# The Area and Perimeter of a Playground

Using this map of a neighborhood playground, find its area in square meters.   Additionally, the neighbors want to build a chain-link fence around the playground.  They can determine the number of meters needed for fencing by finding the perimeter (or distance around) the playground boundaries or the total length of the playground’s dimensions. A math proficient third-grader could produce the following correct responses:

• The area of the playground is 250 square meters, as it is composed of a 10m x 10m square and a 10m x 15m rectangle.
• The perimeter of playground is the total of all of the 5 boundary lengths measured in meters.  It’s my impression that calculators are not allowed on world-class, third-grade tests.  So, one mental or paper-and-pencil strategy for finding the correct total of 70 meters is to multiply 3 times 10 and then add 30 meters to the sum of 15 and 25:  3 x 10 + (15 + 25) is 30 + 40 = 70. The numbers in the problem are “friendly” and encourage mental math.

This problem requires an understanding that area is measured in square units and perimeter is measured in units of length, thus incorporating Goal #8 for measuring lengths.  proficiency with Power-Start Goals # 8 and #9. Let’s move to a slightly higher level in our own thinking about area and perimeter and compare this playground with the playground considered in my previous blog post.

• What is the area in square meters of Playground A?
• What is the area in square meters of Playground B?
• What is the perimeter in meters of Playground A?
• What is the perimeter in meters of Playground B?

In fact, Playground A and Playground B have the same area but the perimeter of Playground B is 5 meters shorter.  If a farmer wanted to economize on fencing for her barnyard and still have the same amount of space, she would probably select the shape of Playground B over Playground A.

### Remember to Start Early

The following quotation comes from page 173 of Learning and Teaching Early Math: The Learning Trajectories Approach (2009), by Douglas Clements and Julie Sarama.

Area is an amount of two dimensional surface that is contained within a boundary.   Area is complex, and children develop area concepts over time.  Even so, sensitivity to area is present in the first year of life, as is sensitivity to number.

The book’s chapter on area goes on to say that many preservice teachers are limited in their knowledge of area and perimeter, and my Indiana experiences teaching math courses for elementary and preschool teachers bore this out.     Here are two types of problems we considered in my math classes for elementary and preschool teachers which require deep understandings of these different processes for geometric measurement.   I remember student comments similar to this one,  “I’ll be teaching kindergarten; so why do I need to know about area and perimeter?”

1. Two homeowners are planning to repaint a 20 ft. by 15 ft. living room, selecting beige for the walls and white for the ceiling.  They have two small cans of white paint left from another painting project and each can covers 200 square feet.  Will there be enough paint? Will they have some left over? Approximately what portion of the second can will be left?
2. The homeowners also want to install crown moulding around the top of the room and select a style priced at \$2 per foot.  How much did the moulding for the room cost?

Now, of course, these two problems not only require basic understandings of area and perimeter but application and analysis probably beyond a third-grader’s knowledge and/or experience.  Even so, we can encourage young children to explore area concepts by covering surfaces, with a layer of tiles, beans, paper, or even packing peanuts.   And, we can encourage them to measure lengths using fingers, steps, sticks, stacking cubes, or rulers as preliminary steps for considering perimeter.  If we give them these experiences while in preschool, then we are laying proper foundations along learning trajectories for area and perimeter.

### Inches and Square Inches are Different Units of Measure

In our everyday lives, we use different units to assign a numerical measurement to different kinds of quantities.  When giving children experiences with area and perimeter, make this distinction between units of square measure and linear measure.  the.  Power-Start Goals #8, #9, and #10 all involve measurement concepts and units of measure.  Goal #10 expects third-graders to measure, estimate, and compare quantities with commonly used units of measure.

• Commonly used units for liquid capacity include cups, quarts, and liters.
• Commonly used units for weight include pounds and kilograms.
• Commonly used units for temperature are degrees Fahrenheit and Celsius.
• Commonly used units of length are inches, feet, centimeters, and meters.
• Commonly used units of area include square inches (in2) and square meters (m2).

And remember that measurement is something that’s always hands-on and most children really enjoy it.  Furthermore, measurement experiences use arithmetic and frequently involve shapes.  Thus, I’m thinking that activities involving measurement can be connected to all ten goals on our list!!

### Composing and Decomposing Areas

The first purpose of the picture below is to remind you that areas can be composed and decomposed, and thus, addition and subtraction can be used to determine areas of the parts or the whole shape.   Also, take away with you the visual reminder of using square units for area and a ruler for lengths.  Use these 3 or  4 years of preschool and the primary grades to get measurement ideas across (experientially, of course).  And here’s another thought —  use blocks and small cubes, as well as materials like beans or packing peanuts to explore measurement of volume.

### Simple Maps

Since our original problem is about the area and perimeter of a playground, I thought I would take this opportunity to mention that young children enjoy using simple maps you might sketch for them — of a room; of a house; of a playground; or of a neighborhood.  The use of maps enhances children’s innate sense of spatial orientation and develops their navigational skills.  Additionally, this map locates playground places using horizontal and vertical coordinates.