6.7.1 Modeling Percent

Number & Operations for Teachers

Modeling the Meaning of Percent

Figure 6.6: Modeling the Meaning of Percent

Fundamentally, the term percent means per-hundred. Figure 6.6 provides a concept model for exploring the meaning of these terms. In the figure, two bars are shown, one 50 units in length and one 30 units in length. One hundred evenly spaced rulings appear along the bottom edge of the 50 unit and 30 unit bars. Consequently, the distance between any two consecutive rulings represents 1% of each bar’s total length. Fifty percent of each bar is shaded black. Comparing the shaded portions of the two bars, it is clear that 50% of 50 is greater than 50% of 30. A careful examination of the percent rulings reveals the basis for this observation: 1% of 50 is greater than 1% of 30. This comparison highlights two important considerations:

· A percentage expresses a part-whole relationship; and

· When comparing percentages, care must be taken to specify the whole(s) to which the percentage(s) refer.

Textbook percentage problems fall into one of three categories:

· Those that seek to identify a percentage;

· Those that seek to identify a part; and

· Those that seek to identify a whole.

Problems of the first sort (e.g., “What percent of 50 is 10?”) may be addressed visually by examining the concept model or logically by using proportional reasoning. For instance,

· 10 is 20% of 50 since the shaded portion spans 20 one-percent rulings; alternatively,

· 10 is % of 30 since the shaded portion spans one-percent rulings; alternatively,

The same strategies apply to problems of the second sort (e.g., “What is 20% of 50?”)

· Because the 20^th ruling along the bottom edge of the 50-unit bar coincides with the 10-unit shaded region, we may conclude that 20% of 50 is 10; alternatively,

· Because rulings along the bottom edge of the 30-unit bar coincides with the 10-unit shaded region, we may conclude that % of 30 is 10; alternatively,

Again, the same strategies used to address problems of the first and second sort apply to problems of the third sort (e.g., “Ten is 20% of what number?”).

· Recognizing that 100% = 5 x 20% and that 20% of the whole equals 10, the whole is identified by placing 5 10-unit bars end to end, resulting in a whole that is 50 units in length; alternatively,

· Recognizing that 100% = 3 x % and that % of the whole equals 10, the whole is identified by placing 3 10-unit bars end to end, resulting in a whole that is 30 units in length; alternatively,

Notice that the proportions used to solve all three sorts of problems are similar. In each case an expression is created involving three known quantities and one unknown quantity. In general, these proportions take the form . Once the proportion is written, the unknown quantity (part, whole, or percent) is obtained by cross-multiplying then dividing both sides by the coefficient of the unknown quantity.

Students may also be taught to solve problems of this sort by converting the percent to a decimal then performing the appropriate arithmetic operation. The first part of this two-step procedure is performed by dividing the given percent by 100. Doing so changes the form of the proportionality statement to . Depending on the sort of problem, this statement is then manipulated to obtain the solution. Table 6.9 illustrates these manipulations in the context of the 50-unit bar seen in Figure 6.6

Problem	Solution
What decimal part of 50 is 10?	[Divide 10 by 50]
What is 20% of 50?
10 is 20% of what number

Table 6.9: Converting Percentage Problems to Decimal Problems