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Sarah Barrett Mars Hill School
5916 West Lake Street
Chicago IL 60644
This presentation can be adapted for intermediate and upper grades.
Students will review and demonstrate an understanding of the three kinds of
angles (by degrees). Triangles and related quadrilaterals will be explored and
their areas found with cut-outs, rearranging parts, etc., to discover and then
apply pertinent formulae. A few days to two weeks of classes will be needed.
Students need a protractor, straight edge, compass, a few sheets of
construction or plain paper, a few poster board pieces at least 8" by 10", and
scissors. The teacher supplies brightly colored pre-cut triangles large enough
for display, (at least one of each kind), a tailor's measuring tape, a
carpenter's rule, tape to post models and student made figures on the wall or
chalkboard, and a supply of construction paper to demonstrate the folding and
cutting of triangles from squares and rectangles.
These steps may be adapted as appropriate for various groups.
1. Review acute, obtuse and right angles by having students model each with
hand and arm formations. List terms with >90o, <90o, or equal to 90o
and label a diagram of each on the chalkboard.
2. Fold a rectangular sheet of paper in half diagonally while asking what kind
of triangles are formed. Have the students do the same folding of their paper
and finger-trace the right angles and then each of the other kinds of angles.
Measure the angles with protractors, name and label the degrees on each, and
draw and label each kind in their personal class notes. (Everything important
goes into class notes.) Post the models described in "Materials" and as many
student samples as practical.
3. Have the students recall how to find the area of rectangles and squares, and
find the area of a sheet of paper, the end of which will then be folded up to
obtain a square. Cut or tear off the excess. Next find the area of the
square. Then fold it diagonally to see what kind of triangles result. Compare
them to those found in step 2 after measuring and labeling their angles as
well. Cut out the triangles and shift them around until a rhombus is formed.
Do the same with the triangles formed inside the rectangle to form a
parallelogram. Post the models for this so that students can see how and why
the area of both the rectangle and parallelogram formed with the same
triangles is equal, and found using the same formula, i.e., L x W = area.
Find the area of the parallelogram and rhombus and compare with the rectangle
and square areas.
All the above figures can be shaped with both the tailor's measuring tape and
with a carpenter's rule for additional visualizing. Students can do the
manipulations described above easily with these tools. Use masking tape to
affix the measuring tape to a chalkboard temporarily to demonstrate with it.
The carpenter's rule has the advantage of rigid segments so that it can also
be used to show polygons with more and more sides up to a duodecagon, and to
elicit the observation that the more sides on a polygon the more closely it
approaches the circle.
4. Return to the triangles found within the rectangle and square. After having
found the quadrilateral areas, ask students how the area of the triangles they
found inside (step 3) can be found. (Half of the rectangle or square they are
in.) Then help them express the formula: b x h divided by 2. Use this to
find areas of several examples, (supplied on worksheets for additional
practice). To find the area of non-right angle triangles, the altitude, or
height must be given, or measured. If it is not already drawn, show students
how to draw a perpendicular from the base to the apex using protractors.
Extend the base on obtuse triangles. The perpendicular, i.e., altitude needed
will fall outside the triangle. It is critical that students see this and
practice it. Using the cut out triangles from step 3 provides the initial
experience for the isosceles and scalene triangles. Be sure to cut some
examples of obtuse triangles as well, for this purpose.
5. Have the students use a circle provided on a worksheet, or draw their own
with a compass, at least 6 or 7 inches in diameter. Use a protractor to
trisect the circumference, marking a point at each 120 degrees, then connect
them with straight line segments, to produce an inscribed equilateral
triangle. Post a few precut display models and student done samples. Have the
students measure and label angles, sides, and find the area.
To do the mini-projects below with best results, use thick enough paperboard.
a) Have students find the midpoints of the equilateral's sides and connect them
producing another triangle within. How many equilaterals result? Cut the
sides of the "outer" triangles and fold the inner sides so that the interior
triangle becomes the base when the vertices of the outside triangles are
pulled up to form a pyramid;
b) Mark the centerpoint of a circle, then trisecting points on the
circumference of a circle at least 8 inches in diameter. Draw line segments
from the centerpoint to the three marked points. Cut alongside the segments
from the circumference in to about half a inch from the center leaving about a
quarter inch on each side of the lines, curving as one approaches the center
to continue alongside the next line. Three equidistant spokes of a "wheel"
about a half inch wide should result. Hold it upright by one spoke and toss.
Optional, but a favorite, is a set of portable triangles constructed with wood
slats. Vertices are formed by attaching ends to each other with bolts which
allow the angles to be changed, but which will retain a set position with slight
tightening. For a surprise effect, attach an additional slat to one of the
triangles to form a rectangle or square. Push the top over to form a
parallelogram or rhombus with the same set of slats.
All the following may be evaluated for assigning grades.
Students will make personal kits of the four kinds of triangles, cut them
out, with protractor measured angles labelled, and will construct a mini-project
which will work as intended only if work assigned has been correctly completed.
Pre-printed centimeter grid worksheets may be used to diagram and find areas of
triangles as directed. These can be used for further practice, and to assess
students' comprehension of the concepts acquired and applied in the activities
below, along with observations of their performance during the activities.