by C. Elkins, OK Math and Reading Lady

This post features 3 more area and perimeter misconceptions students often have. I have included some strategies using concrete and pictorial models to reinforce the geometry and measurement standards. Refer to Geometry Part 7 for 2 other common misconceptions.

**Also, check out some free resources at the end of this post!!**

**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.

Ideas:

- Give the correct definition of a rectangle: A quadrilateral (4 sides) with 4 right angles and opposite sides are equal.
- Give the correct definition of a square: A quadrilateral (4 sides) with 4 right angles and all sides are equal. From this, students should note that
*squares are considered a special kind of rectangle.*Yes, opposite sides are equal – but in this case all sides are equal. - Using square tiles and graph paper (concrete experience), prove that opposite sides of a rectangle and square are equal.
- Move to the pictorial stage by making drawings of rectangles and squares. Give 2 dimensions (length and width) and have students tell the other 2 dimensions. Ask, “How do you know?” You want them to be able to repeat “Opposite sides of a rectangle are equal.” With this information, students can now figure the area as well as the perimeter.
- Move to the abstract stage by using story problems such as this: Mr. Smith is making a garden. It will be 12 feet in length and have a width of 8 feet. How much fence would he need to put around it? (
*perimeter*) How much land will be used for the garden? (*area*). - Measure rectangular objects in the classroom with some square units. Show how to use them to find the perimeter as well as the area using just 2 dimensions. Ask, “Do I need to fill it all the way in to determine the answer?” At the beginning – YES (so students can visualize the point you are trying to make). Later, they will learn WHY they only need to know 2 of the dimensions to figure the area or 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.

Ideas:

- Show objects with a measure of one square unit. Some possibilities:
- One cm or square inch graph paper
- Cut some construction paper to measure 1 foot by 1 foot so students see what a square foot looks like. Put 9 together to form a square yard. Use it with some things in your classroom (door, bulletin board, desk top, table, etc.)
- Look at a map of your town to see 1 square mile in many square blocked neighborhoods.

- Use a ruler, tape measure, or string as another way to measure the perimeter of objects. In this way, they can see these items are not made of squares.
- Show shapes such as these (no squares visible). If you ask students to put them in order from least to greatest by their size, how could this be accomplished? (Remember, some students don’t have conservation abilities and think longer or taller is always more.) The only real way to determine size is to measure the area of the inside.
- Use the often forgotten geoboard and geobands (rubber bands) to create different areas (the squares) and perimeters (the rubber bands). Here is the link to an app for a virtual geoboard: Math Learning Center geoboard app (Note: I couldn’t get the rubber bands to work, so I used the pencil tool to draw the outlines.)

**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.

Ideas:

- The geoboard activity above is a way for students to see that shapes with the same area can have different perimeters.
- See picture below: Using tiles and/or graph paper, assign each group or pair of students a number with the directions to create a shape with that perimeter. Caution: connecting squares must match edge-to-edge (not tip-to-tip) with no holes inside the shape. Once they make different shapes with the same perimeter, they can also determine the area. They will notice shapes with the same perimeter don’t necessarily have the same area. Students enjoy the challenge to see all of the different ways they can make a shape with the given perimeter.
Here are some area and perimeter resources I like:

- The geoboard app mentioned above (free): Math Learning Center virtual geoboard
- K-5mathteachingresources.com (free) Fencing a Garden problem
- K-5mathteachingresources.com (free) Designing a Zoo Enclosure problem
- K-5mathteachingresources.com (free) Square Units
- K-5mathteachingresources.com (free) Jack’s Rectangles problem (area and distrib. property)
- K-5mathteachingresources.com (free) Area of Rectilinear Figures
- K-5mathteachingresources.com (free) Rectangular Robot (perimeter activity)
- Laura Candler (free @ TPT) Mystery Perimeters (great for determining missing dimensions on irregular rectangular shapes)
- Math Coachs Corner ($5 at TPT) Area and perimeter task cards