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1-Algebra Fundamentals

Important Concepts for This Unit

We will review the real numbers, equations, and the coordinate plane. you are probably already familiar with these concepts but it is helpful to get a fresh look at how these ideas work together to solve problems and model real-world situations. The standards that will be assessed in this chapter are

1.2 Can simplify expressions using properties of exponents
1.3 Can simplify expressions involving factoring and expanding
1.4 Can combine and simplify rational expressions
1.5 Can solve equations involving exponents, algebraic expressions, and rational expressions.


Topic, Focus, and Exercises

Khan Academy Videos

Khan Academy Homework (KHW)

 

u1.1 - Real Numbers

 

·   The various subsets of the real number line.

·   The algebraic properties of real numbers.

·   Closed and open intervals of real numbers, and their unions and intersections.

·   Absolute value and distance.

 

Core Exercises: 1, 28, 31, 41, 44, 55, 64, 70, 76

 

Recommended: 1, 13, 18, 22, 28, 31, 32, 33, 40, 41, 44, 45, 48, 55, 59, 64, 70, 76, 87

 

Algebraic expressions with fractions.

Absolute value equations.


 KHW: Practice with nested fractions.

 
KHW: Absolute Value Equations 

u1.2 - Exponent and Radicals

  • The definition of an in the case where n is rational, and where negative.
  • Algebraic properties of exponents.
  • Rationalizing denominators.
  • Scientific notation and significant figures. 

Core Exercises: 3, 44, 67, 70, 76, 86, 88, 100

 

Recommended: 1, 3, 6, 15, 40, 44, 45, 48, 62, 67, 70, 74, 76, 83, 86, 88, 89, 90, 95, 100

Properties of exponents

Exponent properties examples with variables

Fractional exponents

Simplifying radical expressions

Solving equations with radicals

 
 KHW: Practice simplifying expressions with exponents

  KHW: Manipulating fractional exponents

  KHW: Fractional exponent practice

 KHW: Simplifying square roots

 KHW: Adding and subtracting radicals


 KHW: Solve equations containing square roots of variables

 KHW: Extraneous solutions to radical equations

 u1.3 - Algebraic Expressions

  • Definition of and algebraic operations with polynomials.
  • Special product formulas.
  • Factoring expressions by finding common factors and recognition of special cases.
  • Factoring quadratics by trial and error.

Core Exercises: 2, 18, 27, 42, 51, 60, 72, 94, 136

Recommended: 2, 4, 13, 18, 20, 24, 27, 29, 33, 42, 46, 51, 55, 60, 68, 72, 78, 85, 94, 124, 136

Adding and multiplying polynomials

Factoring quadratics

 

 

 
 KHW: Adding and subtracting polynomials

 KHW: Multiplying binomials by polynomials

 KHW: Factoring quadratics with leading coefficient of 1

 KHW: Factoring polynomials using quadratic methods

 KHW: Factoring difference of squares
 
 KHW: Factoring polynomials with special product forms

u1.4  Rational Expressions

  • Finding the domain of an algebraic expression.
  • Simplifying, adding, and subtracting rational expressions, including compound fractions.
  • Rationalizing numerators and denominators.

Core Exercises: 4, 6, 15, 24, 48, 65, 83, 98

Recommended: 1, 4, 6, 10, 12, 15, 18, 24, 25, 31, 39, 48, 54, 65, 83, 86, 98, 99, 25

Rational Expressions

Domain and range

Simplifying rational expressions

Adding and subtracting rational expressions

Multiplying and dividing rational expressions

Solving rational equations

 
 KHW: Domain of a function
 
 KHW: Dividing polynomials by binomials

 KHW: Simplifying rational expressions 

 KHW: Adding and subtracting rational expressions

 KHW: Multiplying and dividing rational expressions

 KHW: Solving rational equations 

 u1.5 - Equations

  • Solving equations using the techniques of adding constants to both sides of the equation, multiplying both sides of the equation by a constant, and raising both sides of the equation to the same nonzero power.
  • Avoiding the pitfalls of accidentally multiplying or dividing by zero, or introducing extraneous solutions.
  • Solving quadratic equations using the techniques of factoring, completing the square, and the quadratic formula, and the use of the discriminant to determine the nature of the solutions to a quadratic equation.
  • Solving generalized quadratic equations.
  • Solving equations involving fractional expressions.

Core Exercises: 1, 13, 32, 45, 58, 70, 80, 90, 123, 129

Recommended: 1, 5, 10, 12, 13, 20, 30, 32, 41, 45, 51, 58, 64, 70, 73, 80, 81, 87, 90, 123, 129, 139

 

 

u1.7 - Modeling with equations

  • Solving applied problems described verbally (word problems)
  • Presenting the solution process in a clear, organized way.

Core Exercises: 3, 7, 20, 37, 42, 46, 56, 70, 85

Recommended: 3, 6, 7, 15, 18, 20, 23, 29, 37, 42, 46, 48, 51, 56, 61, 66, 70, 77, 82, 85 

 

 

  

 u1.9 - The Coordinate Plane 

  • The distance and midpoint formulas.     
  • The relationship between a two-variable equation and its graph. (This is the most crucial concept.)
  • Sketching graphs by plotting points, using intercepts and symmetry.
  • Equations of circles and their graphs, including the technique of completing the square.

           
 Core Exercises: 2, 10, 25, 35, 50, 60, 71, 92, 98, 115

 Recommended : 2, 6, 8, 10, 14, 17, 25, 35, 38, 45, 50, 56, 60, 65, 71, 76, 87, 92, 98, 106, 115, 127

 

 

 
  u1.12 - Modeling Variation

  • Direct variation and proportionality
  • inverse variation
  • Combinations of direct and inverse variation

  Core Exercises: 3, 7, 34, 39, 47, 49, 74, 79, 96 
 
  Recommended:
 3, 4, 7, 13, 18, 24, 31, 34, 39, 46, 47, 49, 50, 62, 71, 74, 79, 86, 92, 96


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