19951996 National Teacher Training Institute
Master Teacher: Barbara S. Massicotte, Vilas School, Alstead, NH
Subject: Science, Mathematics
Grades : P, E
Series: Math Works Episode 9: Measurement: The Difference Between Perimeter and Area
Objectives
Materials
Previewing Activities
Focus for Viewing
Viewing Activities
Postviewing Activities
Action Plan
Extensions
Resources
Frameworks
Worksheets
Overview
This lesson focuses on exploring the relationship between area and perimeter of rectangles. In a threeday lesson plan (120 minutes), students will explore making rectangles of different areas and a constant perimeter, extend their knowledge of rectangular dimensions by using body measurements to form rectangles, and make models of their bodies using Cuisenaire rods with proportional accuracy. Prerequisites for this lesson include students having a general knowledge of rectangles, linear measurement (standard or metric), simple fractions, ratios, multiplication, addition, and graphing.
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Learning Objectives
Students will be able to:
 identify the terms perimeter, area, cubit, dinar, ratio, and proportion
 tell the difference between area and perimeter
 recognize that the shape of rectangles affects the area
 recognize that perimeter and area are measured differently (linear units versus square units)
 compute the perimeter and area of rectangles
 measure their arm span and height accurately in inches or centimeters
 use graph paper to draw their bodies with proportional accuracy and frame their bodies in a rectangle or square
 build models of their bodies with proportional accuracy
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Materials
 measuring tape or yardsticks (meter tapes or meter sticks)  one per student pair
 color tiles  25 per student pair
 pipe cleaners  3 or 4 per students pair
 Cuisenaire rods  one set per student pair
 Plastitak  used for hanging posters  a l x l inch piece per student
 calculators  one per pair
 graph paper  3  4 sheets per student
 ditto paper  3  4 sheets per student
 rulers  one per student
 activity worksheets  one per student
 idea wheel sheets  4 per pair of students
 pieces of string cut into 1foot lengths  1 per pair of students
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Previewing Activities
Before viewing the video Math Works: Measurement: The Difference Between Perimeter and Area, arrange your students in pairs.
Say: "Has anyone ever heard of the word perimeter? What things come to mind when you hear it? What about the word area? Can you think of ways in which this word is used? Hint: use your 'math' mind."
To ascertain your students' knowledge of area and perimeter, pass out four idea wheel sheets per pair of students. Have students write the words area and perimeter in the ovals. Tell them to brainstorm together what these words mean and any connections or ideas that come to mind when they think of these terms. These ideas should be written on the spokes of the wheels.
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Focus for Viewing
Say: "As you watch Math Works, Measurement: The Difference Between Perimeter and Area, I want you to listen carefully to find out what these words mean, how they are different, and what they are used for. I will pause the video at several points to give you time to write down your information on another idea wheel sheet. At the end of the video, check your before and after idea sheet sheets and compare them. Were your ideas correct? What new things did you learn?"
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Viewing Activities
Start the video, Math Works, Measurement: The Difference Between Perimeter and Area at the beginning where Duane and his sister Lisa are discussing how to decorate their mother's birthday cake. Duane runs out of candies needed to outline a star. Lisa convinces him to put the candies around the circular edge. There are more than enough because the perimeter of the star is less than the perimeter (circumference) of the circle.
In order to check vocabulary comprehension, pause the video after the host states: "Let's go back to that word perimeter. The perimeter is the distance around the outside of a shape." Give students time to copy down this definition on their second idea wheel sheet.
Say: "Listen to the man as he goes on to talk about perimeter. Listen carefully to his explanation of how you measure perimeter, what area is and how you measure it. I will pause the video again so you will have time to write down these notes on your idea wheel."
Resume the tape.
Pause after the host says: "Think about rectangles," in order to check comprehension. Students should have written that perimeter is the line outside the shape, and you measure it in linear units such as meters or yards. Area covers the surface. You measure it in square units.
Ask students to think about rectangles. Draw one on the board, four units by two units. Have students draw one on the their graph paper with sides 4 units by 2 units. Ask them to label the perimeter and the area. Some students may have already figured out what the area and perimeter are and they may write this down on their graphs as well. Check comprehension.
Say: "Let's watch the video in order to find out how Lisa and Duane plan a rectangular garden. Find out what makes the perimeter of the garden and how long it is" (a fence 12 meters long).
Resume the tape.
To check comprehension, pause after Duane says to Lisa: "How could it be bigger? Twelve meters is twelve meters" and Lisa responds with: "Yeah, but..." ask students what the perimeter of the garden is. (12 meters of fencing)
Tell students that the host will now explain how to measure area and that they should look for a formula which will help them and write it on their area idea wheel sheet. Resume the tape to help students discover that area = length x width.
To check comprehension, pause the tape after the host says: "Shape makes the difference." Elicit volunteers to tell you the area formula. Ask students to guess what the host meant by saying, "Perimeter or area can change while the other stays the same." Elicit responses.
Say: "Let's see if you are right. Listen also to find the length, width and area of Lisa's and Duane's garden." Resume the tape.
To check comprehension, pause the tape after Lisa says: "I get to plant the things..."
Say: "What was the length and width of the garden? (3 meters by 3 meters) What was the area? (9 square meters) Why did Lisa want to change the shape? (It maximizes the area; the garden is bigger.) What can you say about perimeter and area? (The perimeter can stay the same, but the area can change.) Let's watch the next segment of the video which is called The Magic Carpet. This part reviews what we learned about area and perimeter. Listen carefully. Find out how Ali Baba is tricked by Honest Abdu. Find out what Ali Baba does to get even. Write down the words cubit and dinar on another idea wheel sheet. Find out what they mean."
Resume the tape .
Pause to check comprehension after you hear Ali Baba say to Honest Abdu: "5...4...3...2...1 Lift off!"
Ask the class:
*How was Ali Baba tricked Abdu? (He sold him a carpet with a smaller area than his old one, but with the same perimeter.)
*How did Ali Baba get even with Honest Abdu? (He played the same trick on him.)
*What is a cubit? (A measurement of length. You might want to tell students that this is an ancient measure based on the length of the forearm, varying from 17 to 21 inches in an adult.)
*What is a dinar? (An oriental coin, especially gold coins of ancient Arab countries. Today it is a small monetary unit of Iran equal to one hundredth part of a rial.)
Say: In the next section, we will learn how area stays the same, but the perimeter changes. Listen carefully and take notes on your area and perimeter idea wheel sheet.
Resume the tape. You should be at the part where Duane says to Lisa: "Hi, brat!"
Stop the tape after Duane says: "Shape makes all the difference." Ask the class to explain why Duane was able to fit the ribbon around the six boxes of chocolate rabbits while Lisa couldn't. (Duane arranged the boxes so the perimeter was less than Lisa's even though the area was the same.) Say: "So, we can have different areas with the same perimeter. And, we can have different perimeters with the same area."
Now direct your students' attention to their before and after idea wheel sheets. As a review technique, have them check the accuracy of their information before they see the video compared to after. Also have them review the new information they learn with their partner. Elicit responses orally from student pairs.
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Postviewing Activities
Day 1: Pipe Cleaner Rectangles
Review the concepts of area and perimeter as shown in the video.
Say: "You've seen that we can have rectangles with different areas, but the perimeter stays the same." What examples did you see in the video? Elicit responses from the students. (garden, magic carpets) We can also have rectangles with different perimeters whose areas are the same. What example did you see in the video? (The ribbon, with a constant perimeter, tying up the area of 6 stacked boxes compared to the area of 6 boxes lined up) Can you give me some examples of perimeter in this classroom? Elicit student responses. (Border of a desk, outline of the blackboard) How about area? (area of rug, surface of a book, surface of a desk) Today we are going to investigate further the idea that the area of a rectangle can change even when the perimeter stays the same. I'm going to pass out some pipe cleaners, rulers, color tiles, a worksheet and several pieces of graph paper. (Pass out manipulatives to each group) With your partner, twist 2 pipe cleaners together so the combined length is 20 inches. Bend the pipe cleaners so they form a rectangle with sides 1 inch by 9 inches. Use your ruler to be sure your sides are accurately measured. Take the square inch color tiles and fill in the rectangle.
*How many did you use? (9)
*What is the area of this rectangle? (9 square inches) Record this information on your worksheet and draw this rectangle on your graph paper, being sure to label the sides 1 inch and 9 inches. Now, straighten your pipe cleaners and rebend them so one side of the rectangle you form is 2 inches and the other side is 8 inches. Put the color tiles inside the rectangle.
*How many fit? (16)
*What is the area of this rectangle? (16 square inches) Record this new area on your worksheet, draw the rectangle on graph paper and label the sides. Continue in the same way until you have a rectangle with the same measurement for length and width. Then, look at your data on the worksheet and look at the rectangles you drew on graph paper. What did you notice happening? (As the width increased and length decreased, the area increased.) Write a paragraph explaining what you observed. Here are some questions you might consider answering in your report: What shape rectangle had the smallest area? (The first one we drew. 1 x 9 inches.) What shape rectangle had the largest area? (The 5 x 5 inch square.) NOTE: Allow students to continue working on this project for the period. If necessary, let them have more time to write their paragraph report neatly, making sure they include their worksheet and the graphs they drew. This alternative assessment project is a rich one, for it can be done with increasing sophistication in the higher grades, using mixed/decimal numbers for the sides such as 2.5 x 8. Graphs can be drawn comparing the area and perimeter, resulting in a parabola.
Day 2: Frame Yourself
Tell the class that today they will watch a small part of the video Futures 1, #2 which stars a wellknown math teacher, Jaime Escalante. (Some students might know of him from the movie Stand and Deliver and you might want to have a short discussion with the class about him.)
Say: "This video segment is about fractions. Did you know that they can be expressed as ratios or proportions? Can you give me some examples of fractions? (Elicit examples such as 2/3 of a pie or 5/6 of the class.) Do you know what a ratio or proportion is? Write down your ideas on another idea wheel paper, brainstorming with your partner. Listen carefully to what Escalante says about ratios and write down the information on your wheel. Since this segment is so short, I will play it three times. The first time listen and watch carefully. Take no notes. The second time I will play the video segment with no sound. Watch carefully. The third time I will play it with picture and sound. Then you can take notes.
Fast forward to the part of Futures 1, Aircraft Design near the beginning where Jaime Escalante is teaching fractions and ratios to his class. Because this segment is so short, but crucial to the lesson, rewind the tape and play it again, this time without sound. This technique forces the students to focus and pay attention more carefully, actually "thinking" along and writing the script in their minds. Rewind and play it again a third time (with sound) to help students confirm their "script."
Have students explain what they learned to their partner. Ask two or three students to volunteer their ideas to the class. Possible answers are:
*Fractions can be expressed as ratios, comparing one quantity to another.
*A person's arm span can be compared to his/her height and expressed as a ratio or proportion in the form 1:1.
*The measurement (perimeter) of a person's clenched fist can be compared to the person's foot length in a 1 to 1 ratio.
Pass out pieces of string and let students experiment with measuring their clenched fist and seeing if this measurement is the same length as their foot. Ask students to comment on their findings. Is it the same length? Is it a 1:1 ratio or close to it?
Say: "Today we will be measuring our arm spans and height to see if they are a 1:1 ratio. Work with your partner, using tape measures and yardsticks (or meter tapes and meter sticks) to measure each other." (It might be fun for the teacher to do this activity too. Having an adult model the process helps the students. Also, the ratio from a fullgrown adult might be different from a child.)
Say: "Record your measurements on the graph paper I am passing out which is titled: Frame Yourself. Record arm span first, then height. Express this as a fraction. Are the numerator and denominator the same number or close to the same number? If they are, then you have a 1:1 ratio."
Next, have the students draw the length of their arm span measurement near the bottom of the graph paper, using 1 space for 2 units of length. This line will become the bottom of a rectangle. Have students draw lines up from each end of the length, representing their height. Have them connect the top with a line across (which is their arm span). They should label the length (xaxis) and put down the arm span measurement. Have them do the same for the height (yaxis). If the numbers are the same or close to each other, then their arm span/height ratio forms a square. If the arm span measurement is less than the height, they have made a tall rectangle. If the arm span measurement is greater than the height, they have formed a wide rectangle.
Have students compute the perimeter of their body rectangle and put the information on the graph. Do the same for the area.
Have the students draw themselves in the rectangle as accurately as they can, with arms outstretched touching each side of their rectangle. Remind them to have their head and toes touch the top and bottom of the rectangle. Students may have to do some other measurements, such as feet to waist, shoulders to top of head, and knees to toes so their body is proportionally accurate. Perhaps the art teacher might help you expand the lesson on body proportions.
NOTE: Expectations for this project can be simple or complex, depending on the age and maturity of your students.
Day 3: Make Me a Model
This activity is adapted from EQUALS INVESTIGATIONS UNIT ENLARGING THE BLOCK PERSON.
Students can build a model of themselves using Cuisenaire rods and Plastitak to hold the rods together. They can use the measurements they have collected from activity 2 to help them select the rightsized rods. The graph paper from activity 2 has 1/2 cm. squares and fits nicely with these centimeterbased rods.
Suggestions include helping students notice similar measurements of other body parts. Their forearm measurement is similar to their knee to crotch measurement. If they decide to have their arms down at their waist, elbows are waist length.
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Action Plan
Your school district may be experiencing overcrowding in the classrooms. I know it is happening where I teach. In fact, last year I had 27 sixth grade students and 2 adults in a room with only 505.6 square feet of available space. To justify the need for a larger classroom, I used the information gathered by a former student from a project he did on measuring the area of my room and all the contents. It showed dramatically that each child in that class had less than 9 square feet of personal space, hardly conducive to a great learning environment. Students also wrote reports on how the size of their classroom affected their learning. The final result was that two new classrooms were built.
To further involve children in a project of this nature, students could design floor plans for the new classroom, gathering data on the furniture costs and sizes. They could design a plan which makes the most of the available space for whole group meetings, study areas, a reading corner and project area, whatever the needs of your classroom may be. They could present their findings to the principal or building facilities committee, making the classroom truly their own.
This project may be adapted to designing floor plans for many places, from a children's wing of a library to an activity center at a hospital. Younger children might enjoy designing the "Ultimate in Comfort" dog house. The possibilities are endless.
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Extensions
Science:
Explore growth rates. At what ages do people experience the most growth? Is the growth rate the same for all parts of the body? Research giantism and dwarfism. What causes these conditions? Are body proportions different from the general populace?
Language and Ethics:
Read excerpts from Alice in Wonderland, Gulliver's Travels, The Borrowers, Otis Sprul, or Jack and the Beanstalk. Have children write essays about what it must be like to be proportioned very tall or very small. Discuss prejudice concerning size. Does it happen? Are short people discriminated against? Are tall people discriminated against? What would you do if you witnessed teasing or harassment of a classmate based on his/her size?
Art and Math:
Design a parquet floor made of varying sized rectangles, each section being 1 square foot. How many squares can you have in a 12 by 16 foot room? Figure the cost per square foot by sending your design to a flooring company for a free estimate. Figure the total cost for the room.
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Resources
Additional Video Resources
Math Works Episode 1: Measurement: Finding the Area of Rectangles
The Eddie Files Episode 9: Length and Area
Internet Resources
Shape Surveyor
Test your area and perimeter knowledge at this site from Fun Brain.
Shape Explorer
Make shapes and then figure out their area and perimeter at this interactive site from the Shodor Education Foundation .
Everything You Wanted to Know About Area and Perimeter
Learn about area and perimeter and then test your knowledge at this interactive site.
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NH Framework Correlations
Mathematics
1a. K12 Broad Goal: Students will use problemsolving strategies to investigate and understand increasingly complex mathematical content.
1b. K12 Broad Goal: Students will use mathematical reasoning
2a. K12 Broad Goal: Students will communicate their understanding of mathematics.
2b. K12 Broad Goal: Students will recognize, develop, and explore mathematical connections
3a. K12 Broad Goal: Students will develop number sense and an understanding of our numeration system.
3b. K12 Broad Goal: Students will compute.
4b. K12 Broad Goal: Students will develop spatial sense.
4c. K12 Broad Goal: Students will develop an understanding of measurement and systems of measurement through experiences which enable them to use a variety of techniques, tools, and units of measurement to describe and analyze quantifiable phenomena.
5a. K12 Broad Goal: Students will use data analysis, statistics and probability to analyze given situations and outcomes of experiments.
6a. K12 Broad Goal: Students will recognize patterns and describe and represent relations and functions with tables, graphs, equations and rules, and analyze how a change in one element results in a change in another.
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Worksheets
Worksheet 
Note: Worksheets are in Adobe PDF format. You will need Adobe Acrobat Reader to access them.
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