# 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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## Percentage Review

In beginning the study of problem solving using percentages, the idea of percent is foremost. Students have been using percents for the past few years in school and have done extensive exercises with them. They have changed the percent to decimals and fractions, done calculations with these and rounded the answer, and used pictorial representation involving percents. Spend time reviewing these because what a percent truly represents quite often may have escaped some students. In your review use hundreds grids and number lines as models for percents, fractions and decimals. Remind students that when you have a denominator of 100, you can use the term percent to mean “for each hundred”. When working with a grid, you can ask them to shade in 3 squares, 14 squares, or 40 squares. To reinforce out of 100 ask: How many squares are in the grid? What percent have you shaded? Consider these questions to reinforce 100:

Shade 72 squares. What percent are shaded? What percent are unshaded? Check the total: 72% shaded + 28% unshaded = 100%. What fraction of squares is shaded?

These questions ask the students to compare and verify. Students can make needed connections with fractions and decimals using follow up questions about the grid.It is imperative in questioning that you include a distracter (such as unshaded) occasionally. This is frequently seen on standardized tests and to practice it in class helps the students to read more critically.
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It is important for the students to see what percents such as 100, 50, 25, 0% can be related to in real world. Consider, for example:

The entire class is present in school today. What percent of the students are present? What percent are absent?

With this type of question, you want the students to realize that the whole amount is 100%. That 100% is equal to 1 whole class. No one is absent therefore it is 0% absent. Next, try this for a quick review:

Half of the students in this class walk to school each day. What percent of the class are walkers?

This question gets the students thinking about what 1/2 represents in terms of percent. Students will begin to see a relationship between 1/2 and 50%, if they do not see it already. Many students also will know a variety of equal fractions and percents. They should be comfortable with changing familiar fractions to percents easily. A review of changing percent to fractions and vice versa will be needed at this point.

The class will be at various skill levels with percentages and seeing relationships between problems will not be easy for all of them. But some students can go beyond. If you add to the problem above that there are 24 students in the class and ask: How many of the 24 students are walkers?

With this added question several students will calculate using mental math to get 12. Again you can point out the relationship between half and 50%, and 12. Continue with practice such as this:

One fourth of the class is left handed. What percent of students in this class are left handed?

Conversely use the complement: What percent of the students are right handed in this class
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These questions create benchmarks for students as to what common percents and fractions are related to. Some will throw out there that 6 students are left handed. Let them explain their reasoning to the class. As they see this 1/4 and 25% connection, a question that can be proposed to the class is “Why do we work with percents?” The response that truly answers it is that it allows us to work with fractions yet, give the appearance that we are dealing with whole numbers. Many students are more at ease with whole numbers, and for the most part two digit numbers, which are used to express most percents, than they may be with fractions. Using percent allows the student to avoid the difficulties they incur with fractions, a skill that many have not yet truly mastered. Also using percent is a skill that is real world and is frequently associated with purchasing items at the store or from a catalog involving tax and discount. Budgeting and circle graphs are a usual tool involving percents This real world skill lends itself to creating word problems that provide meaning to the child and hopefully peaking their interest. It also is a necessary skill for the child to develop because it is so vital to adulthood. A set of review problems to work with percent review follows the unit.