Five Success Secrets for Mathematics Teachers via · PDF fileFive Success Secrets for...

Five Success Secrets forMathematics Teachers

via Ancient China & IndiaJonathan J. Crabtree

Mathematics HistorianMathematical Association of Victoria (MAV) Annual Conference 2017

La Trobe University Melbourne Campus, Australia7th December 2017

www.jonathancrabtree.com/mathematics | research@jonathancrabtree.com

Our Two Goals Today

2nd Blend ‘Chindian’ integer ideas with ratios and proportion and play multiplication and division games in which the answers appear in (simpler) ways.

1st Explore China and India’s ancient mathematical ideas on positives and negatives and reveal five success secrets.

Eg. –2 x –3, 12 ÷ –4 and if tim e (2 – 10 ) x (3 – 10)

Consider the ~150 BCE rod numerals of China...

Negatives

Positives

The units and hundreds places had vertical rods while the tens and thousands places had horizontal rods.So, ‘5342 negatives’ appeared as .

The set of vertical and horizontal rod numerals.https://en.w

ikipedia.org/wiki/C

ounting_rods

Zero as a number was neither known nor needed. https://en.w

ikipedia.org/wiki/C

ounting_rods

Rod numerals were used with counting boards.https://en.w

ikipedia.org/wiki/C

ounting_rods

Rod Numerals werealso used in Japanas depicted here

Success Secret #1

The Grammar Flip

(from abstract to concrete)

Most children reply ‘Three negatives’.

‘What’s negative seven minus negative four?’

Most adults reply, ‘Negative eleven’.

‘What’s seven negatives minus four negatives?’

‘Seven’ is a simpler adjective than ‘negative’ and Chinese positives and negatives were nouns!

In ‘negative seven’ we have ‘adjective noun’.

In ‘seven negatives’ we have ‘adjective noun’.

‘What’s positive seven plus negative three?’

Is there a simpler approach to introduce positives and negatives than the number line?

Ground Level ZeroNo Positives (Bumps) and No Negatives (Holes)

Podo the Puppy, loves to dig!

From Ground Level Zero, Podo makes ...

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Separate, or altogether, all my bumps and holes

give me zero!

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True or f alse?

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True or f alse?

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True or f alse?

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True or f alse on Groun d Level Zero?

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True or false? (Podo the Puppy wo n’ t look !)

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3 holes

7 bumps

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‘What’s positive seven plus negative three?’

3 negatives

7 positives

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‘What’s positive seven plus negative three?’

3 negatives

7 positives

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‘What’s positive seven plus negative three?’

So +7 + –3 + = +4

7 bumps (+7 ) and 3 holes (–3) leads to 4 bumps (+4 )

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‘What’s positive seven plus negative three?’

DIRECTIONFINANCIAL POSITIONPOPULATIONTEMPERATURESUFFICIENCYSEA LEVELTIME

North/South, East/West, Right/Left, Up/DownAssets/Debts, Profit/LossBirths/Deaths, Immigration/EmigrationHot/Cold, Above Zero/Below ZeroMore Than Enough/Less Than EnoughAbove/BelowPast the hour/To the hour

Q. How +ve result bad & –ve result good?

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A. Your cancer test!

Positive numbers just count or measure things (units).Negative numbers just count or measure opposite things!

Success Secret #2

Defining, Splitting and Adding Zero

Defining zero.

NowNothing, nil, none, nought, or ...‘Any number subtracted from itself.’ e.g. 5 – 5 = 0

628 CE, Brahmagupta (India).Zero = ‘The sum of a positive number and negative number of equal magnitude.’सम-ऐक्यम् खम् (Brāhma Sphuta-siddhānta, Chapter 18:30a).

NOTE: Negatives were NOT considered ‘less than zero’.

- China had positives and negatives in the 2nd Century BCE, yet their first mathematics text with ‘zero’, appeared in 1247 CE, The Mathematical Treatise in Nine Sections, by Qin Jiushao.

- So, China used positive and negatives for ~1400 years without any concept of zero. The Chinese obviously did not think negatives were less than a non-existent idea!

Negatives and positives of the same size are equal in number yet describes units opposite in nature.

Splitting zero into equal numbers of opposites.

When you don’t have enough for subtraction, just add zero as equal numbers of opposites.For 3 positives minus 7 positives, just add zero!

+3 minus +7 (we’re 4 short.)

After adding zero in the form of + 4 and – 4, we have 7 positives minus 7 positives, and 4 negatives left over, so +3 – +7 = –4

The following integers sum to zero, as per Brahmagupta’s definition of zero by addition, (red/pos and black/neg).

–1 +1 These integers, being equal in size, yet opposite in nature, cancel each other out as zero.

Take away +1 from 0 and –1 remains.

Take away –1 from 0 and +1 remains.

–1 +1

–1 +1

The following integers sum to zero, as per Brahmagupta’s definition of zero by addition, (red/pos and black/neg).

–1 +1 These integers, being equal in size, yet opposite in nature, cancel each other out as zero.

Take away +1 from 0 and –1 remains.

Take away –1 from 0 and +1 remains.

–1

+1

Success Secret #3

How Integer Multiplication (REALLY) Works

a multiplied by b is EITHER a added to zero b times, OR, a subtracted from zero b times, according to the sign of b. (Where a can be either positive or negative.)

2+ x –3 (Via China a nd India)

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2+ x –3

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Two pos itiv es subtr act ed thr ee times!

2+ x –3

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Canc el opp osite s (China ) or subtr act the zeros (India) .

2+ x –3 = 6–

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See the answer!

2– x –3

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2– x –3

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Two negativ es subtr act ed thr ee times!

2– x –3

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Canc el opp osite s (China ) or subtr act the zeros (India) .

2– x –3 = 6+

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See the answer!

–1 +1

Take away +1 from 0 and –1 remains.

Take away –1 from 0 and +1 remains.

–1

+1

Negatives Added N Times to Zero Multiplier Positives Added N Times to Zero81 72 63 54 45 36 27 18 9 +9 9 18 27 36 45 54 63 72 8172 64 56 48 40 32 24 16 8 +8 8 16 24 32 40 48 56 64 7263 56 49 42 35 28 21 14 7 +7 7 14 21 28 35 42 49 56 6354 48 42 36 30 24 18 12 6 +6 6 12 18 24 30 36 42 48 5445 40 35 30 25 20 15 10 5 +5 5 10 15 20 25 30 35 40 4536 32 28 24 20 16 12 8 4 +4 4 8 12 16 20 24 28 32 3627 24 21 18 15 12 9 6 3 +3 3 6 9 12 15 18 21 24 2718 16 14 12 10 8 6 4 2 +2 2 4 6 8 10 12 14 16 189 8 7 6 5 4 3 2 1 +1 1 2 3 4 5 6 7 8 9

–9 –8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +99 8 7 6 5 4 3 2 1 -1 1 2 3 4 5 6 7 8 9

18 16 14 12 10 8 6 4 2 -2 2 4 6 8 10 12 14 16 1827 24 21 18 15 12 9 6 3 -3 3 6 9 12 15 18 21 24 2736 32 28 24 20 16 12 8 4 -4 4 8 12 16 20 24 28 32 3645 40 35 30 25 20 15 10 5 -5 5 10 15 20 25 30 35 40 4554 48 42 36 30 24 18 12 6 -6 6 12 18 24 30 36 42 48 5463 56 49 42 35 28 21 14 7 -7 7 14 21 28 35 42 49 56 6372 64 56 48 40 32 24 16 8 -8 8 16 24 32 40 48 56 64 7281 72 63 54 45 36 27 18 9 -9 9 18 27 36 45 54 63 72 81Negatives Subtracted N Times from Zero Multiplier Positives Subtracted N Times from Zero

Addition of Integers to ZeroSID

E OF PO

SITIVE MU

LTIPLICA

ND

SSI

DE

OF

NEG

ATI

VE M

ULT

IPLI

CA

ND

S

Subtraction of Integers from Zero

Negatives Added N Times to Zero Multiplier Positives Added N Times to Zero81 72 63 54 45 36 27 18 9 +9 9 18 27 36 45 54 63 72 8172 64 56 48 40 32 24 16 8 +8 8 16 24 32 40 48 56 64 7263 56 49 42 35 28 21 14 7 +7 7 14 21 28 35 42 49 56 6354 48 42 36 30 24 18 12 6 +6 6 12 18 24 30 36 42 48 5445 40 35 30 25 20 15 10 5 +5 5 10 15 20 25 30 35 40 4536 32 28 24 20 16 12 8 4 +4 4 8 12 16 20 24 28 32 3627 24 21 18 15 12 9 6 3 +3 3 6 9 12 15 18 21 24 2718 16 14 12 10 8 6 4 2 +2 2 4 6 8 10 12 14 16 189 8 7 6 5 4 3 2 1 +1 1 2 3 4 5 6 7 8 9

–9 –8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8 +99 8 7 6 5 4 3 2 1 -1 1 2 3 4 5 6 7 8 9

18 16 14 12 10 8 6 4 2 -2 2 4 6 8 10 12 14 16 1827 24 21 18 15 12 9 6 3 -3 3 6 9 12 15 18 21 24 2736 32 28 24 20 16 12 8 4 -4 4 8 12 16 20 24 28 32 3645 40 35 30 25 20 15 10 5 -5 5 10 15 20 25 30 35 40 4554 48 42 36 30 24 18 12 6 -6 6 12 18 24 30 36 42 48 5463 56 49 42 35 28 21 14 7 -7 7 14 21 28 35 42 49 56 6372 64 56 48 40 32 24 16 8 -8 8 16 24 32 40 48 56 64 7281 72 63 54 45 36 27 18 9 -9 9 18 27 36 45 54 63 72 81Negatives Subtracted N Times from Zero Multiplier Positives Subtracted N Times from Zero

Addition of Integers to Zero

SID

E O

F PO

SITIV

E M

ULTIP

LICA

ND

S

SID

E O

F N

EG

ATI

VE

MU

LTIP

LIC

AN

DS

Subtraction of Integers from Zero

2– x –3?

Two negativ estaken away thr ee timespr oduces sixpos itiv es in Q3 so 2– x –3 = 6+

Success Secret #4

Multiplication and Division are

(ALWAYS) Proportional Concepts!

Done That Do This Multiplication!

• https://www.youtube.com/watch?v=cnqEGW5MUQk

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Copyright © 2017 Jonathan Crabtree All Rights Reserved

From repeated addition and r epeated subtr action to Propor tion

–2 x –3 = ?

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Whatever we do to t he Unit (1) to make the Multi pli er (–3), we do to the Multi plicand (–2) to make the Product .

We place d thr ee Units and chang ed their sign to make the Multiplier –3.

–2 x –3

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Previous ly w e placed t hree Unitsand c hanged the ir sign to mak e the Multi pli er .

So, you place thr ee Multipl icandsAnd c hange the ir sign to mak e the Product !

–2 x –3

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As +1 is t o –3 so –2 is t o +6

+12 ÷ –4

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As –4 is t o +1 +12 is to...

Success Secret #5

Proportional Covariation Extends from the Integers to the Real Number System.

–2 x –3 +12 ÷ –4

C

C

b

aa x b = c As 1 is to b, so a is to c

a ÷ b = c As b is to 1, so a is to c

Proportional Covariation Applets

https://www.geogebra.org/m/ZBTZd6AF

https://www.geogebra.org/m/v62CqVEN

Q1. +2 × (0 + 3)

2 Positives Added 3 Times onto Zero

makes

6 Red Positives

Q2. –10 × (0 + 3)

10 Negatives Added 3 Times onto Zero

makes

30 Black Negatives

Q3. –10 × (0 – 10)

10 Negatives

Subtracted 10 Times

from Zero makes

100 Red Positives

Q4. +2 × (0 – 10)

2 Positives Subtracted

10 Times from Zero

makes

20 Black NegativesAltogether, we have 106 red positives and

50 black negatives. After cancellation, 56

red positives or +56 remains.

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