Algebraic Relations: Solving Equations with Variables on Both Sides

Contents

Theme
Common Core State Standards
Introduction
Symbolic Representation
Interviews with students
Mathematical Issues
Teaching Strategies
References
Your Experience

Theme

Solutions

Common Core State Standards

6.EE.5, 6.EE.6, 6.EE.7, 7.EE.4a, 8.EE.7, A-CED.1

Introduction

There are many errors students can make when solving equations, including simplification errors, sign errors, "unbalanced" errors (adding different quantities to each side of the equation), and arithmetic errors. A solid foundation of arithmetic can help students understand the rationale behind the symbolic manipulations that go into solving an equation.

Symbolic Representation

Problem Response
2x + 3 = 6x -5

-4x + 3 = -5

-4x = -8

x = -2

Common Student Error: dividing a negative by negative gave a negative answer (shown).

Other possible errors are dividing "backwards": x = 1/2.

Correct answer: x = 2

Solve:

6x-3=2x-19

6x-3=2x-19

6x+2x=19+3

Common Student error: Not recognizing the sign of the variable (shown).
5x -3x +1 = 4x - 3

+3x +3x +3x

8x +1 = 7x - 3

Common Student Error: Students may start to eliminate variables before they combine like terms and might not attend to the equal sign. Students can avoid this error by combining like terms prior to solution steps.

Interviews with students

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Mathematical Issues

There are two standard teaching approaches for solving linear equations.

(1) "Repeatedly doing the same thing to both sides of the equation" until you arrive at the answer.

(2) The "change sides, change signs" approach.

The main idea in both methods is to transform the original equation into a simpler one.

Students’ selection of a solution strategy depends extensively on the strategies employed in lessons by their teachers.

Teaching Strategies

Students with an "operational conception" have difficulty solving equations, so more time needs to be spent on a "relational conception" that helps students to consider how parts of an equation fit together. See the Teaching Strategies section of Algebraic Relations: One and two step equations (x on one side only) for a description of relational versus operational thinking.

● Asking the students to justify their steps forces them to do only steps that they can explain.

● Students should be expected to check the validity of their solutions both for verification and for growth in their confidence.

● Students should explicitly use the reflexive, symmetric and transitive properties of equality to improve their understanding of the use of equal sign.

○ For example, instead of going from x+6= 3x+1 to 2x+1=6, teachers could show the in-between steps by indicating the properties being used:

   x+6=3x+1 subtracting x from both sides, we get 
     6=2x+1 and by the symmetric property of equality, we arrive at 
     2x+1=6. 

○ As another example, if 2x+1=y, then when y=0, 2x+1=0 by the transitivity property.

● The above situation brings up the concern that equation solving is taught as procedures rather than using algebraic properties as indicated above: Results only:

          x + 6 = 3x + 1					           x + 6 = 3x + 1
         -x     = -x						       6 = 2x + 1
              6 = 2x + 1						       5 = 2x	
             -1 =    - 1						          2x = 5
              5 = 2x						           x = 5/2
              5 = 2x	
              _   _
              2   2
          x = 2.5 or 5/2

Some teachers have students show these steps so that they see the steps they are doing; however, the more adept students will show "results only."

● Common errors should be discussed at the appropriate point, both in lessons and in textbooks, especially at the introductory level because such discussion helps prevent formation of bad habits and development of inaccurate constructions on the part of the student. (See Hall, 2002.)

● Students should be encouraged to check equations for the truth of the solution in the original equation.

● Interviewing students will help uncover misconceptions more than just looking at work.

● Students should align their work vertically rather than horizontally to show the development of their ideas toward a solution.

● Going from concrete to abstract will help build understanding of relational solutions of equations.

● Solving single unknown equations by introducing a second variable can translate an algebraic problem to a graphical one. For example, 2x+3=4x+1 can be rewritten as y=2x+3 and y=4x+1. These two functions can then be compared graphically to find the common solution.

● Students need to overcome two levels of representation of unknowns:

- y representing the unknown (first level)

- y representing the unknown in terms of the expression 2x+3 (second level), as in the bullet point above. This requires a higher level of abstraction: students must consider the left side as f(x) and the right side as g(x) and then solve f(x)-g(x) = 0. This gathers all variables and numbers on one side of the equation. From there, students can use the distributive property to combine like terms, resulting in a two-step equation to solve (which should be routine). Students can check the solution using a comparative table or by graphing and finding the point of intersection. Also see the entry on Algebraic Relations: Student Intuition and Informal Procedures.

References

1. Filloy, E., Rojano, T., & Solares, A. (2010). Problems Dealing With Unknown Quantities and Two Different Levels of Representing Unknowns. [Article]. Journal for Research in Mathematics Education, 41(1), 52-80.

Yerushalmy, M., & Gilead, S. (1997). Solving equations in a technological environment. Mathematics Teacher, 90, 156-162.

Falle, Judith “From Arithmetic to Algebra: Novice Student’s Strategies for Solving Equations” p337.

Cortes, Anibal and Kavafian, Nelly “Two Important Invariant Tasks in Solving Equations: Analyzing the Equation and checking the Validity of Transformations”

Linsell, C. (2007). Solving equations: Students' algebraic thinking. New Zealand.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does Understanding the Equal Sign Matter? Evidence from Solving Equations. Journal for Research in Mathematics Education, 36(4), 297-312.

Godfrey, David, & Thomas, Michael O. J. (2008). Student Perspectives on Equation: The Transition from School to University. Mathematics Education Research Journal, 20(2), 71-92.

Vlassis, J. (2004). Making Sense of the Minus Sign or Becoming Flexible in "Negativity". Learning and Instruction, 14(5), 469-484.

Pirie, S. E. B. and M. Lyndon (1997). "The Equation, the Whole Equation and Nothing but the Equation! One Approach to the Teaching of Linear Equations." Educational Studies in Mathematics 34(2): 159-181

McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-SchoolEquation Soving Students' Understanding of the Equal Sign: The Books They Read Can't Help. Cognition and Instruction, 24(3), 367-385.

Filloy, E., Rojano, T., & Solares, A. (2010). Problems Dealing With Unknown Quantities and Two Different Levels of Representing Unknowns. [Article]. Journal for Research in Mathematics Education, 41(1), 52-80.

Rossi, P. S. (2008). "An Uncommon Approach to a Common Algebraic Error." PRIMUS 18(6): 554-558. Stephens, Ana Equivalence and relational thinking: preservice elementary teachers’ awareness of opportunities and misconceptions

Demby, A. (1997). Algebraic Procedures Used by 13-to-15-Year-Olds. Educational Studies in Mathematics, 33(1), 45-70.

Dickson, L. (1989). Equations. In K. Hart, D. C. Johnson, M. Brown, L. Dickson & R. Clarkson (Eds.), Children's Mathematical Frameworks 8-13: A Study of Classroom Teaching. Windsor, England: NFER-NELSON.

Asghari, A. (2009). Experiencing equivalence but organizing order. Educational Studies in Mathematics, 71(3), 219-234.

Gates, L. (1995). Product of two negative numbers: An example of how rote learning a strategy is synonymous with learning the concept. Paper presented at the 18th Annual Mathematics Education Research Group of Australasia Conference. Salma Tahir Macquarie University <mrs_galaxy@hotmail.com> Michael Cavanagh Macquarie University <michael.cavanagh@mq.edu.au Attorps, 2003, p. 1-2).

Your Experience

  • What have you learned about your students’ thinking regarding this topic?
  • What have you learned that is an effective teaching strategy to help students understand this topic?