6.EE.5, 6.EE.6, 6.EE.7, 7.EE.4a, 8.EE.7, A-CED.1
Linear equations are the simplest kind of equations to solve because they are built by a sequence of invertible arithmetic operations with which students are familiar. Linear equations in which the variable only appears on one side are a good starting point for students learning to solve equations because manipulation of variables is minimal; most of the manipulations involve explicit numbers.
|x– 4 = 6||Students usually solve these by using the opposite operation to undo whatever had be "done" to the variable. Equations like -5x = 10 can be problematic because, not understanding the symbolism, students don’t know whether to add 5 or divide by -5.|
5 - 2x = -15
2x – 5 = -15
2x = -10
x = -5
|Rewriting 5-2x as 2x-5 is a common error (shown). Students think that they can use the commutative property with subtraction or don’t associate the – sign with the 2x. Placement of the variable can present problems, and use of algebraic properties to rewrite the problem can be difficult when negatives are involved.|
Solve the equation:
5x – 3/2 = 1
5x -3/2 + 2/3 = 1 + 2/3
5x = 5/3
x = 1/3
Common student errors:
1. Incorrectly add fraction: (inverse for multiplication rather than addition -- shown) 5x-3/2 + 2/3 =1+2/3
2. Incorrectly use “common denominator” (not shown): (5x - 3)/2 = 1.
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● To develop complete understanding of linear equations students need to see equations as a whole, recognize them as an object rather than see them as an accumulation of numbers, unknowns and operations.
● In brief, there are two standard teaching approaches to solving linear equations.
(1) “Doing the same to both side of the equation repeatedly” until you arrive at the answer.
(2) “Change sides, change signs” approach. The main idea in both methods is to transform the original equation to a simpler one until "simpler" actually gives the solution: x=a.
● Students need to master the skill of collecting like terms before working with equations.
● Students’ selection of a solution strategy depended extensively on the strategies employed by their teachers to solve linear equations in their lessons.
Students who rely on "computational thinking" have more difficulty solving equations than students who use "relational thinking." "Relational thinking" means that children are able to take advantages of relationships between numbers to rewrite expressions in ways that make the computation easier. For example, a student thinking relationally might rewrite 47+38 as 50+35, which is somewhat simpler to compute.
● What may seem like "real-life" situations may not be real for all students. For example, the balance model may be confusing for students who have never seen a balance. One must know the students and their experience when selecting a model that will make sense to them.
● Once problems move beyond small positive numbers, the need for formal algebraic techniques becomes apparent to students.
● “Students need to solve equations where the solutions are not whole numbers and that require the use of algebraic methods.” (Board of Studies NSW, 2005, p 86)
● Giving students simple equations and insisting they use algebraic methods to solve them does not encourage algebraic skill development. (See entry on Student Intuition and Informal Procedures)
● 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.
● Common errors should be discussed at the appropriate point, both in lessons and in textbooks, especially at the introductory level because it prevents formation of bad habits and development of inaccurate constructions.
For an entire article on error analysis see Hall in the references.
● 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 not write work in the horizontal style referenced above, but rather the vertical alignment to show progression of the transformations in order of operations.
● Going from concrete to abstract will help build understanding of relational solutions of equations.
Hall, R. (2002). An Analysis of Errors Made in the Solution of Simple Linear Equations. Philosophy of Mathematics Education Journal, 15.
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-School 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 <email@example.com> Michael Cavanagh Macquarie University <firstname.lastname@example.org (Attorps, 2003, p. 1-2).
- 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?