.MATH.CONTENT.
HSS.IC.B.4
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the
use of simulation models for random sampling.
CCSS.MATH.CONTENT.
HSS.IC.B.5
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences
between parameters are significant.
CCSS.MATH.CONTENT.
HSS.IC.B.6
Evaluate reports based on data.
CCSS.MATH.CONTENT.
HSS.CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
CCSS.MATH.CONTENT.
HSS.CP.A.2
Understand that two events A and B are independent if the probability of A and B Occurring together is the product
of their probabilities, and use this characterization to determine if they are independent.
CCSS.MATH.CONTENT.
HSS.CP.A.3
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as
saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability
of B given A is the same as the probability of B.
CCSS.MATH.CONTENT.
HSS.CP.B.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
CCSS.MATH.CONTENT.
HSS.CP.B.8
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and
interpret the answer in terms of the model.
27
CCSS.MATH.CONTENT.
HSS.CP.B.9
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
CCSS.MATH.CONTENT.
HSS.MD.A.2
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
CCSS.MATH.CONTENT.
HSS.MD.B.5
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected
values.
Summative Assessments:
●
●
Quizzes
Chapter Test
Formative Assessments:
●
●
●
●
●
●
●
●
●
●
●
●
check homework
students will do problems on the board
exit tickets
mini white boards
use of technology
Pre-Assessments
do nows
oral questioning
closure
projects
technology based assessments
reflect and correct
28
Enduring Understandings:
● Probability is used to make inferences and predictions.
● The way that data is collected, organized, and displayed influences interpretation.
● The probability of an event’s occurrence can be predicted with varying degrees of confidence.
● Outliers affect the measures of central tendency.
● Constants affect the measures of central tendency.
● Proper experimental design is necessary to ensure unbiased results.
● Graphs produce visual displays of data in meaningful ways.
● Measuring the spread of data is essential for comparing data sets.
Essential Questions:
● What are the different ways in which data can be organized and analyzed?
● In what real-world situations would you encounter conditional probability?
● How does one determine whether events are independent or dependent?
● What is the relationship between theoretical and experimental probability?
● Why is the probability of dependent events different from the probability of independent events?
● When will the mean of a data set differ significantly from the median of a data set?
● Are all the statistics of a data set affected when you transform the values in a data set?
● Where are the values in a normal distribution that rarely occur displayed on a normal curve?
● What should be true of the sample when you conduct a survey?
Instructional Outcomes:
Students will know:
● how to summarize, represent, and interpret data.
● how to make inferences and justify conclusions from sample surveys and experiments.
29
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
how to compute theoretical and experimental probabilities for a sample space.
how to calculate the probability of an event.
how to use lists and tables to state the difference between independent and dependent events.
how to determine the union, intersection, and complement of events.
how to determine the complement and negation of a single event.
how to solve problems involving conditional probability.
how to find the conditional probability of an event.
how to solve problems involving conditional probability.
how to apply the addition and multiplication rules of probability.
how to find the measures of central tendency.
how to find the measures of central dispersion.
how to find permutations and combinations.
how to set up a normal distribution and use it to answer questions.
how to name the type of sample.
how to find the margin of error in a sample.
how to read histograms.
Suggested Learning Activities:
1. “I have, Who has..”: Playing as a team, students will get a set of cards with questions and answers. Each answer will lead to the next
question. If they answer incorrectly, they will not make it through the deck of cards.
2. Human Trouble: Have students answer a math question correctly to get on the board. Student will roll die to move. Play passes to the
next team. Each time a student wants to move, he or she must first answer a question correctly to be able to roll the die. If another
student lands on the same spot, the first student is sent back home.
3. Greedy Pig: To test probably students will be placed in small groups and will have a single die. Each person is allowed to roll as many
times as he/she would like, adding the numbers they roll as they go. However, if a 1 is rolled, their score becomes zero. This is to
30
show that the probably for rolling a 1 is ⅙ every time, but to see students reactions (if they think it gets higher as they are more into
the game).
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
Notes/Comments:
Unit 7: Sequences and Series
Unit Summary: Students would be able to continue, write a rule for and/or, find the sum of a sequence.
Interdisciplinary Connections/Content Area Integrations Including Technology: Financial, Economic, Business, and Entrepreneurial Literacy
CCSS/NJCCCS Number
CCSS/NJCCCS Content
31
CCSS.MATH.CONTENT.
HSA.SSE.B.4
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to
solve problems. For example, calculate mortgage payments.*
CCSS.MATH.CONTENT.
HSF.IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
CCSS.MATH.CONTENT.
HSF.BF.A.1.A
Determine an explicit expression, a recursive process, or steps for calculation from a context.
CCSS.MATH.CONTENT.
HSF.BF.A.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model
situations, and translate between the two forms.*
CCSS.MATH.CONTENT.
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).
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.
Summative Assessments:
●
●
Quizzes
Chapter Test
Formative Assessments:
●
●
●
●
●
●
●
●
check homework
students will do problems on the board
exit tickets
mini white boards
use of technology
Pre-Assessments
do nows
oral questioning
32
●
●
●
●
closure
projects
technology based assessments
reflect and correct
Enduring Understandings:
● Sequences and series are discrete functions whose domain is the set of whole numbers.
● Arithmetic and geometric sequences and series are mathematical patterns that stem from practical situations and be used to model
real-life situations.
● Sequences and series provide the foundation for upper level mathematics, especially calculus. Sequences and series are a direct
result of finding patterns.
Essential Questions:
● How you can you write an expression for sums such as 5 + 10 + 15 + … + 50?
● How can you tell if a sequence is arithmetic or geometric?
● How can you find the sum of the terms of geometric series?
● When does an infinite geometric series have a sum and when does it not have a sum?
● How do you write a recursive rule for a sequence?
Instructional Outcomes:
● how to recognize and write rules for number patterns.
● how to determine if a sequence is arithmetic or geometric.
● how to use sum notation for sequences.
● how to find the next terms of a sequence.
● how to write the explicit formula for arithmetic and geometric sequences.
● how to find the term named in a sequence.
● how to find the sum of an infinite geometric series.
33
●
how to write a recursive formula for a sequence.
Suggested Learning Activities:
1. Foldables: Arithmetic Explicit formula, Geometric Explicit Formula, Geometric Series
2. Vocabulary: Use a template to go through the vocabulary from the unit. The template will include the word, definition,
characteristics, examples and non-examples.
3. Stomp: A review game. Project a Kuta worksheet of multiple choice questions on the overhead. Have two sets of four pieces of paper
(A, B, C, D) on the floor. Split the class into two teams and line up behind the paper and assign each pair of students a number.
Project one question and give all students time to answer. Roll a die and call a number. Those 2 students will race to stomp on the
correct answer.
Suggested Differentiation:
1. Guided Notes: Have guided notes filled out at different levels according to ability.
2. Foldables: Creating tangible products to help students digest information while hitting several of the multiple intelligences.
3. Differentiated Loops: Have problems posted around the room. Have students “loop” to specific questions based on difficulty.
Curriculum Development Resources: Textbook, Kuta Software, Whiteboard, Overhead projector, Math Websites, Chromebook, Calculator
Notes/Comments:
34