# The Craft of Word Problems

## CONTENTS OF CURRICULUM UNIT 04.05.09

- Introduction
- Rationale
- Mathematics Standards
- Strategies/ Role of the Teacher
- Percentage Review
- The Percentage Formula(s)
- Additional Ways to use the Formula (1)
- Extensions of percent
- Conclusion
- Reading List: Electronic Resources
- Print Resources: Teacher
- Print Resources: Student
- Mathematics Standards Appendix
- Notes

### Unit Guide

## Do the Math 100%

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## Strategies/ Role of the Teacher

Strategies for solving all types of word problems are multifaceted. It is not so much the strategy that determines success but rather that a student follows a plan for solving. If students can see relationships between problems that they are successful in solving to new problems they are presented with, then they have a basis for their plan. Generally, solving problems should be broken down into three areas:

- - understanding what the problem says;
- - thinking and doing the math
- - responding and giving the answer

After reading the problem, understanding what to do can be a major difficulty for some students. They often are weak with the vocabulary necessary to translate the problem into mathematical language. Repeatedly using and charting math terms and their meanings are helpful. Student writing in journals where they use the clue words that are needed reinforces their meanings. When they write their own story problems, they gain further familiarity with the terms they need for understanding. For example, the use of the word “of” when combined with “percent of” in mathematical language means indicates multiplication. When using percent, to understand that percent often means out of 100, is as important, as how to represent the percent out of 100 by using a decimal or fraction. Both are vital to solving the problem. Encourage students to reread the problem for understanding and try to relate this problem to ones that have seen and solved before. It is seeing the relationship between problems that often will guide a student towards understanding what to do. With percentage problems, continuous practice will allow the student to identify what information is needed to be solved, the percent, the part of the quantity, or the whole quantity.

As students reread the problem and look for the information that is given to them in the word problem, the plan for thinking how to solve can begin. First uncover the clue words needed to indicate what operation must be done in order to correctly solve the problem. The teacher should encourage the student to circle the clue words and numerical information that they see in the problem. When students see the percent sign they will be reminded that they can use a formula to solve. By continually presenting students with practice problems that require them to see relationships, students learn to relate the clues to new problems that they are trying to work out. When teaching percent for instance, the teacher will introduce methods for solving the problems that students can practice and gain a level comfort. By using formulas students can make sense of the procedures they need to do to solve the problems they are given. Once comfortable with using formulas, they can plug in the numbers from the problem that go into the formula. This unit explores the use of formulas to solve percentage problems and the analysis of this. Use cooperative learning groups and classroom discussion about how to solve to find the answer when appropriate. Students often benefit from small group discussion on how to set up a mathematical sentence and then find the solution.

The response of the student will occur in several ways. Obviously, you want the students to calculate and give the answer. You also want students to be able to write a written response which explains and justifies their method of how they found the solution. This may be asked of them on certain tests. They need to be sure that theirs is a reasonable answer for the question that is being asked. Again proper understanding of terminology helps this process. Students must also be prepared to answer using a grid format as this is frequently asked of them.

As teachers we need to set up successful situations, as well as, challenging situations for all of the students. We need to have the students achieve a measure of success and to challenge without frustration, if possible. Solving words problems and writing a response to the solution are the driving force to mastery of the CAPT goals that students are required to achieve. “… basic skills and conceptual understanding are intertwined, and both are necessary before students can successfully apply mathematics to the solution of problems.”1 CAPT activities in mathematics will require the students to discover the ways to solve real world problems in a performance task. Students will attack the tasks with more confidence when they are able to draw on what they already know and have repeatedly practiced in previous grades.

Frequently when beginning a new type of problem to set up a successful situation, we can choose to model our own strategy for solving a problem and let the students “see” our thinking. Students often want to “help” us, as well. We cannot expect all students in the class to jump successfully into problem solving in today’s diverse ability level classrooms. Therefore, teachers should devote instructional time to discussion and modeling how to solve word problems. It is helpful for the students to see you use an illustration, chart or table to solve the problem if appropriate. When the students can observe your plan as you go through the steps of understanding, thinking and coming up with an answer, it often times helps them to begin to use a plan on their own. Our class instruction must involve effective communication, questions, and discussion in order to engage students in learning and providing feedback.

It can be helpful for the students when attacking the percent problem to draw on individual familiarity, as can be the case with story problems. The teacher can point out examples that students may have had prior experience with, such as going to the mall and purchasing sneakers or music. Students are familiar with paying tax or looking for a discount. Using catalogs for tabulating items can involve percent both with tax and shipping costs. As students get to 8th grade, the level of difficulty of the word problems increases, however, their skill levels may still be weak and can still be a deterrent to their getting the correct answer. Use of the calculator can help to bridge the arithmetic gap. Even though this weakness of skills may occur, they still need to experience the type of word problem that will be challenging them. “… to best serve Connecticut’s students, we encourage educators to adopt the following measures: overall, set higher expectations for all students…in curriculum, provide a more rigorous study of mathematical skills and concepts and their applications in today’s world…”2 They need to be ready to face the more daunting job of preparing for more advanced mathematics courses in high school and taking the CAPT. The exploration in this unit of the differing types of percent problems are typical of what will be asked of students to solve on the CAPT.