Algebraic Properties and Literal Equations

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Objective

SWBAT solve literal equations by applying Algebraic Properties to geometric formulas.

Big Idea

Students understand that the variables in an equation represent real numbers, therefore Real Number Properties apply.

Warm Up

10 minutes

I intend for today's Warm Up to take about 10 minutes for the students to complete and for me to review with the class.  I allow students about three minutes to work on their own. Their task is to write down all of the algebraic properties they can name, then to define a literal equation. After giving them time to write,  I take responses from the students and write the list of algebraic properties on the board.  We then compare the responses to a sheet from the Virginia Department of Education that lists all of the algebraic properties.  It is found at the following website:

http://www.doe.virginia.gov/instruction/mathematics/resources/va_algebraic_properties.pdf

As I review each property, I provide examples to demonstrate the properties. When we get to the definition of a literal equation, we discuss how this name applies to a formula in which all of the variables represent real numbers. Therefore, I say, "real number properties, such as these algebraic properties, apply to operations on literal equations as well as one-variable equations."

Teacher's Note: While reviewing the properties, I stress how all of the operations can be completed using addition or multiplication.  I focus particular attention on the multiplicative inverse, taking an opportunity to re-teach this when reviewing problems that give my students difficulty.  I demonstrate my approach in the video below.

 

 

 

 

 

 

 

 

 

Whole Class Activity

30 minutes

After reviewing the Warm Up, I hand each table a laminated copy of a PARCC High School Reference Sheet that I have created for this activity.  My students are scheduled to take the PARCC Exam in Spring 2015. Based on the current schedule, the assessment will be held three-fourths through the curriculum, and, at the end of the year.  I  found the Reference Sheet at the following website:

http://www.parcconline.org/sites/parcc/files/ApprovedPARCCHighSchoolReferenceSheet_111612.pdf

The main objective of this activity is for my students to be able to solve one of the geometric formulas on the PARCC reference sheet for an indicated variable.  I also want my students to be able to recognize different forms when solving equations. I have created a set of geometric formulas as individual cut-outs from the reference sheet. I place them in a cup for a random draw.  

To start the activity, I draw a formula out of the cup.  I ask the students to solve the formula for a specific variable (that I choose) on their individual white boards. As a student displays his/her work, I ask him/her to identify when an algebraic property is being used to solve the literal equation for the variable.  I will probably probe for an explanation of why it works in this particular situation. As the class listens to explanations, I ask my students to write down different approaches, to help them recognize that flexibility can make things easier (or harder) based on the choice of method. 

For example, if I draw the formula for the Volume of a Sphere first, I will ask my students to solve for r.  Here are two examples of student work:

When comparing the work of these two students, I will point out that in Example 2 the student simplifies the formula further. I'll say, "Of course, that does not mean that Example 1's work is incorrect."  Then we will discuss why both formulas are correct, and, equivalent.

Exit Slip

10 minutes

I use today's Exit Slip as a formative assessment to check for student understanding of literal equations and how to apply the algebraic properties.  I have used names for variables, rather than letters, to reinforce the theme that the properties apply to real numbers and variables that represent them. I plan to distribute the Exit Slip with about 10 minutes remaining in class.  

The work on the Exit Slip should show the students ability to solve for Mass in the Density formula, and use that formula to solve an application problem.  When reviewing the Exit Slip with students, I will discuss that requested calculations can be found by solving for Mass, first, and then making substitutions. However, I will also discuss how substitutions could be made first, and then solved for the Mass. If time allows, we may discuss when it makes sense to use one or the other approach.