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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 = 2x3 • 1 + 2 + 3 + 4 + 5 = 3x5 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 = 3x4/2 • 1 + 2 + 3 + 4 + 5 = 5x6/2 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 = 3x4 • 2 + 4 + 6 + 8 + 10 = 5x6 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 = 2x3 • 1 + 2 + 3 + 4 + 5 = 3x5 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 = 3x4/2 • 1 + 2 + 3 + 4 + 5 = 5x6/2 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 = 3x4 • 2 + 4 + 6 + 8 + 10 = 5x6 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. An odd problem • The sum of the odd numbers is ? • 2 + 4 + 6 = 3x4 • 1 + 3 + 5 = 3x4 -1 -1 -1 or it may be something else Use your method to sum the first 999 odd numbers. Why does this work? Explain that and you are doing mathematics 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 5x5 to 10x10. Explain why it works And you are doing mathematics. Hand Tables Demonstrate How the hands can be used To do numbers 5x5 to 10x10. 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? Add to Subtract! • 786 -567 • 786 becomes 213 • 213 + 567 = 780 • 780 becomes 219 • 219 is the answer. Hmmm… Why? A multiple problem • The sum of the multiples of 3 is ? • 1 + 2 + 3 + 4 = 4x5 /2 • 3 + 6 + 9 + 12 = ? Use your method to sum the first 999 multiples of three. Why does this work? Explain that and you are doing mathematics Old and Easy • • • • • • Think of a number Double it Add 10 Halve your answer Subtract your original number Your answer is 5 Hmmm… Why? A powerful problem • The sum of the powers of 2 is ? • 1+2+4+8=? Use your method to sum the first 999 powers of two. Can you generalise this for the powers of n? Why does this work? Explain that and you are doing mathematics An infinity • 1 + half + a quarter + an eigth + … • 1 + half + a third + a quarter + … • What is your guess? • What is the answer? Consecutive sums • CAN all numbers be sums of consecutive numbers? • 7=3+4 • 26= 5+6+7+8 • 101 = 50+51 • 21 = 7+8+9 = 10+11 We are doing maths if we investigate! Doing mathematics • Is … Thanks