Areas with Pentominoes/Graph Paper

Areas with Pentominoes/Graph Paper/
Geoboards
Reporting Category
Overview
Related Standard of Learning 6.11
Objectives
The student will determine if a problem situation involving polygons of four or fewer sides
represents the application of perimeter or area.
The student will apply formulas to solve problems involving area and perimeter.
Activity A: Areas with Pentominoes
Materials needed
Scissors
Sets of pentominoes in an envelope or plastic bag (see “Patterns for Pentominoes” below)
“Patterns for Pentominoes,” (two pages), one copy for each student
“3-by-5 Puzzle Solutions,” one copy for each student
“4-by-5 Puzzle Solutions” one copy for each student
Instructional activity
1. Students may be in small groups or in a large group. Ask students to carefully cut out a set of
pentominoes from the “Patterns for Pentominoes” activity sheets.
2. Invite students to take some of their pentominoes and see if they fit together like jigsaw puzzle
pieces. Point out that there are usually several ways to combine pentominoes to fill a given area.
For example,
can be made by combining
and
like this
or in several other ways.

3. Have students fill in the 3-by-5 and 4-by-5 areas with pentominoes. Ask them to keep a record of
their work on a separate piece of graph paper. Have them compare their results with the 3-by-5 and
4-by-5 Puzzle Solutions.
4. Ask students how many square units there are in each pentomino. Ask them to find the pentomino
with the largest perimeter and the pentomino with the smallest perimeter. Discuss how shapes with
the same area can have different perimeters.
Sample assessment
Ask students how many square units there are in all 12 pentominoes combined. (60) Have them
design a symmetrical shape with an area equal to 60 square units and then see if it can be completely
filled in with all twelve pentominoes.
Activity B: The Perimeter Is 24 Inches. What Is the Area?
Materials needed
24-inch paper strip (collar)
38 1-inch cubes
1-inch block graph paper
1
2
-inch block graph paper
Instructional activity
1. Organize the students into teams of two or three.
2. Ask the teams to find out how many different rectangular arrays they can make that have a perimeter
of 24 inches. Have them make the arrays with the cubes and have them check the perimeter with a
24-inch collar. What is the area of each array?
3. Once they find a rectangular array, have them draw its representation on the graph paper and write
the perimeter and area for each array.
Activity C: Change The Area
Materials needed
Square geoboards,
Rubber bands
Instructional activity
1. Organize students into small groups.
2. Have students copy this figure on the geoboard and onto dot paper,
labeling its area and perimeter.

3. Have students change the figure to make another shape that has the same area and a larger perimeter,
recording it on dot paper with its area and perimeter.
4. Have the participants change the figure to make another shape that has the same area and a smaller
perimeter, recording it on dot paper with its area and perimeter.
Sample assessment
Have the students make three more shapes that have different perimeters but the same area,
recording them on dot paper.
Have them record their observations in their math journal.
Activity D: Areas of Polygons on the Geoboard
Materials needed
Square geoboard and rubber bands, one set for each student
Instructional activity
1. Have students construct this rectangle on their geoboards and find its area. They should record it and
its area on the geoboard dot paper.
2. Have students construct this right triangle and find its area by first counting squares in the
corresponding rectangle. Record.

3. Have students construct this figure and find its area by counting squares.
4. Have students form a 2 by 4 rectangle and construct this figure within it. They should then find its
area from the area of the rectangle and record.
5. Have students construct this square on their geoboards, find its area, and record.
6. The students should next construct this square on their geoboards, find its area, and record.

7. Have the participants construct this triangle, find its area, and record.
Follow-up/extension
Have students find the area of these triangles using their geoboards. Discuss what they discovered.
Next have students construct this triangle on their geoboard and find its area. Hint: They should start
with the area of the rectangle enclosing it.
The area of the previous triangle was 4 1 square units. Have students construct as many other
2
triangles as they can with the same area, but with different shapes. They should copy each one on
their geoboard dot paper.
Homework
Have each student make a complicated polygon on his/her geoboard and find its area. Then they
should give it to another student to find the area.

One-Inch Graph Paper

One-Half-Inch Graph Paper

Geoboard Dot Paper

Patterns for Pentominoes

Patterns for Pentominoes

3-by-5 Puzzle Solutions

4-by-5 Puzzle Solutions