Summary
The use of manipulatives helps students create various geometric shapes and measure their area.
Materials
Additional Resources
Books
Math on Call, A Mathematics Handbook, by Great Source Education Group; ISBN 0-669-
45770-1
Background for Teachers
This two-day activity involves the composition and decomposition
of trapezoids and irregular polygons. A trapezoid is a quadrilateral
with exactly one pair of parallel sides. An irregular polygon is a closed
figure whose sides are not all the same length.
Before teaching this lesson, students need to know how to find the
area of squares (b x h), rectangles (b x h), parallelograms (b x h), and
triangles (1⁄2 b x h). They should also be familiar with using rulers to
measure in centimeters.
Intended Learning Outcomes
2. Become effective problem solvers by selecting appropriate methods,
employing a variety of strategies, and exploring alternative approaches to
solve problems.
4. Communicate mathematical ideas and arguments coherently to peers,
teachers, and others using the precise language and notation of mathematics.1
Instructional Procedures
Invitation to Learn
Play I Have, Who Has? with your class. You may need to remind
them how to find the area of squares, rectangles, parallelograms and
triangles.
Instructional Procedures
Day One
- Hand out Trapezoid 1 and Trapezoid Shapes 1 to one third
of your students. Do the same thing with Trapezoid 2 and
Trapezoid Shapes 2, and then Trapezoid 3 and Trapezoid Shapes 3.
- Have students cut out the shapes on their Trapezoid Shapes
pages.
- Have them find which shapes fit together to make their
trapezoid.
- Once they know which shapes make their trapezoid, have them
measure (with a ruler and using cm) the base and height of
each of those shapes. Tell them to use their measurements to
find the area of each shape.
- After they have found the areas of their shapes, ask them how
they could use the area of those shapes to find the area of their
trapezoid.
- Discuss ideas.
- Have students figure out the area of their trapezoid.
- In their journals, have students draw a trapezoid and describe
how to find the area of it by dissecting it into familiar shapes.
- Hand out cm graph paper to each student.
- Have them create their own trapezoid using squares, rectangles
and triangles.
- Tell the students to figure out the area of their trapezoids.
- Have students switch their trapezoids with a partner and find
the area of the new trapezoid.
- Have partners compare answers and discuss their findings.
- Have students complete Trapezoid Assessment.
Day Two
- On the overhead, show your students the Irregular Polygon
Overhead. Ask students if they have any ideas of how to find
the area of the polygon.
- After the Day One activity, they should realize they can break
the polygon into triangles, squares, rectangles or parallelograms.
Then they can find the area of each shape, and then add all of
the areas together. That will give them the area of the irregular
polygon.
- Have volunteers come to the overhead and demonstrate how
they could break up the polygon into different shapes.
- With an overhead ruler, demonstrate how to measure each
shape using centimeters.
- As a class, find the area of each shape, and then add the areas
all together to get the area of the irregular polygon.
- Hand out Irregular Polygons 1 to 1⁄4 of your students, do the
same with Irregular Polygons 2, Irregular Polygons 3, and
Irregular Polygons 4.
- Have students draw lines to break their polygons into triangles,
squares, rectangles, and parallelograms.
- Have them measure (with a ruler and using centimeters) each
shape, find its area, and then add them together to find the area
of their irregular polygon. Write that area in their journal.
- Have students cut up their polygon up into the new shapes they
created.
- Have them switch their shapes with a partner.
- Have partners create a new irregular polygon with those shapes
and glue it into their journals.
- Measure each of the shapes, find its area, and then add them
together to find the area of the new irregular polygon.
- Have partners compare results. They should find that the area
of the original polygon and the area of the new polygon are the
same.
- Discuss their findings. Even though the shape changed, the
area remained the same.
- In their journals, have students explain how they find the area
of irregular polygons. Have them use the irregular polygon they
glued into their journals as an example.
- Have students complete the Irregular Polygon Assessment.
Extensions
Curriculum Extensions/Adaptations/
Integration
- For students who have a difficult time using a ruler to measure,
copy their trapezoid and irregular polygons on cm graph paper.
This will be an easier way for them to find the area.
- Show students examples of floor plans for homes on the
Internet (or borrow some from a builder). Explain to the
students that floor plans are just big irregular polygons, and
in order for a contractor to figure out how much material is
needed for the house, he/she has to find the area of that plan.
Have students design the floor plan of their own dream house
on cm graph paper.
Family Connections
- Have family help the students find the area of their dream house
floor plan.
- Have students show someone in their family how to decompose
a trapezoid or polygon into different shapes in order to find the
area.
- Have students find the area of their own house.
Assessment Plan
- Informal assessment includes observation of student work, class
discussion and journals.
- Trapezoid Assessment
- Irregular Polygon Assessment
Bibliography
Research Basis
Ball, D. (1991). Whats all this talk about discourse? Professional Standards for Teaching
Mathematics. National Council of Teachers of Mathematics, 1991.
Deborah Ball defines discourse as described by the NCTM
Standards. A discussion from her classroom, along with entries from
her teaching journal, illustrate how thoughtful discourse can be used to
help students learn to discuss and understand mathematic concepts.
Bryant, V.A. (1992). Improving Mathematics Achievement of At-Risk and Targeted Students
in Grades 4-6 Through the Use of Manipulatives. ERIC Source (ERIC #ED355107).
Retrieved December 10, 2007, from http://eric.ed.gov./
This document presents a study designed to improve mathematics
achievement in grades 4-6 through the use of manipulatives. The
primary goal was to provide mathematics manipulatives that would
assist in helping at-risk and targeted students. Results indicated
improvement on test scores, report card grades, and use of mathematics
manipulatives.