Doing Numbers and Doing Mathematics By Jim Hogan
Doing Numbers and Doing Mathematics By Jim Hogan University of Waikato School Support Services
An average problem • One way to sum the counting numbers is to take the middle number and multiply it by the number of numbers. • 1 + 2 + 3 = 2 x 3 • 1 + 2 + 3 + 4 + 5 = 3 x 5 Use this method to sum the first 999 numbers. We are just doing numbers.
A next problem • We could also take the last number and multiply it by the next one and divide by 2. • 1 + 2 + 3 = 3 x 4/2 • 1 + 2 + 3 + 4 + 5 = 5 x 6/2 Use this method to sum the first 999 numbers. We are still just doing numbers.
An even problem • The sum of the even numbers is the product of two consecutive numbers. • 2 + 4 + 6 = 3 x 4 • 2 + 4 + 6 + 8 + 10 = 5 x 6 Use this method to sum the first 999 even numbers. We are still only doing numbers.
It’s a Curious Incident… • Doing numbers is quite easy. It involves manipulation but basically it is following a pattern. Following someone elses thinking or just repeating your own. • So what is it that I am getting at?
Back to an average problem • One way to sum the counting numbers is to take the middle number and multiply it by the number of numbers. • 1 + 2 + 3 = 2 x 3 • 1 + 2 + 3 + 4 + 5 = 3 x 5 Use this method to sum the first 999 numbers. Why does this work? Explain that and you are doing mathematics
Back to a next problem • We could also take the last number and multiply it by the next one and divide by 2. • 1 + 2 + 3 = 3 x 4/2 • 1 + 2 + 3 + 4 + 5 = 5 x 6/2 Use this method to sum the first 999 numbers. Why does this work? Explain that and you are doing mathematics
Back to an even problem • The sum of the even numbers is the product of two consecutive numbers. • 2 + 4 + 6 = 3 x 4 • 2 + 4 + 6 + 8 + 10 = 5 x 6 Use this method to sum the first 999 even numbers. Why does this work? Explain that and you are doing mathematics
These are SIMPLE examples of what I mean when I refer to “Doing Numbers” and “Doing Mathematics”
Doing Mathematics Is understanding what is going on and being able to explain it to someone. Thinking and Telling Studying mathematics is a great way to develop these abilities.
Really mean n What does n+1 mean to you? What does n-1 mean to you? Why is the product of two consecutive odd numbers always one less than a square number? EG 3 x 5 = 16 - 1 Does this work for even numbers?
Hand Tables Demonstrate How the hands can be used To do numbers 5 x 5 to 10 x 10. Explain why it works And you are doing mathematics.
Hand Tables Demonstrate How the hands can be used To do numbers 5 x 5 to 10 x 10. Explain why it works And you are doing mathematics.
Square Pegs, Round Holes? Which is the better fit: a square peg in a round hole or a round peg in a square hole – formal proof expected. What does better mean?
- Slides: 14