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
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Doing Numbers and Doing Mathematics