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Doing Numbers and Doing Mathematics By Jim Hogan University of Waikato School Support Services

Doing Numbers and Doing Mathematics

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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 - PowerPoint PPT Presentation

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Page 1: Doing Numbers  and  Doing Mathematics

Doing Numbers and

Doing Mathematics

By Jim HoganUniversity of WaikatoSchool Support Services

Page 2: Doing Numbers  and  Doing Mathematics

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.

Page 3: Doing Numbers  and  Doing Mathematics

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.

Page 4: Doing Numbers  and  Doing Mathematics

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.

Page 5: Doing Numbers  and  Doing Mathematics

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?

Page 6: Doing Numbers  and  Doing Mathematics

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

Page 7: Doing Numbers  and  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 = 5x6/2Use this method to sum the first 999 numbers.

Why does this work?Explain that and you are doing mathematics

Page 8: Doing Numbers  and  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 = 5x6Use this method to sum the first 999 even numbers.

Why does this work?Explain that and you are doing mathematics

Page 9: Doing Numbers  and  Doing Mathematics

These are SIMPLE examples

of what I mean when I refer to “Doing Numbers”

and“Doing Mathematics”

Page 10: Doing Numbers  and  Doing Mathematics

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.

Page 11: Doing Numbers  and  Doing Mathematics

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

Page 12: Doing Numbers  and  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?

Page 13: Doing Numbers  and  Doing Mathematics

Hand Tables

DemonstrateHow the hands can be usedTo do numbers 5x5 to 10x10.

Explain why it worksAnd you are doing mathematics.

Page 14: Doing Numbers  and  Doing Mathematics

Hand Tables

DemonstrateHow the hands can be usedTo do numbers 5x5 to 10x10.

Explain why it worksAnd you are doing mathematics.

Page 15: Doing Numbers  and  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?

Page 16: Doing Numbers  and  Doing Mathematics

Add to Subtract!

• 786 -567

• 786 becomes 213• 213 + 567 = 780

• 780 becomes 219• 219 is the answer. Hmmm… Why?

Does this

always

work?

Page 17: Doing Numbers  and  Doing Mathematics

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

Page 18: Doing Numbers  and  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?

Can you

make up

another?

Page 19: Doing Numbers  and  Doing Mathematics

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

Page 20: Doing Numbers  and  Doing Mathematics

An infinity

• 1 + half + a quarter + an eigth + …

• 1 + half + a third + a quarter + …

• What is your guess?

• What is the answer?

Page 21: Doing Numbers  and  Doing Mathematics

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!

Page 22: Doing Numbers  and  Doing Mathematics

Doing mathematics

• Is …

Page 23: Doing Numbers  and  Doing Mathematics

Thanks