Solving Literal Equations and Formulas
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GOAL: I can solve an equation for any given variable in the equation.
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| Text reference: Chapter 2, section 5, pages: 109 ─ 114 |
Illinois learning standard: 8.C.3 |
NCTM standard: XX─XX |
| Lesson Vocabulary |
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Literal Equation
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A literal equation is an equation that has two or more variables in it.
Some examples include:
y = 3x + 2
3x ─ 2y = 10
2x = 7x ─ 16
2P + XP = 15
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| Formula |
Formulas are special types of literal equations.
A formula is an equation that gives the relationship between the variables.
Some examples include:
P = 2•L + 2•W
A = L • W
C = 2 π r
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Strategies and Content Practice |
Notes |
Begin
(prepare) |
Literal Equations and Formulas (Chapter 2, Section 5) contains way too much information to be the subject of just one section. For students to successfully solve the problems in this section, they need to be able to accomplish some pretty challenging and intricate calculations. I've been teaching this section for nearly a decade and have come to the conclusion that this section can (and should) be divided into four distinct learning objectives (lessons). They include...
1) "
Variables in the Dungeon"
2)
Fractional Coefficients
3)
Exponents
4)
Factoring (basic, GCF - greatest common factor)
I have made each of these topics into its own lesson. I devote enough time to each of these lessons to build students' confidence in how to solve these types of problems. A brief explanation of each of these areas follows. The full lessons are linked here and at the left.
Variables in the Dungeon
Until now, students have never been asked to solve an algebra problem where the variable they were trying to solve for (isolate) was in the denominator of a fraction. In pre-algebra and algebra (up to this point) the variable students seek to solve for is either in the numerator of a fraction or there is no fraction at all. Thus students have no prior knowledge solving for variables in the denominator of a fraction, what I call "variable in the dungeon". I call them this because the variable is trapped below and need to be "rescued" from the dungeon (brought up to the numerator) before they can be isolated and solved for.
For example, take this very simple equation from the 8th grade science, "Motion and Forces" book:
;)
In this equation, the Rate is already solved for; the Distance can be solved for by multiply both sides of the equation by Time; but the Time is stuck in the denominator of the fraction. Even if the Distance could be moved to the other side of the equation, the right side of the equation would still NOT be Time... it would be 1 over Time or
which is not the same as Time.
A Super-Hero algebra tool needs to be introduced and used to rescue Time from the denominator so the equation can be solved for Time. This is the topic of lesson 2-5 Part A.
Fractional Coefficients
Fractional Coefficients are fractions that precede variables. The following are some examples:
In the first example of solving for X, the 3/4 needs to be eliminated to get X all alone.
In the second example of solving for Y, the 5/3 needs to be eliminated to get Y all alone.
In the third example of solving for A, the 3/7 needs to be eliminated to get A all alone.
This can be accomplished with a two-step process of multiplying and dividing but there's a more elegant way.
Here again, a Super-Hero algebra tool needs to be introduced and used to get the variables X, Y, and A all alone on one side of the equal sign. This is the topic of lesson 2-5 Part B.
Exponents
Exponent variables present their own set of challenges when solving algebra problems. In this section the focus is on solving for variables that are raised to the second power (squared variables). The concepts taught in this lesson for squared variables apply to higher power exponents like third power variables (cubed variables), forth power variables, etc. The following are some examples of exponent variable problems:
In the first example, the X is all alone by itself on the left side of the equal sign, so all that needs to be done to solve for X is to "unsquare", that is, to take the square root of both sides of the equation.
In the second example, Y has a coefficient in front of it. Before the square root can be taken of both sides of the equation, the coefficient need to be removed.
In the third example, Z is being divided by a coefficient. Before the square root can be taken of both sides of the equation, the coefficient need to be removed.
Yet again, a Super-Hero algebra tools are used to get the variables X, Y, and Z all alone on one side of the equal sign. This is the topic of lesson 2-5 Part C.
Factoring (basic, GCF - greatest common factor)
Nearly half of algebra 1 is devoted to teaching various factoring methods. In this section, students are introduced to the first, most basic form of factoring. This method involved finding the greatest common factor for a binomial or trinomial expression. The following are some examples of basic factoring problems:
The first example gives an equation where P is the variable that is being solved for. The P has to be isolated on one side of the equal sign in order to solve the problem.
The second example gives an expression where the student needs to figure out the greatest common factor that can be taken from each monomial of the polynomial.
In this last part, the Super-Hero algebra tool is the greatest common factor technique. It can be used to get the variable P all alone by itself. It can also be used to reduce the polynomial containing X to its simplest terms. These techniques will be discussed and demonstrated in lesson 2-5 Part D.
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