· Activity
Name: Perimeters, Areas and Fractions
· Objectives:
Students
will discuss the differences between
areas and perimeters, and will chart data.
The students will also find ratios, percents, and fractions from the
pictures. The idea of one and two-dimensional
units, if presented, may be stretched to the next level, three dimensions, by
asking the students what would happen if the blocks were stacked.
· EALR/Standards:
1.1 understand and apply concepts and
procedures from number sense.
1.2 understand and apply concepts and
procedures from measurement.
1.3 understand and apply concepts and
procedures from geometric sense.
2.1 investigate situations.
5.1 relate concepts and procedures within
mathematics.
· Materials:
Pattern
blocks
White Paper (Legal size)
Rulers
Pencil
Coloring utensils (stamp pads can be
used to speed up the drawing part)
· Teacher
Notes
o
Prerequisites for the learner:
None
o
Teacher hints for the activity:
This activity works best on a block schedule. It can be used on regular schedule but it needs a couple of days.
The three example area/perimeter questions for the students to draw are set up so that two of the pictures have the same area and two have the same perimeter. This allows for class discussions on the differences between areas and perimeters and makes the charts a little more interesting.
In deciding what the area and perimeter restrictions will be, you should always try the numbers first, or remember: “odd area = odd perimeter, even area = even perimeter.” This leads to a good extension question of “why does an odd area require an odd perimeter when using the triangle as the base block with an area of 1 and perimeter of 3?” This can also be extended to the Euler formula of V – E + F = 2 for polyhedra, where V = vertices, E = edges and F = faces.
o
Introductory questions:
**Do not give answers to these. Let the students share their ideas with the
class, but do not correct them except for question number two. They will want to revise their answers as
they work through this activity.
If I have 100 feet of fencing, how much
room will my dog have to run?
Is the fencing the area or
perimeter? How about the “room?”
Is there a correlation between area and perimeter?
o
Wrap-up questions:
If I have 100 feet of fencing, how much room will my dog have to run?
If my dog has 100 square feet of room, how much fencing is needed?
What is the minimum fencing? Maximum?
What are the corresponding shapes of the dog pen?
o
Solutions:
Introductory questions: Room depends on the shape of the pen.
Fencing is perimeter and “room” is area.
No simple correlation.
Wrap-up questions: Depends on the shape of the pen.
Minimum would be a circle.
Maximum would be an infinite fractal.
o
Assessment suggestions:
For the drawing of the shapes, give one grade for the entire group. If everyone in the group is correct, then they get all the points. This will encourage the students to have their partners check their work before they do a final drawing.
For the fractions and decimals, verify that the columns add to one. For the percents, they add to 100.
Allow students to post their papers around the room and then have the students verify that their classmates’ pictures and charts are correct. If the students have done everything as a group, have each group pass their papers to another group. This group then verifies the papers, keeping their thoughts secret. The papers are then passed again until every group has verified every paper. At this time, the groups can hand in their results which say which papers had errors. Points can be given for every group that correctly found errors, and for original correctness.
· The
Activity:
Pass
out pattern blocks to groups of four.
Allow a couple of minutes for the students to explore if they have never
used the blocks before.
1) To
begin the activity, have students identify the different shapes and the
relationships between their areas. (It
might be best to leave out the square and the tan rhombus.) These relationships work best if the small
triangle is considered to have an area of one and a perimeter of three. (The idea of square units for area and plain
units for perimeter should also be addressed at this time with emphasis on
one-dimensional perimeter versus two-dimensional area.)
Once
the relationships are known, students may begin creating pictures according to
specific rules. Examples:
1) Area of 15 any perimeter 2) Perimeter of 12 any area
3) Area of 20 perimeter of 12 4)
Area of 20 perimeter of 16
5) Area of 26 perimeter of 16
(Use
the first two as examples and assign the last three)
Students
should make and then draw a picture that fits each rule and label it
appropriately. (The three pictures will
fit on a legal size piece of paper folded into thirds) The group should try to come up with four
different answers.
During
the drawing of the pictures, the following questions should be addressed:
1) Can there be holes in the middle of the
shape?
2) Is there a limit on how many pictures can be
drawn for each rule?
3) Do
two triangles fit together on one side count the same as a blue rhombus?
4) Is area or perimeter more restraining in you
picture?
5) Can the blocks be stacked? What happens if yes?
Once
the students have drawn their pictures and answered the questions, post the
pictures around the room.
2) The
second part of this activity is for students to compare the different pictures
and to look at the color ratios for each rule.
Begin by choosing one rule for the students to compare (i.e. perimeter
12 area 20) then, have the students build a chart to show the number of each
shape used to build each picture that follows the rule. From the chart, students should be able to
find the ratio of the different blocks’ areas.
Questions like the following should be addressed for each of the three
pictures:
1) If we replaced every piece with green
triangles, how many triangles would we use?
2) Using our green triangles as the base
number, in the stamped picture, what are the fractions of the different pattern
blocks?
3) What are the percent ratios?
4) Can the picture be built using only one
color? (Other than green triangles) If
yes, how many different colors work?
This
same idea can be used to find perimeter ratios. On the chart, have students count the number of sides exposed on
the perimeter for each color. Remember
that the long side of a trapezoid counts as two.
Once the area and perimeter charts have been made and the ratios found, a further investigation is to look for relationships between the area ratios and the perimeter ratios.
· Assessment
material:
Verify on each student’s chart that the
columns sum to one for the fractions and decimals, and 100 for percents. Also ask groups to exchange papers and to
verify the new papers.
· Extensions:
Ask
the students: “Why does an odd area require
an odd perimeter when using the triangle as the base block with an area of 1
and perimeter of 3?” Look for a pattern
with the faces, edges and vertices.
Remember that if a shape is made of two triangles, then there are two
faces, fives edges and four vertices.
Three triangles means there are three faces, seven edges, and 5
vertices. This can then be extended to
the Euler formula of V – E + F = 2 for polyhedra, where V = vertices, E = edges
and F = faces.
This lesson can also lead into a lesson on
fractals. Ask the students how they can
use a set area and get the largest perimeter.
Great examples include the Koch snowflake and the Mandelbrot set.