07SD _2_fjsdf sdfsdfjf

ARITHMATIC FROM A MATHEMATICAL PERSPECTIVE James Stevenson

1 of 52

LEARNING ARITHMATIC FROM A

MATHEMATICAL PERSPECTIVE

BY

JAMES STEVENSON

JUNE 2007


2 of 52


3 of 52

TABLE OF CONTENTS INTRODUCTION ..................................................................................................7 OBSTICLES TO LEARNING MATH-----------------------------------------------------------6 PURPOSE FOR STUDYING MATH: ..................................................................14 TWO BASIC RULES...........................................................................................15 LEARNING THE ALPHABET AND HOW TO COUNT........................................16 MODEL THEORY ...............................................................................................17 BREAKING PATTERNS .....................................................................................24 MULTIPLICATION ..............................................................................................36 DIVISION ............................................................................................................63 WHAT IS A FRACTION? ....................................................................................71 DECIMALS .........................................................................................................79 EQUIVILANTS ....................................................................................................81 FRACTIONS TO DECIMAL EQUIVILENTS........................................................83 PERCENT...........................................................................................................89 WORD PROBLEMS............................................................................................94 METRIC ..............................................................................................................97 BASIC GEOMETRY..........................................................................................100 ALGEBRA.........................................................................................................106 INTEGERS .......................................................................................................109 CONCLUSION..................................................................................................113 COPYRIGHT ....................................................................................................115


4 of 52


5 of 52

INTRODUCTION I originally worked as a Petroleum Engineer for twelve years prior to entering the teaching profession. My first teaching positions were in federal and provincial prisons. As a result my approach to math is different than most and different to the way that I was taught When I first started teaching in the prisons, I would have students come to me and prove that they have grade ten math yet they cannot do their multiplication tables. My first response was to blame the school system for passing incompetent people, so I would re-teach these students the way that I was taught. We would make slow but positive progress and all would be happy thinking that we were successful. After being released from prison, they would re-offend and be back in prison again within six to eight months. This time when I interviewed them, they had forgotten all that they had learned. Only this time I would use the excuse that it was the drugs that made them forget every thing it couldnt have been my teaching. Again the inmate would be released and would return within the year. This time he may remember a little more but still had forgotten most. What would we use for an excuse this time? The easiest excuse was that maybe the student was just dumb and would never be able to do math. After this happened many times, it was getting harder to convince the student that this was not true. Slowly, it became obvious that the method that we use to teach math does not work for many. So a different action was required. I reverted to the inverse model theory. Instead of me teaching them the same way over and over again, let them teach me how they do math. If the questions are phrased correctly, all the students had a coping method of deriving the correct answers. I started seeing patterns in their coping methods and I was able to refine some of them for the students. Math has been taught through memorization. We memorize: additions, subtractions, multiplication tables, formulas and procedures as thousands of unrelated rules and data rather than seeing how it is all related into one big model. So stop memorization! Look for the patterns. Looking for patterns is one of the few transferable skills that students will retain after you leave college. Another block to learning math was the students patterns of failure and the threats and stigma attached to failure. The brain just shuts down and goes into protection mode rather than using its power to be successful and creative. Most students have been told that they must work hard to learn math, so they work harder and harder just to prove to them selves that math is hard. By showing that math can be easy we can remove the fear that math has to be hard. The easy lazy method is the only way many will overcome their years of learned failure patterns.


6 of 52

OBSTICLES TO LEARNING MATH The mind can play many tricks on us. Some of the false beliefs and experiences that we have had can prevent us from reaching our potential. To illustrate this lets look at an example. If I were to lay a 2x12 plank that is 15 long on the floor, very few people would have trouble walking from one end to the other. In fact most would have no problem doing it blindfolded. Some could even do somersaults from one end to the other. If we lifted the plank 3 feet off the ground a few people would get nervous and may fall. Most wouldnt do somersaults. If we raised the same plank 50 feet into the air, most people would be nervous about crossing the same plank in any form. What has changed? The plank has stayed the same merely the environment has changed. Fear of falling effects our ability to succeed. Another example is an experiment conducted at UBC in 2006. The purpose of the study was to determine how certain beliefs effect health and whether certain information should be released to the public. The experiment took 160 UBC female students at random. They were broken down into four groups. Each group was given essays to read prior to writing the math entrance exams for UBC. The first essay talked about how females lacked a math gene that was recently discovered.. The second group read about the fact that females did not do as well as males in math was because of the environment in which our society treats females. The third group read that there is no evidence what so ever to indicate whether males or females will score higher on math exams. The fourth and control group read an essay about comparing art. The first group that falsely believed there was a gene responsible for their ability to do math score 50% lower than the control group. These false beliefs were only held for a short while and not reinforced. Imagine how false beliefs that are held for years and have been reinforced for 20 years would effect ones ability to do math. Another example many of us have is public speaking. We can talk to a friend or small group of friends without any problems, But stick a mike and 100 people in front of us and we act like we are just learning to speak. What is the difference? We are saying the same thing. Fear prevents our ability for success. We have all encountered bumps along the math highway. From these we have created our own false beliefs that prevent us from doing as well as we should. Some of these false beliefs have been reinforced by parents, teachers, peers and of course our own feelings of incompetence. Most of our problems are created because teachers insist that we memorize adding, subtracting multiplication tables and division. This was the method that they used to learn math but it is very limited in the understanding of math. Many elementary teachers never understood the concepts of math. They were only capable of memorization to pass tests. Those of us with poorer memories did poorly and thus reinforced our false beliefs. Others may have been traumatized


7 of 52

by other events while growing up. These events sapped our energy for learning. An extreme example of this was one student that slowly learned to add but absolutely could not subtract. We spent several weeks trying to subtract, Once we started making progress, it would be totally forgotten the next day. In frustration, I decided that he should learn multiplication. Over a moderate period of time, the process of multiplication was accomplished and we proceeded onto division. But division requires subtraction. Because it was called division he was able to accomplish division without realizing that he was subtracting. The trauma associated in his life with learning subtraction prevented him from learning subtraction, yet he was able to learn more complex arithmetic because there wasnt the fear associated with it. We all carry some of these debilitating beliefs to some degree or another. We must become aware of them and do our best to minimize or eliminate them. It is like carrying around a lot of extra weight while climbing ladders. PURPOSE FOR STUDYING MATH: Ninety-five percent of people will never ever use math once they have learned arithmetic ( knowing how to add, subtract, multiply, divide and do some simple fractions). The other five percent may use more math in the sciences and technologies. And if they cannot do math in their head they will use a calculator anyway. The discipline of math is merely a tool used to learn how to learn. It is the study of learning the relationships between different factors or functions. It is the ability to learn how to identify and solve problems. This is the invaluable transferable skill that will be used in latter life. It can be used to exercise the mind similar to working out in the gym exercises the body. Many enjoy working out in the gym but nobody is going to apply for a job stacking weights. Pushing weights is to the body as learning math is to the mind. TWO BASIC RULES With the exception of geometry, trigonometry and calculus, all math can be broken down into adding, subtracting, multiplying, dividing and doing fractions. Everything else in math can be broken down into these five principles. These five principles can be broken down into two simple rules. One: knowing how to count; and two; knowing how to be lazy. Being lazy and knowing how to be lazy are two different things. To be lazy one must understand the relationship between things and find short-cuts. If math creates problems for you which one is a problem for you; knowing how to count or being lazy?


8 of 52

LEARNING THE ALPHABET AND HOW TO COUNT. Think about how children learn the alphabet and how they learn to count. Most children can learn the alphabet and how to count to ten by the age of two and a half or three. But most children cannot learn to count to one hundred until they are five or six. The question arises; why is it easier to learn twenty-six characters than it is to learn ten numbers? The alphabet is learned by rote and by rhyming songs. They have no idea what an A, B", or a C are but they can rhyme them off. To learn how to count to one hundred, one would have to learn one hundred numbers until one realizes that they only have to learn to count to ten and the pattern repeats itself. Young childrens brains have not yet developed the ability to recognize the more complex patterns of repletion. We all know and use alphabetical order to file material or look up information that is filed alphabetically. However, if I were to ask, what is the fifth letter after P? What is the letter and how did you find it? Now what is the fifth letter before P? How did you find it? Could you use the same technique? Was it easy? If you had to do this often how would you do it so you could do it faster? If I were to ask what is the fifth number after 10 you would have no problem. And if I were to ask what is the fifth number before 10 you also would have no problem. Why? We see numbers in relationship to one-another. The relationship to the letters is different, yet we all file things or look thing up in alphabetical order. MODEL THEORY We all use models, consciously or unconsciously, to predict an outcome or measure something. This is what math is and does. Models can be useful and good, but unless we recognize the models that we use they can be just as limiting as beneficial. When I worked as a Petroleum Engineer, we used a commonly accepted model to predict the life and productivity of and oil and gas wells. The problem with the current model was that we only directly measured two of about twelve parameters that affect the productivity of a well. We needed all the parameters to do the calculations. Some parameters could be measured from other sources and others had to be estimated. If we estimated correctly, we received a good correlation; if not, we were out tens of millions of dollars. An Engineer that I worked with stated that if the model can be reversed and still be true then maybe it is true; otherwise probably not. He recognized that we were forcing the data to fit the model. Inversely we should make the model fit the data. He created a new model or series of models that gave much more realistic outcomes. The education model forces the students to fit the memorization model, rather than creating a model to fit the student. Many students do not fit the educational model, thirty percent of the students entering grade one do not complete grade


9 of 52

twelve. Of the remaining seventy percent, half will forget most of what they learned within two years. We as educators are trying to force the students to fit a model that does not wok for them. We must rethink the educational model to fit the students rather than the other way around. Most of our students have had bad experiences with math and they have not fit into our model of math education otherwise they would not be returning to upgrade or redo their math. Brain research indicates that when the mind is experiencing stress, parts of the brain not needed for survival shut down and adrenaline is released into the brain to ensure survival and thus minimizing learning. Brain research indicates that the brain is constantly capable of learning. Within the brain, new dendrites grow every time learning takes place. When learning takes place the axioms release endorphins and the body has a feeling of pleasantness and euphoria. The brain craves positive stimulation. It always craves new a challenge; not more dreary memorization. The model that we use to teach math does not stimulate the brain if we get students to memorize timetables, additions, and rules. Constant repetition of long sums and divisions quietly lull the mind to sleep. Internal or external stress is put on the students that dont fit into our educational model of memorization. If the student looked for a new relationship each time there would be a positive stimulation and enhance learning. The majority of students will never use math again once they learn to add, subtract, multiply, divide, and do fractions. The exceptions are those going on to the technologies and the sciences So the question is why do we make students sit in class for twelve years learning something that they will never use? The reason is because math trains people to learn and to see the relationships between things, i.e. Problem solving. So why do we get them to memorize every thing? If students cannot memorize rules they become frustrated and if they do memorize the rules they forget them shortly after the exam. So whats the point? I dont want any student to memorize anything. I want them to learn how to be creative.. To do that one must look for patterns and relationships. This forces the mind to look at things differently and is rewarded when the endorphins are released. Most students are told to work hard but when they do, they get tired and then they make mistakes, then they get frustrated and make more mistakes before they decide and prove to themselves that math is hard. The brain gets stressed and releases adrenaline and treats math as a negative to be avoided. We must look at teaching math from a different perspective. What is meant by knowing how to count and by being lazy? If one wanted to add two numbers, one could start counting from one number to the next. E.g. To add 8+7, we can start counting at 8 until we count 7 more numbers and arrive at 15. But if we want to add 3,284,789 + 2,0325,498, we have to find a short cut or a lazier way because it is going to take us too long to count that high and we would probably make three or more mistakes. If we want to subtract 8-5, we


10 of 52

could start at 8 and count backwards 5 places until we arrived at 3. But it would be easier to ask what do we need to add to 5 to get 8. Either way the answer is three. It is easier to count forward than backwards. To multiply 3 x 4, we could rewrite it as 3+3+3+3=12 or 4+4+4 =12. So multiplication is just a series of additions only it is easier to write 3 x 4 than 4+4+4=. But we need short cuts to make it easier and quicker. Division is just a series of subtractions. For example: if 4 grade one kids returned $13 worth of pop bottles, how would they divide the money?

?4 13 - 4 1 9 -_4 2 5 -_4 3 1 3 +( 1 4 ) = 3

Division is just a series of subtractions. 13 minus 4 leaves 9. 9 minus 4 leaves 5. 5 minus 4 leaves 1. So 4 can be subtracted from 13, 3 times. But what do we do with the $1 left over. We can rewrite divide as . What does the top dot stand for in the sign? What does the bottom dot stand for? The top dot represents the numerator and the bottom dot represents the denominator. The bar represents divide or a number split that many ways. A fraction is the top dot or numerator divided by the bottom dot or denominator. 1 divided by 4. 1 is the numerator and 4 is the bottom dot or denominator 14=1/4 Or $1 is split 4 ways is a quarter. We can now add, subtract, multiply, divide and do fractions just by knowing how to count and be lazy. With the exception of geometry, what else is there to know in math? Knowing when to use a short cut or not comes with experience. It is also a function of how much time is available. An analogy would be if a carpenter had to cut a piece of wood 3 by 2 by with two holes. One is 5/8 and the other . It takes five minutes to make one piece. But if he had to make 300 pieces it would take him 1500 minutes to make all of them. But if he spent 2 hours making a jig that would allow him to make the pieces in two minutes each, he would save 1500 (600 + 120) = 880 minutes. Therefore it is worthwhile spending time developing a jig. If he only had to make four pieces, it would not be worth his while. Making jigs is similar to finding short cuts. Short cuts to adding, subtracting, multiplying and dividing are worthwhile. It is important for the students to find their own jigs or short cuts. If they create them for them selves they can be easily recreated again if they are forgotten. Also creating new short cuts trains the mind to look for new patterns.


11 of 52

BREAKING PATTERNS Patterns can be broken and encourage students to look for new patterns by a simple exercise. First ask the students to make up a six digit number 234587,eg. Then ask for another six-digit number. 452635,eg. Then add another six-digit number of our own, 765412,eg. Then ask the students for another six digit, 852369,eg. Again follow this with one of our own, 147630,eg. Write them in a column and ask them to quickly add the numbers.

234587 452635 765412 852369 147630

The answer is obviously 2452632. Why? What patterns were followed? If one adds rows 1 and 3 the answer is 999999. If one adds rows 4 and 5, one gets 999999 also. Adding these two answers our total is 1,999,998 or 2,000,000 minus 2. If we add that total to row number two (452,635) our answer is obviously 2,452,633. Instead of always adding the columns, lets look at alternatives. If those numbers were too complex to start out with then lets look at a simpler set of numbers.

09 18 27 36 45 54 63 72 81 90

What patterns are apparent here? We can see that if we add 9 to each subsequent number, we get the next number in the sequence. 18+ 9= 27, 27+9=36 etc. Or we could say that this is the nine times tables. This can be used to learn or to remember or double check our nine times tables. We notice the sequence 1,2,3,4,5,6,7,8,9,0 going up the column and again down the next column.

We notice that 09 written backwards is 90 We notice that 18 written backwards is 81 We notice that 27 written backwards is 72 We notice that 36 written backwards is 63 We notice that 45 written backwards is 54


12 of 52

We also note that:: 18 = 1+8=9 27 = 2+7=9 36 = 3+6=9 45 = 4+5=9 54 = 5+4=9

63 = 6+3=9 72 = 7+2=9 81 = 8+1=9

Again we can use this information to check our nine times tables. The digits in our answer must add to 9 We also note that we can add on the diagonal

1+9=10, 2+8=10 3+7=10 4+6=10 5+5=10

6+4=10 etc

Also 8+0=8 7+1=8 6+2=8 5+3=8 4+4=8

Etc But this information is not really useful

Another interesting pattern of the 9 times table is: 0 x 9 +8 = 8

9 x 9 + 7 = 88 98 x 9 + 6 = 888

987 x 9 + 5 = 8888 9876 x 9 + 4 = 88888

98765 x 9 + 3 = 888888 987654 x 9 + 2 = 8888888

9876543 x 9 + 1 = 88888888 98765432 x 9 + 0 = 888888888

987654321 x 9 1 = 8888888888 9876543210 x 9 2 = 88888888888

What would the next numbers in the series be?


13 of 52

Getting back to our 9 times tables; 09 18 27 36 45 54 63 72 81 90

Now quickly add up these numbers. The answer should quickly be seen as 495 why?

Instead of adding 9+8+7+6 etc. we can add groupings of tens

09 18 a 27 b 36 c 45 d 30 54 40 20 63 10 72 81 90

a 9+1=10 b 8+2=10 c 7+3=10 d 6+4=10

= 40 Plus 5 more = 45


14 of 52

A easier method is to: 09 18 27 36

45 5X9=45 54 63 72 81 90 45

45 495 Why does this work?

A quicker method would be to notice that 5 is the middle number and we have 9 of them by looking at them from a different perspective. 09 = 5+4 6=5 + 1 18 = 5+3 4=5 1 27 = 5+2 7=5 + 2 36 = 5+1 3=5 2 45 = 5+0 8=5 + 3 54 = 5+4 2=5 3 63 = 5 -2 9=5 + 4 72 = 5 -3 1=5 4 81 = 5 -4 5=5 + 0 45=45+0 45 =45+0

Other examples of adding a series are:

9+8+7=3 x 8=24 3+4+5=3 x 4=12

1+2+3+4+5=5 x 3=15

An other method of adding this column quickly is to add the first row and the last row to get

09 +90 =99 18 + 81=99 27 + 72=99 36 + 63=99 45 + 54=99 5 x 99 = 495


15 of 52

09 18 27 36

45 54 63 72 81 90

495

LEARNING ADDIDTION

Adding can be simplified for some who do not want nor have the skills to memorize their addition flash cards.

7+8 = 7+7=14+1=15 Or

7+8=8+8=16 1=15 This works for numbers that are close together. For some reason people remember the sum of pairs.

Some prefer to break the numbers into functions or bases of 5s

7 = 5+2 +8 = 5 +3 15 =10+5=15

This is fairly cumbersome but necessary for some. Most can use a base of 10.

7 7 + 3=10 +8 8 3=5

15 15+0 = 15 What must be added to 7 to make 10 ? --- 3 So we must subtract 3 from 8 =5


16 of 52

Or What must be added to 8 to complete the 10 ? ----= 2 So we must subtract 2 from the 7 = 5

7 7 2 = 5 +8 8 + 2 =10

15 15 + 0= 15 OR

8 8 + 2 = 10 + 7 = 7

17 2 = 15

We take the 8+ 2 more to make 10 plus the 7 = 17 minus the 2 that we added in is equal to 15. If one already knows their addition, these are cumbersome methods. However by looking at these coping methods, one can identify patterns of coping to transfer and to apply to more complex problems. We are learning transferable coping skills for future use.

MULTIPLICATION MULTIPLICATION TABLES SHOULD NOT BE MEMORIZED ; BUT SHOULD BE SEEN IN RELATIONSHIP TO EACH OTHER. The students will eventually learn or memorize the times tables because it is easier. However the times tables should be memorized as a secondary method not as a primary method of learning. They need a primary method that they can always revert back to if their memory temporarily fails them. If a student does not know their times tables down pat, I get them to fill out a blank times table form.


17 of 52

The students are asked to fill in the blank spots. They are asked how they completed the table. What relationships did they use and what relationships can they find. Now they know that they can do the times tables on their own. Once they have completed the table, and we know that they can figure them out on their own, we take it and crumple the paper up and throw it away. This gives the student the confidence that they can do times tables. We just have to find quicker and easier methods of recalling the times tables. The nine times tables create a lot of problems, yet they are one of the easiest. Here are some different coping methods of approach:

1) 9 x 7 =?

We know 10 x 7= 70

10 groups of 7 so

9 groups of 7 would be 10 x 7 = 70 7 =63

one group of 7 less than 70 = 63

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 10 20 3 15 30 4 20 40 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 6 30 60 7 35 70 8 40 80 9 45 90

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 11 55 110 12 60 120 13 65 130 14 70 140 15 75 150


18 of 52

2) ALTERNATIVE METHOD

9 x7 ?

9 x7

We subtract 1 from 7 = 6 ?

9 X7 6?

We can now subtract the 6 from the 9 =3 The answer is 63 3) A THIRD METHOD The easiest method for most is the finger method. Start by placing our hands face up and fingers out . Let the Os represent the open fingers and the Xs represent the finger that is folded over.

1 x 9 is formed by folding the left thumb ( X ) in towards the palm. This leaves 9 fingers out. =9 1,2,3,4 5,6,7,8,9

XOOOO OOOOO = 9 1 2 x 9 is formed by folding the first finger in towards the palm This leaves the 1thumb up to the left of the first finger and 8 fingers to the right of the first finger. 1 + 1,2,3, 4,5,6,7,8

OXOOO OOOOO =1 and 8 = 18 12


19 of 52

3 x 9 is formed by folding in the middle finger. This leaves the thumb and first finger or 2 digits to the left of the second finger and 7 fingers to the right of the second finger. 1,2 +1,2, 3,4,5,6,7

OOXOO OOOOO = 2 and 7 = 27 123 4 x 9 is formed by folding in the ring finger. This leaves 3 fingers to the left of the ring finger and 6 fingers to the left. 1,2,3 +1 2,3,4,5,6

OOOXO OOOOO = 3 and 6 = 36 1234 5 X 9 is formed by folding the baby finger down. This leaves 4 fingers to the left and 5 to the right 1,2,3,4 + 1,2,3,4,5

OOOOX OOOOO = 4 and 5 = 45 6 x 9 is formed by folding in the left baby finger. This leaves 5 fingers on the left hand and 4 on the right hand. 1,2,3,4,5 + 1,2,3,4

OOOOO XOOOO = 5 and 4 = 54 7 x 9 is formed by folding in the left ring finger. This leaves 6 on the left and 3 on the right of the right ring finger. 1,2,3,4,5, 6 +1,2,3

OOOOO OXOOO = 6 and 3 = 63

8 X 9 is formed by folding in the right middle finger. This leaves 7 fingers to the left and 2 to the right of the right middle finger. 1,2,3,4,5, 6,7+ 1,2

OOOOO OOXOO = 7 and 2 = 72

9 X 9 is formed by folding in the right pointing finger. This leaves 8 fingers to the left and 1 to the right of the right pointing finger. 1,2,3,4,5 6,78, +1

OOOOO OOOXO = 8 and 1 = 81


20 of 52

7AND 8 TIMES TABLES GIVE THE MOST PROBLEMS TO THE MOST PEOPLE There are several methods that can be used. If we want to multiply.

7 X 8 =? We know that 8 is = 4 X 2; so we can rewrite the equation as:

8 = (2 X 4) X 7 = ? We know that :

4 X 7=28 Now we multiply :

2 X 28 = 56

OR Another method is :

7 X 8 = ? We know our 5 times tables by counting 5 x : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14,15,16,17, 18 5,10,15,20,25,30,35,40,45, 50,55, 60, 65,70, 75, 80, 85, 90

7 x 8 = ? We know 5 x 8 = 40 And we need 2 more groups of 8 2 x 8 = 16 Adding together, we get 40 + 16 = 56 So 7 groups of 8 = 7 x 8 = 56

OR A more awkward but equally valid method to find 7 x 8 =? is similar to finding 9 x 7= 63 (We went 10 x 7 =70 minus 7 gave 63). Likewise 8 x7 = 9 x 7 = 63 minus 7 = 56 We can go from a known multiplication to an unknown multiplication by adding or subtracting one group of the number being multiplied.

OR The 6 Times tables can be based on the 5 times tables also.

6 x 7 = ?

5 x : 1, 2, 3, 4, 5, 6, 7 , 8, 9, 10, 11, 12, 13

5, 10,15,20,25,30,35,40,45,50, 55, 60, 65,

5 x 7 = 35 plus one more group of sevens 1 x 7 = 7

6 x 7 = 35 +7 = 42


21 of 52

OR A method most people find useful once they get the hang of it, is to put both hands face up with fingers pointing at each other. On both hands: The thumbs represents 10 The pointer fingers represents 9 The middle fingers represents 8 The ring fingers represents 7 The baby fingers represents 6 Left Hand Right Hand To multiply 7 x 8, we touch together the 7 finger ( ring finger) on our left hand to the 8 finger( middle ) on the right hand. This leaves 5 fingers below the 7 & 8 fingers. This gives us our 5 towards the answer. We have 3 fingers above the 7 finger on the left hand and we have 2 fingers above the 8 finger on the right hand. We now multiply the 3 x 2 = 6. The 5 and the 6 are combine to yield the answer 56. The description is harder than the answer. 1 3 x 2 1 2 2 3 = 6 5 Fingers

Touching

7 x 8 = 56

Thumb 10

Pointer 9

Middle 8

Ring 7

Baby 6

10 thumb

9 Pointer

8 Middle Finger

7 Ring Finger

6 Baby Finger

Thumb 10

Pointer 9

Middle 8

Ring 7

Baby 6

10 Thumb

9 Pointer

8 Middle

7 Ring

6 Baby


22 of 52

OR

8 x7 ?

8 x7 can be rewritten with a 10 off to one side as:

10 8 -2 X7 -3

5 , 6 We start by subtracting 8 from 10 leaving 2 And by subtracting 7 from the 10 leaving 3 Now multiply the 2 X 3 gives us the 6 Next, we subtract the 3 from the 8 or we subtract 2 from the 7. Both will give us the 5 The result is 56 This methods works for any number. However if we are using a two-digit number, we use 100 instead of 10.

100 94 6 X97 3

91 + 18 Since 94 is a double digit we use 100 as our base. First we subtract 94 from 100 giving us 6 Next we subtract 97 from 100 giving us 3 We multiply the results of our subtracting 6 x 3 = 18 Now we subtract the 3 from the diagonal number 94 3 = 91 Or we can subtract the 6 from the other diagonal 97 - 6 = 91 As we can see both subtractions on the diagonal will yield the same answer = 91 The final answer is 9,118 The question is why does it work every time? What limitations does it have ?


23 of 52

If the numbers are larger than the base of 10 or 100 we add rather than subtract. Ie.

12 +2 x 14 +4

16 8 12 is 2 more than 10 14 is 4 more than 10 We add 4 to the 12 OR add 2 to the 14 either will give 16 We now multiply the 2 x 4 to give us the 8 We combine the answers to give 168 If one number is above the base and the other is below, we add or subtract To illustrate:

8 x14 = ?

8 -2 x14 +4

12 -8 120 -8 =112 8 is 2 less than 10 -2 14 is 4 more than 10 +4 We add -2 to 14 to give 12 Or we add +4 to 8 which also gives 12 We now multiply the (-2) x (+4) which yields -8 The 12 represents the number in the 10s column or 120 We now combine the 120 and (-8) = 120 8 = 112 So 8 x 14 = 112 So if we wanted 96 x 108, we can use the same principle.

96 -4 108 +8

104 -32 96 is 4 less than 100 108 is 8 more than 100 We add on the diagonal 108 + (-4) =112 Similarly , we could add on the diagonal 96 + 8 =112 The 104 represents the 100s column or 10400 We now combine the 10400 and the (-32) to yield 10368


24 of 52

THE 11 TIMES TABLES

The 11 times tables are very easy. Everybody can do 7 x 11 = 77 but if we ask:

27 x 11 = ? Normally we write

27 x11 27

270 297

Here is where we deviate. If we add the first column as per normal and then skip over to the third column we get

27 x11 27

270 2__7

This leaves us with a 2 and a ,7 the same number we started with 27. Next we add the middle column which is a 2 and a 7, the same number we started with 27. So why not just add the 2 to the 7 = 9 and stick it in the middle of 2 7

2 9 7 = 297 3+4=7

34 x11 374

However

67 6+7=13 is not x11

6137

67 x11 67 670 737


25 of 52

When the two digits add to 10 or more the tens digit carries over to the hundreds column.

6+7=13 67

x11 737

Any number can be multiplied by eleven using this process.

245 26495 4+5 =9 2+4=6 x11 x11 2695 2695

MULTIPLES OF 11 CAN BE DONE IN OUR HEADS

23 2 x 23 = 46 4106 X22 x11 x11 506

Since 22 is 2 x 11, doubling the 23 gives us 46. And 46 x 11 = 506 23 3 x 23 =69 6159 x33 x11 x11 759 By tripling the 11 we get 33 and by tripling the 23 we get 69. 69 x 11 = 759 MULTIPLES OF 25 CAN BE DONE IN OUR HEADS When we want to multiply by 25, we should look at what 25 represents rather than taking 25 at face value. If we dont like the number s we have; change them into one of equal value that are easier to work with. 25 has the same value as 254100 = OR 1004/125.10025 x==

36

x25 = ( 100 4 )= 25 36 x 100= 3600 4 = 900 ?


26 of 52

OR

Another way of looking at the same question is:

36 x25 x( 25 100 ) = .25 = x 36 = 9 9x100 =900 When 25 is divided by 100, we get .25 or 1/4. A of 36 is 9 and multiply the 100 back in to give 900 If we want:

36 36 x26 x25 ? 900 900 + 36 =936

Multiply the 36 by 25 =900 and add one more group of 36s 900 + 36 =936 SIMILARILY 24 times a number

36 36 x24 x25 ? 900 900 36 = 864 Multiply 36 by 25 and subtract one group of 36.

MULTIPLYING BY 75 OR 50

Multiplying by 50 or 75 is the same as multiplying by 25 only 2 or 3 time bigger.

36 36 x75 x25 x3 ? 900 x3 = 2700

OR

36 36 x = 27 x 100 = 2700 x75 ?


27 of 52

MULTIPLING BY 5 5 can be rewritten with an equivalent value of 2105 = A large number can be simplified by multiplying by 10 and dividing by 2 46824 x 10 = 468240 x 5 2 234120 234120 Inversely dividing by 5 can be achieved by multiplying by 2 and dividing by 10 Dividing by 5 is the same as multiplying by 1/5 So 1/5 = 2/10 234120 5 = ? 234120 10 =23412 x 2 =46824

OR

123456 5 = 123456 10 = 12345.6 x 2 = 24691.2 MULTIPLYING AND DIVIDING BY 125

648 x125 ? If we divide the 125 by 1000 we get .125 which is =1/8 We multiply 648 x 1/8 =81 then x 1000 =81000

4512 x 1/8 =564 x1000=564,000 x 125 9753 9753 x 1/8 = 1219 1/8 x 1000 = 1,219,125 x125


28 of 52

Similarly if we want to divide by 125 , we would multiply by 8 and divide by 1000 8/1000=125

872.1125234234125 == OR 18728234 =X then divide by 1000= 1.872

FINDING THE SQUARE OF ANY NUMBER THAT IS A MULTIPLE OF 5 Finding squares of numbers that are multiples of 10s is easy ; just square the first number. For example 9001009103)103(30 2222 ==== To find the squares of 5, 15, 25, 35, 45, 55, ..95,105, 115. We take the number preceding the 5 and multiply it by the next number on the number line Eg. 122525122543352 =+=+= x 0r 3x4=12 plus 2552 = gives 1225 2054452 == plus 20252552 ==

72255729885 22 =+== 1322551321211115 22 =+==

FINDING THE SQUARE OF ANY DOUBLE DIGIT NUMBER

312 = 31X 31 = ? To find the square of any two-digit number we remember : (A +B)2 = (A + B)(A+B) = A2 + 2AB +B2 (31)(31) = (30 + 1)( 30 + 1) = 302 + 2(30)(1) +12 = 900 +60 +1 =961 4356113042251)1)(65(265)165(66 2

2 =++=++=+= With practice these can be done in ones head. To keep the numbers simpler, it may be easier to subtract rather than add. 4096113042251)1)(65(265)165(64 =+=+==


29 of 52

MULTIPLYING DOUBLE DIGIT NUMBERS THAT ARE CLOSE TO EACH OTHER: To multiply two numbers that are close to each other and have a difference between each other that is equal, we use the difference of squares formula. (A + B)(A B) = A2 B2

43 40 + 3 A=40 B=3 43=A+B x37 40 3 37= 40 3 A - B (A B)(A + B) ( 40 3 )( 40 + 3) = 402 32 = 1600 9 = 1591

69 70 1 x 71 70 + 1 (70 + 1)( 70 1) = 702 12 = 4900 1 = 4899

DIVISION

Division is the number of times that the divisor can be subtracted from the dividend 13 4 = ? This can be written as:

?4 13 -4 1 once 9 - 4 2 times 5 - 4 3 times 1 The 1 still has to be divided; so we rewrite it as 1 4 = In the what does the top dot represent? It is for the numerator or the 1 goes on top What does the bar represent? It means divide or split What does the bottom dot represent? It represents the denominator or the 4

1 4 can be rewritten as 14

, which we call one quarter.

This requires too much work but it is possible for anyone to do.


30 of 52

An easier method must be found. We know that we can write division three different ways.

1 4 = 4 1 = 14

We know that if we have six bottles of beer out of a dozen, we can write it as: 6 112 2

We have all had enough beer to know that this is the correct answer What we did was divide 6 by 12 to give us OR we reduced the first fraction 6/12 to a lower or simpler fraction of by dividing the numerator and divisor each by 6 . Similarly we can rewrite

3754848 1212 303 464 484864 16 4 1

=

3754

64 4848

4848

64 can be reduced by 4 and 1212

16 can be reduced by 4 also etc. etc. Instead of dividing by a large number we can gradually reduce the divisor and the dividend by equal increments This is only possible if the divisor and the dividend are reducible by the same factors. We may only be able to reduce the divisor by one or two steps, but even that makes for an easier division. If the divisor is a prime number then we need another method.


31 of 52

TO DIVIDE BY A PRIME NUMBER : To divide a number like:

19 875735 There is no obvious easy method to do this because 19 is a prime number and cannot be factored by definition.

Therefore we can draw a line like a ruler and mark both ends with 0 to19 on the bottom and 0 to 10 on the top

0________________5________________10 0 9.5 19

We look at the first number in our dividend and we see that it is an 8. We determine where the 8 would fit on the bottom scale and that would correspond to a 4 on the top scale.

0____________4____5________________10 0 8 9.5 19 4 19 875735 76 11 11 is our next number

Next we position the 11 on the bottom scale and find the it corresponds to a 6 on the top scale

0____________ 5____6___________10 0 9.5 11 19

46

19 875735 76

115 114 1

0___1_________ 5_______________10 0 1 9.5 19


32 of 52

Since the 1on the bottom is less than the 1 on the top scale, we choose the 0

460 19 875735

76xxx 115 114 173

This leaves us with a 17 to put on the bottom scale and we find that corresponds with a 9 on the top scale

0____________ 5___________ 9___10 0 9.5 17 19

4609 19 875735 76xxx 115 114 173 171 2

0__1__________ 5___________ ___10 0 2 9.5 19 46091. 6/19 19 875735 76xxxx 115 114 173 171 25 19 6


33 of 52

This method is not any different from the regular long division except that we now have a method guesstimating which numbers to use. How and why does it work?

WHAT IS A FRACTION?

What does say? says:

1. 1 2 =1/2 2.

1/ 22 1

3. 0.5 =1/2 4. 50% =1/2 5. 1:2 ratio 6. 1 part out of 2 7. 1 over 2

says all seven of these things and we want to be able to use any of the seven different transformations to make it easier. E.g. 10 x = 10 x (1 2) =10 x 1 = 10 2 = 5 By translating math into English, this says 10 times 1 ( numerator) divided by 2 (denominator) When we multiply two whole numbers together we get a bigger number. When we multiply a number by a fraction we get a smaller number. Why? x = 1 4 = x 1 =1/4 2= 1/8 In English this says 1 divided by 4 equals multiplied by 1 (numerator) and divided by 4 (denominator) However this is clumsy so lets try an easier method. We get the same results if we multiple numerator times numerator and divide by the product of denominator times denominator. x = 1 x 1 =1

( 4 x 2) = 8 = 1/8 Visually, if we have 1 unit and divide it into 4 we get; 1 2 3 4 0 2/4 4/4 0 1


34 of 52

If we divide the s in half we get 0 1 0 1/8 3/8 5/8 7/8 8/8

Multiplying mixed numbers is similar to multiplying double-digit numbers. E.g. 10 x 25 = 10 x ( 20 + 5 ) = (10 x 5) + (10 x 20) = (50) + ( 200) = 250 10 x 2 = 10 x (2 + ) = (10 x 2 ) + (10 x ) = (20) + ( 5) = 25 OR We can change the fraction into a decimal and multiply by the number. can be rewritten as decimal 5 or .5 . So 10 x 2.5 = 25 Same thing only different. However it is often easier to change the mixed numbers to improper numbers. E.g. 5 x 7 = (5 x 4 =20 +1 = 21/4s ) x (7 x 2 = 14 +1 = 15/2s)

= 21 154 2 = 315

8 = 39 3/8

OR

1 35 33 4 = 16 15

3 4 =

since 3 x 4 = 4 x 3 we can rewrite 16 153 4 as 16 15

4 3

16 154...... ....... 34 3 = 4 x 3 = 12

It is easier to cancel out our numbers as we go to keep the numbers smaller. If the numbers are convenient, here is an alternate method.

12127

318

413 =x OR

(A+b)(C+d)= where A=3, b=1/4. C=8, and d=1/3

12127

1211224)

4)(

31()

41)(8()

31)(3()8)(3()

318)(

413( =+++=+++=++


35 of 52

DIVIDING FRACTIONS If we divide two numbers into each other the result is a smaller number. However if we divide a number by a fraction the result is a larger number. Why? 10 2.5 10 = 40 What this says is that if we have a 10-dollar bill and ask to have it changed or divide it into s we should get 40 quarters back. Since there are 4 quarters in 1 loonie and there are 10 loonies in a ten dollar bill there must be (10 x 4 ) = 40 quarters in a 10 dollar bill. Larger smaller Divisor answer

10 5 = 2 10 4 =2.5 10 3 = 3 1/3 10 2 = 5 10 1 = 10 10 = 20 10 1/4 = 40 10 1/8 = 80 smaller larger divisor answer

ADDING AND SUBTRACTING FRACTIONS

Rather than teaching the addition of fractions as something new, lets see what we already know. How much money do you have if you have:

1 quarter .25 25/100 3dimes .30 3/10 30/100 3nickles .15 3/20 15/100 7 cents .07 7/100 7/100 77 cents .77 77/100

Adding up the coins we get 77 cents. No problem if we convert them to decimals or hundredths, we can add them. If we see them as separate fractions, we must convert them to a common denominator., which is what we do when we convert to decimals. And add the numerators. Same thing only different. When we subtract fractions the same rules apply as adding. 5 = 5 2/4 -3 = -3

2


36 of 52

Only sometimes the numerator of the minuend may be smaller than the numerator of the number to be subtracted. In this case we just borrow from the whole number as we do in regular subtraction. E.g.

5 = 5 = 4 +(4/4) + = 4 5/4 -3 = -3 2/4 = -3 1

The easiest method is to add the common denominator to the numerator of the minuend and subtract 1 from the whole number of the minuend. This results in the numerator being large enough to subtract without being negative. This is the same as getting change back from $5.25 when we spend $3.50. We change the $5.00 bill to 4 loonies and 4 quarters plus the other $0.25 giving us 4 loonies and 5 quarters. Some may find an alternate method easier but many dont. It requires the use of negatives

5 = 5 -3 = -3 2/4 2 = 1 5 minus 3 = 2 and minus or 2/4s = negative . The result is 2 minus =1 DECIMALS

If we divide 13.00 4 = 3.25 OR 13 4 = 3 What is the difference? Which is worth more? Why do we write it one-way one time and different the next time? What are the advantages and disadvantages of either?

3.25 can be rewritten or reduced as 25 5 13 3 3100 20 4

= 3 A decimal is a fraction with a denominator of 10, 100, 1000, 1,000, 10,000, 100,000, . Etc EXEPT for rule #2 : we are too LAZY to write in the denominator. It becomes redundant when we know the denominator will be a factor of 10. I call them metric fractions. Even though there is no such designation. Our monetary system is based on decimals, which are factors of tens. $3.25 tells us that we have three dollars and 2 dimes or 2/10 of a dollar plus 5 cents or 5/100s of a dollar. The 2 represents the 1/10 dollar place the 5 represent the 1/100s of a dollar place etc..


37 of 52

An easy method of reading decimals is to put your pencil on the decimal point and draw two lines ; one vertical below the decimal point and one horizontal to the right of the decimal point and add the same number of zeros as we have numbers to the right of the decimal point 3.25 === 3 25 = 3 25 = 3 25 00 00 100 and is read three and twenty five hundredths To multiply decimals, we multiply the decimal by a factor of10 and divide the answer by the same factor of 10. 8 8 8 x0.5 x(0.5 x 10)= 5 x5 40 10 = 4.0 To divide by a decimal, we multiply BOTH the divisor and dividend by the same factor of 10.

?

2.5 1.25 ( ) ? 0.52.5 10 1.25 10 25 12.5x x

( ) ( )?? ? 10.2

0.45 4.590 0.45 100 4.590 100 45 459.0 45 459.0x x EQUIVILANTS

A fraction, a decimal, and a percent are all equivalent representations of the same value. = 0.25 = 25% are all equal or equivalents Each has its own advantages and disadvantages. We must be able to convert back and forth easily.

= 1 4 =0.25

4 1 = 0.25 =25%

0.25 = 25100

= 14

=25%

25%= 25100

= 14

= 0.25


38 of 52

Fractions can be converted to decimals by dividing the numerator by the denominator (which is the definition of a fraction) Decimals can be converted to fractions by an easy method. That is to put your pencil on the decimal point and draw two lines ; one vertical below the decimal point and one horizontal to the right of the decimal point and add the same number of zeros as we have numbers to the right of the decimal point. The numbers to the right of the decimal form the numerator and the horizontal line is the divide by . The denominator is formed by the vertical line which represents the 1and the zeros that are added on to form 10s, 100s, 1000s, etc.. A percentage can be converted to a fraction by changing the% sign to 1/100s and then we have a fraction. Eg 25% is equivalent to 25/100 4/1 . The percentage can be converted directly into a decimal by removing the % sign and dividing by 100 or by moving the decimal two places to the left.

25% 41

1002510025

FRACTIONS TO DECIMAL EQUIVILENTS There are some common and very useful fraction to decimal equivalents that are well worth knowing. But it is important to see the pattern rather than memorize the list.

We start with = 1 2 = 0.5

2 1 =0.5000 A half of is = x = and a half of 0.5000= 0.500 x = 0.2500 A half of is = x =1/8 and a half of 0.2500 = 0.250 x = 0.125 A half of 1/8 is = 1/8 x =1/16 and a half of 0.125 = 0.125 x = 0.0625 3/8 is found by 3 x 1/8 = 3/8 and 3 time 0.125 = 0.375 OR 3/8 is found by 1/8 + = 3/8 and 0.125 + 0. 250 = 0.375 5/8 is found by + 1/8 = 5/8 and 0.125 + 0.500 = 0.625 is found by + = and 0.250 + 0.500 = 0.750 7/8 is found by + 1/8 = 7/8 and 0.750 + 0.125 = 0.875


39 of 52

The next group starts with 1/3 = 1 3 = 0.33333

3 1.0000 = 0.33333333333 repeating

2/3 is formed by doubling 1/3 2 x 1/3 = 2/3 and doubling 0.33333 = 0.66666 1/6 is found by halving 1/3 and halving 0.33333 x 1/3 = 1/6 0.333333 x = 0.166666 1/12 is found by halving 1/6 and halving 0.166666 1/6 x = 1/12 0.166666 x = 0.0833333 1/9 is found by finding a third of 1/3 and a third of 0.3333 1/3 x1/3 =1/9 0.333333 x 1/3 =0.111111 2/9 is found by doubling 1/9 and doubling 0.111111 1/9 x 2 = 2/9 2 x 0.111111 = 0.22222 49 = 4 x 1/9 4 x 0.111111 = 0.444444 etc

1/11 is found by 0.0909090909

11 1.0000000000 0.09090909 repeating

2/11 is found by doubling 1/11 2 x 1/11 = 2/11 2 x 0.090909 =0.18181818 3 x 1/11 = 3 11 3 x 0.090909 =0. 27272727 4 x 1/11 = 4/11 4 x 0.090909 = 0.36363636 etc. 1/9 = 0.111111 = 0.11,11,11, factors of elevens and 1/11 = 0.090909 = 0.09,09,09 factors of nine

IN ORDER

1/32 0.03125 1/12 0.08333 1/16 0.0625 1/11 0.0909090 1/8 0.125 1/9 0.111111 0.250 1/6 0.166666 3/8 0.375 2/11 0.18181818 0.500 2/9 0.222222 5/8 0.625 1/3 0.33333 0.750 2/3 0.66666 7/8 0.875 5/6 0.833333 We also notice that 1/12 = 0.083333 which is the same as 0.83333 = 5/6 only ten times larger.

If we multiply 1/12 x 10 = 10 512 6

= 1/12 = 0.08333 x 10 =0.83333


40 of 52

We can now do all the fractions or divisions of: , 1/3s,1/4s, 1/5s, 1/6s, 1/7s,(to follow) 1/8s, 1/9s, 1/10s, 1/11s, 1/12s, 1/6s, and 1/32s. 1/7s are unique and fascinating: First we take the 7 and double it. 1/7 =0.14 we double the 14 = 1/7 =0.1428 We now double the 28 and add the1 from the numerator, which gives us 56 + 1 = 57 1/7 = 0.142857 and this number repeats itself forever. 1/7 = 0.142857142857142857.etc 2/7 can be found by doubling 1/7. 2 x .142857142857 2/7 =0.2857142857. 3/7 can be found by tripling 1/7 or adding 1/7 + 2/7= 3/7 = 0.42857142857etc 4/7 , 5/7 and 6/7s can be found by adding the multiples of 1/7s. however if we look at the pattern we find a easier method is: 1/7 = 0.1428571428 2/7 = 0.2857142857 3/7 = 0.42857142857 4/7 = 0.5714285714 5/7 = 0.71428571428 6/7 = 0.85714285714 The same sequence repeats itself over and over again. The only difference is that we start the sequence at the first smallest number 1 and 2/7 with the second smallest number in the sequence at 2 and 3/7 with the third smallest number in the sequence 4 And the 4/7 at the fourth smallest number in the sequence at 5 and so on and so . We can quickly convert other fractions to decimals by combining

? 0.31255 1 116 5 0.250 0.0625 0.3125 16 516 4 16

= = + = + = PERCENT

The etymology of percent is : PER means divide or each e.g. miles per hour; dollars per hour; etc CENT means one hundred 100 So per cent means divide by 100. or for each 100

So why do we write 43 percent as 43% not 43100

? Rule number #2. People just

got lazier and lazier.


41 of 52

43100

43 100

43

100

43 100

43

100

43 100

43 100

43 100

43 0 0 = 43%

43% = 43100

Initially there are only three types of questions that we can ask in percentages. What is 75 percent of 12 9 is what % of 12 and 9 is 75% of what number?


42 of 52

To do this we use ratios to make one fraction the same as another fraction.

We also put the fractions in a box. 9 7512 100

=

9 is 75 %

12 of 100

The box always contains the inserts: is, of, %, and 100 in the same order. By reading the questions and noticing where the words of, is, % fall in the question. We insert the numbers appropriately. The 100 always goes in the lower right box. There should always be two other numbers in the question to appropriately fill in boxes and we can calculate the third number. Now we just make the two fractions equal by finding a common denominator. Or by cross multiplying the two numbers on the diagonal and by dividing by the third number present. E.g. What (x) is 75% of 36? The x or unknown goes in the is box and the 75 goes into the% box. The 36 is next to the of box and goes there.

X is 75 %

36 of 100

The 36 is multiplied by the 75 (the two numbers on the diagonal). And divide by the third number ( 100) 36 x 75 = 900 100 = 9 9 is 75% of 36 OR What (X) percent is 9 of 36?

9 is X %

36 of 100

9 x 100 = 900 36 = 75 %


43 of 52

OR We can reduce the fraction: 9 1 ? 1 25 25 25%36 4 100 4 25 100

xx

== = = == and fin out how many s in 100 WORD PROBLEMS Most people have problems with word problems because there are too many variables to memorize. So we need a easy method of handling word problems. There are seven steps that can be used too solve almost all word problems.

1 RTFQ 2 What is given? 3 What are we looking for? 4 Draw a diagram. 5 Change the numbers 6 Write and equation and solve 7 Revert back to the original numbers and solve

#1 RTFQ means Read The F ing Question. F stands for Following or Full. The student has their choice as to what F stands for. #5 means choosing numbers that are easy to work with and are easy to visualize. EG. If a tank contained 285 liters when it was 5/8s full. How much would it contain if it were full?

1 RTFQ 2 How much would it contain if it were full? 3 We have 285 liters and 5/8 full 4

600 =300

5/8 =285


44 of 52

5 If we change the numbers from 285 to 300 and 5/8 to we can see in the changed numbers would give us. If the tank had 300 when it was half full it must have 600 when it is full. The answer must be 300 ? =600 Now we have to determine what procedure to use that will give us our answer. We only have four choices: add, subtract, multiply or divide.

300 600 subtraction wont give us the correct answer 300 + 600 addition wont give us the correct answer 300 x 600 multiplication wont give us the correct answer 300 = 600 we can see that the correct procedure would be to divide

6 285 5/8 = 285 x 8/5 = 456

METRIC

Metric has been used in Canadas monitory system for over a century now. Our monitory system uses groups of tens. There are 100 cents in a dollar and 10 cents in a dime and 10 dimes in a dollar. The metric system for all forms of measurement was adopted in Canada in the mid seventies. The French originally created the metric system about 300 years ago to simplify measurements in science and trade. A meter was originally defined as one, ten millionth the distance from the North Pole to the equator. A more precise and consistent method of measuring this distance is to measure the time that light travels over the one meter. This time is

1299,792,458

s of a second. Or, the speed of light is measured at 299,792,458

meters per second. The prefixes kilo, hecta, deca, deci centi, and milli are used for all three base units. The base units are the meter, liter and gram. Each is broken into factors or multiples of tens. The different denominations of dollars all have different slang equivalent terms. 1000 100 10 1 1/10 1/100 1/1000 Grand c-note sawbuck $ dime cent mil Kilo Hecta Deca Meter Deci cent milli Liter Gram


45 of 52

A kilometer is 1,000 meters A meter is 1,000 millimeters or 100 centimeters or 10 decimeters A kiloliter is 1,000 liters. A liter is 1,000 milliliters. A kilogram is 1,000 grams A gram is 1,000 milligrams A milliliter is also equal to 1 cubic centimeter (1cm3) by definition. A gram is the mass or weight equal to one milliliter of water at Standard Temperature and Pressure There is a direct relationship between linear measurement, volume and mass (weight). If we know one measurement, we can calculate the other two measurements, unlike the Imperial system. How many cubic inches in a peck? To convert metric units of measurement, all we have to do is move the decimal point the appropriate number of places. To convert 234 centimeters (cm) to kilometers, we look at the scale and determine that kms are five positions to the left of cms. So we move the decimal place five places to the left. 234 cm 234. cm = 0.00234 km 5 5 5 4 3 2 1 0 Kilo Hecta Decca Meter Deci cent milli

0 1 2 3 4 To convert 23.12 hectometers, we move the decimal place four places to the right (in the direction from hectometers to centimeters four units) 23.12 hm = 23.1200. 231,200 cm . . 4 4 Similarly, liters and grams are converted by moving the decimal place left or right depending on the relative direction the prefix is to be converted. BASIC GEOMETRY Rather than memorizing all sorts of formulas, lets look at the similarities among the different geometric shapes. We need to calculate perimeter, circumference, area, and volume. Perimeter is the distance around an object, regardless of how many sides or shape that the object has. Circumference is the perimeter of a circle or the distance around a circle. If we wrapped a string all the way around the geometric figure from start to finish and stretched the string out, that would be the total measured distance around the figure. The distance between the two points on the string (from start to finish) would be a one-dimensional measurement. E.g.


46 of 52

feet , centimeters, meters, inches, etc Perimeters are found by adding the total of all sides together. The area of an object would be the number of square units that would fit into each geometric figure. Area is found by multiplying the number of units in one direction by the number of units in a perpendicular direction. Since we are measuring in two directions, we will have two-dimensional measurements. The exponents on the units will always be a 2. E.g. cm x cm = cm 2, feet x feet = feet 2, M x M = M2 two-dimensional

Volume is the number of cubes that can b put into a three dimensional geometric figure. Volume is measured by multiplying the number of measuring units in three different perpendicular directions. E.g. cm x cm x cm = cm3 , ft x ft x ft = ft3 , M x M x M = M 3 three dimensional PERIMETER AREA VOLUME One-dimensional two-dimensional three dimensional S5 S1 + S2 + S3 + S4 =P S1 X S2 = A A x S5 = V S1 x S2 x H = V S1 S4

S2 CUBE

H S1 S2 S1 +S2 +S3 +S4 = P S1 X S2 = A A X H = V RECTANGLE (S1 x S2 ) x H = V

PERIMETER AREA VOLUME S1 + S2 + S3 = P A xH = V S3 S2 S1 x S2 = A S1 TRIANGLE = a rectangle

S3

S3 S4


47 of 52

D1 D2 D4

D3

The perimeter of the box with dimensions equal to the Diameter of the circle is 4D. But we dont want the full perimeter of the box because we dont go all the ways to the corners. We follow the curve from D1, to D2, to D3, to D4, and back to D1. So instead of 4D we use 3.14 D = D = circumference

The area of box #1 is R X R = R2 .

And if we add the area of the other four small boxes we would have an area of 4 R2 But we dont want the

shaded areas of the boxes . So instead of 4R2 we

use 3.14R2 OR R2

Diameter

R 1

R


48 of 52

ALGEBRA Algebra is just somebody too lazy to write out all the words that they want to use is solving a problem. E.g. Q: Johnny went to the store and bought four apples. The four apples cost one dollar. How much did each apple cost? A: Johnny bought four apples.

Four apples cost one dollar One apple cost one quarter. OR Quicker and easier 4 apples = $1 1 apples = 1 4 = OR easier still 4 Apples = $1 4 A = 1 1 Apple = A =

We went from 69 letters in the first solution to 7 symbols in the last or algebraic solution. That is all that algebra is; just somebody too lazy to write out all the extra words. Algebra is like a foreign language that has to be translated back into English. When that is done; it is easy. X usually represents some unknown number. So every time we see X , we can translate it to read some unknown number. Algebraic equations are balanced similar to a teeter-totter. If the weight or value on one side is equal to the opposite side, then the teeter-totter stays balanced otherwise one side goes up and the other side goes down. We want to keep algebraic equations balanced. E.g. 5X + 7 = 23. Translated into English the equation says, 5 times some unknown number plus 7 more is equal to 22. What is the unknown number? To solve this, we want to know what some unknown number X represents. We want to put the variable or X on one side of the teeter-totter and every thing else on the other side. If we take 7 off one side of the teeter-totter then one side will go up and the other side down. So we have to subtract 7 from both sides. But another method of seeing the same thing is to ask What is attached to the X and how? We see that the 7 is attached by a + and if we want the 7 on the opposite side we use the opposite sign of + which is negative a -.on the opposite side of the equation.


49 of 52

We want the 7 to change sides ; so we change signs 5X + 7 = 22 5X = 22 7 = 15 5X = 15 The 5 is still attached to the X by multiplication, and we want it on the opposite side . So the opposite of multiplication is division. So we divide the opposite side by 5 5X = 15 X = 15 5 = 3 X = 3 5(3) + 7 = 22 5 times 3 plus 7 more = 22 The X is isolated on the one side of the teeter totter and the equivalent is on the other side of the teeter-totter. ( X + 2 )(X +3) =X2 + 5X + 6 Translated into English, this says some number X plus two more multiplied by the same number plus three more. This is no different than 22 x 23 = (20 + 2) x (20 + 3) = 400 + 40 +60 + 6 = 506 written horizontally rather than vertically. 22 20 + 2 x23 x 20 + 3 66 60 + 6 440 400 +40 506 400 + 100 + 6 = 506 It is the same thing only different. INTEGERS Integers are positive and negative numbers. One must learn to be able to interchange the plus sign + and the positive sign. The rules for multiplication and division of positive and negative integers are the same. Translating math into English, we have: + x + = + I have some = + have is + and some is + - x + = - I dont have some = - dont is - and some is + + x - = - I have none = - have is + and none is - - x - = + I dont have none = + dont is - and none is - if you dont have none , you must have some


50 of 52

To add or subtract two or more numbers with the same sign, we just add or subtract as per normal and maintain the same sign. + 22 - 22 + 33 - 33 + + 33 + - 33 - + 22 - -22 + 55 - 55 + 11 - 11 How ever to add two numbers of opposite signs, we change the sign of the smaller number and subtract the smaller number from the larger and keep the sign of the larger number.. Eg. + 33 + 33 - 33 - 33

+ - 22 - + 22 + + 22 - - 22 + 11 - 11 This procedure makes the signs the same and we add or subtract as per normal. And inversely to subtract two numbers of opposite signs; we change the sign of the number to be subtracted and add the two numbers together. + 33 + 33 - 33 - 33 - - 22 + + 22 - + 22 + -22 + 55 - 55 An alternate method of looking at adding and subtracting positive and negative integers is to combine the two signs into and do as required.eg. + 33 + 33 - 33 - 33 - - 22 - x - = + + 22 - + 22 - x+= - -22 + 55 - 55 A negative times a negative is a positive and a negative times a positive is a negative.


51 of 52

CONCLUSION Math is a process and there is more than one approach or process that will achieve the correct answer. There are no right or wrong methods of doing math; only easier and harder methods. Not all brains operate in the same manner as others or the same manner each time. This booklet was designed to break people out of a model that did not work well or at all for them. Hopefully it is designed to stimulate others into finding newer approaches and a better understanding of math. The methods described in this booklet are mostly from students and have been modified and rewritten to show a consistent pattern. If one can recognize the oneness or the similarity of addition, subtraction, multiplication, division and fractions; then one can see the oneness or simplicity of all aspects of math. If you have additions, modifications, or improvements please email me at [email protected] COPYRIGHT This material is copyrighted but may be used or copied by anyone with the exception of the material being sold for profit. In which case permission may be requested by contacting [email protected].


52 of 52

07SD _2_fjsdf sdfsdfjf

Documents

Transcript of 07SD _2_fjsdf sdfsdfjf