6.NS.7a, 6.EE.5, 6.EE.8, 7.EE.4b
Inequalities represent a difficult step up from equations for many students. The procedures for solving inequalities are are deceptively similar to those for solving equations, but there are some differences that students often overlook. Teachers can help students recognize these differences and become proficient at solving inequalities by providing multiple solution methods.
|Can x=3 be a solution to a system of inequalities?*||Yes, because the solution to 5x-10>0 is x>2 and x=3 satisfies this.|
|Is there a value for x that will make (2x-6)(x-3)<0 true?**||
(2x-6) > 0 and (x-3) < 0, so x > 3 and x < 3
(2x-6) < 0 and (x-3) > 0, x < 3 and x < 3
|Let p and q be odd integers between 20 and 50. For these values, is 5p-q > 2p+15 always true, sometimes true or never true?**||
3p – 15 > q
If p = 21, then 3*21 -15 = 48, so it would be true unless q = 49. So, it is sometimes true.
|Is there a value for x that will make the following statement true? (6x-8-15x)+12 > (6x-8-15x)+6**||
(something) + 12 > (something) + 6
This is always true
- (Tsamir & Bazzini, 2001)
- (Lim, 2006)
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When introducing inequalities, students will try to connect what they already know about how to solve equations or inequalities; however, they do not connect the correct previous material to the new material. Students tend to apply strategies for solving equations when solving inequalities. They use their intuitive techniques here and this causes errors. It is important to explicitly draw their attention to the differences between an equation and an inequality.
Students often depend on algebraic methods for solving equations and inequalities and use strategies interchangeably (and incorrectly). This misconception is understandable because equations and inequalities "look" similar. Students struggle with solving quadratic inequalities, especially if they are provided with only one method of solving them.
Typical student errors when solving quadratic inequalities are:
• When multiplying equations on both sides by a negative number, students forget to change the direction of the inequality sign.
• Students have a hard time accepting their final solution when solving inequalities if the solution results in a single answer, the empty set or the set of all real numbers.
• Students have difficulties translating words such as “at least” or “not more than” into mathematical symbols using inequalities.
When introducing inequalities, it can be helpful to distinguish between different classes of equations and expressions:
● Identities: cos^2 x + sin^2 x = 1
● Non algebraic equations: [Integral]f(x)dx = x^2 + C
● Equations with more than one unknown: 3x+4y = a
● Trivial equations: x=2
● Functions: f(x)=2x+1
● Inequalities and expressions: 3x+2 > 4x +5
Developing meaning for symbolic representations of equations or inequalities (including linear, second, and third degree) help students to understand what the solutions represent or even judge if an answer makes sense. Designing lessons where equations and inequalities are compared and contrasted will promote better understanding of the equal sign in students.
Providing more than one solution method for solving inequalities also helps students in the long run to become more flexible solving problems in different topics. Students who are introduced to more than one method for solving quadratic inequalities, for example, cope better when they are stuck because they can try another method. The context of a problem can help students realize that the solution to an inequality should be > or < than the similar equation. Different strategies include:
● the graphical representation method
● the sign-chart method
● the logical connectives method (creating a system of quadratic equations that are connected to each other by “or” or “and”)
Graphing each side of inequality as a separate line and doing a logical comparison can help promote an understanding of the result. If students are already familiar with graphing functions, then solving quadratic inequalities with graphic methods can help them to understand the solution better, especially the weaker than average ones. The graphical method for solving quadratic inequalities also helps these students to answer questions related to the functions graphed.
By focusing on graphical methods of solving equations and inequalities, students see representations of equations and inequalities that look different, and they are less likely to incorrectly apply strategies from one type to the other. Teachers should provide more than one method to solve inequalities and help students to choose a method that they understand well.
Hattikudur, S., & Alibai, M.W. (2010). Learning about the equal sign: Does comparing with inequality symbols help? Journal of Experimental Child Psychology 107, p. 15–30.
Lim, K. H. (2006). Characterizing students' thinking: Algebraic inequalities and equations. Paper presented at the Annual meeting of the Psychology of Mathematics Education in North America.
Tsamir, P., & Almog, N. (2001). Students' strategies and difficulties: the case of algebraic inequalities. International Journal of Mathematical Education in Science & Technology, 32(4), 513-524.
Tsamir, P. & Bazzini. L. (2001). Can x=3 be the solution of an inequality? A study of Italian and Israeli students. Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, Netherlands.
Tsamir, P., & Reshef, M. (2006). Students' preferences when solving quadratic inequalities. Focus on Learning Problems in Mathematics, 28(1).
Verikios, P. & Farmski, V. (2010). From equation to inequality using a function-based approach. International Journal of Mathematical Education in Science and Technology. 41(4). p. 515-530.
- 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?