Illuminations: Sorting Polygons

# Sorting Polygons

 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.

### 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

### Instructional Plan

Teacher Notes

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

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.

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.

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

 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?" 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.

### NCTM Standards and Expectations

 Data Analysis & Probability 6-8Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations. Geometry 6-8Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

1 period

### Activities

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