Chapter 18 To accompany Helping Children Learn Math Cdn Ed, Reys et al. ©2010 John Wiley & Sons Canada Ltd.

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 points, connect top point (12 o/clock) to 5 points, clockwise. Continue connecting every 5 points with a straight line. (12, 5) Star

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 Row 0 Row 1 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?

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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