Equations
The background taught for the expressions will be the foundation upon which solving equations will be built.
The equal sign has become a stumbling block in mathematics curriculum, mainly because the education system has pushed so hard for students to memorize steps, arrive at an answer, and bubble it in on a scantron. We have lost the sense of process and understanding. J.J. Madden illustrates this problem in the article, "What Is The Equals Sign?" "Another dysfunctional thought-model connects the equals sign we write on paper with the equals key on a calculator, which we press when we want the calculator to show us the answer?" 5 I hope to alleviate this issue by spending quality time looking at equivalent expressions. In his article Madden points out that students who come into Algebra with the notion that there should be a, math problem = answer, are not incorrect. The equation 7 + 4 = 11 is a true statement and an accurate use of the equal sign. However, students need a broader interpretation of the equals sign. They need to see the relationship between values and expressions in order to grasp the meaning of the equals sign in all contexts.
To solve equations students must learn two more Rules of Arithmetic, the Identity Rule and the Inverse Rule. However, before directly teaching these rules I want to help students get a clearer idea of what an equation is. A clear explanation of the change between expressions to equations is required. Students have now spent a considerable amount of time learning that an expression is a recipe for calculation. The equation now takes two expressions and states that they are equal, and then asks us to find the value of the variable that makes this true.
When describing the goal for solving the equation students will use the words, "What value of (given variable) makes the equation true?" I will then give students equations they are able to compute mentally. Some examples are 5x = 30, x + 7 = 25, 3x + 4 = 10. Rather than following the usual rules and procedures for solving equations, I want students to make the connection that x has a value. Then I will put up the problems ¾ x – ½ = 17 ½ and 2.7x + 13.08 = 26.58. Two problems students are not able to solve mentally. How do we figure out the value of x now? After some discussion we will look at the Identity Rule and Inverse Rule.
The Identity Rule can cause some confusion for students if its purpose is not made clear. It tells us that there is a number such that, when it operates on any other number, it does not change it. It is clear to most students that The Identity Rule for Addition states that a + 0 = a. The rule allows us to build the foundation for understanding inverses.
The Inverse Rule for Addition states, for any number a, there exists a number, -a, such that a + (-a) = 0. After a few samples, students can easily answer a questions like, 7 plus what number gives you 0? Or, what is the additive inverse of – 9? Regardless of their ability to answer these questions I think it is important that the idea is continually reinforced with a number line or visual model. This concept is easy to memorize in isolation, but can be difficult to apply when working with equations. Students will need to practice applying the rule first to simple equations and then continue with more advanced equations. For example x + 7 = 9, students will add -7 to both sides, (x + 7) + (-7) = 9 + (-7). Students will then simplify using the Commutative and Associative Rules, x + (7 + (-7)) = 2, x + 0 = 2, x = 2.
A little more difficult for students to understand is the Multiplicative Inverse Rule, which tells us that a rational number "a" multiplied by its reciprocal equals 1, (a)(1/a) = 1. Few students grasp multiplication of fractions in the primary grades and therefore it is a challenge to understand (a) (1/a) = 1. Some review of multiplying fractions is necessary for them to see the connection. This will provide an excellent opportunity to review the unit fraction and the idea that multiplying by 1/d is the same as dividing by d. Since this is not a new topic it is a great place for group discussion. Students will work in groups to discuss why (5)(1/5) is the same as 5 divided by 5 and why (25)(1/5) is the same as 25 divided by 5. Though this type of computation should be second nature by 8 th grade, it is not. Extra time spent here will prevent confusion when solving equations, especially those containing fractions. To see how well the students have mastered the multiplicative inverse concept I will ask questions like these: Why does zero not have a reciprocal? What is the reciprocal of 0.001?
While conceptual understanding is crucial for students to be successful in math, I also value procedures and organization. To practice using the properties to solve multi-step equations, student set up their work in a clear and organized manner. As we begin to look at some problems as a class, students will set up the two-column paper. The left is for the computation and the right is for identifying the property (same as we did when we simplified expressions). Students will write the equation and draw a line down from the equal sign to show the balance of the equation. It is useful to keep the equal sign working its way down the equation in order to continue to reinforce the idea of balance and equivalent expressions.
Once students are able to find a value for the variable, their solution needs to come in some form of a statement, such as "when the value of x is 7, this equation is true." It is important for students to stay connected to what this statement means. Therefore they must plug the value of the variable back into the original equation to be sure the statement is true. Lots of discussion is important at this point. If a student gets an answer that doesn't work, it is a great opportunity to discuss what the solution would need to be if that was the correct value. Also, as students check their answer by plugging it back into the equation, they should be asked to read expression as the recipe for calculation. For example, if a student solves the problem 2(3x + 8) = 40 and states the answer, "When the value of x is 4, this equation is true." They should then be able to go back and check by saying, "Three times 4 is 12, and 8 more is 20, then doubled makes 40."
As the problems increase in difficulty, students will find variables on both sides of the equations. This is one area it is so easy to allow students to just follow the procedure to solve rather than discussing what it means. I would like to see students gain the ability to explain the idea of equivalent expressions. Often, students are taught to get variables on one side only and solve. Yes, this works, however it is important that they understand that each side contains an expression. The goal is to find the value of the variable that can be plugged into both expressions to find equal answers. While students will learn to solve equations by getting the variable one side, I think it will be beneficial to reinforce this concept of the expressions being equal when checking the answer. See Appendix A for sample problems.
Word Problems
8 th graders see word problems as the last two problems on the page, optional and often not worth the effort. When asked, students feel justified in their answer, "I didn't understand, I tried but couldn't figure it out, etc…" The more ambitious student recognizes that the first 20 problems on the page were solved using single step addition equations and therefore the word problem must follow this pattern. The diligent, committed to completing homework, plug numbers and variables into the format and hope they chose location correctly.
My degree is in bilingual education. So my strength is working with English language learners in the content of the classroom. Math is often seen as a second language, so I have learned skills necessary to teach word problems to non-native math speakers. The word problem can be broken down and understood totally apart from the computation process. Time spent doing this will help break the habit students have of filling in random numbers and operations regardless of meaning. To begin, students will work with a wide variety of word problems that look at situations students can relate to and understand. Students practice by drawing pictures, charts, graphs, to represent the information given. Initially they will work on identifying what information is given, what information is the question asking, and what is the relationship between different knowns and unknowns in the word problem. Students will be formally assessed on their ability to simply understand all the components of the word problem. See examples in Activity Three.
Once students begin to develop the habit of reading word problems for meaning, we will move to writing the equations that solve the word problems. This is a difficult transition, however if students have mastered the previous sections of the unit, they have the background knowledge necessary to build the equations. As suggested in the article "Arithmetic to Algebra" by Roger Howe, students need to practice translating a wide variety of word problems into both algebraic and arithmetic expressions. The idea in this article is that students will benefit from translating the same types of problems into algebraic notation and arithmetic notation. The simple problems will teach students to set up equations and the more challenging problems will show them the advantage of using the equations. "By experiencing the strong connections between the two, they can come to appreciate the maxim that 'algebra is generalized arithmetic'. Three key benefits of teaching the problems in parallel are: all levels of problems work both arithmetically and algebraically, the arithmetic and algebra are comparable in structure, and as the problems advance in difficulty the advantage of the algebraic form becomes more obvious. 6
Students with the ability to read the word problem, interpret the information, translate it into expressions that in turn create an equation, solve for the unknown value, and identify what the unknown value represents, have mastered the multi-step problem and are in an ideal place mathematically to move forward in their learning.
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