Squares Cubes Roots TN

Teacher Notes

“One thing we’ve seen is that the best learning experiences come when people are actively engaged in designing things, creating things, and inventing things - expressing themselves." (DMLCentral) 

The above statement is a good summary of what this lesson offers to students. Students greatly enjoy assembling the cubes, combining the different colours, arranging the disposition of the squares and cubes, the angle and elevation of the photo i.e. inventing their own creative design. The presentation requirement offers a further opportunity for personalisation, extending the potential for creativity, innovation and ownership.

What to expect 

Classes continue to surprise me by how varied their pictures and presentations are from one group to another. Students’ make their own visual and mental representations that, as a result, are more likely to stand a chance of being integrated into their existing knowledge, memory and understanding.  

Patterns and Conjectures

It is important to emphasise and encourage students to look for patterns and strategies when doing the squares and roots investigation. The aim is, through experimentation and collection of data, that students then experience the sense of surprise and wonder that comes with realising:

  • if they subtract consecutive square numbers they get consecutive odd numbers
  • the difference of two consecutive squares always gives their sum 
    e.g. 4² - 3² = 7 (4+3=7), 3² - 2² = 5 (3+2=5) etc. 
  • the difference of two squares always gives the product of their difference mutliplied by their sum. 
    e.g. 8² - 4² = 48 (8-4=4, 8+4=12, 4x12=48), 9² - 6² = 45 (9-6=3, 9+6=15, 3x15=45) etc. 
  • even numbers can be obtained using square numbers which are half the even number targeted e.g. 6² - 4² = 20 (6+4 is half of 20), 4² - 2² = 12 (4+2 is half of 12), 7² - 5² = 24 (7+5 is half of 24) etc. - a specific example from the previous pattern.
  • see the student example below for other interesting patterns.

Student Exemplars

Example of some students' creative approaches and strategies to the difference of two squares problem:

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