Venn Diagrams. The Sorting Polygons page in the Student Activity Book has two overlapping regions and looks similar to a Venn Diagram. Strictly speaking, Venn Diagrams for the three sets of triangles pictured in Figures 2, 3, and 4 would look like those in Figure 5. Note that these diagrams have different configurations for intersecting sets (Figure 2), disjoint sets (no elements in common as in Figure 3), and sets where one is a subset of the other (Figure 4). If Venn Diagrams are included in your local standards, sketch the configurations as shown in Figure 5 on the board for each game.
Classify Polygons Using Properties. Have students work with a partner, so that each pair of students can work with two sets of 15 Power Polygons™.
- Sort your polygons into two or three groups based on properties. Think about number and size of sides, number and size of angles, and whether sides are parallel.
- The groups can overlap in the same way that the right triangles and triangles with two equal sides (isosceles triangles) do.
- The groups can have no shapes in common as with the obtuse and right triangles.
- All the shapes in one group can also belong to the second group. For example, all the equilateral triangles also belong in the group with at least two equal sides (isosceles triangles).
Give pairs five or ten minutes to sort their polygons. As students work, talk with them about their choice of groups. How they choose to sort the shapes is up to the students at this point. However, it is important that they be able to name the properties of the shapes in each group and that they included the correct shapes based on the properties they chose.
Ask questions to find out which properties they chose to use to sort the shapes:
- What properties do all the shapes in this group have in common? In the second group?
- Why is [shape name] in that group? Show me how it has the property (or properties) for that group.
- Is [shape name] in both groups? Why or why not?
Have one or two groups share their sorting procedure.
Sort and Name Shapes Based on the Number of Sides. Identify or encourage a group to sort the Power Polygons™ by side length. Discuss this sorting process with the class.
- How many groups do you have? (3 groups)
- What are the names of the polygons in each group? (triangles, quadrilaterals, hexagons)
Model making the three shapes in Figure 6. Have partners make the shapes using their second set of polygons.
- Think of each of the three shapes you made with two pieces as just one shape. Right now we can call them the new blue, green, and brown shapes. Add these shapes to your groups.
- To which group does each of the new shapes belong? How do you know?
Students will see that the new blue shape has six sides, so it is a hexagon. The new green and brown shapes have five sides, so they will form a new group. Call the shapes in this group pentagons.
- Make another pentagon using two or more of the polygons. Add it to the pentagon group. What shapes did you use to make your new pentagon? (Possible response: I used the orange square [Shape B] and the green triangle [Shape N].)
Naming and Classifying Polygons Using Properties. Display a blue rhombus (Shape M).
- Is this shape a polygon? What properties tell you that it is a polygon? (Yes, it has straight sides that meet at the corners.)
- A regular polygon must have all equal sides and all equal angles. Is the blue rhombus a regular polygon? Why or why not? (No. The sides are all equal, but the angles are not.)
- Look at the shape charts from Lesson 9. Find other names for this shape. (quadrilateral, parallelogram, rhombus)
- What do we usually call this shape? What did you call it in other grades? (rhombus)
- What properties does it have that tells you it is a rhombus? (It has 4 equal sides. Opposite sides are parallel. Opposite angles are equal.)
Discuss names for shapes in everyday language and in math class. In everyday language, we usually use the shape name that gives us the most information. Calling the shape a rhombus tells us more than if we just called it a quadrilateral. Calling it a quadrilateral just tells us that it has four sides. Calling it a rhombus tells us that it has four equal sides. In mathematical language, we know that it belongs to all the following groups: polygons, quadrilaterals, parallelograms, and rhombuses.
Display the red trapezoid.
- What names can we call this shape? (polygon, quadrilateral, trapezoid)
- Does it belong on the parallelogram chart? Can we call it a parallelogram? Why or why not? (No, because it doesn't have two pairs of parallel sides.)
Play Mystery Shapes. Have partners work with a Sorting Polygons page from the Student Activity Book and two sets of Power Polygons™. Tell students that they are going to play a new version of the Mystery Properties game called Mystery Shapes. This time they will name the shapes that belong in Box A and Box B, instead of the properties.
Display the Sorting Polygons page.
- Watch carefully as I place shapes one at a time into the two boxes. As I place the shapes in each box, find the shapes and place them on your page.
- All of the shapes in A will share the same name. All of the shapes in B will share another name.
- Some shapes will have both names. They will go into the middle section where A and B overlap.
- Shapes that have neither name will go outside both the dotted and solid lines.
- Your job is to discover the two names. Do not guess out loud. Raise your hand when you think you know both names. When many hands are up, I will ask one person to place another shape with that name in the correct set. That person will call on someone else to place a shape with that name in the other set.
Begin placing shapes as shown in Figure 7. This game is a quick review of regular polygons. Box A has regular polygons and B has polygons that are not regular. Note that some shapes are made with two Power Polygons™. Encourage students to do the same to find new examples of polygons.
Once students have solved the mystery, ask:
- Can a polygon be in Box A and Box B at the same time? Why or why not? (No, a shape can't be both regular and not regular at the same time.)
- What other polygons can we add to B? (Any polygon but the yellow square [A]).
- What other polygons can be added to A? (The yellow square or a square made of two large green triangles [E].)
Continue playing the Mystery Shape game using the polygons shown in Figure 8 (triangles and regular polygons) and Figure 9 (parallelograms and rhombuses). Place polygons in the boxes one shape at a time until many hands are up and then have students add polygons to the appropriate boxes. Be sure to display shapes that go outside both boxes, so students can eliminate some shapes. Remind students to refer to the class charts from Lesson 9. Use the discussion prompts below following each game. Encourage students to use properties to justify their answers.
For triangles and regular polygons (Figure 8), ask:
- How did you discover the mystery shapes? (Possible responses: I recognized the regular polygons from the last game. All the shapes in A are triangles.)
- How do you know they are triangles? (They all have three sides.)
- What shapes are in both A and B? (Triangles with all sides equal or equilateral and with all equal angles.)
- What properties do equilateral triangles have that put them in both A and B? (They have three sides, so they go with the triangles. They have all equal sides and all equal angles, so they also go in B.)
Is a square also a rectangle? The geometric reasoning that is necessary to answer the discussion questions and Questions 8–9 in the Student Guide are not trivial for many fourth-graders. For example, one student explained her feelings to researchers about the fact that a square is a rectangle. She said that she believed it in her head, but not in her heart. Students' success with these questions depends on their experience with geometric concepts in previous grades and lessons.
To be able to classify a square as a rectangle, students must be able to reason about the relationships between shapes. For example, they may think, “A rectangle is a quadrilateral with four right angles, and opposite sides are parallel. A square has all those properties, so it must be a rectangle, too.” This reasoning is at Level 2 in the Van Hiele Levels of Geometric Thought (described in the Mathematics in this Unit section). Such thinking depends on successful experiences of visualization of shapes at Level 0 and analysis of shapes by their properties at Level 1. Geometric reasoning at Level 2 allows students to classify shapes into hierarchies: squares are rectangles, rectangles are parallelograms, parallelograms are quadrilaterals, and quadrilaterals are polygons. The activities in previous grades and lessons in which students identified and analyzed shapes by their properties prepare students for this level of thought.
For parallelograms and rhombuses (Figure 9), it is important to show a parallelogram in Box A that is not a rhombus, such as a shape made with the red trapezoid [K] and the small green triangle [N], and a quadrilateral that is not a parallelogram, such as a trapezoid, outside both boxes.
- How did you discover that the mystery shapes in A were parallelograms? (Possible responses: I looked on the charts. All the shapes in A looked like the ones on the parallelogram chart.)
- What properties do they have? (They have four sides. They all have two pairs of parallel sides.)
- If they have four sides, why didn't you say quadrilaterals? (I saw the trapezoid on the outside. It is on the quadrilateral chart, but not the parallelogram chart.)
- Why isn't a trapezoid a parallelogram? (Only two of its sides are parallel.)
- How did you decide on the name for the shapes in B? (They are the shapes on the rhombus chart.)
- Are all rhombuses parallelograms? Use properties to tell how you know. (Yes. All rhombuses have two pairs of parallel sides.)
- How are rhombuses special parallelograms? What is different about the shapes in A and the ones that are in both A and B? (All four sides are equal in the rhombuses.)
