Strategies
Journaling
Geometry involves a lot of new vocabulary for most students. By using a math journal, I want to encourage my students to write, retain, and synthesize the new words they will learn during this unit. The journal will be interactive, as we will be adding to it and referring to it during the unit. Most of the time, when the students are expected to add vocabulary to their notebooks, it will be important for them to draw diagrams to illustrate the meanings, as well as write the definitions. Vocabulary that will be included in this unit can be found in the Background Knowledge section.
Using Graphic Organizers to Classify Polygons
"Graphic organizers are powerful ways to help students understand complex ideas. By adapting and building on basic Venn diagrams, you can move beyond comparison and diagram classification systems that encourage students to recognize complex relationships." (4) This kind of document will be a work in progress, so as the students learn more about the characteristics of the different kinds of polygons, they can add more detail to their graphic organizer. It will also provide a resource for them to refer to as they work through questions on their own.
The first kind of graphic organizer I will use with my students is a Venn diagram. A Venn diagram is used to show similarities and differences between information. In math we can use a Venn diagram to show how polygons share characteristics.
When I begin to diagram quadrilateral classifications, I will start with a giant circle or oval that will encompass all of the other circles within it. This will be labeled "quadrilaterals" as all of the figures we will put in the diagram will be four-sided polygons. Within this circle I am going to have the students draw some non-specific quadrilaterals. Then we will go through each category of quadrilaterals.
I want to start with the trapezoids. This circle or oval will not overlap with any other figures or circles. The students will draw a circle for the trapezoids and label it with the name and defining characteristic (one set of parallel sides). The students will draw out three different kinds of trapezoids within the circle and label them correctly. There should be an isosceles trapezoid, right trapezoid, and a general trapezoid that does not have congruent sides or a right angle. As we will move on to parallelograms next, I would take this time to point out another way that trapezoids are related to symmetry, which is that each trapezoid can be obtained by cutting a parallelogram in half by a line through its center. The corresponding trapezoids created will be congruent under a rotation around the center of the parallelogram.
The next section the students will create is for the parallelograms. This circle or oval will need to be rather large in size because there are several different kinds of parallelograms. The circle or oval should be labeled and the defining characteristic of the parallelograms should be written underneath the heading (two sets of parallel sides). Then the students will draw an example of a general parallelogram that is not an example rectangle, or rhombus. For the curriculum used in North Carolina, a parallelogram is not considered a trapezoid. For this unit, when a trapezoid is discussed, it is considered a quadrilateral with only one pair of parallel sides.
Within the region parallelograms, the students will create two more overlapping ovals. One will be labeled rectangle and the other rhombus. The defining characteristic for the rectangle is that it has four right angles (or you may want your students to write down that the sides are perpendicular to each other). The rhombus category has all sides congruent. For each circle the students will draw an example of each of these kinds of parallelograms. In the section where the two circles overlap, the square is located. Its defining characteristics are that it exemplifies both rectangular and rhombic qualities. The students will then draw a picture of a square to complete this section of the Venn diagram.
The last section is for the kite. This section will need to overlap with the rhombus, as a kite with two sets of congruent, adjacent sides can also be a rhombus. The students will write that the kite is defined by having two sets of adjacent sides congruent. The students will draw an example of this kind of quadrilateral and complete their Venn diagram on quadrilateral classification.
The Venn diagram that the students created will be a document that they will use in this unit and for the remainder of the geometry section. Students will continue to add more information to their Venn diagram when they explore the different diagonals of each figure and how that defines that kind of quadrilateral.
The students will also use their diagrams to create written relationships between the different quadrilaterals. The students will write comparison sentences that begin with sentence starters such as; "the rhombus is like the rectangle because ___." They will also create true and false statements about the figures to share with their classmates. An example of a true or false statement that they might create would be, "all rhombi are squares." In this case the students would have to prove or disprove this statement. By drawing an example of a rhombus that is not a square, the other students could prove that that statement is false.
Student Exploration
In order to meet the needs of all of the varying abilities in my classroom, I like to provide activities that promote student exploration. Students work with manipulatives and in groups to work through problems. I will use this approach when investigating reflectional and rotational symmetries of quadrilaterals.
After learning what it means for a figure to have rotational or reflectional symmetry the students will be given the task to find how many symmetries there are in the different kinds of quadrilaterals. Students will work in pre-selected partners for this task. I believe that students should be paired up with someone that complements their learning style. For a task that is exploratory in nature, it does not benefit the students to have a partner that will go too fast or too slow for them. In their partner groups, the students will be given several different kinds of material to complete the task.
Each group will have construction cut outs of each different kind of quadrilateral (square, rectangle, rhombus, isosceles trapezoid, kite, and parallelogram). They will also get overhead sheets labeled with the varying degrees of a circle. They can use these overhead sheets to draw their figures to test for rotational symmetry. The students will work together to prove the kinds of symmetries each figure has and write down how they know. When the groups are finished, they will share together what they discovered. When the class has come to an agreement on the symmetries of all of the figures, I will share with them the diagram (see Figure 1.1), and they will record the information in their journals.
Utilizing Technology
An integral part of my mathematics program is the integration of different technologies. For this unit, I want to incorporate the use of a geometry computer program. Geometer's Sketchpad is a licensed computer program that allows students to create geometric shapes, as well as perform transformations. (5) If this program is not available at your school, as it is not at mine, you can download a free version of a similar program called GeoGebra. (6) My students will be using this program to create symmetry patterns using quadrilaterals.
Another technology that is available in my classroom is the use of the Promethean Whiteboard. (7) We will use this to have students share what they have created using GeoGebra.
Comments: