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**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

<Blank for Now>

# 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?