Exploring Maximum and Minimum Perimeters of Planar Figures
(11)
Objectives: Students will discover ideas about perimeter; that
different planar figures made up of unit squares have varying perimeters;
there are maximum and minimum perimeters that can be found using a given
number of unit squares.
Grade Level: Fourth Grade
(9)Materials: 24 1x1 unit squares (that are the same size as the
squares on the grid paper) for each student; colored pencils
(2)Note: It is important that students have some understanding
of length, width, and fractions (fractions are only in Challenge 1, unless
students come up with figures where squares cover only part of the squares
on the grids).
(12)Time Frame: This activity is designed to cover one 90 minute
period, or two 45 minute periods.
(2)Before beginning the activity, discuss the unit square. Think
of the unit square as a square where each edge of the square is equal to
one unit. (10)Students will use the unit squares to come up with
possible solutions to the problems posed.
Exercise 1
Place your 24 unit squares on your grid paper so that they create a figure. (15)You must use all 24 unit squares. The squares must touch one another in some way (you can't have any squares that don't touch any other squares). Now take a colored pencil and color in the squares on the grid that were covered by the unit squares. Look at the outer edges of the figure. How many of the grid lines surround the colored section? For example,
this figure is surrounded by 22 grid lines.
(1)(4)(5)Now, try some more figures using all 24 unit squares.
Color in the grid where you place your figures. Count the number of grid
lines that surround the outer edges of the colored figure. Draw your figures
in the grids below. If you need more grids, there are extra ones at the
end of the activity. Try to create some figures (still using all 24 unit
squares) that are surrounded by the fewest grid lines; try to make
some figures that are surrounded by the most grid lines.
Grid 1
Grid 2
Grid 3
Grid 4
Grid 5
Create a table with your data.
Grid # Sketch the Figure # of Grid Lines
Surrrounding the Figure
1.
2.
3.
4.
5.
Which figures are surrounded by the fewest grid lines? Why do
you think that is?
Which figures are surrounded by the most grid lines? Why so you
think that is?
(3)CHALLENGE 1
Create the following figures on your grid paper. Color them in.
Figure 1:
Figure 2:
Figure 3:
How might you figure out the total number of grid lines that surround
each of the above figures?
How many grid lines surround each figure? Fill in the table below.
Table continued:
(1)(4)(5)Create your own figures similar to these. Continue labeling
the figures; the next figure will be Figure 4. Add them to the table above.
How are the number of grid lines in figures 1, 2, and 3 different from
the number of grid lines in the figures that you created? Why do you think
that is?
How are the number of grid lines in figures 1, 2, and 3 similar to the
number of grid lines in the figures that you created? Why do you think that
is?
(1)(2)(3)(4)(5)(6)CHALLENGE 2 (as a class)
Divide students into groups of two. (One student can draw the figures,
the other student can fill out the chart). Every two groups will receive
the same number of unit cubes. Groups will receive 8, 12, 16, 20, 28, 32,
36, etc. number of unit cubes (or any number that has several factors).
Each group will create new figures and measure their perimeters in the same
manner that we have been doing. When you regroup as a class, each of the
two groups that have the same number of unit cubes can share their results.
Make a chart on the board so that students can see the results altogether.
(8)This way they can compare results to find out which solid (or
open) figures have the least/greatest perimeters. Ask the students what
they can determine from the results.
FOLLOW-UP
(6)(8)Discuss the term perimeter. A good way for students
to remember the definition of perimeter is to point out that it has the
word rim in the middle of it.
(13) After the class discussion:
What have you learned from this activity?
What does the term perimeter mean?
How do you think you might be able to use this information to figure
what size fence you need to put around your backyard?
After students have an understanding of perimeter, this same activity
can be modified to lead them into the concept of area.
(7)Some Possible Responses
The following are some possible responses to some of the questions asked
in the activity. These are by no means all of the possible responses.
Exercise 1
Figure A:
Figure B:
Figure C:
Figure D:
Which figures are surrounded by the fewest grid lines? Why do you think that is?
Figure D is surrounded by the fewest grid lines. This is because it has
the least number of cubes on the outer edges of the figure.
Which figures are surrounded by the most grid lines? Why do you think that is?
Figure C is surrounded by the most grid lines. This is because it has
every cube on the outside of the figure.
Challenge 1
| Figure # | # of grid lines surrounding the figure |
1 |
48 |
2 |
36 |
3 |
60 |
Use Grids Like Above:
Student Examples:
(Extensive Examples
of Student Work Will be Added Soon)