1) 25 Colored Squares

Date: Fri, 24 Jan 1997 19:41:14 -0800
From: Laurel Drake ldrake3@ua1vm.ua.edu
To: uc@mathforum.org
Subject: area and perimeter

I do an activity with 7th grade students (I think this will work as well
with older students).  I give 25 flat colored squares to each student or
each small group of students.  I ask them to start with 12 of the squares
and answer these questions using the 12 squares:

1)  use all 12 of the squares to form as many rectangles as you can.  You
must use all 12 squares to form each rectangle. Make a table with 3
columns:  dimensions of rectangle, area of the rectangle, perimeter of
the rectangle.  (So for the 1x12 rectangle,the area is 12 and the
perimeter is 26).

2) Now make a similar table for 16 squares, 20 squares, 24 squares and 25
squares.

3)Discussion questions:  Can a rectangle have a larger numeric value for
area than for perimeter?  Can a rectangle have a larger numeric value for
perimeter than for area?

Subsequent activity:  Use paper with 1 cm x 1 cm grid marked to cut out
the rectangles for the 24 squares (1x24, 2x12, 3x8, 4x6).  Let them paste
the rectangles on another piece of paper and label them with area and
perimeter.  Ask, as the perimeter decreases for the same size area, how
does the rectangle change shape?  Also, if a farmer wanted to build a 24
sq. ft. area and wanted to minimize the amount of fence to buy, which of
these rectangles would he choose as his model?  Why?

I did this activity with my class and they seemed to like it.  Hope it
helps you.

Your math friend,
Laurel Drake

2) Fences, Rugs, and Gardens
Date: Sat, 25 Jan 1997 17:04:47 -0800 (PST)
From: Judith Haemmerle jhaemmer@whitecap.psesd.wednet.edu
To: Jeannette Flores (uc@mathforum.org)
Subject: Re: teaching area and perimeter

Jeanette, I hope you got some good suggestions.  Here is my contribution.

Even in urban Seattle, I get good results with "perimeter is fences, area
is rugs".  In fact, you can find ads in the paper for "area rugs".  This
is a lot more concrete than footsteps and all that.

 Design some imaginary fences
(get a fence book out of the library - Sunset publishes a good one) to go
around your room, bring in a flyer from a local
hardware/lumber store (our locals are Eagle, Home Depot, etc - their
flyers usually have fence prices) and price it out. Then "cover"  the
floor with "grass".  Start by laying sod (get a square foot price), and
cost it out.  Then sprinkle grass seed - get one bag and use the square
foot coverage from the bag to calculate cost for seed (using the room
area, of course) and compare seed and sod.

If this is working, you can go on to design a whole garden.  Raised
bedsneed to be outlined with 2x6's, so you'll need the perimeter of each.
Then you will have to figure out how much fertilizer it takes to cover the
area in your raised beds.  If you really want to get intyo it, try
covering the beds with 3" of compost!  How many square yared do you need.

Of course, it would be great to have an actual garden plot, but you can
have fun simulating the whole thing on the floor of your room with masking
tape.  design paths and a drip irrigation system.  Add pictures of the
things you are growing.  Then when they have the basics down, divide the
room up and assign teams garden plots to lay out with chalk or tape again
- free choice designs or challenges like Elizabethan knot gardens
(circles!  Circumference!  Pi!)

Gardeners on your faculty will cheerfully bring in catalogs and other
props.  Gardeners are mostly fanatics (can't you tell?) and will probably
be full of more ideas.  And this is the perfect time of year, too.

If this is more of a project than you want, yooou could invest in a 1'
wide roll of something cheap - waxed paper, shelf paper.  Mark out the
perimeter of your room with it, marking off
feet as you go.  Then have them figure out how much of it it will take to
cover the floor.

Good luck!  Let me know how it goes.

Judith Haemmerle
Billings Middle School
Seattle, WA

3) Geoboards
Date: Sat, 25 Jan 1997 22:11:50 -0500
From: "Guy F. Brandenburg" guyfbran@erols.com
Organization: DCPS&NCTM
MIME-Version: 1.0
To: Jeannette Flores (uc@mathforum.org)
Subject: Re: teaching area and perimeter

There are great manipulatives known as geoboards, kind of like pegboardw
with pegs in the holes. Most of the math supply houses have them. You will
want one that can go on the overhead projector. You use rubber bands to
connect the pegs to form various geometric figures. One way to explain the
difference between area and perimeter is to talk about a floor: the moulding
that goes around the outside of the floor is a good example of a perimeter
(="around-distance" in Greek). The tiles or carpet or floorboards that you
stand on is the area.  General math students in high school can be really
difficult. I've been there. Somehow you have to connect, show how the math
is useful and INTERESTING. In my opinion, it is a waste of your time and
theirs to try to teach them the stuff that they should have learned in
elementary school but didn't. They don't really want to hear it. The least
imaginative among them will (if you let them) spend an entire period adding
fractions wrong (example: 1/2 + 2/3 = 3/5 or some such thing). I hope you
have a decent textbook. There are good ones and bad ones. Teach them some
algebra, geometry, probability, programming, and statistics. 

Good luck.

 4) Fields and Fences

Date: Sun, 26 Jan 1997 17:56:24 -0800 (PST)
From: Dexter Horton dhorton@cks.ssd.k12.wa.us
To: Jeannette Flores (uc@mathforum.org)
Subject: Re: teaching area and perimeter

        I teach grades 3/4 and have worked as high as grade 6.  I have
always used geo-boards and graph paper or dot paper.  I always teach both
together.
        I have the students make a field around a pasture.  How long is
the wire around the field, and how many squares of grass will there be?
Wire can be strung with square corners only, no diagnals.
        I have given a group color tiles to make their fields.  The number
of *inches* around the outside is the perimeter and the number of tiles is
the area.
        There's nothing really unique here, but I think it works because I
teach multiplication by having the students make arrays with the same
color tiles and graph paper.  We do multiplication, area and perimeter and
then go back to multiplication.  Somewhere in the middle of this, the kids
discover that it's all the same, with different stories told to describe
how you're arranging the graph or tiles.
        In time, students learn to visualize math.  I havn't taken *Math
in the Minds Eye* yet, but I love the title, and I have taken the 2
courses for younger students.

        -Dex