# Geometry Part 7: Area and Perimeter

by C. Elkins, OK Math and Reading Lady Today’s topic is the measurement of area and perimeter.  Even though these may be considered measurement standards, they are highly connected to geometry (such as the attributes of a rectangle). Check out previous posts in my Geometry series (Composing and Decomposing) for other mentions of area and perimeter.

Misconceptions provide a window into a child’s thinking.  If we know the misconceptions ahead of time, we can steer our teaching and directions to help students avoid them. I will go through several misconceptions and some strategies and/or lessons that might address them. Misconceptions #1-2 appear in this post. Misconceptions #3-5 will be featured in next week’s post.

Misconception #1:  A student hears this:  “We use area to measure inside a shape and perimeter to measure around a shape.”

• Problem:  The student doesn’t know how to apply this definition to real situations which require the measurement of area and/or perimeter.
• Problem:  The student may think, “Since perimeter measures the outside edge, then area means to measure the inside edge.”
• Problem:  Students confuse the two terms.

Ideas:

• Brainstorm with students (with your guidance) examples of the need for area and perimeter. Use these scenarios as you solve concrete or pictorial examples.
• Area:  garden, room size, ceiling tiles, carpet, rug, floor tiles, football field, tv screen, wallpaper, wall paint, etc.
• Perimeter:  picture frame, fencing, floor trim, wallpaper border, bulletin board border . . .
• Show this diagram which emphasizes the concept that area measures the entire inside surface using squares of different sizes (cm, inch, foot, yard, mile), while perimeter measures the rim / edge / around an object or shape usually using a ruler, tape measure, or string. • Try this project: Use graph paper and one inch tiles (color tiles) to concretely make shapes with a given area.  Example, “Build a rectangle with an area of 24 tiles.” Specify there can be no holes in the rectangle — it must be solid. Students can experiment while moving the tiles around. Then trace the rectangle and cut it out. With this practice, students are also focusing on arrays and multiplication.  Did they find 4 ways? (2 x 12, 3 x 8, 4 x 6, 1 x 24)? Did they find out a 3 x 8 rectangle is the same as an 8 x 3 rectangle (commutative property)?
• Note:  Make sure students stay on the given lines when tracing their shapes and cutting. I found this to be difficult for many students even though I said explicitly to “Stay on the lines when you trace.”
• NO – this is not a solid rectangle. No holes allowed.

• Similar to the above:  Use the tiles and graph paper to create irregular shapes with a given area – meaning they don’t have to be rectangular. Give the directive that tiles must match at least one edge with another edge (no tip to tip accepted).  And, same as above — no holes in the shape. You can even assign different areas to each small group.  Compare shapes – put on a poster or bulletin board.
• Using the same shapes made above, determine the perimeter.  I suggest placing tick mark to keep track of what was counted. With this lesson, students will hopefully realize shapes with the same area do not necessarily have the same perimeter. Advise caution when counting corners or insets – students usually miscount these places. • Try this project:  Design a bedroom with dimensions of (or square feet of): ___________ Include a bed, and at least one other piece of furniture or feature (nightstand, dresser, desk, shelf, chair, closet, etc.)
• The student can use smaller scale graph paper with 1 square representing 1 square foot.
• Let them have time to “work it out” and practice perseverance as a rough draft before making their final copy.
• Together, use the square foot construction paper pieces to determine an appropriate size for a bed and other furniture. Students must think of it from an overhead perspective. When I did this with a class, we settled on 3 ft. x 6 ft bed for a total of 18 square feet. We did this by laying the square foot papers on the floor to see in real life what this size looked like.
• Label the Area and Perimeter of each item in the bedroom.
• The items in the bedroom could also be made as separate cut-outs and arranged / rearranged on the “floor plan” to see all of the different ways the room could look.
• Students in 3rd grade might want to use whole units, while 4th and up might be able to use half-units. • On a test, area answer choices would include ” ____ square inches” or “inches squared” or “inches²”  Answer choices for perimeter will omit the word “square.”

Misconception #2:  A student hears this: “To find area, multiply the length times the width.”

• Problem:  Student doesn’t know their multiplication facts.
• Problem:  Student doesn’t know which dimensions are the length and width.

Ideas: • A good guide:  The length is the longest side. The width is the shortest side.
• The length and width are two adjoining sides (not opposite sides).
• Show how to partition a rectangle into squares.  If a rectangle had a dimension of 4″ x 2″, then show them how to make 4 columns and 2 rows.  Watch how they do this.  For many students, they would draw 4 vertical lines inside to make the columns and 2 horizontal lines to make the rows.  This would obviously result in a 5 x 3 set of squares – but students don’t always check.  Do they know — “To divide a rectangle into 4 columns, I only need to draw 3 lines.” Now, counting squares isn’t the most efficient method – but one that might help students who struggle with the concept or with multiplication facts. • Some rectangles are too large to draw squares in. If students don’t know their multiplication facts, the rectangle can be partitioned into 2 (or more) smaller rectangles using facts they do know.  Then the area of each smaller rectangle is added together to find the total area. See picture below for an example.  Again, students have to connect to geometry to understand that opposite sides of a rectangle are equal.
• Example:  A rectangle has dimensions of 8 x 7.  Using facts more readily known, break one of the dimensions into 2 addends (such as breaking 7 apart into 5 + 2). Partition the rectangle into 2 rectangles and use the 5 + 2 to label one side (instead of 7).  Using the concept of the distributive property, the student is calculating this:  8 x 7 = 8 (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 sq. units. Misconception #3:  A student only sees 2 given numbers on a picture of a rectangle and doesn’t know whether to add them or multiply them.

• Problem:  The student doesn’t know the properties of a rectangle that apply to this situation — that opposite sides are equal in measurement.
• Problem:  The student doesn’t see how counting squares can help calculate the area as well as the perimeter.

Misconception #4:  A student hears this:  “Record your measurement for area as square inches and the measurement for perimeter as inches.”  Note: This applies to use of units such as cm, feet, meters, yards, miles, and so on.

• Problem:  The student doesn’t understand the difference between square and non-square measurements.

Misconception #5:  Students think there may be a relationship between area and perimeter. They may think all shapes with the same area have the same perimeter.

• Problem:  This means if one shape has an area of 12 square inches, and the perimeter is 16 inches, they might think all shapes with an area of 12 have perimeters of 16 inches.

Next post:  Ideas to address Misconceptions #3, 4, and 5 and links for area and perimeter activities!!     STAY TUNED