Rectangles and Squares Activity

Teacher Lesson Plan

This activity is aligned to NCTM Standards - Grades 6-8: Number and Operations, Algebra, Geometry, Measurement, Problem Solving, Reasoning and Proof, and Communication and California Mathematics Standards Grade 7: Number Sense #1.4, Measurement and Geometry #1.1 and Mathematical Reasoning #1.1, 1.3, 2.2, 2.4, 2.5, 2.8.

Building a bridge between technologically representing the squares and rectangles and the actual physical squares and rectangles is important. Visiting the problem using both techniques addresses a variety of learning styles, offers the interest of bringing the abstract into the concrete and includes the interest of interacting with the computer while investigating, discovering, forming hypotheses, drawing conclusions and benefitting from the quick feedback a computer provides. Once the students have had these experiences it is important to arrive at a synthesis by spending the time necessary to internalize the concepts. Manipulatives can provide space for group work, computers for individual explorations, and synthesis can take place during a full-class discussion followed by each student demonstrating their individual understanding using the writing process.

Using Technology

    Exploring the areas of squares and rectangles:

Open Squares and Rectangles by Nathalie Sinclair.

1. Explore area of squares and rectangles.
2. Note the lengths of the two sides - what happens when they are equal?
3. Note the effect of changing the angle.
4. Note where the dots (of the isometric dot paper background) are in relation to the vertices of the square or rectangle.

   Summarize your explorations. What did you find out?

    Directed tasks #1:

1. Make a square with an area of 1 sq. unit. What is the length of the sides?
         Can you make a rectangle that has an area of 1 sq. unit?
2. Make a square with an area of 4 sq. units. What is the length of the sides?
         Can you make a rectangle that has an area of 4 sq. units?
3. Make a square with an area of 9 sq. units. What is the length of the sides?
         Can you make a rectangle that has an area of 9 sq. units?

   Can you hypothesize anything from that short exploration? Explain.

    Directed tasks #2:

1. Adjust the angle to measure 45 degrees. Extend the length and width so that they reach the dots. What figure have you made and what is the area?
         Can you make a rectangle that has an area of 2 sq. units with length and width both integer values?
2. Can you make a square with an area of 3 square units? Explain what you did.
         Can you make a rectangle that has an area of 3 sq. units with length and width both integer values?
3. Can you make a square with an area of 5 square units? Explain what you did.
         Can you make a rectangle that has an area of 5 sq. units with length and width both integer values?
4. Can you make a square with an area of 6 square units? Explain what you did.
         Can you make a rectangle that has an area of 6 sq. units with length and width both integer values?

    Charting the investigation:

We have considered squares and rectangles with areas of 1, 2, 3, 4, 5, 6, and 9. Fill in the following chart:

What patterns do you notice?

1. Which numbers are prime?
2. Which numbers are square?
3. What is the first number that is not prime but whose root is irrational?
4. What would be the next such number?

    Directed tasks #3:

1. Can you construct a rectangle with an area of 6 sq. units with length and/or width of non-integer values?
         Explain what you did.

Let's consider the following:

It is interesting to note that the squares with side lengths equal to 1, 2, and 3 and corresponding areas of 1, 4, and 9 have side lengths that are rational numbers. Squares with areas of 2, 3, 5, 6, 7, and 8 have side lengths that are irrational numbers.

Using Manipulatives

    Introducing the activity:

Pass out the following materials for each student:

• geoboard
• rubber bands
• isometric dot paper

    Using geoboards:

Instruct the students to construct squares with areas of:

1. 1 square unit
2. 2 square units
3. 3 square units
4. 4 square units
5. 5 square units
6. 6 square units
7. 7 square units
8. 8 square units
9. 9 square units

    Findings:

On the isometric dot paper, record your results from the geoboard investigation.

Synthesizing the Activity

At this point, students have investigated the problem using technology (manipulating the square/rectangle) and manipulatives (using geoboards).

Ask the students to summarize their investigations using the following vocabulary:

• square
• side length
• rectangle
• width
• length
• area
• rational number
• irrational number

Extensions/Resources

    Game

"Give me a number, any number, and let's see whether we can write it as a fraction!"

Square Roots and Other Radicals
Square Roots

Integers, Rational Numbers, Irrational Numbers - Ask Dr. Math FAQ
Pi and Irrational Numbers - Ask Dr. Math Archives
Rational and Irrational Numbers - Ask Dr. Math Archives

Roots by George W. Bright and Susan E. Williams