## Sorting Polygons

- Lesson

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.

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 closable 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.

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.

Venn Diagram with Two Intersecting Circles

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*.

Venn Diagram with One Circle Inside Another

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.

### 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.

### NCTM Standards and Expectations

- Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.

- Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.

### 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.