scope & sequence Linear Equations..... 2 Linear Inequalities ..... 2 Systems of Linear Equations and Inequalities ..... 3 Absolute Value Concepts of Function..... 3 Graphs of Functions

scope & sequenceHigh School Math (Grades 9 - 12)

See the breadth and depth of the content

covered and the order in which Uzinggo’s

interactive, engaging tutorials are taught in:

• Algebra I• Geometry• Algebra II

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ZingPathTM

A ZingPath is a group of fun, interactive tutorials that Uzinggo uses to teach an idea or a concept much like a lesson plan. Upon completion of a ZingPath, students receive reward points and badges.

ZingPathTM

ZingPath TM Tutorial

What is a... Uzinggo / tutorial typesMath tutorials consist of five different types:

1. Concept Development

These activities introduce concepts through engaging, real-world scenarios and develop these concepts using an inquiry-based approach.

2. Dynamic Modeling

These tutorials provide learners the opportunity to manipulate variables and observe dynamic changes with interactive 3-D objects.

3. Skills Application

These activities help learners apply and extend their knowledge and practice essential mathematical skills.

4. Problem Solving

These activities engage learners with a guided problem-solving process to apply and enhance their mathematical understanding.

5. Visual Proofs / Animations

These activities provide learners with visual justification of formulas, theorems and relationships.

Uzinggo scope & sequenceMath Table of Contents

Algebra 1 ZingPaths

Numeracy ................................................................................... 1Units and Accuracy .................................................................... 1Exponents and Radicals ............................................................. 1Setting Up Equations and Formulas .......................................... 1Graphs of Linear Equations ....................................................... 2Solving Linear Equations ............................................................ 2Linear Inequalities ...................................................................... 2Systems of Linear Equations and Inequalities ........................... 3Absolute Value Equations .......................................................... 3Concepts of Function ................................................................. 3Graphs of Functions ................................................................... 4Composition and Inverses ......................................................... 4Linearity, Slope, and Intercepts ................................................. 4Linear Relationships and Their Graphs ...................................... 5Plots and Charts ......................................................................... 5Statistical Calculations ............................................................... 5Polynomial Expressions ............................................................. 6Quadratic Equations .................................................................. 6Quadratic Functions ................................................................... 6Parabolas .................................................................................... 6Graphing Quadratic Functions .................................................. 7

Geometry ZingPaths

Points and Lines ......................................................................... 8Planes ......................................................................................... 8Angles ........................................................................................ 8Fundamental Axioms and Theorems ......................................... 8Parallel Lines .............................................................................. 9Triangles and Their Parts ........................................................... 9


Geometry ZingPaths Contd.

Triangle Congruency and Similarity ........................................... 9Triangle Side Lengths ................................................................. 9Proportion and Similarity ......................................................... 10Median, Altitude, and Bisector ................................................ 10Applying Transformations ........................................................ 10Representing Angles ................................................................ 11Trigonometric Ratios and Circles ............................................. 11Polygon Fundamentals ............................................................ 11Quadrilaterals ........................................................................... 11Basic Perimeter and Area Calculations .................................... 12Area and Perimeter of Polygons .............................................. 12Area and Circumference .......................................................... 12Nets and Cross Sections .......................................................... 13Volume ..................................................................................... 13Surface Area ............................................................................. 13Sets ........................................................................................... 13Counting Principles .................................................................. 14Permutations and Combinations ............................................. 14Probability ................................................................................ 14Probability Calculations ........................................................... 14

Algebra 2 ZingPaths

Polynomial Expressions and Factoring .................................... 16Polynomial Operations ............................................................ 16Graphs and Polynomials .......................................................... 16Solving Rational and Radical Equations ................................... 16Lines in the Cartesian Plane ..................................................... 17Angles and their Measures ...................................................... 17Trigonometric Ratios ................................................................ 18

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Uzinggo / scope and sequence for math


Algebra 2 ZingPaths Contd.

Graphs of Trigonometric Functions ......................................... 18Problem Solving with Trigonometric Functions ....................... 18Vector Concepts ...................................................................... 18Vector Operations .................................................................... 18Inner Product and Linear Dependence .................................... 19Vectors and Line Equations ..................................................... 19

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ALGEBRA 1

Zing-Path Tutorial Description

Num

erac

y

Prime Factorization • How to write out a composite number as the product of its prime factors is given.

l

The Greatest Common Factor of Numbers

• How to find the greatest common factor of two or more numbers is explained.

l

Estimating the Square Root of Non-Perfect

Squares

• The square roots of non-perfect squares are estimated by determining the square roots of the nearest perfect squares in order to find the appropriate frames for paintings in an art exhibit.

l

Properties of Rational & Irrational Numbers

• Students identify and approximate irrational numbers by distinguishing them from rational numbers through decimal expansions and geometric reasoning.

l

Sums and Products of Rational and Irrational

Numbers

• Students determine whether the sums and products of rational and irrational numbers are rational or irrational. l

Uni

ts a

nd A

ccur

acy Operations on Numbers in

Scientific Notation• Understanding of number relationships and the basic properties

of operations of numbers in scientific notation is applied to solve problems.

l

Ratio and Rate • The definitions of ratio and rate, and the properties of ratio are given.

l

Scientific Notation and Significant Figures

• Students learn how and why to utilize significant figures and scientific notation.

l

Exp

onen

ts a

nd

Rad

ical

s

Exponents and Their Properties

• The properties of exponents to evaluate expressions are applied.

l

Properties and Rules of Radicals

• Expressions with fractional exponents are written in radical form and vice versa, and radical expressions are simplified.

l

Adding and Subtracting Square Roots

• Students add and subtract radical expressions, then simplify them.

l

Sett

ing

Up

Eq

uati

ons

and

For

mul

as

Proportion and Its Properties

• It is determined if two ratios form a proportion and solved for the unknown value of a proportion.

l

Percentage of Mixtures • Students determine unknown information based on amounts, percentages of mixtures, or combined mixtures.

l

Translating Problems Into One-Step Equations

• Students translate word problems (involving all four basic operations) into one-step equations.

l

Simple Interest* • Students solve simple interest problems while visiting a bank. l

Distance Problems: Two Travelers Starting At The

Same Time

• Application problems involving rate and distance using a problem solving plan is explained. l

Isolating a Quantity of Interest

• Students determine the quantity of interest in real-life examples and isolate it.

l

ALGEBRA 1


Gra

phs

of

Line

ar E

qua

tion

s

Determining the Coordinate Plane

• Students define the Cartesian plane, write the coordinates of the points on that plane, and identify its quadrants.

l

Directly Varying Quantities and Their Graphs

• Directly varying quantities given graphs, tables, or statements are identified and problems involving direct variation are solved.

l

Graphs of One-Step Linear Equations

• Students graph one-step linear equations by finding missing x or y values and plotting ordered pairs on the coordinate plane.

l

Graphs of Two-Step Linear Equations

• Two-step linear equations are graphed by finding missing x or y values and plotting ordered pairs on the coordinate plane.

l

Slope of a Line• Students learn how to use a coordinate system to specify locations

and to calculate the slope of a line from a graph in an activity similar to a well-known strategy game.

l

Determine the Relationship between two Lines by Comparing their

Slopes and y-intercepts

• Students interpret the y-intercept of a line geometrically by shifting a line that passes through the origin. l

Solv

ing

Lin

ear

Eq

uati

ons

Solving One-Step Linear Equations

• One-step linear equations in one variable are solved using addition, subtraction, multiplication, or division.

l

Solving Two-Step Linear Equations

• Two-step linear equations in one variable using addition, subtraction, multiplication, or division are solved.

l

Solution Sets of Linear Equations

• Students determine if linear equations of the form ax + b = cx + d have exactly one solution, no solutions, or infinitely many solutions.

l

Solve Linear Equations in One Variable.

• Students use inverse operations to solve linear equations of the form ax + b = cx + d, where a is not equal to c.

l

Approximate the Solutions to Linear Equations

• Students use graphs to approximate the solutions to linear equations and observe that the approximation might not be precise.

l

Solve Linear Equations with Indeterminate

Coefficients

• Students solve one variable two-step linear equations with indeterminate coefficients by using inverse operations. l

Checking That Solutions are Reasonable

• Students learn how to determine if a solution to a linear equation that models a situation is reasonable according to the situation.

l

Line

ar In

equa

litie

s

Graphing Linear Inequalities in One

Variable

• Graphing linear inequalities in one variable on the number line is explained. l

Solving One-Step Linear Inequalities

• One-step linear inequalities using addition, subtraction, multiplication or division are solved.

l

Negative Numbers and Linear Inequalities

• Students observe that multiplying both sides of an inequality involving algebraic expressions by a negative number changes the inequality symbol.

l

ALGEBRA 1

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ALGEBRA 1


Line

ar In

equa

litie

s

Solve Two-Step Linear Inequalities

• Students solve a one variable two-step linear inequality with numerical coefficients by using inverse operations.

l

Graphing Linear Inequalities in Two

Variables

• Graphing linear inequalities in two variables on the coordinate plane is explained. l

Writing a Linear Inequality That Corresponds to a

Graph

• Students write linear inequalities from their graphs.l

Syst

ems

of L

inea

r E

qua

tion

s an

d

Ineq

ualit

ies

Writing a System of Linear Inequalities that

Corresponds to a Graph

• Students write a system of linear inequalities from their graphs.l

Solving Systems of Linear Equations Graphically

• Systems of two linear equations are solved graphically. l

Solving Systems of Linear Equations Using the Elimination Method

• Systems of two linear equations are solved using the elimination method. l

The Altitude and Orthocenter in a Triangle

• Students learn the definitions of altitude, base of an altitude, foot of an altitude, and orthocenter of a triangle, and calculate the coordinates of the orthocenter when given the equations of the lines containing the sides of the triangle or the altitudes.

l

Graphing Systems of Linear Inequalities

• Students graph systems of linear inequalities in two variables. l

Ab

solu

te V

alue

E

qua

tion

s

Absolute Values and their Properties

• Absolute value and its properties, which are applied to problems of varying difficulty are explained.

l

Absolute Value Equations, Graphs, and Intersection

Points

• Students use graphical solution methods to solve an equation of the form |ax + b| = |bx + c|. l

Solve Absolute Value Equations with Graphing

Technology

• Students approximate the solutions of an absolute value equation by using graphing technology. l

Con

cep

ts o

f Fu

ncti

on

Fibonacci Sequence • Students learn about the Fibonacci Sequence and use it to solve mathematical problems.

l

Evaluation of algebraic expressions

• Algebraic expressions are evaluated using substitution. Through playing a tropical island game, students learn to evaluate algebraic expressions using substitution and to simplify them using the order of operations.

l

The Basics and Properties of Sets

• How to determine empty, universal, finite, infinite, equal, and equivalent sets using Venn diagrams, the listing method, and/or set builder notation is explained.

l

The Concept of Relation• Using a real-life example involving songs and musicians, the

concepts of relation, the domain (input) and range (output) of a relation are explored, and how to represent a relation by listing ordered pairs, mapping, and graphing is explained.

l

ALGEBRA 1


Con

cep

ts o

f Fu

ncti

on The Concept of Function• A function, the domain (input, independent variable) and range

(output, dependent variable), and the properties to determine a function are identified.

l

The Domain and Range of a Function

• How to represent a function using tables, graphs, the verbal rule, and equations is explained.

l

Determining Whether a Relation is also a Function

• Students use various representations of relations such as tables, mapping diagrams, equations, verbal rules, and graphs to show what makes a relation a function.

l

Gra

phs

of

Func

tion

s

Different Forms of Representation for a

Relationship

• Various forms of representations for a relationship are explored.l

Reading Values from the Graphs of Functions

• Calculations and observations are performed based on the graphs of functions, inverse functions, and the composite of two functions.

l

Finding Solution Sets Based on Graphs of

Functions

• Calculations and observations are performed based on the graph of two functions and find the zeros of a function. l

Modeling Real Life with Graphs of Functions

• Students interpret graphs of real-world situations by examining the sections where the graph is increasing, decreasing, and constant.

l

Algebraic versus Graphical Solution Methods

• Students learn that solving an equation algebraically leads to an exact solution, while graphical methods might lead to an approximate solution.

l

Choosing between Graphical and Algebraic

Methods

• Students choose between graphical and algebraic solution methods to equations based on whether exact results are necessary.

l

Com

pos

itio

n an

d

Inve

rses

Fundamental Concepts on the Composition of

Functions

• Students determine if the composition of two functions exists, and use multiple representations to evaluate the composition of two functions.

l

Finding and Using the Rule of Composition of Two

Functions

• The rule of composition of two functions is calculated.l

Fundamental Concepts on the Inverses of Functions

• Students learn the fundamental concepts of the inverses of functions.

l

Line

arit

y, S

lop

e, a

nd

Inte

rcep

ts

The Concept of Linearity • Given a verbal description, students determine if a relationship is linear by creating a table or a graph.

l

The Concept of Slope• The concept of slope is introduced as a rate of change between

dependent and independent variables, and as a geometric concept.

l

Writing Linear Equations Using Slope – Intercept

Form

• Students write the equation of a line given the slope and y-intercept, the slope and a point, or only two points. l

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ALGEBRA 1


Line

arit

y, S

lop

e, a

nd

Inte

rcep

ts

Interpreting the y-intercept of a line

with tables, graphs, and equations.

• Students interpret the meaning of the y-intercept of a linear function using a data table, symbolic representation, and a graph.

l

Interpreting the x-intercept of a line

with tables, graphs, and equations.

• Students interpret the meaning of the x-intercept of a linear function using a data table, symbolic representation, and a graph.

l

Line

ar R

elat

ions

hip

s an

d

Thei

r G

rap

hs

Find the equations of linear graphs

• The equation of a line is written in y = mx+b form, from the graph of the line.

l

Finding a General Equation of a Line

• Students use points on a line to write a general equation of a line.

l

Parallel and Perpendicular Lines

• The slope of a line that is parallel or perpendicular to a given line is explained.

l

Writing the Equations of Parallel and Perpendicular

Lines

• The equation of a line given a point on the line and the equation of a parallel or perpendicular line is found. l

Plot

s an

d C

hart

s

Tally Charts • Data is collected, and it is constructed and interpreted in a tally chart.

l

Line Plot • Data is collected and interpreted on a line plot. l

Circle Graphs • Circle graphs are constructed for representing and displaying data.

l

Stem and Leaf Plots • Stem-and-leaf plots are constructed and interpreted. l

Box and Whisker Plots • Students use box-and-whisker plots to represent and display relationships among collected data.

l

Linear Model of Data• Students use a spreadsheet to create linear models for data

that are not exactly linear, and then interpret the slope of the trendline.

l

Find the Appropriate Graph

• The appropriate graphical display (bar graph, line graph, and/or pie chart) for a given set of data and contextual situation are determined.

l

Stat

isti

cal

Cal

cula

tion

s

Mean, Median, and Mode • Students apply their understanding of calculating mean, median, and mode to construct data sets under certain restrictions.

l

Calculate Mean, Median, and Mode

• When to use (and calculate) the mean, median, or mode of a given data set is determined.

l

Accuracy and Precision • Students learn the definitions of accuracy and precision through examples.

l

ALGEBRA 1


Poly

nom

ial E

xpre

ssio

ns

Characteristics of Polynomials

• Polynomials are classified by degree and their terms, coefficients, standard form, and sums of coefficients are identified.

l

Identifying Terms and Factors

• Students define term, factor, sum, and product as they relate to algebraic expressions and identify the terms and factors of a given algebraic expression.

l

Factoring Algebraic Expressions

• Second- and third-degree algebraic expressions are factorized using several factoring techniques.

l

Factoring with Difference and Sum Formulas

• Students factor algebraic expressions that are in the form of the difference of two squares, the difference of two cubes, and the sum of two cubes.

l

Qua

dra

tic

Eq

uati

ons

Introducing the Quadratic Equation in One Variable

• Students recognize quadratic equations and rewrite them in standard form, identifying the leading coefficient, linear term, and constant of the equation.

l

Solving Quadratic Equations by Factoring

• Quadratic equations are solved using factoring. l

Solving Quadratic Equations using Quadratic Formula and Discriminant

• The relationship between the number of solutions of a quadratic equation and its discriminant is analyzed and the quadratic formula is used to find the solution.

l

Solving Quadratic Equations by Completing

the Square

• Quadratic equations in one variable are solved by completing the square. l

Roots and Coefficients of a Quadratic Equation

• The relationship between the roots and coefficients of a quadratic equation is identified.

l

Quadratics with Parameters

• Students find the value(s) of a parameter in a quadratic equation based on different given conditions.

l

Qua

dra

tic

Func

tion

s

Introducing the Quadratic Function and Its Graph

• The concept of quadratic functions is explored using real-life examples.

l

Evaluating Functions • Students evaluate functions at numbers or expressions and find the change in a function over an interval.

l

The Range of a Quadratic Function

• The range of a given quadratic function, where the domain is either all real numbers or restricted to a specified interval is determined.

l

Para

bol

as

Finding the Equation of a Parabola

• Students find the equation of a parabola when two or three points are given.

l

A Quadratic Function Given in General and

Vertex Form

• Quadratic functions are converted from vertex form to general form and vice versa. l

Visualizing the Parabola• How changes in the equation of a quadratic function affect the

parabola and the relationship between the discriminant value and the number of x-intercepts are explored.

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ALGEBRA 1


Gra

phi

ng Q

uad

rati

c Fu

ncti

ons

Approximating Zeroes of Quadratic Functions with

Graphs

• Students approximate the zeroes of quadratic functions by visually inspecting their graphs, and by using graphing technology.

l

Graphing a Quadratic Function: Vertex Form

• Students graph a quadratic function given in vertex form by calculating the vertex, axis of symmetry, orientation, and x- and y-intercepts.

l

Graphing a Quadratic Function: Intercept Form

• Students graph quadratic functions in intercept form. l

Graphing a Quadratic Function: General Form

• Students graph a quadratic function in general form. l

How Two Parabolas Intersect

• The intersection points of two parabolas are found and the set of x-values that represent where one parabola lies above the other is determined.

l

GEOMETRY


Poin

ts a

nd L

ines

Definitions of Points, Lines, and Collinear Points

• Students define points, lines, and collinear points. l

The Definition of a Line Segment

• Students define a line segment and learn that two points determine a line segment.

l

Length and Congruency of Line Segments

• Students find the length of line segments using the coordinates of their endpoints.

l

The Definition of Ray and Its Models

• Students define a ray and observe that two points determine a ray. l

Plan

es

The Definition of a Plane• Students define a plane and learn that three non-collinear points,

two non-coincident lines, or a line and a point not on the line determine a plane.

l

Definition of Open and Closed Half-Planes

• Students learn the definitions of open and closed half-planes. l

Points, Lines, Planes and Their Relationships

• Whether given statements about points, lines, and planes related to visual models are correct is determined.

l

Plotting Points in the Cartesian Plane

• Students plot points in the Cartesian plane and draw two-dimensional figures by plotting their vertices.

l

Ang

les

Angle and Types of Angles • The relationship between pairs of complementary, supplementary, adjacent, and congruent angles is described.

l

Degree and Radian • The concepts of degrees and radians and convert between the two angle measures are defined.

l

Definition of Congruent Angles

• Students define congruent angles as angles with equal measure. l

Fund

amen

tal A

xiom

s an

d

Theo

rem

s

The Definition of Space • Students define a three-dimensional Euclidean space. l

The Relationship Among Points, Lines, and Planes

• Students describe lines as collections of points and planes as collections of lines or collections of points.

l

Basic Elements of Geometry

• Whether given statements about points, lines, and planes related to each visual model are correct is determined.

l

Forms of Proofs • Students learn three different proof formats: two-column, flow chart, and paragraph.

l

The Angles Formed by the Lines Intersected by a

Transversal

• Students explore angles formed by parallel and nonparallel lines intersected by a transversal and determine how to use these angles to identify parallel lines.

l

GEOMETRY

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GEOMETRY


Para

llel L

ines

Euclid's Postulates • Students learn about Euclid’s five postulates, which form the foundation for Euclidean geometry.

l

Parallel and Perpendicular Lines

• The slope of a line that is parallel or perpendicular to a given line is explained.

l

Writing the Equations of Parallel and Perpendicular

Lines

• The equation of a line given a point on the line and the equation of a parallel or perpendicular line is found. l

Tria

ngle

s an

d T

heir

Par

ts

Definition of a Triangle • Students define a triangle and its interior and exterior angles and identify the basic primary elements of a triangle.

l

The Secondary Elements of a Triangle

• Students identify the secondary elements (medians, altitudes, and angle bisectors) of a triangle.

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Classifying Triangles by Sides or Angles

• Students classify triangles according to their angles and to its sides.

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Interior and Exterior Angles of a Triangle

• The properties of the interior and exterior angles of triangles are observed.

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The Side Angle Relationship in Triangles

• Students discover the relationship between the side lengths and angle measures of a triangle and apply this relationship.

l

Proof: The Exterior Angle Measures of a Triangle

Sum to 360 Degrees

• Students prove that the sum of the measures of the exterior angles of a triangle is 360 degrees. l

Types of Triangles • Different triangles are analyzed and classified by varying the side lengths and angle measurements.

l

Tria

ngle

Con

gru

ency

and

Si

mila

rity

Congruent Triangles• Students define congruency for triangles and learn the side-

angle-side, angle-side-angle, and side-side-side congruency theorems.

l

Let’s Find the Congruent Triangles

• Congruent triangles are analyzed by matching them in pairs. l

Definition of Triangle Similarity

• Students define the similarity of triangles. l

Let’s Find the Similar Triangles

• Similar triangles are analyzed by matching them in pairs. l

Tria

ngle

Sid

e Le

ngth

s

Triangle Inequality Theorem

• Students discover and prove the triangle inequality theorem, and apply the theorem to determine if it is possible to form a triangle when given the lengths of three line segments.

l

The Reverse Triangle Inequality Theorem

• Students learn about and apply the triangle inequality and reverse triangle inequality theorems to find a range for a side length of a triangle.

l

GEOMETRY


Tria

ngle

Sid

e Le

ngth

s

Proof of the Pythagorean Theorem

• Pythagorean Theorem is prooved by examining the relationship between the side lengths of a right triangle.

l

Using the Pythagorean Theorem to Solve

Problems

• To solve practical problems the Pythagorean theorem is used.

l

Prop

orti

on a

nd S

imila

rity

Euclidean Relationships• Students discover Euclidean relationships, which explore how

the height of a right triangle drawn to its hypotenuse divides the triangle.

l

Basic Proportion Theorem • Students prove the basic proportion theorem and its converse. l

Thales’ Intercept Theorem• Students prove Thales’ intercept theorem, which states that the

line segments created when parallel lines intercept two or more intersecting lines have proportional length.

l

Thales’ Second Intercept Theorem

• Students explain Thales’ second intercept theorem. l

Med

ian,

Alt

itud

e, a

nd B

isec

tor

The Median and Centroid in a Triangle

• Students learn the definitions of median and centroid in a triangle, discover and prove that medians cut each other in the same ratio, and draw the medians of a triangle using that property.

l

The Coordinates of the Centroid in a Triangle

• Students derive a formula for the coordinates of the centroid of a triangle and use the formula to find the coordinates of the centroid when provided with the coordinates of the triangle’s vertices.

l

The Interior Angle Bisector of a Triangle

• Students define the angle bisectors and incenter of a triangle, explore the properties of angle bisectors, and use these properties to solve problems.

l

The Altitude and Orthocenter in a Triangle

• Students learn the definitions of altitude, base of an altitude, foot of an altitude, and orthocenter of a triangle, and calculate the coordinates of the orthocenter when given the equations of the lines containing the sides of the triangle or the altitudes.

l

Comparing the Length of the Altitude, Angle

Bisector, and Median in a Triangle

• Students determine the relationship between the lengths of the altitude, angle bisector, and median drawn from the same vertex of a triangle.

l

Ap

ply

ing

Tra

nsfo

rmat

ions

Transformations • Students explain rotation, reflection, translation, and glide reflections of figures on the plane.

l

Application of Translation • How can a geometric figure be translated is given. l

Drawing the Reflection of a Figure

• The reflection (flip) of a figure is drawn over a given line. l

Symmetry of a Figure • A pattern is completed by forming congruent shapes using various symmetries of a given figure.

l

Introducing Tessellations • Students define tessellations and explore which regular polygons, or combinations of regular polygons, can tessellate the plane.

l

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GEOMETRY


Rep

rese

ntin

g

Ang

les

Angles in Standard Position

• Its determined that two angle measures for an angle, one positive and the other negative.

l

The Reference Angle • The definition of a reference angle and how to use it in determining trigonometric ratios is explained.

l

Co-Terminal Angles • The co-terminal angles and principal angle of a given angle is found.

l

Trig

onom

etri

c R

atio

s an

d C

ircle

s

Trigonometric Ratios in Right Triangles

• Sine, cosine, tangent, and cotangent for an acute angle are calculated.

l

Trigonometric Ratios of Special Angles

• The properties of a right triangle are used to find the sine, cosine, tangent, and cotangent ratios of 30°, 45°, and 60° angles

l

Trigonometric Ratios on the Unit Circle

• The ratios and values of sine, cosine, tangent, and cotangent are calculated and represented based on principles of the unit circle.

l

Arc Length in a Circle• To calculate the length of an arc in a circle, the radius of the

circle, or the measure of a central angle necessary formulas are used.

l

Area of a Sector • The area of a sector in a circle, the radius of a circle, and the measurement of a central angle are calculated.

l

Arcs and Angles of a Circle• The relationship between the arcs and angles of a circle as a

result of changing the locations of points A, B, and C on the circle is observed.

l

Poly

gon

Fun

dam

enta

ls

Definition of a Polygon• Students learn formal and informal definitions of a polygon,

define concave and convex polygons, and define the interior and exterior regions of a polygon.

l

The Number of Diagonals in a Polygon

• Students find the number of diagonals from a vertex of a convex polygon and use this to find the total number of diagonals in a convex polygon.

l

Regions of Polygons • Students define convex and concave regions, polygonal regions, and convex and concave polygonal regions.

l

Interior Angles of Polygons

• The effects on the interior angles of polygons are observed when the number of sides and the positions of the corners of the polygons are changed.

l

Sum of the Exterior Angles of Polygons

• The sum of the exterior angles of regular and non-regular convex polygons by shrinking the polygon is found.

l

Qua

dri

late

rals Definition of a

Quadrilateral• Students define a quadrilateral and its interior and exterior

angles.l

Classification of Quadrilaterals

• Changes in the classification of a given quadrilateral are analyzed and observed when the variables of the side lengths and angle measures are changed.

l

GEOMETRY


Qua

dri

late

rals

Parallelogram and Its Properties

• Students define a parallelogram, explore its properties and their proofs, and use these properties to solve problems.

l

Rhombus and its Properties

• Students define a rhombus, explore its properties and their proofs, and use these properties to solve problems.

l

Trapezoid and Its Properties

• Students define a trapezoid, explore its properties and their proofs, and use these properties to solve problems.

l

Isosceles Trapezoid and Its Properties

• Students define an isosceles trapezoid, explore its properties and their proofs, and use these properties to solve problems.

l

Bas

ic P

erim

eter

and

A

rea

Cal

cula

tion

s

Area of Composite Shapes • The areas of complex polygons by dividing them into triangles and rectangles are explored.

l

The Relationship Between Perimeter and Area

• The smallest or largest possible perimeter given a fixed area or perimeter to solve a variety of problems is determined.

l

Area of a Parallelogram • The area of a parallelogram is found using the area formulas for rectangles and triangles.

l

Area of Trapezoids • The formula for trapezoids' area is found using the area formulas for triangles and parallelograms.

l

Are

a an

d P

erim

eter

of

Poly

gon

s Perimeter and Area of a Square and a Rectangle

• Students derive the formulas for the area and perimeter of squares and rectangles, and practice using these formulas.

l

Perimeter and Area of a Parallelogram

• Students derive the formulas for the area and perimeter of a parallelogram, and practice using these formulas.

l

Perimeter and Area of a Triangle

• Students derive the formulas for the area and perimeter of a triangle, and practice using these formulas.

l

Perimeter and Area of a Rhombus

• Students derive the formulas for the area and perimeter of a rhombus, and practice using these formulas.

l

Perimeter and Area of a Regular Polygon

• Students derive the formulas for the area and perimeter of squares and rectangles, and practice using these formulas.

l

Are

a an

d

Circ

umfe

renc

e

Ratio of a Circle's Circumference to Its

Diameter

• Problems involving the circumference of circles are solved.l

Calculating the Circumference of a Circle

• The circumference of an object is calculated using the diameter or radius.

l

Formula for the Area of a Circle

• The formula for the area of a circle is derived from the formula for the area of a parallelogram.

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GEOMETRY


Net

s an

d C

ross

Se

ctio

ns

Drawing 2D Views of a 3D Object

• Two-dimensional back, front, right, and left views of three-dimensional objects using unit cubes are formed.

l

Completing the Missing 2D View of a 3D Object

• The 2D view of a 3D object formed by unit cubes is drawn when four different 2D views are given.

l

Object Formed by Rotating a Rectangle

• Students observe how the shape of a rectangle changes when there are changes to its width, length, angle, direction, or axis.

l

Volu

me

Observing Changes in the Volume of Square Prisms

• The changes that occur in the volume of a square prism by changing the variables of the square prisms’ area of the base, height, and incline of a right square prism is observed.

l

Problem Solving Involving Volumes of Prisms

• How to apply the mathematical problem solving process to solve problems involving the volume of a prism is explained.

l

Observing Changes in the Volume of Quadrilateral

Pyramids

• The changes that occurs in the volume of a quadrilateral pyramid when the area of the base, height, incline change is observed. l

Observing Changes in Volume of Cylinders

• The changes that occur in the volume of a cylinder when its height, radius, and incline change are observed.

l

Formula for the Volume of a Cone

• The formula is derived for the volume of a cone from the formula for the volume of a pyramid.

l

Formula for the Volume of a Sphere

• The formula for the volume of a sphere is derived from the formula for the volume of a pyramid.

l

Surf

ace

Are

a

Observing Changes in the Surface Area of Regular

Prisms

• The relationship between the surface area and side lengths of a regular prism is observed. l

Observing Changes in the Surface Area of Square

Pyramids

• The changes in surface area of a square pyramid are observed as a result of changing the variables of the pyramid’s height and base side length.

l

Observing Changes in the Surface Area of Cylinders

• The relationship among the height, radius, and surface areas of a cylinder is observed.

l

Observing Changes in Surface Area of Cones

• The changes in the surface area of a cone when the height and the radius are changed are observed.

l

Sets

Union and Intersection of Sets

• The union and intersection of sets and their properties are explored.

l

Subsets of a Set• Subset, proper subset, superset, and power set are defined,

and then how to find the number of subsets in a set under given conditions is explained.

l

Complement of a Set • The complement of a set and its properties are explored. l

GEOMETRY


Sets

Con

td. Difference of Two Sets • How to find the difference of two sets and its properties are

explained.l

Problems Involving Sets • How to use data in a Venn diagram to find a missing value is explained.

l

Cou

ntin

g P

rinc

iple

s

Counting Problems: Number of Parallelograms

• The number of parallelograms is found by determining the number of combinations and applying the counting principle by multiplication.

l

Fundamental Counting Principle

• The number of possible outcomes for a compound event using a tree diagram or the fundamental counting principle is determined.

l

Problems Involving the Number of Subsets of a

Set

• The combination formula is used to find the number of subsets of a set according to the given conditions. l

Factorial Notation • The concept and notation of factorials are identified. l

Counting Principles: Digits • The fundamental counting principle is applied to simple and compound digit problems

l

Perm

utat

ions

and

Com

bin

atio

ns

Permutations with Repetition

• The concept of permutations is explored and developed with repetition.

l

Permutations and Their Properties

• The permutation formula is applied to solve problems. l

Combinations • All possible arrangements of a set of up to four objects are determined using a tree diagram or a systematic list.

l

Combinations and Their Properties

• All possible arrangements of a set of objects are found using a list or formula, for which order is not important.

l

Circular Permutation Problems

• Solving problems involving circular permutation. l

Prob

abili

ty

The Concept of Probability • The concept of probability, its complement, less likely, more likely, impossible and certain events are explained.

l

Overlapping and Mutually Exclusive Events

• The probability of overlapping and mutually exclusive events is identified and founded.

l

Conditional Probability

• Using real-life scenarios involving drunk driving accident rates and cell phone usage, the concept of conditional probability, how to interpret conditional probabilities in real-life contexts, and how to calculate conditional probabilities by using the formula P(A and B)/P(B) is explained.

l

Prob

abili

ty

Cal

cula

tion

s Playing with Probability • Compound independent events to compare probabilities in order to determine fairness in a game are used.

l

Find the Given Probability • Probability experiments of two or more independent events using dice or coins from a given probability are created.

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GEOMETRY


Prob

abili

ty C

alcu

lati

ons

Con

td.

Experimental and Theoretical Probabilities

• It is discovered that the experimental probability of an event approaches the theoretical probability as the number of trials increases.

l

Analyze Experimental Probability Using Graphs

• The results of theoretical and experimental probability are analyzed by conducting an experiment and interpreting a graph of the data.

l

Probability Using a Tree Diagram

• How the probability of simple compound events using a tree diagram can be determined is explained.

l

ALGEBRA 2


Poly

nom

ial E

xpre

ssio

ns a

nd F

acto

ring

Characteristics of Polynomials

• Polynomials are classified by degree and their terms, coefficients, standard form, and sums of coefficients are identified.

l

Operations on Polynomials • Students add, subtract, and multiply polynomials. l

Identifying Terms and Factors

• Students define term, factor, sum, and product as they relate to algebraic expressions and identify the terms and factors of a given algebraic expression.

l

Factoring Algebraic Expressions

• Second- and third-degree algebraic expressions are factorized using several factoring techniques.

l

Factoring Advanced Algebraic Expressions

• Students factor algebraic expressions that are in the form of the difference of two squares, the difference of two cubes, and the sum of two cubes.

l

Poly

nom

ial

Op

erat

ions

Polynomial Long Division • A polynomial is divided by another polynomial using long division. l

Polynomial Synthetic Division

• Synthetic division is used to determine the quotient and remainder from a given polynomial function and linear polynomial divisor.

l

The Remainder Theorem • The remainder theorem is used to find the remainder when dividing a polynomial P(x) by a divisor D(x) = (x[sup]n[/sup] – a).

l

Gra

phs

and

Pol

ynom

ials

A Quadratic Function Given in General and

Vertex Form

• Quadratic functions are converted from vertex form to general form and vice versa. l

Solving Quadratic Inequalities by Graphing

• Students solve a quadratic inequality by graphing the related quadratic function.

l

Visualizing the Parabola• How changes in the equation of a quadratic function affect the

parabola and the relationship between the discriminant value and the number of x-intercepts are explored.

l

Graphing a Quadratic Function: Vertex Form

• Students graph a quadratic function given in vertex form by calculating the vertex, axis of symmetry, orientation, and x- and y-intercepts.

l

Graphing a Quadratic Function: Intercept Form

• Students graph quadratic functions in intercept form. l

Graphing a Quadratic Function: General Form

• Students graph a quadratic function in general form. l

Approximating the Zeroes of Quadratic Functions

with Graphs

• Students approximate the zeroes of quadratic functions by visually inspecting their graphs, and by using graphing technology. l

Solv

ing

Rat

iona

l an

d R

adic

al

Eq

uati

ons

Multiplying and Dividing Rational Expressions

• Rational expressions are multiplied and divided. l

Rational Equations and Extraneous Solutions

• Students learn that algebraic solution methods for rational equations can yield nonexistent solutions.

l

ALGEBRA 2

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ALGEBRA 2


Solv

ing

Rat

iona

l and

Rad

ical

Eq

uati

ons

Eliminating Extraneous Solutions to Rational

Equations

• Students learn that a first step in solving a rational equation is to eliminate extraneous solutions by determining which values cause division by zero.

l

Solving Rational Equations • Students solve rational equations in one variable by converting them to polynomial equations in one variable.

l

Solving Radical Equations• Students solve one variable radical equations of the type a times

the square root of bx+c equals d algebraically, where a, b, c, and d are rational numbers.

l

Choosing between Graphical and Algebraic

Methods

• Students choose between graphical and algebraic solution methods to equations based on whether exact results are necessary.

l

Approximating the Solutions to Rational

Equations with Graphing Technology

• Students approximate the solutions to a rational equation in one variable using graphing technology. l

Line

s in

the

Car

tesi

an P

lane

Determine the Relationship between

Two Lines by Using their Equations

• Students determine if two lines are parallel, coincident, or intersecting by looking at their equations. l

Finding an Equation for Points That are Equidistant

from a Line

• Students write equations for the set of points that are equidistant from a given line. l


from Two Intersecting Lines

• Students find an equation for points that are equidistant from two intersecting lines. l

The Distance between Two Parallel Lines

• Students derive a formula for the distance between two parallel lines.

l


to Two Parallel Lines

• Students derive a formula for the equation of a line that is equidistant from two given parallel lines. l

Finding an Equation for a Line That is Equidistant

from Two Points

• Students derive a formula for the equation of a line that is equidistant from two given points. l

Ang

les

and

the

ir

Mea

sure

s

Angles in Standard Position

• Its determined that two angle measures for an angle, one positive and the other negative.

l

Co-Terminal Angles • The co-terminal angles and principal angle of a given angle is found.

l

The Reference Angle • The definition of a reference angle and how to use it in determining trigonometric ratios is explained.

l

Degree and Radian • The concepts of degrees and radians and convert between the two angle measures are defined.

l

ALGEBRA 2


Trig

onom

etri

c R

atio

s

Trigonometric Ratios in Right Triangles

• Sine, cosine, tangent, and cotangent for an acute angle are calculated.

l

Trigonometric Ratios on the Unit Circle

• The ratios and values of sine, cosine, tangent, and cotangent are calculated and represented based on principles of the unit circle.

l

Arc Length in a Circle • To calculate the length of an arc in a circle, the radius of the circle, or the measure of a central angle necessary formulas are used.

l

Area of a Sector • The area of a sector in a circle, the radius of a circle, and the measurement of a central angle are calculated.

l

Trigonometric Ratios of Special Angles

• The properties of a right triangle are used to find the sine, cosine, tangent, and cotangent ratios of 30°, 45°, and 60° angles

l

Gra

phs

of

Trig

onom

etri

c Fu

ncti

ons

Graphing Sine Functions • Characteristics of the sine function and how to sketch the graph of its transformation is explained.

l

Graphing Cosine Functions • How to graph a cosine function and variations involving cosine expression are explained.

l

Graphing Tangent Functions

• The characteristics of the tangent function and how to graph a function that involves tangent expression is explained.

l

Graphing Cotangent Functions

• Characteristics of the cotangent function and how to graph a function that involves cotangent expression is explained.

l

Prob

lem

Sol

ving

wit

h Tr

igon

omet

ric

Func

tion

s The Angle of Inclination of a Line in the Cartesian

Plane

• Students learn that the slope of a line is the tangent of the angle of inclination of the line, and use the angle of inclination to determine the sign of the slope of the line.

l

The Angle between two Lines, Angles of Inclination

or Slope

• Students find the angle between two lines using their angles of inclination or slopes. l

Finding the Period of a Trigonometric Function

• The period of a function is defined and the formulas are discovered to find the period of the some of the trigonometric functions.

l

Vect

or C

once

pts

Introducing Vectors on the Cartesian Coordinate

Plane

• Students define vectors, scalars, equal vectors, opposite vectors, and zero vectors. l

Vectors and Modeling Situations with Vectors

• Students define and explore vectors, scalars, equal vectors, opposite vectors, and the zero vector.

l

Magnitude of a Vector • Students determine the magnitude of vectors by finding the length of their representative directed line segments.

l

Vect

or O

per

atio

ns Vector Arithmetic with Coordinates

• Students learn how to add and subtract two vectors, and multiply a vector by a scalar when given the coordinate representations of the vectors.

l

Adding Vectors Geometrically and

Algebraically

• Students are introduced to vector addition and, using real-life examples, add two vectors geometrically by using the parallelogram and head-to-tail methods, and algebraically by finding the resultant vector coordinate-wise.

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ALGEBRA 2


Vect

or

Op

erat

ions

C

ontd

.

Subtracting Two Vectors Geometrically and

Algebraically

• Students are introduced to vector subtraction and, using real-life examples, subtract two vectors geometrically and algebraically. l

Magnitude of a Vector and Scalar Multiplication

• Students learn how to calculate the magnitude of a vector and how multiplication by a scalar affects the magnitude of a vector.

l

Inne

r Pr

oduc

t an

d L

inea

r D

epen

den

ce

Euclidean Inner Product • Students learn and apply the Euclidean inner product of vectors and its properties.

l

Magnitude of a Vector with the Euclidean Inner

Product

• Students determine the magnitude of vectors, scalar multiples of vectors and the sums and differences of vectors by using the inner product.

l

Linear Dependence of Vectors

• Students learn about linear independence, linear dependence, and linear combinations of vectors.

l

Standard Basis Vectors and Cartesian Coordinate

Plane

• Students learn that any vector on the coordinate plane can be represented uniquely as a linear combination of the standard basis vectors.

l

Vect

ors

and

Lin

e E

qua

tion

s

Vector Equations of Lines and Their Graphs

• Students represent lines with vector line equations and graph lines using vector line equations.

l

Parallelism and Directional Vectors

• Students observe the relationship between the directional vectors of parallel and intersecting lines, and use the vector equations of lines to determine if they are parallel.

l

General Line Equations with Normal Vectors and

Inner Products

• Students use the Euclidean inner product to write the general equation of a line given a point and its normal vector. l

Translation Between Vector, Parametric And

General Equations

• Students define vector and parametric equations of a line and translate vector equations of a line to parametric equations and parametric equations into general equations.

l

Translation from General, to Parametric, and to

Vector Equations

• Students translate from general to parametric equations of a line, and then from parametric to vector equations of the line. l

Finding The Angle Between Two Lines With

Directional or Normal Vectors

• Students find the angle between two lines using both directional vectors and normal vectors. l

Documents

scope & sequence Linear Equations..... 2 Linear Inequalities ..... 2 Systems of Linear Equations and Inequalities ..... 3 Absolute Value Concepts of Function..... 3 Graphs of Functions