• • • • • • • • # From Percentages to Algebra

## Using Authentic Problem Context

We developed From Percentages to Algebra to answer the question, "When will we ever use this?" Percentages are the most common measure of performance in diverse settings. Percentages are so useful that the word is among the top 1,000 words used in the English language. Every child sees a percentage when he or she receives a grade on an exam. Every food label reports the percentage of each daily nutritional requirement. All high schools and colleges work to reduce their dropout percentage. Store sales are advertised as a percentage off of the price. In basketball the percentage of free throws made is a standard statistic.  Every election poll reports the percentage of people who will vote for a specific candidate.

## Summary of Examples: Percentages Everywhere

There are a total of 15 examples. They are designed to explore different ways of working with percentages. All of the examples are embedded in scenarios that involve decisions. 1. Choose the Better Deal: The first example starts with the most common textbook application of percentages, a price discount. The example also includes a fixed dollar discount. The student will compare the two and determine which is the better discount for different priced meals. This is facilitated with the introduction of an algebraic expression to be evaluated. The student then uses an algebraic equation to determine when the two discounts save the same amount of money.

2. Making the Grade: This example uses grades on tests that are reported as percentages, this allows for comparison of grades with a different number of questions. In this example, the student has received a score for his first exam. He wants to figure out was score he needs on the next exam to earn a specific letter grade. An algebraic equation is used to determine the minimum grade required to achieve his desired goal.

3. Free Throw Percentages: This example discusses how a competing team might use the free throw percentages for the Detroit Pistons as guide as to whom to foul late in the game. The major part of the example presents fictional data on free throws that the student uses to calculate the player’s free throw percentage.

4. Dropping Out – When 0% is best: In the previous examples a higher percentage was better. In this example the focus is on the percentage of students who drop out before completing high school. A smaller percentage is better. Students are asked to calculate percentages to determine which dropout prevention program is better.

5. Special Ops: In this example, two officers discuss the challenges of volunteers passing two rigorous weeks of different types of training for a special mission. The student will evaluate whether it is better to have the training with the higher failure rate first or last. This includes an economic analysis of the order of training. Students will use algebra to determine how many volunteers are needed to enter training in order to meet the need for 36 soldiers who have passed both weeks of training.

6. Growing a business – Compound Percentages: Growing a business is one of the few applications in which percentages can exceed 100%. In this example students will evaluate two different marketing programs. One program increases the number of customers by a percentage. Multiple weeks of growth illustrate the concept of compound percentages. The other marketing program adds a fixed number of customers each week. Algebra is used to estimate the number of customers several weeks later as the two programs are compared.

7. McFadden Restaurant – Equal Percentage Up and Down: This example discusses the monthly sales of a restaurant. The example addresses the misconception that if sales increase by 20% one month and then decrease by 20% the next month, the final number is the same as the starting number. It is not. This example also uses algebra to determine total revenue used to calculate the franchise fee.

8. Managing by Percentages: This example is actually a collection of several small examples. In each case the decision maker is allocating a resource based on the percentage demand for different products. In several examples, the resource is space in the store. In another example, it is advertising dollars. This collection introduces the student to a commonly used demographic factor, the percentage of women and men shoppers. It also discusses the mix of customers who prefer organic fruits and vegetables and are willing to pay more organic foods.

9. Multiple Flavor Ice Cream Sales: In previous examples there were only two possibilities. In this example the manager is deciding how much of each flavor of ice cream to stock. There are four flavors to choose from each with a different percentage of demand. This example involves an economic analysis of the alternative plans for stocking flavors. Students will need to work with numbers that do not always produce whole numbers of a product. Algebra is introduced to determine a break-even point for ordering a whole liter of ice cream even if all of it cannot be sold.

10. Clothing sizes – Multiple Percentages: This example continues the development of contexts with more than two percentages. The primary decision is how many items of each size should be ordered. The student is introduced to the concept of a pie chart that is used to show and compare percentages of sizes for men and women. One complication is that the calculations often result in non-whole numbers. However, only a whole number of items can be ordered. The problem also involves working with pairs of percentages, sizes and colors.

11. Compound Growth of APP Users: In this example, two high schoolers develop a successful game app. The number of users grows by 25% each month. The example involves compounding percentages to determine the number of users several months from now. The example involves extensive economic analysis. The developers are selling advertising in order to raise money to pay for living on campus rather than commuting. This example includes a number of graphs that the student is asked to read and interpret.

12. Compound Percentage Decline of Homeless Veterans: Homelessness is a national problem that is being worked on by both federal and state agencies. This example presents two alternatives for reducing homelessness among veterans. One program reduces homelessness by 17% a year. The other helps a fixed number of veterans each year. The example explores the relative effectiveness of the two programs over a multi-year time period. Graphs and algebraic models are an important aspect of this example.

13. Weighted Averages and Grades: This is an extension of the earlier example involving grades. In this example the various components of the grades are not equally weighted. In addition, there will contexts with three elements to calculating the grade. An algebraic equation is used to determine the minimum percentage needed on the final element to bring the overall grade above a specific threshold.

14. Open24 Slushy Sales – Weight Values by Percentage: This example presents data on the demand for different sized iced drinks. Calculating overall performance of the store involves taking a weighted sum of multiple sized drinks. The store manager is trying to decide between two different ad campaigns that can impact sales of these iced drinks.

15. Congressional Districts – Combining Percentages: This example explores how the design of a congressional district could affect the likelihood of a Republican or Democrat winning the congressional election. The region is made up of 12 geographic units with different percentages of Republican and Democratic voters. The state must group these 12 units into four congressional districts. The student will evaluate two different designs for the four districts. The student is then challenged to create a different plan that is more fair to both sides.

All 15 examples use percentages to make decisions in a meaningful context. In designing these examples, we strove to include the use of other important math skills in a natural way. Every example presents data in a table format. A number of the latter examples also include line graphs and pie charts. Seven of the examples introduce algebraic expressions and equations to determine a specific value of interest. At the end of all of the examples, we present a simple project idea for collecting data related to the context of the example.