Algebraic Relations: Solutions Involving Zero

Contents

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

Theme

Foundational Knowledge

Common Core State Standards

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

Introduction

Students are often confused by the appearance of 0 in an equation and frequently interpret it to mean that the solution of the equation is also zero. For example, if they arrive at 0x=7 (see below), they might conclude that x=0.

Symbolic Representation

Problem Response
1. Solve:

2x -3 = 2x + 4

Students get 0x = 7. 7/0 is undefined, but students often write zero.
2. Solve:

4x + 8 = 2(2x+4)

Students arrive at 0x = 0 and often think that the answer is just zero; however, the original equation, 4x + 8 = 4x +8 is an identiy, which means the solution set is the set of all real numbers.
3x + 3 = 2x + 3 x = 0, zero is the only solution that makes this true.

Interviews with students

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Mathematical Issues

Thus they specify two possible types of equation: a conditional equation; and an identity and, for example, 2x + 1 = 6 would be a conditional equation, but 2(2x + 1) = 4x + 2 would be an identity.

● These equations emphasize the need for reasoning and checking the solution to see if their solution is correct.

Teaching Strategies

  • If students get 0x=7 (for example) and conclude that x=7/0=0, teachers could ask them whether they can they take 7 items and share them with 0 people as a way of explaining why division by zero is undefined.
  • 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 for confidence improvement.
  • Connecting the solutions to real-life situations such as sharing cookies can help with understanding.
  • The checking process becomes important in contextual problems because the meaning that is associated with the check will confirm the correct interpretation of the solution.
  • Students should not align work horizontally (e.g., 8x+4=28 8x=24 x=3). To show the progression of ideas leading to a solution more clearly, they should align their work vertically.

References

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?