There are several great math websites which might help you and your students with geometry and measurement standards such as area, perimeter, volume, surface area, angles, etc. The ones I am recommending are interactive and often customizable. Check them out!! (Each title can be clicked to take you directly to the linked website.)
Geoboard by The Math Learning Center: I love the concept of geoboards to help children create polygons and measure area and perimeter. However, most teachers have ditched their physical geoboards. They are often in boxes relegated to the basement storage areas. I get it, though. They take up a lot of shelf space in the class, there aren’t enough rubber bands to go around (aka geobands), the kids misuse them or break them, they don’t stretch far enough, the pegs get broken, etc.
I think you will LOVE this app. Check out the little “i” on how to get the most use out of it, but it has 2 variations for the board size and you can show it with/without gridlines or numbers. There are different colored bands which you drag to the board and stretch to whichever pegs you need. You can shade in areas, copy, and rotate (which is helpful to see if 2 similar shapes are equivalent). There is also a drawing palette in case you want to freehand something or draw lines (and with different colors as well).
What are the possibilities with this?
Use with primary students to create squares, rectangles, and other polygons. The teacher can elicit different responses with directions such as: Make a square. Make a different size square. Make a trapezoid. Are any of our trapezoids the same?
Creations can sometimes be recorded on dot paper – although I would reserve this for less-complicated shapes.
Count the pegs around the shape to determine perimeter. The teacher might ask students to create a rectangle with a perimeter of 10 (or 12, or another amount). How many different ways are there? Be cautious with diagonal connections because they are not equivalent to vertical or horizontal connections. Think of how you can get students to discover this without just telling them.
Show the gridlines to help students determine area. Initially, students may just count the squares inside the shape. Guide students to more efficient ways to figure this (multiplying, decomposing into smaller sections, etc.).
This app is also great for creating irregular shapes in which students may decompose into smaller rectangles or triangles. Then check them with the standard formulas.
Volume: You can use the gear symbol to select the size (l, w, and h) of the rectangular prism, or use the default ones shown. Then there are 3 tools used to fill the rectangular prism: individual cubes, rows of cubes, or layers of cubes. I prefer using the layer tool to support the formula for volume as: area of the base x height. The base is the bottom layer (which can be determined by looking at the length x the width). The height is the number of layers needed to fill the prism. Once you compute the volume, enter it and check to see if it is correct.
Surface Area of Rectangular Prism:To calculate the surface area, you must find the the area of each face of the prism. Again, you can customize the size using the gear tool. I prefer this as the shapes shown randomly often are too small to see. Yes, there is a formula for surface area — but conceptually we want students to note the surface area can be thought of in three parts. With a click on each face, this app opens (or closes) a rectangular prism into the 6-faced net making it easier to see the equal sized sections:
Area of the front and area of the back are the same
Area of the top and area of the bottom are the same
Area of each side is the same
Be sure to explore what happens when the prism is a cube.
3.Surface area with Desmos: This link provides an interactive experience with surface area, using a net. This time, the three visible faces of the prism are color coded, which helps with identifying top / bottom; front / back; and side / side. The prisms on this site are also able to be changed so students can see how altering one dimension affects the surface area.
These three may be more relevant to middle school math standards. Check them out!! Also take a look at the “Resources” link (left side of web page). There are plenty of other good links for arithmetic standards as well – too many to list here. You may have to create a log-in, but it’s FREE!
Enjoy! Do you have other websites to recommend? Let us know.
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)
6 x 4
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 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)?