Strategies
Word problems should not be taught in isolation. Hence, I plan on teaching them throughout the year. Word problems force students and teachers to take a break from routine problems. Polya asserts the definition for a routine problem is one that a student solves by plugging numbers into a formula and starts crunching away or one that requires substitution in a step by step manner. Both approaches lack originality and mental activity. However, routine problems are essential and help to build a strong mathematical background. It is important to remember that routine problems need to be supplemented with problems that require higher level skills. Polya makes an interesting analogy to only teaching routine problems in the classroom. He remarks that demonstrating only the mechanical operations is using less brain power than when using a cookbook. He believes that kitchen recipes leave more room for creativity and judgment of the cook than mathematical recipes (Polya, 1988, page 172).
Specifically, the unit will be taught during the first semester because this is when the students are beginning to learn how to formulate expressions and equations. I am anticipating the unit to take about two to three weeks to complete. Then, the second dimension of two variables will be in the first part of second semester. However, the unit may be shortened or lengthen to fit the needs of students' individual needs. For example, adjusting the number word problems dealing with particular concepts would be a common adaptation.
Investigating and analyzing various word problems in seminar helped to develop many useful strategies to implement in the classroom. Primarily, an underlying strategy throughout the entire unit is the idea of beginning with simple word problems and gradually building a foundation that will ideally help students solve complex scenarios and utilize higher level thinking processes. In addition to building their word problem solving abilities, I hope students' confidence will rise as a product of their knowledge. Confidence will help students attack problems on standardized tests, which are purposely not routine problems. The breakdown strategy will hopefully cater to students who possess lower reading levels. Simple word problems may involve only one translation. In order to take into account many abilities, mathematical terminology will heavily be stressed in the beginning of the unit. An emphasis on vocabulary will help to build a strong foundation needed to breakdown word problems. A word wall will be created by the students to aid vocabulary development. The wall will encompass words associated with all four operations and will be located on the side wall for the first semester in order for the students to reference while formulating equations and solving problems.
The purpose of the word wall is to help students identify the operation suggested in word problems. However, an individual word in a problem may suggest one operation, but the overall meaning calls for another.
Another strategy implemented during the unit will be referred to as false positioning. The method is similar to a mental math approach. The unit will still maintain a focus on setting up and solving equations, but I wanted to offer another technique in order to see if students may grasp the approach better. I feel that false positioning may help students to internalize mathematical concepts and in turn develop a true meaning of what the problems are asking them to do. For example, consider the common ticket problem.
You are selling tickets for a concert at your local community college. Student tickets cost $5 and general admission tickets cost $10. If you sell 100 tickets and collect $575, how many student tickets and how many general admission tickets are sold?
The false positioning method would have an individual consider if you sold all of one kind of ticket and then deduct to find the true amount. In the above example, you may want to think if only student tickets are sold the revenue would have been $500, but the problem is stating that $575 is made. So we know that to account for the difference of $75 another kind of ticket was sold, general admission. The difference between the two types of tickets is $5, so for each general admission ticket that is sold instead of a student ticket; you gain $5 in revenue. When divide $75 by $5 you get 15 general admission tickets were sold, and in turn 85 students tickets were sold. The false positioning method turns a two variable problem into a one variable problem.
Lastly, the strategy of drawing tables will be utilized in the unit. Drawing tables is especially helpful when solving a mixture problem. These types of problems normally frustrate students tremendously. My hope is that a table will organize the important information in such a manner that will aid in attacking the problem. Consider the question,
How much of a 10% vinegar solution should be added to 2 cups of a 30% vinegar solution to make a 20% vinegar solution?
A table, as shown below would be illustrated in class in order to help students organize the information given in the problem:
Cups of solution% vinegarTotal Vinegar solution10% solutionX.10.10X30% solution2.302(.3)MixtureX + 2.20.2(X + 2)
(table 07.06.10.01 available in print form)
Now the student can create the equation .10x + 2(.3) = .2(x + 2) and use the concepts of Algebra I to solve for the one unknown quantity of how many cups of 10% vinegar solution is needed for the desired 20% vinegar solution.
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