|The Investigation poses questions to generate interest in various
mathematical topics from the text and encourages students to formulate and investigate their
own conjectures. One use of the investigations is for term papers in which students report on
their conjectures and the patterns they find. |
Click on the Read Me file below to open the investigation in a Word file:
Read Me - Difference of Squares Instructions (Word Format)
The Math Investigator is a data collection software program that may be used to collect data for the investigation. You may type answers onto the Word file or copy
and paste in data from the Investigator.Click here to launch the Difference of Squares Investigator
Math Investigator 2.3
DIFFERENCE OF SQUARES on the Math Investigator determines the ways, if any, that a number can be written as the difference of two square numbers.
The number 65 is the difference of the two square numbers, 81 and 16:
81 - 16 = 65
Square numbers can be illustrated as square arrays of tiles. The difference between 81 and 16 is illustrated below by the shaded L-shaped region. (39.0K) Starting Points for Investigations
- Investigate the whole numbers less that 50 to determine which numbers can and cannot be written as the difference of two square numbers. You may wish to print these numbers and their differences from the computer to find patterns. Form conjectures based on your observations. Test your conjectures on other numbers.
- Select some odd numbers and for each one find pairs of squares whose difference equals the odd number. Find a pattern and form a conjecture. Test your conjecture on large odd numbers.
- Summarize your observations and conjectures. Apply your conjectures to determine if numbers such as 1024, 2076, or 2077 are the difference of two squares. Use the Math Investigator to check your predictions.
- One way to see that 72 can be written as the difference of two squares is to represent 72 as a 4 by 18 array of tiles, make two cuts, and rearrange the pieces into an L-shape to show that 72 = 121 - 49, as shown below . Can such cuts in a rectangular array of tiles be made to form an outer and inner square if 72 is represented as a 3 by 24 array of tiles? Try this method on some numbers that can not be represented as the difference of two squares.