Travel Triangles

Sarah Barrett                  Mars Hill School
                               5916 West Lake Street
                               Chicago IL 60644
                               (312)287-0019

Objectives:

     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 >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.
   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.