Grade Eight
Strand: Geometry/Measurement
Topic: Polygons

Instructional Notes

• Geometry represents the world around us. It is therefore important to keep making connections constantly between activities carried out in class and the outside world.

• It is important to continue the manipulation of solid objects so that the students can develop a better understanding of geometry concepts.

• Activities involving the classification of triangles will allow the students to familiarize themselves with the properties of triangles. Discuss with students the reasons for two classification systems. (One is based on the lengths of the sides and the other on the sizes of the angles.)

• This activity can be carried out by the entire class or it can be done in groups.

• The polygons for the suggested activity should include irregular shapes, a variety of different quadrilaterals and triangles, as well as pentagons, hexagons, etc.

• You may wish to extend the activities and assist students in relating mathematics to everyday life by having them identify, compare, and debate the merits of shape in present and past architectural construction methods and decoration features such as the golden rectangle. The golden rectangle is one where the ratio of its length to its width is about 1.618:1. The resulting rectangles were considered to be appealing to the eye and many architectural designs and artistic works use this ratio. A link can be made with ratio and proportion.

• Have students draw five different rectangles and then devise a numerical measure of "squareness" that would allow them to rank their rectangles from the one most like a square to the one least like a square. Have them justify their choice. (CCT, COM)

• One possible pair of irregular polygons that are similar but not congruent are two rectangles where the dimensions of one are twice those of the other.

G/M-16 Russian nesting dolls can also be used as an example of similar shapes.

• G/M-7 Have students use pattern blocks, linking cubes, or graph paper to construct similar polygons. Ask them to record in a journal their knowledge about similarity. Have them draw two similar triangles, measure the corresponding angles and the corresponding sides, and determine any relationship that may exist.

• G/M-19 Help students to make the link between scale drawings and ratios. This is a good place to integrate the objectives in the two strands.
  
  
• G/M-20 Enlarge and reduce polygons using a photocopier and then have students find the scale factor.

• Look at the scale factors that are used in road maps and geographical maps. Ask the students to find an appropriate scale factor to use in making a drawing of the classroom.

• At the Saskatchewan Royal Museum in Regina, display cases show representations of various regions. Many of these displays use the scale 1:10 which means that 1 cm represents 10 cm or 10 cm represent 1 m. Have students find out the length of an average adult gopher and determine what the length of a gopher would be in the display case.
  
  
• G/M-23 A pantograph can be constructed as seen in the diagram to the left.
  QRST must be kept as a parallelogram; i.e., QR = TS; QT = RS. P is the pencil. In this case, it is best if the pencil can be fixed at this point by making the hole just big enough for the pencil to fit in tightly.
  Q, R, S, and T are all loose joints and T should have a pointer beneath it. Use a nut and bold at this joint and file the bolt to a point. U has to be held fixed when the pantograph is in operation so either a screw or a sharp point is needed.
  How to use: Fix U to a drawing board, place the diagram to be copied under T and a sheet of plain paper under P. Carefully trace the outline with the pointer at T and the enlarged drawing will follow at P.

  What would happen if you fixed P and put the pencil at U?
  What about fixing U, putting the pointer at P and the pencil at T?
  
  

• G/M-25 At the Middle Level, it is sufficient to agree that shapes are congruent if there is a way to carry 'this one' over to 'that one' and see if they match. We can also talk of translating, rotating, and reflecting objects so as to try for such matchings.
  
  
• G/M-26 Constructions can be done using paper folding, mira, set squares, compass and straightedge, or appropriate software. After learning about each of these methods, students should be allowed to choose the method they prefer. They should be expected to describe their process orally, in an informal manner, but need not be required to write any step-by-step explanations for their constructions. (COM)

• Activities that involve reflections and rotations reinforce the concepts of congruency and symmetry.

• The students can demonstrate translations, reflections, and rotations in arts education and physical education activities.

• Pentominos are all 2-dimensional figures formed by the combination of five adjacent congruent squares. They can be used to examine the following concepts:
  • congruency: the students construct their own pentominos (12 in all);
  • symmetry: find the axes of symmetry of the different pentominos;
  • tessellation: translate, reflect, or rotate a pentomino to create a design;
  • problem solving: use one or more pentominos to make up rectangles and squares; and,
  • measurement: find the perimeter and the area.

• For more information, refer to "Pentominos Revisited", by Barry Onslow, in Arithmetic Teacher 37(9), 5-9.

• G/M-36 The last example can be linked to the Algebra strand where students graph on the coordinate plane.