Dear Matt,

I’m teaching a Geometry course and I’m having trouble coming up with engaging activities when teaching the sections on quadrilaterals and their properties. I think students have learned some of this in middle school but they don’t seem to remember it so class gets boring. Any ideas?

Looking for suggestions

Response:

Dear Looking:

Since Minnesota’s 2007 Mathematics Standards are far less repetitive than most standard textbooks, your students may have less experience with geometric shapes than you expect, depending on the schedule for your district to make this transition, and how faithfully teachers have used the standards as the curriculum focus rather than a textbook.

In these newest standards, here are the places students might have experienced the geometry of two dimensional shapes and their properties:

• Grade 4: simple polygons (triangles and quadrilaterals) are identified and described
• Grade 6: angles are explored
• Grade 7: scaling, similarity, shapes on the coordinate plane, especially transformations
• Grade 8: Pythagorean Theorem, parallel and perpendicular lines

Thus students in your geometry course will have many building blocks for the study of shapes, but minimal experience with the properties of shapes. This concept of the properties of geometric figures is a major focus of the high school geometry standards. Therefore, whatever activities you plan should have as their primary goal helping students think deeply about the properties of the shapes, which properties distinguish one figure from another, and which properties shapes have in common.

Here are some principles for selecting activities that will engage students and help them understand properties of geometric shapes. Since talking is often a component of thinking, and your goal is to help each student think about the mathematics, having students work in pairs or groups is always an effective strategy. Even in a very crowded classroom two students can put their desks together in order to discuss an idea or solve a problem. This allows them a safe venue for exploring, and gives maximum practice with language and terminology. Any kind of physical movement is also excellent. Even the simple act of standing up gives students a burst of oxygen to their brains.

Though you mention quadrilaterals, these ideas mostly apply to all geometric shapes. You can adjust as desired if you will be examining only quadrilaterals. You could begin the unit by asking students to engage in a KWL activity: discuss in your group (or with your partner) what you already Know about shapes (or quadrilaterals), and what you Want to learn. Then a culminating or summary activity becomes a further discussion of what students did Learn. This could be done to summarize a daily lesson as well as at the end of the unit.

Sorting activities provide an effective means of both some hands-on work and mental processing, coupled with language to aid thinking and vocabulary-building. For example, you could create a set of shapes on cardstock for each pair or group, including shapes with the following properties: Open and closed shapes, shapes with curves, line segments, or both, convex and concave shapes (which determine whether all diagonals are inside the figure or some are outside), and simple or complex shapes (determined by whether the shape has one or more than one region inside). Some should be regular (with all congruent sides) and others non-regular. Some should have symmetries, others not. Include non-typical, unexpected shapes such as chevrons, Norman windows, zig-zags. Each shape should be labeled with a letter for easy identification and discussion. Such a set of shapes may already be available in your school or district. Some middle school teachers have “Shape Sets” that you could borrow. Once you have a set of these shapes (about 15 will make for interesting sorts) you can give them to students for a rich activity. (The first set of students to use the shapes can cut them apart; then they can be stored in envelopes for later use.)

A mix of pre-determined and student selected sorts makes an excellent activity. You can select some such as closed and non-closed, just specifying the letters to go in each “pile”, and asking students to find what each group has in common. You can see that this will help them start focusing on properties, particularly if they record their generalizations. A sort could include regular and non-regular polygons or those that have symmetries and those that do not. An especially valuable task is sorting the shapes that are polygons and those that are not, without initially telling students one set is polygons. An effective follow up to this is to have each group or pair of students create a “working definition” of polygon based on the properties they identify in each group. A class discussion that examines each group’s proposed definition gives far more depth to their understanding than simply copying a formal definition from the text or whiteboard. This activity will extend beyond one regular class period, perhaps with the various sorts the first day and the group presentations and discussion of the polygon definition the second.

Another culminating sorting activity is to ask students to create a “mystery sort”. This is one that they create that other groups will then try to determine. This could be a “walk around the room” activity, with students recording at each station what they think are the properties used for each group’s set of shapes. You could also add label cards, for example using the labels “one pair of congruent sides”, and “concave” you could sort all the shapes into four groups: those with one property or the other or both or neither.

A new and major resource for Minnesota teachers is the Mathematics Framework developed by SciMathMN and released online in Summer 2011 at: Frameworks-Polygons. This wonderful resource includes an overview of the standard/benchmark, possible student misconceptions, a classroom vignette, a list of teaching resources, ideas for assessment, differentiation, and parents. The resources for the benchmark related to properties of polygons include activities with technology resources such as Geometer’s Sketchpad. Also included as a resource in the Frameworks is NCTM’s Illuminations website: NCTM Illuminations which includes an activity called Diagonals to Quadrilaterals. This approaches the study from the various possible relationships of the diagonals. This site also contains links to other websites which may give you additional ideas.

Often students do not get much practice drawing shapes based on their properties. A good activity for this is to begin with blank paper, asking each student to fold it in 4 (or more) sections and draw a shape in one section. They then exchange papers, and draw another example of the original shape that differs in at least one way from the first. Pairs return papers, and discuss the differences. A third shape is drawn which differs from the first two, and this continues until the sheet is filled.

Another activity involves asking students to create a concept map for geometric shapes. A handout would include all possible terms. These would be cut apart, and pasted by the students onto a large chart showing relationships, subsets, etc with drawings. A related opportunity here is to explore the fact that there are two definitions for trapezoid. Students love to explore this idea, (which seems slightly subversive to students, and highlights the reality that mathematical definitions are made by people) and discover its implications on a concept map.

Finally, a great summary activity for a unit exploring geometric shapes is to create a version of the card game “I Have, Who Has”. You will need to create a set of linked cards, with enough for each student to have at least one card. On each card is an answer and a question, in this form: I Have XXX, Who Has YYY? All students begin standing holding one (or more) card(s), and the teacher or a designated student begins. For this topic, examples might be

I have a circle. Who has a quadrilateral that is both equilateral and equiangular?
The student with the following card would respond:
I have a square. Who has an equilateral five-sided figure?
The response should be
I have a regular polygon. Etc
The last person standing should have a card that ends with:
Who has a set of points equidistant from a single point?
This brings the activity to the circle, which began the activity. As before, the focus is on the properties of the shapes.

I hope this gives you some ideas for activities that focus on properties and provides a place to start. Once you begin, you will find additional resources and ideas linked to those presented here.

Happy moving, talking and thinking!

Matt

Matt Mentor, a wise and experienced teacher, offers advice about teaching mathematics topics to beginning teachers. Of course, experienced teachers can join in as well.

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