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

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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

Pre-activities

Students will brainstorm ways we measure.

Activities

Day 1

1. This will be an exploring day. The students will be given individual sets of tangrams to work with. A good site for some tangram puzzlers for the students can be found at http://tangrams.ca/inner/makeset.htm. This site also gives the teacher a way to make paper sets if they do not purchase the individual sets.
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.
4. 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.
5. 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. Therefore the answer is _ centimeters or whatever unit of measure you were using. 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 http://www.mathgoodies.com/lessons/vol1/perimeter.html, 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. It is the outside. 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

Day 6

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

http://www.shodor.org/interactivate/discussions/SurfaceAreaAndVolume/
This is a good discussion on surface area and volume.
http://www.shodor.org/interactivate/discussions/ShapeExplorer/
Provides steps to help teacher/student with area and perimeter.
http://www.mathgoodies.com/lessons is a good source for teachers to create worksheets to use for area, perimeter, and volume.
http://tangrams.ca/inner/makeset.htm would provide tangram information - how to make them and puzzles to create with them.

Related websites

Area Explorer: http://www.shodor.org/interactivate/activities/AreaExplorer/. Game for students to explore area and perimeter.
Perimeter Explorer:
http://www.shodor.org/interactivate/activities/PerimeterExplorer/. Game for students to explore area and perimeter.
Shape Explorer: http://www.shodor.org/interactivate/activities/ShapeExplorer/. Game for students to explore area and perimeter.
Triangle Explorer: http://www.shodor.org/interactivate/activities/TriangleExplorer/. Game for students to explore area of different types of triangles.
Surface Area and Volume: http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/. Game for students to explore surface area and volume.
Lesson on perimeter for teacher use: http://www.shodor.org/interactivate/lessons/Perimeter/
Lesson on area for teacher use: http://www.shodor.org/interactivate/lessons/Area/
Lesson on length, perimeter, and area for teacher use: http://www.shodor.org/interactivate/lessons/LengthPerimeterArea/

Lesson on surface area and volume for teacher use: http://www.shodor.org/interactivate/lessons/SurfaceAreaAndVolume/
Discussion for teacher on area and perimeter: http://www.shodor.org/interactivate/discussions/ShapeExplorer/
Discussion for teacher on surface area and volume: http://www.shodor.org/interactivate/discussions/SurfaceAreaAndVolume/
Worksheets/Questions: http://www.mathgoodies.com/lessons. This site offers some questions for students as well as the capability for teachers to generate worksheets.
Site for students to play games dealing with area and perimeter: http://www.funbrain.com/poly/index.html
A site for tangram patterns and puzzles: http://tangrams.ca/inner/makeset.htm

Comments

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