Since percents are ratios, the concept of percents is deeply rooted in proportional reasoning. The Common Core State Standards require students to use proportional relationships to solve percent problems. Teachers can approach percent problems by using proportional relationships with ratios or equations; this post outlines a path using ratios.
Before teaching percent change, students should have prior knowledge of percents. Students should be able to find a percent of a quantity as a rate per 100. And students should also be able to find the whole when given a part and the percent. Both of these concepts are included in CCSS 6.RP.A.3; however, teachers may prefer to informally assess these skills prior to introducing percent change. The "Calculate This!" activity requires students to determine the fraction, decimal, and percent represented by the shaded portion of a figure. (Note: The document is organized to be printed double-sided with answers on the back of the corresponding figure.) The "Estimate This!" activity requires students to estimate percent problems and sort the estimates into four categories including less than 50%, between 50% and 100%, between 100% and 200%, and more than 200%.
Percent change is a common real-world application. Percent increase applications include tax, markups, fees, gratuities and commissions. Percent decrease applications include discounts or markdowns. A procedural approach to the concept of percent change can quickly overwhelm students either through the series of steps or in the confusion of how to duplicate a "shortcut" on their own. For example, a 5% discount on a $45 hoodie purchased with 9.5% sales tax could have a process emphasis with the following steps: 5% of $45=$2.25; $45-$2.25=$42.75; 9.5% of $42.75=$4.06; $42.75+$4.06=$46.81 total OR 95% of $45=$42.75; 109.5% of $42.75=$46.81 total. And when students are asked to determine the discounted price or the amount of tax, they struggle differentiating the steps to extract the specific solution. This is a great reminder that process must first be grounded conceptually. Perhaps a geometric investigation using color tiles with focus on percent change in area could provide students with a visual reference point when encountering the hoodie problem.
One common percent change misconception that becomes problematic in the midst of percent error is the misunderstanding that the percentage of increase from an original area to a new area is the same percentage of decrease from the new area back to the original area. Teachers can informally assess students via a journal prompt and then provide remediation as necessary by using the RallyCoach structure for
Kagan cooperative learning.
Percent error shows how inaccurate an experimental measurement is. If the experimental measurement is more than the actual measurement, describe the inaccuracy by what percent increase the experimental (new) measurement is compared to the actual (original) measurement. If the experimental measurement is less than the actual measurement, describe the inaccuracy by what percent decrease the experimental (new) measurement is compared to the actual (original) measurement. The Mars Research Lab Activity poses a suspected issue in one plant that prompts students to analyze samples to determine percent error for each color of candy within the sample. This lab activity can be used independently or as the research for a culminating project.
These activities highlight the Common Core State Standard 7.RP.A.3 included in MATH-7.
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