In A (Kola) Nutshell Workbook Book 4: Workbook by Linda James Foreword by Théophile OBENGA, Ph.D.

Math in a (Kola) Nutshell Series Book 4 - Workbook By Linda James Copyright ©2006 All rights reserved. No part of this shall be reproduced, stored in a retrieval system, or transmitted by any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission from the author. Original Copyright 2001 James, Linda C., Math in a (Kola) Nutshell, Book 4

Printed in the United States of America ISBN # 1-929772-08-4

To order additional copies or other books in the Math in a (Kola) Nutshell series, reach Ms. James at:

E-mail: lindajames711@sbcglobal.net Detroit, MI 48206

(313) 865-6528

Design, typesetting and editing by PrintWrite

(313) 527-2525

members.aol.com/sj030862/Publish0 E-mail: SJ030862@sbcglobal.net

Other Math in a

(Kola) Nutshell Topics:

Algebra

Basic Facts

Computation

Geometry

Trigonometry

Statistics

Calculus

Mathematics Is Nothing

But Rhythm

The African Medicine

Wheel

Illustrations and calligraphy by Ron Scarborough Graphics by Forrest Parks

For Mathematics

All dictionaries, without exception, tell us what Mathematics is doing, the study of number, form, arrangement, and associated relationships, using rigorously defined symbols. However the basic meaning of the word Mathematics is never given. Why?

To make things clear and simple, the ancient Greek root math means “to learn by inquiry.” Hence mathēmata, that is, “that which is learned,” “a lesson,” and mathēmatikos, “disposed to learn,” “a mathematician,” and finally ta mathēmatika, “things to be learned,” “lessons,” “mathematics.”

Therefore, if I learn from somebody disciplines such as music, theology, painting, oratory, history, politics, etc., I am learning mathēmatika, that is, mathematics, in the Greek sense.

In our days, the word mathematics is exclusively reserved to disciplines such as arithmetic, algebra (from Arabic al-jabr), differential and integral calculus, geometry, trigonometry, etc.

In KMT or Ancient Egypt, King Amenemmes III, a 12th Dynasty ruler, reigned from 1842 to 1797 B.C. During his reign, Egyptian scholars came to define mathematics as follows:

“Right method (tp-hsb) of investigating nature to know all that exists, all mysteries and all things secret.” (the exact title of Rhind Mathematical Papyrus, the ancient mathematical textbook, now found in the British Museum).

In all objectivity, the first strict definition of mathematics, in world history, came from Ancient Egypt, before Greek scholars like Thales, Pythagoras, Euclid, Archimedes, etc., were born.

According to the Greek historians themselves, Thales, Pythagoras, Plato, Eudoxe, etc., did study mathematics and astronomy in Ancient Egypt. Aristotle, himself, wrote that mathematical arts were first established in Ancient Egypt.

From the School of Alexandria, in Egypt, once again, knowledge reached Rome, that is, knowledge of philosophy, mathematics, literature, geography, medicine, etc.

Arabs and Moors will build later on the first universities in Europe,

precisely in Spain at Seville, Cordoba, Salamanca, etc. It is urgent to open our eyes and mind to the cultural history of the world,

and more important issues, beyond modern paranoia and its discourse on “gender,” “sex,” “race,” “feminism,” “deconstruction,” etc.

It is quite obvious that KMT civilization or Ancient Egypt did apply mathematics in a practical manner in building temples, palaces, pyramids, erecting obelisks (tekhenu), sculpture, irrigation, transportation, even in rituals and ceremonies. The Egyptian mind was very accurate, precise, balanced, likening themselves after their goddess, Maat.

One daily example of the application of mathematics in KMT is where beautiful women would walk down the road on a very hot and sunny day wearing a mathematical cone, shaped at the head, so that perfume inside the cone would drip down purposely in intervals like an hour clock. In this manner, what would normally be a hot and sweaty situation would turn into a sweet-smelling, beautiful woman.

In order to truly understand mathematics as well as any subject, one must return to the source, to antiquity. One must try to know the basic meaning of words, notions, and concepts, and their historical and cultural evolution.

Linda James’ book, Mathematics in a (Kola) Nutshell, clearly reflects such an effort of understanding. Being a fine instructor on mathematics, her book will be useful to many of us who struggle to improve our educational systems. We must therefore salute and welcome Linda’s contribution. Hotep!

Théophile OBENGA, Ph.D.

7

Introduction

Learning is remembering what you already know…

Dr. Malidoma Some ̀̀

Math in a (Kola) Nutshell is a practical, easy to use book that teaches you how to

solve math problems in simple steps and, often, in one step.

The name, Math in a (Kola) Nutshell reflects one of the many uses of the

kolanut, a very bitter nut found in West Africa. The kolanut has cultural and

dietary significance and its bitterness has medicinal value. The idea behind this

math book derives from its medicinal worth.

Although there are many helpful math study guides on the market today,

Math in a (Kola) Nutshell presents a powerful way of learning math by tapping

into the psyche, through reminding one of the natural patterns of rhythm that are

found in each and every person. As a result, the subconscious mind signals to the

conscious mind a reminder. These natural patterns are then recalled and from

them mathematical concepts are derived.

For example, when dividing a number by 9, one can use the following

patterns:

25 ÷ 9 = 2 (the first number) and the remainder is 7 (the sum of 2 + 5)

41 ÷ 9 = 4 (the first number) and the remainder is 5 (the sum of 4 + 1)

8

Another pattern is that which was used in Ancient Kemet (Egypt) Halfing and

Doubling. For example, to multiply 16 × 35

As a last example, here are the MAGIC 9’s

Notice the ten’s column of the MAGIC 9’s is 1,2, 3, 4, 5, 6, 7, 8, 9. The one’s

column is 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.

Throughout the book, Math in a (Kola) Nutshell, you will be amazed to

discover for yourself innovative math patterns that allow you easier and more

enjoyable ways of doing math.

Math in a (Kola) Nutshell features cultural diversity, a historical

perspective on mathematics, a clear way of transitioning from basic math to

algebra (The lack of transitioning is a major reason why 33% of students fail first

16 × 35 take half of 16 and double 35

Results 8 × 70 =

560, therefore 16 × 35 = 560

18 × 25 take half of 18 and double 25

Results 9 × 50

450, therefore 18 × 25 = 450

9 × 0 = 0

9 × 1 = 9

9 × 2 = 18

9 × 3 = 27

9 × 4 = 36

9 × 5 = 45

9 × 6 = 54

9 × 7 = 63

9 × 8 = 72

9 × 9 = 81

9 × 10 = 90

9

year algebra.). Math in a (Kola) Nutshell contains vocabulary that reflects the

etymology of words. For example, numerator is a Latin word that means

‘counter’, so we add or subtract (count) the numerators. On the other hand, the

word denominator is a Latin word that, when broken up, the ‘de’ means down and

the ‘nomin’ means a name or place. Consequently, by the very definition, we

never add or subtract denominators, a very common error.

Whether you are looking for a thorough understanding of basic math,

algebra, geometry, trigonometry, data and probability or word problems, you will

find it in Math in a (Kola) Nutshell, a book that includes the standards laid down

by National Council for Teachers of Mathematics (NCTM).

Just what makes the usage of math necessary? First of all, math is used in

our daily activities such as checkbook balancing, comparative shopping,

measuring curtains, windows or rugs. Secondly, exams, such as high school state

assessments, ACT, SAT for college entrance, or exams used in hiring people for

jobs are timed and require necessary and vital mental math, shortcuts, and

conceptual understandings found in this book. Certainly people who have built

wealth use mathematics to help the process.

Math in a (Kola) Nutshell is a must for any student, that special child in

your life, or adult learner. Therefore, I dedicate this book to anyone who uses and

needs math ——— EVERYONE IN THE COSMOS.

Sincerely,

Linda James

10

11

12

13

14

15

Basic Math

The kolanut is largely found in West Africa

(also found in Jamaica). One major use of

the kolanut is medicine. The kolanut is a

great stimulant and antiacid.

Basic mathematics is a stimulant and

foundational medicine for all mathematics.

Observe the patterns found in basic

mathematics and all mathematical concepts

will be found.

16

17

1. Multiply 5 × 5. 2. Multiply the 10’s digit by

the number that comes after it.

35 × 35 1225

To multiply a number by 25, divide the number by 4 and add two zeros. 25 × 16 = 400

16 ÷ 4 = 4

To multiply a number by 50. Divide the number by 2 and add to zeros. 50 × 14 = 700

14 ÷ 2 = 7

To multiply a number by 11

1. Write the ones digit. 2. Add the one and tens

digit. 3. Write the tens digit.

11 × 18 = 198

1 + 8 = 9

To check an addition problem 1. Add all the numbers in the addend until a single

digit is obtained. 2. Add all the numbers in the sum until a single

digit is obtained. If the numbers are equal, the answer is correct.

42 + 25 67

Examples

4 + 2 + 2 + 5 = 13 = 1 + 3 = 4 6 + 7 = 13 = 1 + 3 = 4

18

1. 85 x 85

2. 65 x 65

3. 95 x 95

4. 25 x 25

5. 105 x 105

6. 25 x 16 =

11. 50 x 14 =

7. 25 x 28 = 12. 50 x 18 =

10. 25 x 24 = 15. 50 x 22 =

8. 25 x 12 = 13. 50 x 24 =

9. 25 x 20 = 14. 50 x 16 =

15. 11 x 73 =

14. 11 x 41 =

12. 11 x 12 =

13. 11 x 24 =

11. 11 x 18 =

1. 42 + 25 67

2. 38 + 17 55

3. 62 + 29 91

4. 42 + 42 84

5. 76 + 28 104

Check these Problems

19

(India) Examples

I. IV. II. V. III.

Dividing a 2 digit number by 9 R7 1. The answer is the digit (5). 2. The remainder is the sum of

5 + 2. If the digit’s sum is 9 or more, the answer is the next number and the remainder is the sum of the numbers added until a single digit is obtained.

Base 10

1. Write how far away the

numbers are from 10. 2. Multiply 1 × 3. 3. Subtract 7 - 1 or 9 - 3.

To remember the multiplication facts of three, add them until a single digit is obtained. 3 6 9 12 15 18 21 24 27 12 = 1 + 2 = 3, 15 = 1 + 5 = 6, etc. Then subtract like this, using the difference as the ones digit and the subtrahend as the tens digit. 3 6 9 3 6 9 3 6 9 - 0 - 0 - 0 - 1 - 1 - 1 - 2 - 2 - 2 3 12 21

6000 28733127

5 9 9 103 1 2 72 8 7 3

−=

1. 9 - 1 7 - 3 6 3

×

Base 100

1. Write how far away the

numbers are from 100. 2. Multiply 2 × 4 = 8 and

write a zero in the 10’s digit.

3. Subtract 96 - 2 or 98 - 4.

98 - 2 96 - 4 9408

×

59 52

20

I. 8000 - 2365

600 - 143

II. 9 x 8

8 x 8

9 x 6

7 x 6

III. 98 x 97

IV.

V. Multiplying by 3 3 6 9 3 6 9 3 6 9

96 x 95

9 2 4 9 6 2 9 3 1 9 4 8 9 5 7

21

1 + 1 = 2, SHOE

2 + 2 = 4, DOOR

3 + 3 = 6, STICKS

4 + 4 = 8, GATE

5 + 5 = 10, HEN

22

6 + 6 = 12, ELVES

7 + 7 = 14, GREEN

8 + 8 = 16, BEANS

9 + 9 = 18, SKATING

23

1 × 1 = 1, SUN

2 × 2 = 4, DOORS

3 × 3 = 9, LINES

4 × 4 = 16, ICE CREAM

5 × 5 = 25, HIVE

24

6 × 6 = 36, STICKS

7 × 7 = 49, DIMES

8 × 8 = 64, POUR

9 × 9 = 81, FUN

25

To compute (add, subtract, multiply, or divide) numbers that are 2-digit or larger, using play money is a very good and practical way to learn computation. It is essential to know, by memory, the basic facts in order to efficiently and successfully learn to compute. It is also essential that one knows place value. Place value is the value of a number when it is in a certain place or position. For example:

235 means 200 + 35 + 5 641 means 600 + 40 + 1 159 means 100 + 50 + 9 333 means 300 + 30 + 3

Also, each place (from right to left) is 10 times bigger than the one to the right of it. For example:

1 ten = 10 ones 1 hundred = 10 tens 1 thousand = 10 hundreds

Let’s first add numbers that do not involve exchanging (carrying or regrouping). Example 1)

1. Add the 2 ones and the 5 ones 2. Add the 5 tens and the 3 tens

52 +35 87

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

26

Example 2) 362 +115 477

1. Add 2 ones and 5 ones 2. Add 6 tens and 1 ten 3. Add 3 hundreds and 1 hundred

Now let’s add numbers that involve exchanging (carrying or regrouping). Practically, it is less bulky to carry 1 ten dollar bill rather than 10 one dollar bills. It is easier to carry 2 ten dollar bills than it is to carry 20 one dollar bills. Likewise, it is easier to carry 3 ten dollar bills than 30 one dollar bills, etc. Therefore, the rule is whenever we have 10 one dollar bills, we will exchange them for 1 ten dollar bill and add this one ten dollar bill to the other ten dollar bills. This process of exchanging ten one dollar bills for 1 ten dollar bill and adding the ten dollar bill to the other ten dollar bills is called regrouping or carrying.

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

27

Example 1) 56 +28

Since ten ones = 1 ten, we will exchange the circled ten one dollar bills for 1 ten dollar bill and add the one ten dollar bill to the other ten dollar bills. The result is the ten dollar bills increase from 7 to 8 and now we only have 4 one dollar bills.

The problem looks like this: 1 This one means 1 ten dollar bill is added to 56 the other tens. +28 84 This 4 means that 10 one dollar bills were taken from 14 ones and 4 one dollar bills remained.

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

28

Example 2: 363 +218 581 Result: The result is that 10 one dollar bills are exchanged for 1 ten dollar bill, after adding 3 ones and 8 ones. Then 1 ten dollar bill is added to 6 ten dollar bills and 1 ten dollar bill. Finally, 3 hundred dollar bills and 2 hundred dollar bills are added.

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 100

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 100

$ 100

$ 100

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 100

$ 100

$ 10

$ 10

29

Example 3: 1 1 409 +398 807

The results is that 10 one dollar bills are exchanged for 1 ten dollar bill, after adding 9 ones and 8 ones. Next, 10 ten dollar bills are exchanged for 1 one hundred dollar bill, after adding 9 tens and 1 ten. Finally, 1 one hundred dollar bill is added to 4 hundreds and 3 hundreds. Results:

* Notice there are no tens remaining.

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 100

$ 100

$ 100

$ 10

$ 10

$ 10

$ 10

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 10

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 100

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

$ 1

30

1. 31 + 23

2. 47 + 30

3. 20 + 18

4. 55 + 44

5. 62 + 27

6. 140 + 29

7. 234 + 311

8. 711 + 246

9. 508 + 120

10. 834 + 124

11. 34 + 58

12. 29 + 11

13. 56 + 19

14. 44 + 36

15. 85 + 29

16. 326 + 147

17. 152 + 283

18. 496 + 287

19. 653 + 477

20. 809 + 446