## Grid and Percent It

- Lesson

In this lesson, students use a 10 × 10 grid as a model for solving
various types of percent problems. This model offers a means of
representing the given information as well as suggesting different
approaches for finding a solution. This lesson is adapted from "A
Conceptual Model for Solving Percent Problems," which originally
appeared in *Mathematics Teaching in the Middle School*, Vol. 1, No. 1 (April 1994), pp. 20-25.

This lesson presents a method of solving percent problems that focuses on the basic concept of *percent,*
that of "parts per hundred." A 10 × 10 grid, which is a common model
for visualizing percents, is extended in the following examples to
solve various types of percent problems.

Before using percents to solve problems, students should have experiences in shading 10 × 10 grids to represent given percents. Distribute the Grid Worksheet to all students. (These sheets can be laminated so that students can shade the grids with dry-erase, water-based, or grease markers.) Ask students to represent various percents on these grids. In addition to whole-number percents, you should also include decimals and fractions, such as 63.25% and 18½%.

In addition, given shaded grids, students should be able to determine the percent for a shaded amount. For instance, students should recognize that the grids below represent 1%, 9%, and 78%, respectively. You may wish to display some shaded grids on the overhead grid. Again, include decimals and fractions, which can be done by shading just a portion of a square.

Once the student recognizes that 100 percent is represented by one whole square (i.e., a *unit square*)
and that one percent is represented by one small square, or
one-hundredth of a unit square, then 10 × 10 grids can be shaded to
illustrate percents less than 1 percent and greater than 100 percent.
The figures below show that when ¼ of a square is shaded, this
represents ¼%, and when an entire unit square plus another
34 small squares are shaded, this represents 134%.

There are two other preliminary activities that can set the stage for solving percent problems.

- Let the unit square (10 × 10 grid) represent given amounts, and
then determine the value of one of the small squares (1 percent). To
help students determine the value of 1 percent of the unit square, they
can think of sharing the given amount equally among the 100 parts of
the unit square. Dividing by 100 can be done by mental computation. As
examples, if the unit square represents 400 people (that is, if
400 people represents 100% in some situation), then each small square
represents 4 people. Similarly, if the unit square represents
85 pounds, then each small square has a value of 0.85 pounds; and, if
the unit square represents 162 days, then the value of each small
square is 1.62 days.
Successfully determining the value of one square (1%) is key to solving percent problems. To prepare students for this task, it may be helpful to pose questions that involve convenient fractional parts of the grid. For example, if the unit square represents 400 people:

- How many people would be represented by half of the unit square?

… by one-fourth of the unit square?

… by twenty small squares?

… by ten small squares? - What part of the unit square would represent 200 people?

…100 people?

…40 people?

…4 people?

- How many people would be represented by half of the unit square?
- Assign a value to one or more of the small squares and then
determine the value of the unit square. For example, if one of the
small squares represents 0.75 centimeters, then the unit square has a
value of 75 centimeters; if 12 of the small squares have a value of 30,
then one small square has a value of 2.5 (30 ÷ 12 = 2.5) and the unit
square has a value of 250; and, if 160 small squares have a value
of 384, then one small square has value 384 ÷ 160 = 2.4, and the unit
square has value of 240, as shown below.

The following seven percent problems can be solved using the unit square grids. The first three problems use a two-part approach: first, students are asked to use a percent grid to represent the given information; then, students use the sketch to answer a question related to that information. This approach emphasizes the importance of thinking through the given information before attempting to obtain an answer. Although this approach is not demonstrated with the last four problems, it could be used for them, as well.

Problems 1-3 are typical percent problems. Problem 4 involves percents greater than 100. Problems 5‑6 deal with percent increase, a situation that often causes anxiety in students. Finally, problem 7 uses the grid method to solve a problem involving a percent discount.

- Twenty percent of a company's 240 employees are classified as minorities.
- Use a percent grid to show that 20 percent of a business's 240 employees are classified as minorities.
- How many employees are classified as minorities?

[The given information can be represented by letting the unit square represent 240 employees and shading 20 percent of the square.

Students may notice immediately from the shaded grid that one shaded column represents 1/10 of 240, or 24, so two shaded columns—which is 2/10, or 20%—represent 48.

Alternatively, if the whole square represents 240 employees, then one small square represents 2.4 employees, so 20 squares represent 20 × 2.4 = 48. Therefore, 48 employees are classified as minorities.]

- Twenty-five acres of land are donated to a community, but
the donor stipulates that six acres of this land should be developed as
a playground.
- Use a percent grid to represent this situation.
- What percent of the land is to be used for playground?

[If the unit square represents 25 acres, we need to determine how many small squares represent 6 acres. Each small square represents 0.25 = ¼ acres, so 4 small squares represent 1 acre, and 24 small squares represent 6 acres. Or, since each small square has a value of 0.25 acres, then 6 ÷ 0.25 = 24 squares are needed to represent 6 acres.

The playground is represented by 24 shaded squares, so 24 percent of the land is to be used for the playground.]

- In Hazzard County, 57 of the schools have a teacher-to-student
ratio that meets or exceeds the requirements for accreditation. These
57 schools represent 38 percent of the schools in the county.
- Use a unit square sketch to represent this situation.
- How many schools are in Hazzard County?

[The unit square represents the number of schools in the county, and 38 small squares (38 percent) represent 57 schools.

Because 38 small shaded squares represent 57 schools, each small shaded square represents 57 ÷ 38 = 1.5 schools. So the unit square represents 100 × 1.5 = 150, which is the number of schools in the county.]

- The school population in Provincetown is 135 percent of the
school's population from the preceding year. The new student population
is 756. How many students did the school have the previous year?
[The current population can be represented by 135 small squares, so each small square represents 756 ÷ 135 = 5.6 students.

The school population for the preceding year is the value of one unit square, so the number of students the previous year was 100 × 5.6 = 560.]

- The first-year profit for a small business was $32,900, and it
increased 76 percent during the second year. What was the profit the
second year?
[One 10 × 10 grid represents the first-year profit of $32,900, and the 76 small squares in the second grid represent the increase.

So, all 176 shaded squares represent the second-year profit. Since each small square has a value of $32,900 ÷ 100 = $329, the second-year profit was 176 × $329 = $57,904. (The increase in profits from the first year to the second year was 76 × $329 = $25,004.)]

- A new company entered the New York Stock Exchange in July, and
by December, the price of each share of stock had risen 223 percent
to $32.50 a share. What was the cost of each share of stock in July?
[The shaded unit square on the left below represents the cost of a share of the stock in July, and the remaining shaded squares represent the 223 percent increase. The total 323 shaded squares represent the $32.50 value of each share of stock in December.

The value of each shaded square is 32.50 ÷ 323 ≈ 0.1006. Thus, the unit square (100 percent), or July price, is equal to 100 × 0.1006 = $10.06. (This answer is rounded to the nearest cent.)]

- A sweater is on sale at 30 percent off the listed price of $95.99. What is the discounted cost of the sweater?
[The $95.99 price of the sweater is represented by one unit square, and the discount is represented by 30 small shaded squares.

One of the small squares has a value of $95.99 ÷ 100 = $0.9599.

The grid suggests two methods of solution:

- Determine the amount of discount by finding the value of 30 small squares, which is 30 × $0.9599 = $28.797, and then subtract this amount from $95.99; the discounted price of hte sweater is 95.99 ‑ 28.80 = 67.19.
- Determine the discounted price directly by finding the value of the 70 small unshaded squares, which is 70 × $0.9599 = $67.193. To the nearest cent, the discounted price of the sweater is $67.19. (Note that the discounted price of the sweater was found by rounding in the typical way. However, many stores round in a way that is more favorable to them, so the actual discounted price of the sweater might be $67.20.)]

### References

- Bennett, Albert B., and L. Ted Nelson. Mathematics for Elementary Teachers: An Activity Approach. 3d edition. Dubuque, Ia.: Wm. C. Brown Publishers, 1992.
- Bennett, Albert B., and L. Ted Nelson. "A Conceptual Model for Solving Percent Problems." Mathematics Teaching in the Middle School, Vol. 1, No. 1 (April 1994), pp. 20-25.

### Learning Objectives

Students will:

- Shade 10 × 10 grids to represent percent, and determine the percent of a grid that is shaded.
- Determine the value represented by one square in a 10 × 10 grid, and determine the value of a grid given the value of one square.
- Use a 10 × 10 grid to solve percent problems.

### NCTM Standards and Expectations

- Work flexibly with fractions, decimals, and percents to solve problems.

- Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line.

- Develop meaning for integers and represent and compare quantities with them.

### Common Core State Standards – Mathematics

Grade 7, The Number System

- CCSS.Math.Content.7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.