doing numbers and doing mathematics by jim hogan university of waikato school support services
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Doing Numbers and
Doing Mathematics
By Jim HoganUniversity of WaikatoSchool Support Services
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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 = 3x5Use this method to sum the first 999 numbers.
We are just doing numbers.
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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 = 5x6/2Use this method to sum the first 999 numbers.
We are still just doing numbers.
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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 = 5x6Use this method to sum the first 999 even numbers.
We are still only doing numbers.
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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?
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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 = 3x5Use this method to sum the first 999 numbers.
Why does this work?Explain that and you are doing mathematics
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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 = 5x6/2Use this method to sum the first 999 numbers.
Why does this work?Explain that and you are doing mathematics
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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 = 5x6Use this method to sum the first 999 even numbers.
Why does this work?Explain that and you are doing mathematics
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These are SIMPLE examples
of what I mean when I refer to “Doing Numbers”
and“Doing Mathematics”
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Doing Mathematics
Is understanding what is going onand
being able to explain it to someone.
Thinking and TellingStudying mathematics is a great way to develop these abilities.
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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
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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?
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Hand Tables
DemonstrateHow the hands can be usedTo do numbers 5x5 to 10x10.
Explain why it worksAnd you are doing mathematics.
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Hand Tables
DemonstrateHow the hands can be usedTo do numbers 5x5 to 10x10.
Explain why it worksAnd you are doing mathematics.
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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?
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Add to Subtract!
• 786 -567
• 786 becomes 213• 213 + 567 = 780
• 780 becomes 219• 219 is the answer. Hmmm… Why?
Does this
always
work?
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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
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Old and Easy
• Think of a number • Double it• Add 10• Halve your answer• Subtract your original number• Your answer is 5
Hmmm… Why?
Can you
make up
another?
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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
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An infinity
• 1 + half + a quarter + an eigth + …
• 1 + half + a third + a quarter + …
• What is your guess?
• What is the answer?
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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+11We are doing maths if we investigate!
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Doing mathematics
• Is …
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Thanks