## Rectangles and Parallelograms

- Lesson

Students use dynamic software to examine the properties of rectangles
and parallelograms, and identify what distinguishes a rectangle from a
more general parallelogram. Using spatial relationships, they will
examine the properties of two-and three-dimensional shapes. This
Internet Mathematics Excursion is based on an E-example from the NCTM *Principles and Standards for School Mathematics*.

To introduce this excursion, distribute the Rectangle versus Parallelogram Activity Sheet to each member of the class.

Rectangle versus Parallelogram Activity Sheet

Ask students to carefully examine the two shapes on the handout and brainstorm their similarities and differences. Elicit oral responses about the attributes unique to both the rectangle and parallelogram. While the class-wide discussion is occurring, students should record the information in the corresponding boxes on the activity sheet.

After the students have recorded the similarities and differences brainstormed by the entire class, divide them into pairs or teams of three. They should work together to categorize the attributes listed on the activity handout into groups. For example: can they categorize or group the shapes’ attributes according to length of sides, number of sides, number of angles, measure of angles, etc?

Once the teams have categorized the information, distribute the Things are Shaping Up Activity Sheet to each student.

Things are Shaping Up Activity Sheet

Explain that the students will manipulate the dynamic rectangles and parallelograms in the interactive applet by dragging the corners (vertices) and sides (edges). They should look at the shapes on the handout and mentally manipulate them before trying the activity online. In small groups, they should share their ideas surrounding this mental exercise.

Pose the following questions to students:

- Do you think it will be possible to transform the shape?
- Will the rectangle retain its attributes?
- Will the parallelogram retain its attributes?

Once they have had the opportunity to think about the manipulation, students will go to the Web site and use the applet to recreate the shapes listed on the Things are Shaping Up Activity Sheet.

As the students successfully re-create each shape, they should record a brief description describing the process they used to attain the goal. The teacher can share each of the teams’ solutions and model their problem solving strategies throughout the activity. If students are having difficulty with specific shapes, they can also record the challenges being faced. The key element to this activity is for students to clearly describe the process they use to manipulate the shapes.

The closing should be structured so that students can review and
pull together what they have learned. Include questions or tasks that
encourage students to reflect on their work. For example, you could
have students consider questions similar to the questions (as found in
the *Questions for Students*
section) after they have finished the activity. In so doing they will
consolidate what they have learned. Furthermore, this will provide an
opportunity for you and the students to assess what they have learned
and what they still want or need to understand. This will help you plan
further instruction.

After completing the online activity, encourage students to reflect on the Things are Shaping Up activity. You could closing questions such as the following:

- What was the first attribute you noticed that was similar between the two shapes?
- What was the first attribute you noticed that was different between the two shapes?
- What attributes stayed true to each shape even through the manipulation process?

**Questions for Students**

1. You can rotate or stretch the shapes, but will they retain particular features?

[When rotating the shapes, the size of the shapes, including their sides and angles, does not change.]

2. What is alike about all the figures produced by the dynamic rectangle?

[Opposite sides are parallel and congruent. All four angles are right.]

3. What is alike about all the figures produced by the dynamic parallelogram?

[Opposite sides are parallel and congruent. Opposite angles are congruent.]

4. What common characteristics do parallelograms and rectangles share?

[Opposite sides are parallel and congruent.]

5. How do rectangles differ from other parallelograms?

[The angles in a rectangle must be right.]

**Teacher Reflection**

- Were the students able to recreate the same shape in different ways?

### Learning Objectives

Students will:

- Identify, compare, and analyze attributes of rectangles and parallelograms through physical and mental manipulation.

### NCTM Standards and Expectations

- Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.

- Make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

### Common Core State Standards – Mathematics

Grade 3, Geometry

- CCSS.Math.Content.3.G.A.1

Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Grade 4, Geometry

- CCSS.Math.Content.4.G.A.2

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Grade 5, Geometry

- CCSS.Math.Content.5.G.B.3

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

Grade 5, Geometry

- CCSS.Math.Content.5.G.B.4

Classify two-dimensional figures in a hierarchy based on properties.