Academic Systems Algebra Scope and Sequence
Academic Systems Intermediate Algebra
Lesson | Concept | Objectives | Hours |
Topic EII: Essentials of Algebra | |||
EII.A Real Numbers and Exponents (reviews 1.1, 1.3, 6.1) |
Real Numbers and Notation | a) Number line and notation: sets,
ordering symbols, grouping symbols, exponents, and absolute value b) Operations on signed numbers c) Properties of real numbers: commutative law, associative law, distributive law, additive and multiplicative identities, inverses |
1 hour |
Integer Exponents | a) Nonnegative integer exponents b) Properties of exponents |
1 hour | |
Lesson Total | 2 hours | ||
EII.B Polynomials (reviews 2.1, 6.2, 7.1,7.2) |
Polynomial Operations | a) Algebra building blocks:
constants, variables, terms and coefficients b) Evaluating polynomials c) Adding and subtracting polynomials d) Multiplying and dividing polynomials |
1 hour |
Factoring Polynomials |
a) Factoring out the greatest common
factor b) Factoring by grouping c) Factoring trinomials d) Factoring a difference of 2 squares e) Factoring sums and differences of 2 cubes f) Factoring using a combination of methods |
1 hour | |
Lesson Total | 2 hours | ||
EII.C Equations and Inequalities (reviews 2.2, 2.4) |
Linear | a) Solving linear equations b) Solving linear inequalities |
2 hours |
Lesson Total | 2 hours | ||
EII.D Rational
Expressions (reviews 8.1, 8.2) |
Rational Expressions | a) Negative integer exponents b) Writing rational expressions in lowest terms c) Multiplying and dividing rational expressions d) Adding and subtracting rational expressions |
1 hour |
Rational Equations | a) Solving equations that contain rational expressions | 1 hour | |
Lesson Total | 2 hours | ||
EII.E Graphing Lines (reviews 3.1, 4.1, 4.2) |
Graphing Lines | a) The coordinate system b) Graphing lines |
1 hour |
Finding Equations | a) Slope and intercepts b) Finding the equation of a line |
1 hour | |
Lesson Total | 2 hours | ||
EII.F Absolute Value (new material) |
Solving Equations | a) Solving |x| = a b) Solving |Ax + B| = a c) Solving |Ax + B| = |Cx + D| |
2 hours |
Solving Inequalities | a) Solving absolute value inequalities | 1 hour | |
Lesson Total | 3 hours | ||
Topic 9: Rational Exponents and Radicals |
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9.1 Roots and Radicals | Square Roots and Cube Roots |
a) Definition of square root and
cube root b) Radicand, radical c) Principal square root d) Multiplication and division properties e) Simplifying a square root or a cube root of a whole number f) Simplifying a square root or a cube root of simple monomial expression |
1 hour |
Radical Expressions | a) Simplifying radical expressions b) Like radical terms c) Simplifying a sum or difference of radical expressions d) Multiplying radical expressions e) Conjugates f) Rationalizing the denominator g) Solving radical equations |
2 hours | |
Lesson Total | 3 hours | ||
9.2 Rational Exponents | Roots and Exponents | a) The nth root of a
number b) Definition of a^{1/n} and a^{m/n} c) Properties of rational exponents |
1 hour |
Simplifying Radicals | a) Simplifying radicals | 1 hour | |
Operations on Radicals |
a) Multiplying radical
expressions b) Dividing radical expressions c) Adding and subtracting radical expressions |
2 hours | |
Lesson Total | 4 hour | ||
Topic 10: Quadratic Equations |
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10.1 Quadratic Equations I | Solving by Factoring | a) The standard form of
a quadratic equation b) Putting a quadratic equation into standard form ax^2 + bx + c = 0; a ≠ 0 c) Solving quadratic equations of the form ax^2 + bx = 0 by factoring d) Solving quadratic equations of the form ax^2 + bx + c = 0 by factoring |
1 hour |
Solving by Square Roots |
a) Finding square roots Roots b) Solving quadratic equations of the form ax^2 = b c) Solving quadratic equations of the form (ax + b)^2 = c |
1 hour | |
Lesson Total | 2 hour | ||
10.2 Quadratic Equations II | Completing the Square | a) Solving quadratic
equations of the form x^2 + bx + c = 0 by completing the square b) Solving quadratic equations of the form ax^2 + bx + c = 0, a ≠ 1, by completing the square |
1 hour |
The
Quadratic Formula |
a)
Introducing the quadratic formula b) Using the quadratic formula to solve quadratic equations of the form ax^2 + bx + c = 0 c) Using the discriminant of a quadratic equation to determine the nature of the solutions of the equation |
1 hour | |
Explore | a) Solving Equations | 1 hour | |
Lesson Total | 3 hour | ||
10.3 Complex Numbers | Complex Number System |
a) Definition of complex
numbers b) Powers of i c) Operations on complex numbers |
2 hours |
Lesson Total | 2 hour | ||
Topic 11: Functions and Graphing |
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11.1 Functions | Functions and Graphs | a) Definition of a
function b) Function as an ordered pair of numbers c) Finding function values given a formula d) Function notation: y = f(x) e) Graphing simple functions f) Domain and range of a function g) The vertical line test |
1 hour |
Linear Functions | a) Graphs of linear
functions b) Graphs of absolute value functions |
1 hour | |
Quadratic Functions | a) Graphs of quadratic
functions b) Intercepts of quadratic functions |
1 hour | |
Explore | a) Functions and Graphs | 1 hour | |
Lesson Total | 4 hour | ||
11.2 The
Algebra of Functions |
The Algebra of Functions |
a) The sum and
difference of functions b) The product and quotient of functions c) The composition of functions |
2 hours |
Inverse Functions | a) Finding
the inverse and the equation of the inverse b) Defining one-to-one and checking whether a function has an inverse that is a function c) Graphing inverse functions |
2 hours | |
Lesson Total | 4 hour | ||
Topic 12: The Exponential and Logarithmic Functions |
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12.1 Exponential Functions | The Exponential Function | a) Recognizing and graphing an
exponential function b) Applications of the exponential function c) The algebra of exponential functions |
2 hours |
Lesson Total | 2 hour | ||
12.2 Logs and Their Properties |
The Logarithmic Function | a) Converting from exponents to
logarithms & logarithms to exponents b) Recognizing and graphing a logarithmic function |
1 hour |
Lesson Total | 1 hour | ||
12.3 Applications of Logs | Natural and Common Logs |
a) Base e and natural logarithms Logs b) Finding logs in base 10 and finding powers of 10, using a calculator c) Finding logs in base e and powers of e, using a calculator d) Change of base formula |
1 hour |
Solving Equations | a) Solving exponential equations b) Solving logarithmic equations |
1 hour | |
Lesson Total | 2 hour | ||
Topic 13: More Nonlinear Equations and Inequalities | |||
13.1 Nonlinear Equations | Solving Equations | a) Solving polynomial equations by
factoring b) Solving quadratic-type equations by factoring or by substitution |
1 hour |
Radical Equations | a) Solving
b) Solving a c) Solving d) Solving equations that contain rational exponents |
1 hour | |
Explore | a) Graphing | ||
Lesson Total | 3 hour | ||
13.2 Nonlinear Systems | Solving Systems | a) The solution of a nonlinear
system b) Solving nonlinear systems by graphing c) Solving nonlinear systems by the addition method d) Solving nonlinear systems by the substitution method |
1 hour |
Explore | a) Graphing Systems | 1 hour | |
Lesson Total | 2 hour | ||
13.3 Inequalities | Quadratic Inequalities | a) Solving quadratic inequalities | 2 hours |
Rational Inequalities | a) Solving rational inequalities | 1 hour | |
Explore | a) Graphing Inequalities | 1 hour | |
Lesson Total | 4 hour | ||
Topic 3: Introduction to Graphing |
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3.1
Introduction to Graphing |
Plotting Points | a) The xy-plane b) The x-axis and y-axis c) The origin d) Ordered pairs e) The x-coordinate (abscissa), the y-coordinate (ordinate) f) Plotting ordered pairs of numbers g) Labeling the four quadrants h) Determining the quadrant in which a point lies i) The signs of the coordinates in each quadrant |
1 hour |
Rise and Run | a) Subscript notation b) Geometric interpretation of rise and run c) Algebraic definition of rise and run |
1 hour | |
The Distance Formula | a) Pythagorean Theorem b) The distance formula c) The equation of a circle |
0.5 hours | |
Explore | a) Plotting Points | 0.5 hours | |
Lesson Total | 3 hour | ||
Topic 4: Graphing Linear Equations and Inequalities |
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4.1 Graphing Equations | Graphing Lines I | a) Definition of a
linear equation in two variables b) Recognizing linear equations in two variables c) Solutions of linear equations d) Graphing a linear equation by plotting ordered pairs |
2 hours |
Graphing Lines II | a) Equations and graphs
of horizontal and vertical lines b) The intercepts of a line c) Graphing a linear equation by finding the intercepts |
1 hour | |
Slope of a Line | a) Definition of the
slope of a line b) Positive slope, negative slope, zero slope, undefined slope c) Graphing a line given a point and the slope d) Parallel and perpendicular lines |
2 hours | |
Lesson Total | 5 hour | ||
4.2 The Equation of a Line | Finding the Equation I | a) Finding the equation
of a line given a point on the line and the slope of the line b) The point-slope form of the equation of a line c) Finding the equation of a line given two points on the line |
1 hour |
Finding the Equation II | a) The slope-intercept
form of the equation of a line b) Finding the equation of a horizontal line c) Finding the equation of a vertical line d) Finding the equation of a line parallel or perpendicular to a given line |
2 hours | |
Lesson Total | 3 hour | ||
4.3 Graphing Inequalities | Linear Inequalities | a) Ordered
pairs as solutions of linear inequalities b) Graphing inequalities |
2 hours |
Lesson Total | 2 hour | ||
Topic 5: Solving Linear Systems |
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5.1 Solving
Linear Systems |
Solution by Graphing | a) The solution of a
linear system b) Graphing linear systems c) Systems with a unique solution d) Systems with no solutions e) Systems with an infinite number of solutions |
1 hour |
Solution by Algebra | a) Solving linear
systems by the substitution method: one solution, no solution, and an infinite number of solutions b) Solving linear systems by the elimination method: one solution, no solution, and an infinite number of solutions |
1.5 hours | |
Explore | a) Exploring Graphing Linear Systems | 0.5 hours | |
Lesson Total | 3 hour | ||
5.2 Problem Solving | Using Linear Systems | a) Number problems b) Interest problems c) Coin problems d) Mixture problems |
2 hours |
Lesson Total | 2 hour | ||
5.3 Systems of Inequalities | Solving Linear Systems | a) Solving systems of linear inequalities by graphing | 1 hour |
Explore | a) Exploring Solving Linear Systems | 1 hour | |
Lesson Total | 2 hour | ||
Topic 6: Exponents and Polynomials |
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6.1 Exponents | Properties of Exponents |
a) Definition of
exponent, power, and base b) Multiplication property c) Division property d) Powers raised to powers e) Products raised to powers f) Quotients raised to powers g) The zero exponent |
2 hours |
6.2
Polynomial Operations I |
Adding and Subtracting |
a) Definition of
polynomial, term, and coefficient b) Evaluating a polynomial c) The degree of a term and a polynomial d) Writing the terms of a polynomial in descending order e) Definition of a monomial, binomial, and trinomial f) Recognizing like or similar terms g) Combining like or similar terms h) Polynomial addition i) Polynomial subtraction |
2 hours |
Multiplying and Dividing |
a) Multiplying a
monomial by a monomial b) Multiplying a polynomial by a monomial c) Dividing a monomial by a monomial d) Dividing a polynomial by a monomial |
2 hours | |
Lesson Total | 4 hour | ||
6.3
Polynomial Operations II |
Multiplying Binomials | a) Multiplying binomials
by the "FOIL" method b) Perfect squares, product of the sum and difference of two terms |
1.5 hours |
Multiplying and Dividing |
a) Multiplying a
polynomial by a polynomial b) Dividing a polynomial by a polynomial |
1.5 hours | |
Explore | a) Multiplying Binomials | 1 hour | |
Lesson Total | 4 hour | ||
Topic 7: Factoring |
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7.1
Factoring Polynomials I |
Greatest Common Factor | a) Finding the greatest
common factor (GCF) of a set of monomials b) Factoring a polynomial by finding the GCF when the GCF is a monomial |
2 hours |
Grouping | a) Factoring a
polynomial by finding the GCF when the GCF is a binomial b) Factoring a polynomial with four terms by grouping |
2 hours | |
Lesson Total | 4 hour | ||
7.2
Factoring Polynomials II |
Trinomials I | a) Factoring trinomials
of the form x^2 + bx+ c; x^2 + bxy+ cy^2; |
1 hour |
Trinomials II | a) Factoring trinomials
of the form ax^2 + bx + c, a ≠ 1, by trialand-error b) Factoring trinomials of the form ax^2 + bx + c, a ≠ 1, by grouping c) Solving quadratic equations by factoring |
2 hours | |
Explore | a) Exploring Factoring | 1 hour | |
Lesson Total | 4 hour | ||
7.3 Factoring by Patterns | Recognizing Patterns | a) Factoring a perfect
square trinomial b) Factoring a difference of two squares c) Factoring a sum and difference of two cubes d) Factoring using a combination of methods |
2 hours |
Lesson Total | 2 hour | ||
Topic 8: Rational Expressions |
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8.1 Rational Expressions I | Multiplying and Dividing | a) Determining when a
rational expression is undefined b) Writing a rational expression in lowest terms c) Multiplying rational expressions d) Dividing rational expressions e) Simplifying a complex fraction |
1 hour |
Adding and Subtracting | a) Adding rational
expressions with the same denominator b) Subtracting rational expressions with the same denominator |
1 hour | |
Lesson Total | 2 hour | ||
8.2 Rational Expressions II |
Negative Exponents | a) Notation b) Scientific notation |
1 hour |
Multiplying and Dividing | a) Reducing a rational
expression of the form (a -b)/(b-a) b) Multiplying rational expressions c) Dividing rational expressions d) Simplifying a complex fraction |
1 hour | |
Adding and Subtracting | a) Finding the least
common denominator of two or more rational expressions b) Adding rational expressions with different denominators c) Subtracting rational expressions with different denominators d) Simplifying a complex fraction |
1 hour | |
Lesson Total | 3 hour | ||
8.3 Equations with Fractions |
Solving Equations | a) Solving equations
with rational expressions b) Solving for an unknown in a formula involving a rational expression |
2 hour |
Lesson Total | 2 hour | ||
8.4 Problem Solving | Rational Expressions | a) Ratio and proportion b) Distance problems c) Work problems d) Variation |
2 hours |
Lesson Total | 2 hour | ||
Topic F3: Proportional Reasoning II |
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F3.1 Ratio and Proportion | Ratios | a) Notation b) Equivalent ratios c) Rates d) Applications |
2 hours |
Proportions | a) Setting up a
proportion b) Solving a proportion c) Similar triangles d) Applications |
3 hours | |
Explore | a) Inverting Ratios b) Similar Figures |
1 hour | |
Lesson Total | 4-6 hours | ||
F3.2 Percent | Definition | a) Percent as a fraction b) Percent as a decimal c) “Benchmark” percents d) Applications |
2 hours |
Converting | a) Converting among
fractions, decimals and percents b) Percent increase and decrease c) Applications |
3 hours | |
Solving Percent Problems |
a) Setting up and
solving proportions b) Setting up and solving other equations c) Applications |
3 hours | |
Explore | a) A Survey b) Computing Tips c) A Percent Question |
1 hour | |
Lesson Total | 6-8 hours | ||
Topic F5: Geometry |
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F5.1 Geometry I | Geometric Figures | a) Point, line, line
segment, ray b) Polygons c) Measuring angles d) Angles: acute, right, obtuse, straight e) Relationships between angles: complementary, supplementary, adjacent, vertical f) Applications |
3 hours |
Explore | a) Tiling with Polygons
b) What’s the Sum? |
1 hour | |
Lesson Total | 4hours | ||
F5.2 Geometry II | Perimeter and Area | a) Perimeter of a
polygon b) Area of a polygon c) Area of a rectangle, square, parallelogram, triangle, trapezoid d) Circumference of a circle e) Area of a circle f) Perimeter and area of composite figures g) Applications |
2 hours |
Surface Area and Volume |
a) Surface area of a
solid b) Volume of a solid c) Surface area and volume of a rectangular prism d) Surface area and volume of a cylinder e) Volume of a cone f) Volume of a sphere g) Composite figures h) Applications |
3 hours | |
Explore | a) Don’t Fence Me In b) Why π? c) Packaging Products |
1 hour | |
Lesson Total | 4-6 hours | ||
F5.3 Geometry III | Triangles and Parallelograms |
a) Angle sum of a
triangle b) Congruent triangles c) Isosceles and equilateral triangles d) Right triangles e) The Pythagorean Theorem f) Parallel lines and transversals g) Properties of parallelograms h) Applications |
2 hours |
Similar Polygons | a) Similar polygons b) Similar triangles c) Applications |
3 hours | |
Explore | a) Congruent Triangles b) Door to Door |
1 hour | |
Lesson Total | 4-6 hours | ||
Topic F6: Interpreting Data |
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F6.1 Units of Measurement | US/English Units | a) Measures:
length, weight, time, volume b) Converting from one unit to another c) Adding and subtracting measurements d) Applications |
2 hours |
The Metric System | a) The metric system:
length, mass, volume b) Fahrenheit and Celsius scales c) Converting between US and metric units d) Applications |
3 hours | |
Explore | a) Sizing it Up b) Choosing Units |
1 hour | |
Lesson Total | 4-6 hours | ||
F6.2 Interpreting Graphs | Data and Graphs | a) Reading data from
graphs b) Graphing data c) Pictographs d) Bar graphs e) Circle graphs f) Line graphs g) Applications |
2 hours |
Explore | a) Collecting and
Picturing Data b) Graphs in the News |
1 hour | |
Lesson Total | 2-3 hours | ||
F6.3
Introduction to Statistics |
Statistical Measures | a) Mean b) Median c) Mode d) Box-and-whisker plots e) Applications |
2 hours |
Explore | a) Collecting Data b) Where’s the Middle? |
1 hour | |
Lesson Total | 2-3 hours |