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In Search of Perfect Squares

  • Lesson
  • 1
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Number and OperationsGeometryMeasurement
Victoria Miles
Buzzards Bay, MA

In this lesson, students use geoboards to explore the relationships between the area of a square and its side length. They also gain a numeric and geometric understanding of squaring a number and envision what the square root of a number looks like.

Project the Perfect Squares Overhead. Tell students you are thinking of a very specific quadrilateral. Then show the squares. Discuss with students how squares are the most specific of all the quadrilaterals, with four congruent sides and four right angles. Talk about the measurements you can make with a square: length of a side, perimeter, and area. Analyze how knowing any one of these can help you find the others.

overhead Perfect Squares Overhead 

Have students guess why these figures are called perfect squares. Then have students consider whether there are other perfect squares. Accept all responses, without revealing the answers.

appicon E‑Example 4.2 (Virtual Geoboard)

Project the virtual geoboard. Explain to students that you are going to use a geoboard to measure the length of sides as well as perimeter and area of squares. Point out that a geoboards us rubber bands and pegs that are in perpendicular rows and columns. You may also need to explain to students or remind them of the following:

  • A unit length on a geoboard is the distance between two horizontal or two vertical pegs.
  • Area is the number of square units inside a shape.
  • Perimeter is the distance around the shape.

Clear the virtual geoboard. Stretch an elastic horizontally or vertically over 3 pegs. Ask students how many pegs the elastic is stretched across. One of the most common misconceptions with geoboards is the issue of pegs versus length. Give other examples to reinforce that the number of pegs is not the same as the length. Ask, "Can you identify the length shown by the elastic?" [2 units.] Construct this length into a square. Ask students what the area of the square is. [4 square units.]

To reduce the number of flying rubber bands, let students know that they will receive a certain number of rubber bands on their geoboards, and at the end of class, they will be required to return the geoboards with the same number of rubber bands. When students pass in their geoboards at the end of the lesson, the rubber bands should be placed back on the geoboards in the same way they were received. You might consider storing your geoboards as depicted below.

 3089 perfect squares LP1 

Distribute one geoboard to each student. Tell students to measure out lengths of 3 units and 5 units. Check for understanding.

Pass out the Perfect Squares Activity Sheet. Students work with partners to answer the questions. Check that they are using the geoboards correctly. Listen to students as they determine area. Are they counting the number of squares inside the rubber bands, or are they multiplying length by length?

pdficon Perfect Squares Activity Sheet 

Summarize key concepts by projecting the square below onto a geoboard. Ask students questions about side length and area. Review exponent and square root notation. For example, 32 = 9, and √9 = 3.

3089 perfect squares LP2 


Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., and Phillips, E. D. (2005). Looking for Pythagoras (Connected Mathematics 2). Developed as part of the Connected Mathematics Project. Upper Saddle River, NJ: Pearson Prentice Hall. 


Assessment Options

  1. Have students answer the following questions and hand in their answers before they leave class.3089 perfect squares LP3 
    • How could you prove that this figure is a square?
      [It has 4 congruent sides and 4 right angles.]
    • What is the side length of this square?
      [6 units.]
    • What is the area?
      [36 units2.]
    • How could you find the side length of a square if you know the area, but you don't have a geoboard?
      [Find the square root of the area.]
  2. Use the activity sheet as a formative assessment.
  3. Have students write a journal entry to summarize what they learned in class today. They should describe how they used their geoboards and how they measured the squares. In addition, they should define perfect squares and list the first 20 perfect squares.


  1. Having students construct squares of non-integer lengths on the geoboard prompts them to think about a new "unit length" on the geoboard other than the distance between two horizontal pegs.
  2. Have the student devise a way to use a geoboard to represent a square with a side length of 1½ units. Ask, "What is the area of the square?" You may need to provide students with a hint, such as, "You may need to design a new measure to represent a unit length."
     3089 perfect squares LP4 
  3. Ask students, "What do you think you could make with a 3D-geoboard?" [Prisms.] Have students use centimeter cubes to construct a cube with edge length 5 units. Then have them measure the surface area and volume of this cube. Ask, "If 1, 4, 9, 16, 25, …, are 'perfect squares,' what are some 'perfect cubes'?"
  4. Move on to the next lesson, How Irrational! 

Questions for Students 

1. If you know the side length of a square, how can you determine its area?

[Multiply the side length by itself, or square the side length.]

2. If you know the area of a square, how can you determine its side length?

[Take the square root of the area.]

3. What are you constructing on your geoboard?
[A square.]
4. How do you know it is a square?
[It has 4 congruent sides and 4 right angles.]
5. If the area of a square is 64, how could you determine its side length?
[I would think – what number, multiplied by itself, equals 64, or find the square root of 64. 8 units would be the length.]
Teacher Reflection 
  • Did students make the connection between the area of a square and square numbers? How do you know?
  • Were students able to relate square root of a number A with the side length of a square of area A? What evidence helped you assess students' understanding of the geometric meaning of square root?
  • At what point, if any, did you feel rushed? What part of the lesson could be shortened?
  • Which students were frustrated? What could you do next time to lower frustration levels?
  • Which students did not seem invested? What could be done to better engage them?
  • What type of pre-assessment could be done to differentiate the instruction to meet the needs of all learners?

How Irrational!

In this lesson, students use geoboards to construct non-traditional, "tilted" squares whose side lengths are irrational numbers. This lesson addresses standards in both Number Sense and Measurement.

Learning Objectives

Students will:

  • Investigate perfect squares and square roots using manipulatives.
  • Square numbers and find the square root of perfect squares without a calculator.

NCTM Standards and Expectations

  • Develop and analyze algorithms for computing with fractions, decimals, and integers and develop flue
  • Use geometric models to represent and explain numerical and algebraic relationships.
  • Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

Common Core State Standards – Mathematics

Grade 8, Expression/Equation

  • CCSS.Math.Content.8.EE.A.2
    Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.