# Geometry Websites

by C. Elkins, OK Math and Reading Lady 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.)

1. 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.

2. “Cubes” at NCTM’s site (Illuminations):  This one is perfect for volume and surface area.

• 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.

# Division Basics Part 1: Arrays and Area Model

by OK Math and Reading Lady Division seems to be the hot topic with classes I have been visiting lately, so I thought I’d focus on that for now. Let’s look at some of the basics.  Students as young as first grade actually start thinking about division when working on fraction standards such as:  Determine fair share — equal parts. Most students have had practical experience with dividing sets of objects in their real life to share with friends, classmates, or family (cookies, pizza, crayons, money, pieces of paper). So now our job as teachers is to relate this real-life experience with the division algorithm.

Much like multiplication, there are different aspects of division children should get familiar with:

• Arrays
• Equal Groups
• Repeated Subtraction
• Number lines
• Skip counting

In this post, I will break down the benefits and uses for arrays (and the related area model) to help children understand division (and how it is related to multiplication). There’s a fun FREE game (Block-It) at the end of the post.

What is an array?  An array is a rectangular model made up of rows and columns.  When an array is constructed, the factors are represented by the number of rows and columns.  So, do your students know the difference in a row and column?  (Rows go horizontally, while columns are vertical.)  These are important math terms students should be using.

• Give students experience constructing arrays with manipulative objects (tiles, chips, cubes, etc.):
• You can be specific, such as: “Build an array using a total of 12 tiles. Put them in 3 rows.  How many columns did you create?” In this scenario, there is only 1 way to show this array. Students would be modeling 12 ÷ 3 = 4. Twelve is the dividend (the total amount you started with). The # of rows is the divisor (how it was divided).  The quotient is the result (in this case the # of columns).
• You can also be a little more open ended such as:  “Build an array using 12 tiles. Is there more than one way to do this?” If students are given the opportunity to explore, they hopefully find arrays such as 3 x 4; 4 x 3; 2 x 6; 6 x 2; 1 x 12; or 12 x 1. Students would be modeling 12 ÷ 4; 12 ÷ 2; 12 ÷1, etc. • Give students experience drawing arrays:
• You can be specific or open-ended (as above).
• Children can free-hand draw or use grid paper.  If using grid paper, then these can be cut out and displayed as “Different ways to divide 12.” • Give students experience using pre-drawn arrays:
• Students should label the sides of the array with numbers.
• Use the numbers shown to determine the fact family.  Example:  3 x 4 = 12; 4 x 3 = 12; 12 ÷ 3 = 4; and 12 ÷ 4 = 3
• After the array is made, ask questions or explore more such as:
• How many 3’s are in 12? (count the columns)
• How many 4’s are in 12? (count the rows)
• Circle the rows and / or columns to see the groups more easily.
• Help children make up story problems to match the array:  “I have 12 desks that I need to arrange in 3 rows. How many desks will be in each row?” or “I need to put 12 books equally onto 3 shelves. How many books will go on each shelf?

# Geometry Part 8: Area and Perimeter (cont’d)

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.

# 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.”

# Geometry Part 3: Composing and Decomposing

by C. Elkins, OK Math and Reading Lady 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).

Refer to Geometry Part 2: van Hiele levels to determine if the activities you are choosing are appropriate for Level 0, 1, or 2 students.

One Inch Color Tiles: 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)?

# Volume: Concrete activities to increase understanding (Grades 3-6)

I have worked with several groups of 4th graders lately to build rectangular prisms as a way of learning more about volume. Typically students know the formula (length x width x height), but often lack the strategies or spatial ability to solve problems seen only in picture (2D form). Concrete (hands-on) experiences help cement knowledge when abstract formulas may pose difficulty. And . . . it’s always fun to “play” while doing math!! Here are some observations regarding students’ difficulties:

• Students often resort to counting the visible cubes, not realizing there are others on the back side – which can’t be seen on a 2D representation.
• Students are unsure which dimensions are the length, width, and height.
• Students lack multiplication skills.
• Students don’t know the purpose of finding volume (other than counting the cubes).
• Students are often confused when constructing prisms when one of the dimensions is 1. They weren’t sure this was even a possibility until they saw what it looked like (after building it!).

Some possible solutions:

Give students multiple opportunities to build 3D rectangular prisms:

1. Length is the longest side on the base. Width is the shortest side on the base. The height is how tall it is.
2. Use this variant of the L x W x H formula:  (Area of the base) x Height or (L x W) x H. With this mindset, the students need to find the length and width dimensions first. Finding the area of the base first helps them visualize the bottom layer. Then the height just means the number of total stacks or layers (with all of them matching the area of the base).
3. Give students specific dimensions such as (5 x 3) x 2.
• Using connecting cubes, build the base (bottom layer) first and determine the area (5 x 3 = 15).
• Then build 1 more layer just like it so there is a total of 2 layers (the height).
• Through this experience, students learn what a 5 by 3 base looks like . . . and that each layer of the height has the exact same area. It’s actually several layers stacked on top of each other.
• To complete this prism, compute the area of the base (5 x 3) and then multiply it by the height (2). So (5 x 3) x 2 = 30 cubic units. • This experience shows why the measurement is stated as cubic units (because cubes were used).
• Students may also see another way to solve the problem is to add the area of the base 2 times (15 + 15). Of course, multiplication is more efficient, but seeing the addition solution helps them realize each layer is the same.
• Get this FREE Prism Building Activity and FREE Volume game for building rectangular prisms from me (click on links). For the game, you just need the recording sheet and 3 dice per pair of students.
• Don’t have connecting cubes? Check with KG or 1st grade classes!

# Using Color Tiles to Measure Area of Irregular Shapes

by C. Elkins

I have heard from a few 4th grade teachers that a new standard is difficult for their students to grasp. It is 4.GM.2.2: Find the area of polygons that can be decomposed into rectangles.

I have a couple of suggestions which help students with a concrete-pictorial-abstract progression approach to this problem (which is more developmentally appropriate).

1. I have attached an activity which involves the use of 1” color tiles to partition off irregular shapes and then determine the area of each smaller rectangle. It’s free and in 2 parts:
2. I located an excellent reference which shows pictorially how to do this step-by-step. It also includes good information about perimeter. It is:  Area and Perimeter