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Sorting Polygons

  • Lesson
6-8
1
GeometryData Analysis and Probability
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Location: Unknown

Students identify and classify polygons according to various attributes. They then sort the polygons in Venn Diagrams, according to these attributes. Extensions to fundamental ideas about probability and statistics are also included. This lesson was adapted from an article written by Carol G. Williams, which appeared in the March‑April 1998 edition of Mathematics Teaching in the Middle School.

Teacher Notes 

Included are patterns for twenty-six polygons, as found in the Shapes activity sheet.

1072 polygons

These polygons are numbered so that teachers can pull out certain ones for use in an activity and so that students can identify a particular polygon more easily. Teachers can make durable figures by copying the polygons onto card stock and then laminating them.

pdficon  Shapes Activity Sheet 

Each group of students needs one set of polygons, and the teacher should have a special set of the polygons made from transparencies. Each individual set can easily be stored in a reclosable plastic bag.

Below are overhead masters and images of the Venn diagrams that teachers need to copy and laminate for the students and also make into transparencies for themselves.

pdficon  Various Venn Diagrams (One Circle, Two Circles, Three Circles, Nested Circles) 

The terms and phrases in the Sorting Cards activity sheet should also be made into transparencies and cut into cards for use by the instructor. Alternatively, the instructor may use an overhead marker to write the following terms on the transparencies.


  OPPOSITE SIDES PARALLEL
  OPPOSITE SIDES CONGRUENT
  AT LEAST ONE OBTUSE ANGLE
  AT LEAST ONE RIGHT ANGLE
  ALL SIDES CONGRUENT
  ALL ANGLES CONGRUENT
  TWO CONSECUTIVE SIDES CONGRUENT
  PARALLELOGRAM
  QUADRILATERAL
  REGULAR POLYGON
  OPPOSITE ANGLES CONGRUENT
  PENTAGON
  HEXAGON
  OCTAGON
  RHOMBUS
  ISOSCELES
  TRAPEZOID
  CONCAVE POLYGON
  CONVEX POLYGON
Terms and phrases for describing sets 

Activities 

One simple activity is to place the Venn Diagram with One Circle overhead on the overhead projector and put a phrase, such as "all sides congruent," on the circle. Ask students to separate all the polygons, placing them either inside or outside their circle. When the groups have finished, the teacher can ask different groups to state the number of a polygon that they have placed in their circle, and the class can agree of disagree and present reasons to support their comments.

Using the Venn diagrams with more circles increases the level of difficulty. One choice for the Venn Diagram with Two Intersecting Circles overhead might be "quadrilateral" for one circle and "opposite sides parallel for the second. Besides highlighting the idea that all parallelograms are quadrilaterals, this choice lends itself to using the terms set and subset. Instructors can show a proper subset using the Venn Diagram with One Circle Inside Another Circle overhead. This activity allows teachers to use quite a bit of set notation and terminology.

The terms and phrases chosen by the teacher from the Sorting Cards activity sheet make the activity more or less difficult. If the identifying phrase is "opposite angles congruent," student can discuss how one determines opposite angles or opposite sides, or whether that term has meaning for polygons other than quadrilaterals. Polygon 16 can be used to address difficulties that students have with qualifiers, such as at least and all, as in the phrases "at least one obtuse angle" and "all angles congruent." The phrase "two consecutive sides congruent" will elicit discussion about whether this term means any two consecutive sides congruent or every pair of consecutive sides congruent. The class can proceed to discuss whether the phrase itself should be modified.

Teachers can create another activity by showing students a Venn diagram with the polygons already sorted according to categories known only to the teacher. The students must determine the correct phrases that apply to each circle. Alternatively, a group of students can devise and display a secret sort criterion that the other groups must try to discover.

You may wish to introduce the Shape Sorter tool for students to practice sorting using an online tool.

Extensions
  1.  One can design activities that use sets of polygons to reinforce ideas about probability and statistics. For instance, when holding a bag containing the diagrams, a teacher can ask, "What is the probability of drawing at random a regular polygon from this set of polygons?" A more difficult question is "Given that you have selected a triangle, what is the probability that you have selected an isosceles triangle?"
  2. Another task asks students to sort the polygons into groups determined by the number of sides. They construct a bar graph showing the frequency of three-sided polygons, four-sided polygons, and so on. Having students draw a circle graph showing the percent of polygons in each category will enrich their understanding of percents and angle measurement.
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Learning Objectives

Students will:

  • Precisely describe, classify, and understand relationships among types of two-dimensional objects using their defining properties
  • Use Venn Diagrams to sort polygons according to certain attributes
  • Understand and apply basic concepts of probability

Common Core State Standards – Mathematics

Grade 7, Stats & Probability

  • CCSS.Math.Content.7.SP.C.5
    Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.