CHAPTER 18 Number Theory Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al. ©2012 John Wiley & Sons  

Focus Questions Why study number theory topics in elementary school? What number theory topics are appropriate for students in elementary school? How does number theory complement the teaching and learning of mathematics in elementary school? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

What is Number Theory? Number theory is a branch of mathematics mainly concerned with the integers that has been a topic of study for centuries. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Number Theory in Elementary School Mathematics: Odds and Evens 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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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… Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

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 Continue connecting every 5 points with a straight line. 12 points, connect top point (12 o/clock) to 5 points, clockwise. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Other Number Theory Topics: Polygonal Numbers Polygonal or figurate numbers are numbers related to geometric shapes. Notice the perfect squares. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Other Number Theory Topics: Modular Arithmetic In some elementary books, modular arithmetic is called clock arithmetic. 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? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Other Number Theory Topics: Pascal’s Triangle What would be the numbers in Row 7? Row 8? Row 9? 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 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Row 0 Row 1 Pascal’s Triangle is named for the Frenchman who lived in the 17th Century. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Other Number Theory Topics: Pythagorean Triples A Pythagorean triple is a triple of numbers (a, b, c) such that a² + b² = c². Pythagoras, the person given credit for the Pythagorean Theorem, lived in Greece in about 500 BC. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Other Number Theory Topics: Fibonnaci Sequence Fibonacci was an Italian who lived in the 13th Century. His famous sequence also was known early in India. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . Do you see how it is generated? Look at the picture of the finger bone. Have you seen these numbers before? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Summary Number theory looks at the mathematical properties of numbers. Through this chapter, you observed many conjectures that are well within reach of elementary students to explore and to make convincing arguments about the validity of the conjectures. Number theory often provides a way to both practice and challenge students. In solving math puzzles, students are practicing skills, developing number sense, and using problem-solving strategies. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012