Transcript Number Theory

Chapter 18
To a c c o m p a n y H e l p i n g C h i l d re n L e a r n M a t h C d n E d , R e y s e t a l .
© 2 0 1 0 J o h n Wi l e y & S o n s C a n a d a L t d .
Guiding Questions
1. Why study number theory topics in
elementary school?
2. What number theory topics are appropriate
for students in elementary school?
3. How does number theory complement the
teaching and learning of mathematics in
elementary school?
What is Number Theory?
• Number theory is a branch of mathematics
mainly concerned with the integers.
Number Theory
• Number theory is a prime source to show that
numbers can be fascinating.
• Number theory opens the doors to many
mathematical conjectures.
• Number theory provides an avenue to extend
and practice mathematical skills.
• Number theory offers a source of recreation.
Number Theory and Patterns
• Where are the perfect
squares?
• Where are the odd numbers?
• Find the sum of the upright
diamonds (such as 1 and 3, 2
and 6). If you do this in order,
what is the pattern of the
sums?
• Find the sum of the numbers
in each row. Find a short cut
for finding the sum.
• Find another pattern and
describe it.
Number Theory in Elementary School
Mathematics: Odds and Evens
• Classifying numbers as odd or even is one of
the first number theory topics children
encounter.
Which of the following conjectures are true? Explain
your reasoning.
The sum of two odd numbers is even.
The sum of two even numbers is even.
The sum of three odd numbers is odd.
The sum of any number of odd numbers is odd.
The sum of any number of even numbers is even.
The sum of two odd numbers and an even number is even.
Number Theory in Elementary School
Mathematics: Factors and Multiples
• A factor of a number divides that number with
no remainder. What are the factors of this
rectangle?
• A multiple of a number is the product of that
number and any other whole number.
• Multiples of 5 are 5,10,15,20…
Number Theory in Elementary School
Mathematics: Factors and Multiples (cont.)
• Greatest Common Factor is the largest factor
that is a factor of both multiples.
• Least Common Factor is the smallest number
that is a multiple of both numbers.
Number Theory in Elementary School
Mathematics: Primes and Composites
• A prime number is a whole
number greater than 1 that
has exactly two factors, 1
and itself.
• A composite number is any
number with more than two
factors.
Notice that the
number 5 is a
prime number
since it can only
be represented as
either a 5x1 or a
1x5 rectangle.
Number Theory in Elementary School
Mathematics: Primes and Composites (cont.)
• The fundamental theorem of arithmetic is the
following:
– Every composite number may be uniquely
expressed as a product of primes if the order is
ignored. This is called prime factorization.
Number Theory in Elementary School
Mathematics: Primes and Composites (cont.)
• The most commonly used method in
elementary school to find the prime
factorization of a number is the factor tree.
Number Theory in Elementary
School Mathematics: Divisibility
• A number is divisible by another number if
there is no remainder.
• Today, divisibility rules provide opportunities
to discover why a rule works or to discover a
rule.
Other Number Theory Topics:
Relatively Prime Pairs of Numbers
Two numbers are relatively prime if they have
no common factors other than 1.
• Star patterns can be used to investigate
relatively prime and not relatively prime pairs
of numbers:
(12, 5) Star
12 points, connect top
point (12 o/clock) to 5
points, clockwise.
Continue connecting
every 5 points with a
straight line.
Other Number Theory Topics:
Polygonal Numbers
Polygonal or figurate
numbers are numbers
related to geometric
shapes.
Notice the
perfect squares
or arrays of
square numbers.
Other Number Theory Topics:
Modular Arithmetic
• Modular arithmetic is sometimes called clock
arithmetic because it is based on a limited number of
integers, just like the clock is based on the integers 1
to 12.
• In mod 8 (see the “clock” below), we use the
numbers 0, 1, 2, …7. What do you think the sum of 6
and 7 would be? Try it on the clock and think of
starting at 6 o’clock and adding 7 hours.
What time would it be?
Other Number Theory Topics:
Pascal’s Triangle
1. What would be the
numbers in Row 7? Row
8? Row 9?
2. What patterns do you
see?
3. Find the sum of the
numbers in each row
beginning with row 1 and
ending with row 6. What
do you think would be the
sum of row 7, of row 8, of
the row 20?
1
11
121
1331
14641
Row 0
Row 1
Other Number Theory Topics:
Pythagorean Triples
• A Pythagorean triple is a triple of numbers
(a, b, c) such that a² + b² = c².
• There are many ways to generate Pythagorean
triples and many patterns in the triples.
Other Number Theory Topics:
Fibonnaci Sequence
• Some sequences of numbers are famous
enough to be named. One of these is the
Fibonnaci sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
• Do you see how it is generated? The first two
terms are ones; thereafter, any term is the
sum of the previous two terms.
Other Number Theory Topics:
Fibonnaci Sequence (cont.)
Look at the picture of the finger bone. Have you
seen these numbers before?
The Earliest Beginnings of Number
Theory
• The roots of number theory can be found
thousands of years ago throughout the
ancient world.
• Looking only at the names we have included in
this chapter, you can find influences from
many countries.
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