Inside, outside, and all around

Students will distinguish between perimeter, area, and volume. They will use tangrams and graph paper to create two-dimensional figures that will be measured for area and perimeter. By creating layers of centimeter cubes, the students will explore the concept of volume.

A lesson plan for grades 4–5 Mathematics

By Angeli Jarman

Learn more

• Tangrams This site as a template for teacher to make their own set of tangrams.

Related pages

• Tangrams: While reading Grandfather Tang's Story by Ann Tompert, students will be using tangrams to create the animals that the fox fairies are turning themselves into in this story.
• Area and perimeter unit: This unit will introduce and practice the concepts of area and perimeter. This unit uses resources of Shodor Education Foundation, Inc. Permission has been granted for the use of the materials as part of the workshop “Interactivate Your Bored Math Students.”
• Modeling volume: This activity helps the students see how the volume of something includes the third dimension (width or depth) which is different from area. This activity also helps the students "prove" that the volume formula actually works. Students will already know that the volume of a rectangular prism is found by multiplying the object's length, width, and height. By using the blocks as models of volume, the students should come to realize that volume can be calculated simply by multiplying the area of the base by the height of the rectangular prism. Thus, they will come to realize that there is no need to try and fill the entire box with the tiny 1cm cubes, they can simply fill the bottom (to see how many cubes are there) and figure out how many rows there will be and multiply.

Related topics

• Learn more about area, geometry, hands-on, mathematics, perimeter, surface area, tangrams, and volume.

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Learning outcomes

Goals:

• Students will distinguish between perimeter, area, and volume.
• Students will explore how one factor affects the other.

Objectives:

• Students will calculate the perimeter of shapes and express in appropriate measurement units.
• Students will calculate the area of shapes and express in appropriate measurement units.
• Students will calculate the volume of shapes and express in appropriate measurement units.
• Students will describe their findings as to how one affects the other.

Teacher planning

Time required for lesson

2 Weeks

Materials/resources

• graph paper
• coloring pencils
• notebook paper and pencils
• math journal
• string and tape and rulers
• centimeter cubes
• individual sets of tangram pieces (These can be made out of paper or purchased from Bender-Burkott)
• overhead projector set of tangrams (If you do not wish to purchase these, they can be made from transparent folders that come in colors of blue, yellow, red. They are very inexpensive. The colors will show through on an overhead projector. I like to use different colors so that the students can see the difference in the shapes)
• 3×5 index cards

Technology resources

• computers with internet access
• overhead projector

Activities

Day 1

1. This will be an exploring day. The students will be given individual sets of tangrams to work with.
2. The students will first attempt to make the original square with their tangram pieces. After the students have had time to attempt this on their own, the teacher will then make the original square on the overhead using teacher tangram set.
3. The discussion will then revolve around perimeter for Day 1. The students will then trace the square on graph paper. They should be able to slide it onto their graph paper or they could have graph paper on their desk to be building on. We will then calculate the perimeter.
4. The teacher will then create several shapes using tangrams on the overhead and demonstrate finding their perimeter. The students will then be asked to make at least 5 more shapes using their tangrams. They will trace each on graph paper and determine the perimeter.

Day 2

(Teacher will demonstrate this method with overhead tangram set)

1. The students will be asked to choose one of the shapes made from tangrams on Day 1 (not the square). They will use string to outline the shape. Tape string down at vertices. Cut to the correct length.
2. Then they will remove the string and measure it. If you used centimeter graph paper, you will need to measure the string in centimeters. My goal here is to help the students see that the string is really one-dimensional. We will repeat this process with another shape created from Day 1.
3. The students will then be asked to reflect on what they learned about perimeter in their math journals. Using Math Goodies, the students will answer the 5 questions involving perimeter.
4. The students will then write 5 questions of their own using the same format on 3×5 index cards. (One question per card and they should solve on the back) They will be directed tohttp://www.funbrain.com/poly/index.html to play the game dealing with perimeter.

Day 3

1. Quickly review perimeter. Brainstorm ways we use perimeter (putting up a fence, framing a picture, etc.). Today we will deal with the inside. Tell them this is area. We will use our graph paper shapes from Day 1. Using their colored pencils they are to number each centimeter square that is inside their shape. We will number only whole squares. Pieces of squares will be colored as we pair them with other pieces to make whole squares. (Each of the little pieces that are paired together to make one square will be colored the same color. This will make it easier to count the area.
2. They will then complete this process with all of their shapes that were traced on graph paper. When this has been completed we will talk about how to write the answer. We will discuss that measurement is involving two dimensions - length and width. Therefore we write it in units square.
3. The students will reflect in their math journals about what they have learned concerning area.

Day 4

1. Access shodor.org’s Perimeter Explorer. Today the students will be preferably using the computer lab so that the students would all have access to this site. You could also use the game Area Explorer.
2. Using the help choice under the title will be helpful in understanding the rules and how to play each of these games. The students will need to have pencil and paper with them to complete this activity. They will divide their paper into three columns labeled Area, Perimeter, and Correct.
3. The student will adjust the area and then calculate the perimeter. They will need to record their answer each time before checking it on the computer. Then in the correct column, record yes or no. They will need to complete 25 trials. Check scores and record.
4. Students will then analyze their data and make some generalizations as to how area and perimeter are interrelated. They will need to respond in their math journals.

Day 5

1. Students will be given access to centimeter cubes. The teacher will then give them shapes to create matching the description.
• Example: Create a shape having an area of 25 centimeters squared. (It takes them a few times to realize that will be using 25 cubes since you can’t cut them in pieces. But I never tell them. They have to figure it out.)Create a shape having a perimeter of 14 centimeters. Now fill it and calculate the area.
2. Continue with similar problems until the teacher is satisfied with student understanding. Using the computer lab again, have students go to Shape Explorer. (This site has great directions by clicking How under the title.)
3. Repeat process as in Day 4 with 25 trials. The students will need to make 4 columns today: Perimeter, Correct, Area, Correct. They will need to record their answer and then check score. Respond in the correct column yes or no.

Day 6

Today we will introduce volume. Quickly review perimeter and area.
1. Have students to each make a 4×4 square using centimeter cubes. Discuss the perimeter and area of this. Now have the students to work in groups of four. Each will stack their 4×4 square on top of each others to make one.
2. Discuss with them how many cubes are in each layer. Therefore, how many cubes total? Explain that this is the volume. It is what is inside just like area, but this time there are three dimensions - length, width, and height. Therefore, we write our answer in units to the third.
3. Practice a few more like this. Students will respond in their math journals as to what they have learned about volume.

Day 7

1. Students will return to the computer lab and go to Surface Area and Volume. This game deals with three-dimensional figures. The “How” section is a great tool for directions. The students will also need the accompanying worksheet to complete this activity. The teacher may want to make the worksheet in advance from this site.
2. The students will adjust the width, depth, and height of the rectangular prism. They will need to keep a record of their 25 trials. This page should include 5 columns: Width, Depth, Height, Volume, (Blank for now).
3. After the students have filled in the chart, we will discuss the findings. The last column will be for formula. In our discussions, we will get to how volume is calculated (LxWxH). Then they will check their volume from the game by using the formula. Math journal responses will be made.

Day 8

1. The students will need this Triangle Explorer Exploration Questions worksheet. Teacher may want to have it prepared in advance for the students.
2. The students will be drawing triangles on graph paper to answer these exploration questions. Then the students will go to Triangle Explorer. Directions can be found at Help. The students will make 25 trials recording their answers as Type of Triangle (Scalene, Isosceles, Equilateral, Right), Area, and Correct.

Assessment

CATEGORY: 1. Excellent 2. Good 3. Satisfactory 4. Needs Improvement

Mathematical Errors:

1. 90-100% of the steps and solutions have no mathematical errors.
2. Almost all (85-89%) of the steps and solutions have no mathematical errors.
3. Most (75-84%) of the steps and solutions have no mathematical errors.
4. More than 75% of the steps and solutions have mathematical errors.

Use of Manipulatives:

1. Student always listens and follows directions and only uses manipulatives as instructed.
2. Student typically listens and follows directions and uses manipulatives as instructed most of the time.
3. Student sometimes listens and follows directions and uses manipulatives appropirately when reminded.
4. Student rarely listens and often “plays” with the manipulatives instead of using them as instructed.

Mathematical Concepts:

1. Explanation shows complete understanding of the mathematical concepts used to solve the problem(s).
2. Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s).
3. Explanation shows some understanding of the mathematical concepts needed to solve the problem(s).
4. Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written.

Mathematical Reasoning:

1. Uses complex and refined mathematical reasoning.
2. Uses effective mathematical reasoning.
3. Some evidence of mathematical reasoning.
4. Little evidence of mathematical reasoning.

Strategy/Procedures:

1. Typically, uses an efficient and effective strategy to solve the problem(s).
2. Typically, uses an effective strategy to solve the problem(s).
3. Sometimes uses an effective strategy to solve problems, but does not do it consistently.
4. Rarely uses an effective strategy to solve problems.

1. Explanation is detailed and clear.
2. Explanation is clear.
3. Explanation is a little difficult to understand, but includes critical components.
4. Explanation is difficult to understand and is missing several components OR was not included.

Supplemental information

Comments

This plan was adapted from a series of activities, lessons, and discussions from Project Interactivate which is a part of www.shodor.org.