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.
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.
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.
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.
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.”
24 sq. inches (8 long by 3 wide)
24 sq. inches (6 long by 4 wide)
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.
24 square inches. Each edge matches another full edge.
NO – no holes or tip-to-tip allowed. A full edge must touch another full edge (not half as shown in the top and bottom).
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.”
Composing and decomposing 3D shapes should help your students become more familiar with their attributes. Here are a few activities to help. With emphasis on hands-on methods, examining real 3D shapes may help students find edges, vertices, and faces better than pictorial models.
Nets of 3D shapes are the least expensive way to get a set of 3D objects in each child’s hand, especially since most classrooms just usually have 1 set of plastic or wooden 3D shapes.
Build cubes and rectangular prisms using blocks or connecting cubes.
Construct / deconstruct prisms using toothpicks, straws, coffee stirrers, craft sticks, or pretzel sticks as the edges. For the vertices, use clay, playdough, gum drops or slightly dried out marshmallows.
Lucky enough to have a set of tinker toys? Or Magna Tiles? (We got our grandson some Magna Tiles and he loves them! These tiles have magnetic edges which can hook together in an instant. Creating a cube, rectangular prism, pyramid, etc. is easy! They are kind of expensive, but very versatile and creative.)
Teach students how to draw 3D shapes. When composing a 3D shape, a student becomes more aware of the 2D faces, the edges, and the vertices they are drawing. Plus, if needed the student can draw the 3D shape on paper to assist them if taking a computer based assessment. Here is my tutorial (below), but I’ll also include a couple of good websites in case you are 3D challenged. Click HERE for the pdf of the templates below.
Observe how students count the edges, vertices, and faces. If they are randomly trying to count them, they likely will be incorrect. When needed, show them how to be methodical with their counting (ie: When counting the edges of a cube, run your finger along the edge as you count. Count the top 4 edges, then the bottom 4 edges, then the 4 vertical edges = 12.)
One of my favorite lessons regarding decomposing shapes is when teaching students (5th grade and up) how to measure surface area.Click HERE for the free pdf guide for creating the rectangular prisms shown below. It includes a blank grid so you can create your own (all courtesy of http:illuminations.nctm.org using their “dynamic paper” lesson). Continue reading →
Composing and decomposing geometric shapes (2D and 3D) should be centered around concrete and pictorial methods. In this and upcoming posts, I will illustrate some methods using various manipulatives and line drawings which help students take a shape apart or put shapes together. If you refer back to Geometry Part 1: The Basics, all grade levels KG-5th have standards dealing with this issue. Some of the experiences I plan to share will also help students relate to multiplication, division, fractions, area, and other geometry concepts (such as rotations, reflections, slides).
1. Can you make a larger square out of several individual squares?
Level 0 students will be using the visual aspect of making it look like a square.
Level 1 students will be checking properties to see if their squares are indeed squares (with the same number of tiles on each side).
Level 2 students will be noticing they are creating an array (ex: 3 x 3 = 9) and perhaps learning about squared numbers. 3 squared = 9. They might be able to predict the total number of tiles needed when given just the length of one side.
2. How many rectangles can you make using 2 or more squares? (Level 0-1)
Level 1: Are the green and blue rectangles the same size (using properties to determine)?