4.6 Algebraic Fractions

Lesson Plan 4.6: Algebraic Fractions

Objectives:

• Understand how to simplify algebraic fractions by factoring and employing algebraic manipulations.

Suggested Lecture Breaks:

• MWF: You have two days. Split the material where it feels natural to you.
• MW/TR: You have one day.

Suggested Lecture Notes:

• Begin the class by asking students what an ``algebraic fraction" is, which should be familiar from their RG. Then put up the following definition on the board:
  • An algebraic fraction is the quotient of two polynomials, where the denominator is not zero.
• Work through an example of simplifying an algebraic fraction. One example might be:
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{x^5(x^2-4)}{x^2(x+2)(x+3)}}

• Point out that whenever we cancel common factors we must place assumptions on the value(s) of the variable(s). To this end, define restriction:
  • A restriction is a number for which an expression is undefined.

• Work through some examples of adding/subtracting algebraic fractions. Be sure at least one of these examples requires simplifying an algebraic fraction that involves factoring a quadratic in the denominator. Here are some suggested examples:
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{1}{5} + \dfrac{1}{y}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{2}{3x^2} - \dfrac{3}{7x}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{x+1}{3x+7} + \dfrac{5x-2}{4}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{x^2-2x+3}{x^2+7x+12} - \dfrac{x^2-4x-5}{x^2+7x+12}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{4x}{x^2+x-12} - \dfrac{3}{x^2-9}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{6}{x-2} + \dfrac{x+3}{2-x}}
• Next, work through a couple examples of simplifying algebraic expressions involving multiplication and/or division. Again, emphasize that we must place assumptions on the value(s) of the variable(s) when we cancel common factors. Often we will simply place a statement in the instructions such as, ``Assume any factors you cancel are not zero" to address this issue. Here are some suggested examples:
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{5p}{6q^2}\cdot\dfrac{3pq}{5p}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{3y^4}{4z}\cdot\dfrac{8y^3z}{6y^5}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{x+3}{x+4}\div\dfrac{4x+12}{2x+8}}

Notes

• As you work through the examples, explain why we must find a common denominator and how we use factoring to help us in this process.
• Reinforce that using the least common denominator will result in less of a need to simplify in the end, but any common denominator will do.
• Many times students want to multiply out the common denominator; please discourage this by pointing out that it just causes more work later.
• Be sure you relate this topic back to the previous lessons on factoring quadratic expressions and the lessons from the first week involving equivalent fractions and algebraic manipulation of fractions.

Comments on the handout:

• Questions 1 & 2: These questions should be routine for students. If you find that you are running out of time, you can ask students to do questions 1 and 2 at home. Otherwise, you should allow students to work in class on these first two question a maximum of 10 minutes.
• Because this is the last section that will be tested on the final, you might want to offer detailed solutions to students or plan on spending some time on it at the beginning of the next class period, if time allows.