Return to Mathematics IndexTravel Triangles

Sarah Barrett Mars Hill School

5916 West Lake Street

Chicago IL 60644

(312)287-0019Objectives:

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.Materials:

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.Strategy:

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 >90^{o}, <90^{o}, or equal to 90^{o}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. Two options: 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. Instant boomerang! 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.Performance Assessment:

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.