Investigating Perfect Squares
Here we with to explore the successive perfect squares of integers for any patterns that
may exist among them to help you in mentally estimating and calculating the perfect
integral squares. To help with your investigation, I suggest that you keep track of the
numbers and their squares on a number line or in tables.
Part A
 First, explore the squares of numbers ending in zero up to 100. What pattern do
you notice?
o Make a conjecture about such perfect squares and test your conjecture
with other numbers ending in zero.
o Prove your conjecture using place value (or otherwise), being sure to
identify and define any variables very carefully.
 Second, consider the squares of numbers ending in five up to 100. What pattern
emerges here.
o Make a conjecture about such perfect squares and test your conjecture
with other numbers ending in five.
o Prove your conjecture using place value (or otherwise), being sure to
identify and define any variables very carefully.
 Now consider the sequence of squares for single digits 1 through 9. Look at the
sequence of units digit in the sequence of perfect squares.
o What digits never occur in the units digit of the perfect square?
o Which digits do occur in this position?
o Is there any pattern to their occurrence?
o How would you describe such a pattern with respect to
the digits 5 and 0 in the units place of the perfect square?
o Make a conjecture about the units digit of the perfect squares of two-digit
numbers and verify that it works for any random two-digit number.
 Notice that the four possible unit digits in the perfect squares (other than five
and zero explored above) can be produce only by the same pair of unit digits in
the root of the square. For example, roots with unit digits on each side of zero
(i.e. 9 and 1) are the only roots whose squares end in 1.
o Explore roots with the other three unit digits. What are the unit digits of
their perfect squares?
o What does the distribution of these (root, perfect square) pairs look like on
the number line with respect to the roots ending in five or zero?
o Now explore the differences between squares ending in the same units digit
from the intermediate square ending in five or zero. How does this
sequence of differences relate to the root of the intermediate square?
Make a conjecture about this relationship.
o Use your conjecture to predict the perfect squares of (root, perfect square)
pairs that you don’t know. Then verify your results using a calculator.
 Your conjecture will give you a mental arithmetic technique for finding perfect
squares, but it may not be exact enough to formulate into an algorithm. Identify
the limitations of this approach and evaluate its usefulness. If you believe that
your conjecture is reliable enough to work, then prove your conjecture.
Type I Portfolio Assignment
IB Mathematics SL
Exploring Perfect Squares
Reg Noland, Instructor/Author
Page 1 of 2
AP Calculus AB
Part B
The following exploration will provide you a second way of mentally squaring numbers.
I will illustrate the process of producing squares around 100 and then leave you to
speculate about other squares and to prove the algorithm.
 Let’s consider a two-digit number close to but less than 100, such as 96.
96 differs from 100 by 4. 4 less than 96 is 92. 4 squared is 16. The perfect
square of 96 is a combination of these two numbers: 92 in the thousands
and hundreds places and 16 in the tens and units places or 9216!
o Use this algorithm to compute the squares of the numbers 91 through 99
then use a calculator (or otherwise) to verify your results.
o Speculate on why this algorithm works.
o Formulate the algorithm as a theorem using place value, being careful to
identify and define any variables, then prove your theorem.
o What mental arithmetic special cases would we need to consider for
numbers within three of 100 or less than 91? Explain.
 Let’s consider a two-digit number close to but more than 100, such as 107.
107 differs from 100 by 7. 7 more than 107 is 114. 7 squared is 49. The
perfect square of 107 is a combination of these two numbers: 114 in the ten
thousands, thousands, and hundreds places and 49 in the tens and units
places or 11449!
o Use this algorithm to compute the squares of the numbers 101 through
109 then use a calculator (or otherwise) to verify your results.
o Speculate on why this algorithm works.
o Formulate the algorithm as a theorem using place value, being careful to
identify and define any variables, then prove your theorem.
o What mental arithmetic special cases would we need to consider for
numbers within three of 100 or more than 109? Explain.
 Now generalize the algorithm for numbers on either side of 100.
 Extending your knowledge
o Consider the squares of numbers around 50. Apply the algorithm to
these numbers. Does it work?
o What variation in the algorithm must you apply to make it work?
o Consider the squares of numbers around 200. Apply the algorithm to
these numbers. Does it work?
o What variation in the algorithm must you apply to make it work?
o Consider the squares of numbers around 300. Apply the algorithm to
these numbers. Does it work?
o What variation in the algorithm must you apply to make it work?
 Now generalize the algorithm for any number. See if you can produce a
general theorem for this generalized algorithm and prove it.
 Comment on the limitations and usefulness of this mental math trick.
Type I Portfolio Assignment
IB Mathematics SL
Exploring Perfect Squares
Reg Noland, Instructor/Author
Page 2 of 2
AP Calculus AB