MOW, MOW, MOW THE LAWN
Grades 6 - 8

This lesson will help students see that they can find the area of a rectangle by covering it with squares, or by using a formula. Students will find relationships between areas and perimeters of rectangles. Students will extend their area formulas to parallelograms and trapezoids.

"Math Works 101: Measurement: Finding Areas of Rectangles
Math Vantage 109: A World of Quadrilaterals"

Students will be able to:

• compute area of rectangles and label as square units
• compute perimeter of rectangles
• determine the relationship between area and perimeter: maximize area with a
• constant perimeter, minimize perimeter with a constant area
• write a formula for the area of a rectangle
• compute area of parallelograms and trapezoids

Per group of 4:

• 15 square post-it notes,
• 1 meter/yard stick,
• 10 feet of adding machine tape,
• one 12 inch English/metric ruler
• bag of 24 tiles
• 24 toothpicks

Per student:

• geoboard and 2 rubber bands
• 2 sheets of centimeter graph paper
• worksheet

Per classroom:

• cloth cut to one square meter
• 2 equal length ropes
• 4 sticks to make a large rectangle (with diagonals length of rope)

• area
• perimeter
• rectangle
• quadrilateral
• trapezoid
• parallelogram
• parallel

"Write down everything you know about quadrilaterals. Did anyone write 4 sides, polygon (many sided closed figure), rectangle (polygon with 2 pair of parallel sides, and 4 ninety degree angles), trapezoid (one pair of parallel sides), parallelogram (opposite sides parallel), rhombus (equal sides), square (equal sides, 90 degree angles)?

"Does anyone mow lawns? How do you determine how much you charge? Will you charge the same amount for every yard? Why not? Pass out centimeter graph paper. Show the picture of a yard. Draw your own house surrounded by a yard. Compare your drawings with the others at your table. Which would cost the most to mow? Which would cost the least? Why? " Have the students count the squares to compute the area of their yard. Ask if there is an easier way to find the area other than counting?

List other uses for area: replace the sod, carpet or other flooring. "When did your family compute area?"

Point out the cloth of the square meter in the classroom. Play 20 questions. "Look around the room to find something that appears to be bigger than a square meter. What is something that appears to be smaller than a square meter? Think of an object. Tell us your estimate of its dimensions. We will guess what you are thinking of."

To give students a specific responsibility while viewing, say,"We are going to watch a video. Draw a rectangle around any of the words you listed that you hear in the video. Listen for the use of the word formula."

BEGIN tape as Shirley and Bobby are talking to Mr. Freeman about why he is having someone new mowing his lawn, to learn about using the area to determine fee for mowing.

PAUSE tape when you hear "a penny a square meter," to point out the size of a square meter to the class. "What is a square meter?" "Point out the fabric square meter hanging in the classroom again. "There are 150 square meters in this yard."

RESUME tape until you hear Mr. Freeman say, he used a formula.

PAUSE tape to give students a specific responsibility while viewing. "Write down every formula you can think of. If you see a formula on the video that you wrote down, draw a rectangle around it. How can you measure a rectangle? How can you find out the meaning of formula? How can you find the formula needed? "

RESUME tape. PAUSE after Shirley counts out square post-it notes to try this activity with your students. "Let's use these squares to measure something at your table, like the cover of your math book. How many squares does it take to cover the front of your math book? Do we have to be exact? Cover the front of one person's book with square post-it notes. Do we have to count each square, or is there a short cut? "

RESUME tape to hear host describe a square meter. FAST FORWARD until Shirley and Bobby are through talking to the neighbor about mowing his lawn, because that does not enhance the lesson. RESUME as they begin to measure: 322, 323, 324. PAUSE to reflect on this method with your students. "What do you think Shirley and Bobby are doing wrong? Will this work? Is there a better way? "

RESUME tape until the man on crutches chooses the older boy. PAUSE to give students a specific responsibility while viewing, say "What do you think of the older boy's method? How is his method better or quicker? "

RESUME tape until you hear Shirley say "you can find the area of a rectangle by multiplying the number of squares in one direction times the number of squares in the other direction" to emphasize this point with your students.

"What is the major discovery? They can multiply! Write down what you think Shirley said for finding the area of a rectangle. "

RESUME tape until you see the formula.

PAUSE to emphasize the formula. "What is the formula? What is l and w? What units do you use? "

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Activities

Pass out bags of 24 color tiles. Show a rectangle with area 24. Record its dimensions on the spreadsheet. Now make all possible rectangles with the 24 square tiles, and record all combinations on the spreadsheet. Answer the question on the worksheet. What could we measure in square tiles? Desks, pencil bag, paper?

Pass out 24 toothpicks. Display a rectangle of perimeter 24. Record its dimensions in the spreadsheet. Complete the worksheet.

Display square foot, square inch. What could we measure? Send each group to measure something in the room with one of these units (square post-it notes, square tiles, square inches, square feet, square meter) Now use a meter stick to measure it. Which way is easier? Tape measure?

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RESUME tape through cartoon with witch. "How do you find area? What is the better technique instead of counting each tile? " RESUME tape until end. "96 m x 92 m. How do we use a tape measure, if it is not long enough? Shirley and Bobby will get $90 for mowing this lawn. Must be huge!

Show video MATH VANTAGE 109: A World of Quadrilaterals Start at Ellen Wicket talking with the paleontologist, saying 5x5 grid, 25 square feet. Grid is superimposed, to tell where everything lies. Fast Forward to longitude and latitude to show other uses of rectangular regions.

RESUME tape. PAUSE at 3-D Tetra Game and FAST FORWARD until Ellen is at carpet to review perimeter. "Is perimeter measured in square units? No, in feet, meters, linear units. "

RESUME tape to see a square foot. Point out the grass while muting the video. "How big do you think this is? One square foot."

PLAY tape through. Discuss area of parallelogram and trapezoid.

Pass out geoboards. Make a parallelogram on the geoboard with one rubber band. Make sure opposite sides are parallel. With the other rubber band, make the parallelogram into a rectangle. Find the area of the rectangle by counting the square inside it. "Can you find a shorter way to find the area of the parallelogram? Will it be the same? What two numbers can we multiply?"

Make a small trapezoid on the side of the geoboard. Use the other rubber band to make the same trapezoid next to the first, upside down, like we see in the video. "Can you find the area of the parallelogram? Notice the trapezoid is one half the parallelogram. What is a short way to remember how to find the area of the trapezoid?" Add the top and bottom. Multiply by how far apart they are. Divide by two because we found the parallelogram.

Make irregular shapes on the geoboard, and count the interior pegs and pegs on the perimeter. Try to derive a formula for area on a geoboard (Pick's formula)

Trace around your foot on square graph paper. "What is the area of your foot?" (number of squares). Use string to outline your foot. This is the perimeter. Staple it on to the drawing of your foot. "Is the length of the string the same as the number of squares? " No, these are two different kinds of measurements. One is perimeter, the other area.

Students take the classroom Polaroid camera outside, along with a sheet of paper with a rectangle drawn on it for each person. Each group of 4 takes a picture that includes rectangles. To give students a specific responsibility regarding their photo, say "While the film is developing (quickly) draw what you think you took a picture of in the box." Bring photos and rectangles back to the classroom. Hang up on the bulletin board.

Have students contact people in the construction industry (carpenters, painters, landscapers, interior decorators) to find out if they use the area of rectangles, perimeters, etc. Report back to class the next day. One could become a guest speaker.

Literature
Write the title of a book you have read on each of the square post-it notes. Find a book in your school library that is a square.

Art
Ancient Egyptian artists placed a grid of equal squares over a drawing they wished to enlarge. They copied the drawing, square by square, to a grid of larger squares. Draw a grid of centimeter or inch squares over a picture from a magazine. Number each of the pieces on the back. Cut up the picture along the grids, and give each student or group a square to enlarge on square paper. Put the pictures together by lining up the numbers, and compare to the original.

Social Studies/History
1. Instead of using feet or meters, Egyptians measured length in cubits, which was the length of the forearm of an early pharaoh. Make a cubit out of adding machine tape. (20.67" or .525 m). Measure the classroom dimensions in cubits.

2. Old Russian units were measured from fingertip to fingertip when the arms are outstretched. This was called 1 sagene (176.4 cm or 707/16"). Find out what 1 sagene would be for you, and make it out of adding machine tape.

3. Research Hero's formula for finding the area of triangles if you only know the sides. Explain the extension to find the area of any quadrilateral if you only know the sides.


Multicultural Activities
Shonga people in Mozambique build houses on a rectangular base. To make good square corners, use 2 ropes of equal length tied in the middle for the diagonals. Try to pull the ropes tight to get the 90 degree angles.

Try this alternate method: put 4 poles on the ground to make a rectangle. Measure the diagonal with rope. Measure the other diagonal, and adjust the poles until the diagonals are equal.

Science
Research "magic." Build series and parallel circuits with batteries and bulbs. Examine friction and surface area, area on a sail.

Discuss laying on a bed of nails: the more nails the better. Examine the speed an ice cube melts sitting on the table compared to piled in a bucket. This relates to surface area exposed to air. What crystals in nature are quadrilaterals?

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Resources:

Lumpkin, Beatrice and Dorothy Strong. Multicultural Science and Math Connections. Portland, Maine: J Weston Walch, 1995.

Suydam, Marilyn N. A Teacher's Guide to Math Works. Bloomington, Indiana: Agency for Instructional Technology, 1985.

Math Vantage Project . Spatial Sense Unit. Lincoln: Nebraska Mathematics and Science Coalition.

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Master Teacher: Rhonda Wanger