Friday, October 25, 2030





Elementary Mathematics




Tips and Tricks Elementary School Mathematics

Preface

When I started on this document, I had the intention to write something useful and helpful to the juniors of the elementary schoolers. However, I soon realized that it was a non-doable task as the language proficiency of such schoolers would be inadequate to understand the abstraction of the topics I intent to cover. Hence, I changed my mind and am now writing something that educators ( parents, tutors, teachers ) could use and help transfer the concept to these schoolers.

In addition, I would add that I am not trying to teach nor cover the syllabus that are intended but am trying to add to specific topics especially time-tables. My aim is to help increase the schoolers mind-agility when dealing with numbers for I believe that it is this lack of ability that will hamper them when they are dealing with abstraction later on in life.

Numbering System

The majority of countries in the world use the Decimal numbering system. Under this system numbers are represented with the combination of these digit/numbers : 0, 1, 2 , 3, 4 , 5 , 6 , 7 , 8 , 9. To help understanding of the definitions of these digits I will resort to using visual aids.

From the visual aids you can see that there is only 1 true number and that is one ( 1 ). The higher ( in value ) numbers are just collections, groups, packaging or combinations of 1.
For example, 4 is just merely a group of 4 ones ( 1111 ). I am not trying to unteach what kids learn from educators or schools but just trying to point out that the numbers are just groups.
This is very important when I touch on Arithmetic further down this document.

Representation of a number

A number in this system is represented as a row or strings of digits eg. 573893 The value of each digit is calculated as such:
The number 573893 therefore has a value of five hundred seventy three thousand eight hundred and ninety three.

What is zero

Zero has no value and it’s not even a number.
Zero is enigmatic. As a digit it is just a place holder but, as a number it has value but even that value is non-deterministic and most academics steer away from defining its value, so we will just leave it after stating the followings:
  • any number + 0 = that number
  • 0 + any number = that number
  • any number multiply by 0 = 0
  • 0 multiply by any number = 0
The result of any number divided by zero is not determined so don’t ever do this.
0 divided by any number is defined to be zero ( I do not agree with this but am not able to give a better answer so accepted that grudgingly )
Don’t even bother to understand the statements above just remember them.
Zero is a plug-gap for a non- perfect numbering system.
These operations with 0 do not exist in real life see footnote.











Sunday, September 1, 2030

Mathematics - Rules of Addition & Multiplication

Arithmetic Operations

Addition

Addition involves combining groups of 1 into a bigger group. Examples :
4 + 5 is 1111 + 11111 = 111111111 ( 9 )
6 + 7 is 111111 + 1111111 = 1111111111111 ( 1 3 )

Rules of Addition

Let’s perform a simple addition. I want to drop oranges into a fruit basket and count the total when I am done. If you do not have enough oranges or a fruit basket you can roll small paper balls and use a cup or a bowl.

Pick up the oranges 1 group, (start with the left group first), at a time and throw them into the fruit basket. When you are done count the number of oranges in the basket. If your answer is 9 then well done you got it right. Next, again starting with the same groups of oranges and an empty basket perform the same action except that this time you start with the group of 3 oranges. Count the oranges in the basket when you are done.
Did you get 9 oranges ? What have you learned from this simple exercise?

Rule 1

When you perform addition, it does not matter which group or number you start with first. In the end you will still get the same answer. In the language of Mathematics, it is known as the Commutative law of Addition.
Big and sophisticated sounding , but bah! don’t bother just store the name somewhere in your brain. Even if you should forget, it is alright. I bring it out just in case it comes out in any of your conversation.

6 + 3 = 9 111111 111 111111111
3 + 6 = 9 111 111111 111111111
As illustrated the groups of 9 (1s) are non-distinguishable. The numbering system makes them look unalike - try putting them together 36 or 63 (disaster).

This rule applies to any number of groups or numbers that you are adding.
5 + 6 + 2 = 13
6 + 2 + 5 = 13
2 + 5 + 6 = 13
You can discover for yourself by making several groups of oranges and then dropping them in the basket and counting the total. In other words, order does not matter when you are performing addition.
If you still have any difficulty understanding this rule think of it this way.

Perhaps your class has 35 students and each day they enter the class in groups or individual and at different time. This routine will change from day to day, and each day the group size may be different and the order in which they enter the classroom will also vary. However, the total number of students ( total count ) will be the same ( barring any not coming to school.). Students entering the class is akin to putting oranges into a basket.

Rule 2

When performing Addition, you may consider splitting up each or all of the groups involved into smaller groups and then perform the summing or adding. Or, you can even shuffle the small groups around, then do the adding.This rule is a combination of the Associative and Distributive rule. Again, don’t lose sleep over the terms just understand the application. In fact, during day to day application the rules fade into oblivion.

Always make groups of 10 whenever possible (This is my rule )


Multiplication

Multiplication is just repetitive addition. It is a special case of addition and that is why all the rules of addition is exactly applicable to Multiplication. Schools inadvertently lead students into thinking that Multiplication is an independent operator.
6 x 7 = 7 + 7 + 7 + 7 + 7 + 7
The multiplicand is added a number of times equal to the value of the multiplier, both terms I believe you know so I will not extend on those.

Difference between Addition and Multiplication

When we add several numbers in normal addition the numbers tend to be different in values and rather random eg. 4 + 7 + 12 + 3 whereas when we multiply, a number (the multiplicand) is added several times eg. 7 + 7 + 7 + 7 + 7 + 7. This is the only difference.

Hence there is nothing difficult or scary about multiplication. Multiplication sounds and act different from Addition just because you have been pre-programmed to remember the result of such Additions from very young and you have a vast store of such results in your long term memory. Addition unlike Multiplication tend to be very random and there are just too many combinations to memorize and hence never done. I debate if remembering such results do us good today because all of us including little kids are walking around with calculators ( handphone ). However, I hesitate to recommend not remembering tables as our examination system still depends on such.

Tuesday, August 13, 2030

Multiplication tables

Note: if your kid is still young and not too inclined to numbers then it is best to prepare beads, balls , small marbles etc and use them to demonstrate what I am depicting on the examples below. We are into Heuristic.

1 times table

any number x 1 = that number
1 x any number = that number
6 x 1 = 1 + 1 + 1 + 1 + 1 + 1 ( sum 6 groups of 1 ) = 6 ( 1 group of 6 )
1 x 6 = [ 111111] = 1 1 1 1 1 1 ( 6 groups of 1 ) = 6

10 times table

Any number x 10 = that number + the digit 0 (in front)
6 x 10 = 60 [ 10 + 10 + 10 + 10 + 10 + 10 = 60 ]
9 x 10 = 90 [ 10 + 10 + 10 + 10 + 10 + 10 +10 + 10 + 10 = 90 ]
The zero is there to help us know that the 9 & 6 are in the tens place and have positional value of 10.

10 x any number = that number + the digit 0 in front
10 x 8 = 80 = 8 + 8 + 8+ 8 + 8 + 8+ 8 + 8 + 8 + 8
= [8 + 8 + 8+ 8 + 8 + 8+ 8 + 8] + [ 8 + 8 ]
= [8 + 8 + 8+ 8 + 8 + 8+ 8 + 8] + [ 2+2+2+2 + 2+2+2+2 ]
Transfer the 2(s) over to the 8(s) to make 10(s)
= [10 + 10 + 10 + 10 +10 + 10 +10 + 10 ] = 8 x 10 = 80
Note : I break up the 8(s) into 2(s) because 8 + 2 = 10.
10 x 3 = 30
4 x 10 = 40
Hence, you can infer that there is no need to learn the 10 times table.
This is very important - emphasize it.

11 times table

Any number from 1-9 x 11 produces a twin-digits result
3 x 11 = 33
8 x 11 = 88
11+11+11+11+11+11+11+11
remove 1 from each group of 11
= 10 + 10 +10 + 10 +10 + 10 +10 + 10 + [ 1 1 1 1 1 1 1 1 ]
= 80 + 8 = 88 (Once again, I create 10(s).)
11 x any number from 1-9 produces a twin-digits result
11 x 3 = 33
11 x 8 = 88
8 + 8 + 8+ 8 + 8 + 8+ 8 + 8 + 8 + 8 + 8 = 8 + 8 + 8+ 8 + 8 + 8+ 8 + 8 + [ 8 + 8 + 8 ]
= 8 + 8 + 8+ 8 + 8 + 8+ 8 + 8 + [ 2+2+2+2 2+2+2+2 + 8 ]
Transfer the 2(s) to the 8(s) to produce 10(s)
= 10 + 10 + 10 + 10 +10 + 10 +10 + 10 + 8
= 88
Up till now none of the time-tables need any form of learning. Help your kid understand what I have written and if needed demonstrate with actual objects like beads, balls or whatever you have at hand.

Thursday, July 11, 2030

5 Times table

5 times table

This table is also a very important table.
Odd numbers - 1, 3, 5, 7, 9
Even numbers = 2 , 4, 6 ,8
Take half of these numbers :
if odd subtract 1 then divide by 2
if even just divide by 2
5 is odd so 5-1 → 4 / 2 → 2
4 is even so 4/2 → 2
You may have a problem because your kid has not learned division yet.
Simple, demonstrate to her/him with real objects.
Since 5 is odd show 5 then remove 1 and divide the remaining
( 4 ).To do the dividing just pick up 1 object at a time and place each on each side of the dividing line until all 4 are placed with 2 on each side.
Note: Do not follow your instinct and pick up 2 objects to place on each side of the line. Your child has not learn division so he/she may not know how many to pick up (imagine say 11 objects - she will not know that the final result will be 5. By placing 1 object at a time he/she will come to know that the result is 5 and that will help your child learn division in the process.
Any number x 5:
Follow the halving process described above then add a 5 in front if the number is odd
otherwise add a 0.
9 x 5 = ?
half the nine then add 5 : ( 9 - 1) /2 = 45
8 x 5 = ?
half of 8 = 4 and since 8 is even the result is 40
The commutative law of Addition is applicable to Multiplication so the result
of 5 x any number = any number x 5

Tuesday, June 11, 2030

Heuristic

I will take a rest from tables for a while and demonstrate how Heuristic can be used to solve
all multiplication problems without resorting to remembering tables.
Let’s work out the following problems :
6 x 7 , 9 x 8 , 7 x 4 , 11 x 3 , 12 x 5
6 x 7 = 1111111   1111111    1111111

1111111   1111111    1111111

We create 10(s). We need a group of 3 for every group of 7.
There are 2 ways :
  • break down groups of 7(s) to create groups of 3(s)
  • borrow groups of (3) then subtract the amount borrowed after the fact

Variation of method 1

combine the 6 groups into 3 groups
1111111     1111111     1111111     11111111111111     11111111111111
1111111     1111111     1111111     11111111111111 ( 1 group from 2 smaller groups )
Break the 3 groups into groups of 10(s)
1111111111    1111     1111111111     1111     1111111111     1111
Combining the left-behinds and arranging the groups we have
1111111111    1111111111     1111111111     111111111111
Finally break the recombined group into 10
1111111111    1111111111    1111111111     1111111111     11
Answer = 42
9 x 8 = 8 + 8 +8 + 8 +8 + 8 +8 + 8 +8
= 8 + 8 +8 + 8 +8 + 8 + 8 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2
= 10 + 10 + 10 + 10 + 10 + 10 + 10 + 2
= 72
9 x 8 = 8 x 9 = 9 + 9 + 9 + 9 +9 + 9 +9 + 9
( I chose the bigger number to be the multiplicand )
borrowing 8 (1s) we get 10 + 10 + 10 + 10 + 10 + 10 +10 + 10 = 80
return the 8(1s) or 8 we get 80 - 8 = 70 + 10 - 8 = 70 + 2 = 72
Using the variant of the above method
9 x 8 = 8 + 8 +8 + 8 +8 + 8 +8 + 8 +8
= [ 8 + 8 ] [ 8 + 8 ] [ 8 + 8 ] [ 8 + 8 ] + 8
= [10 + 6 ] [10 + 6 ] [10 + 6 ] [10 + 6 ] + 8
= [ 10 + 10 + 10 + 10 ] [ 6 + 6] [6 + 6 ] + 8
= [ 10 + 10 + 10 + 10 ] [ 10 + 2] [10 + 2 ] + 8
= [ 10 + 10 + 10 + 10 + 10 + 10] [ 2 + 2 + 8 ]
= [ 10 + 10 + 10 + 10 + 10 + 10 + 10] [ 2 ]
= 72
As you can conclude, it is possible to find the answer to any multiplication by using the Heuristic methods presented above. However, they do take a bit of time to finish. Hence, in the next section I will introduce a new method - its less wordy and easier to manage.

Before I do that, I have to introduce 2 groups of numbers.
3 x 3 or 111+ 111+ 111 = 9
3 x 2 or 111+ 111 = 6
Let’s calculate 9 x 8:
we will split 9 into 3 + 3 + 3
and 8 into 3 + 3 + 2
Next let’s draw a rectangle which is 3 columns by 3 rows
Another way , Heuristic
12 x 5 = 5 x 12

Since 11 + 11 + 11 + 11 + 11 = 1111111111
We have 6 groups of 10 = 60
In Mathematical parlance we write this heuristic as
5 x 12 = 5 x 10 + 5 x 2
= 50 + 10 = 60


Sunday, May 12, 2030

2 Times Table

2 is an interesting number.
An Even number is defined as a number which when divided by 2 does not leave any remainder
or
any number or group of 1(s) is even if all the ones can be group as 2(s) without leaving any 1 dangling without a partner.
6 → 111111 → 11 11 11 → even
7 → 1111111 → 11 11 11 1 → not an even number otherwise known as odd ( dangling 1 - in red )
Any number when multiplied by 2 will produce an even number. The order of the multiplication is not important and does not have to be the last. And, so long as at least one multiplication-by-2 is carried out the result will be an even number. Since multiplication by 2 is actually adding the multiplier 2 times, any dangling 1 will be paired with another to produce a group of 2 .
In other words, multiplying by 2 adds out the 1 in an odd number.

Any number multiplied by 2 is just a case of adding the number to itself.

6 x 2 = 6 + 6 = 12 ( always group to 10(s) when possible - 6 + 4 + 2 = 10 + 2 = 12 )
9 x 2 = 9 + 9 = 18 ( 9 + 1 + 8 = 10 + 8 = 18 )
This is so because multiplication is just repeated addition.
In this instance the number is added 2 times to produce the result.
Before moving on ensure that your child knows the 2 Time-tables to perfection.
Just so you know, the 2-table has a wraparound (on the unit digit ) after 5.
6 x 2 → 1 x 2 = 2 and a 10 : 6 x 2 = 1 2
7 x 2 → 2 x 2 = 4 and a 10 : 7 x 2 = 1 4
6 wraps to 1

9 wraps to 4
This is an alternative way to remember the 2 -table- numbers after 5 :

Friday, April 12, 2030

3 and 4 Table

3 Times table

Any number x 3 = the number x 2 + the number

4 Times table

Any number x 4 = that number x 2 then either multiply the result by 2 or add the result to itself to get the answer
e.g. 7 x 4 = 7 x 2 = 14
either 14 x 2 = 28 ( this is effective if you train your child beyond 12 x 2 ) or 14 + 14 = 28
So far, we have covered , 1 , 2 , 3 ,4 ,5 10, 11 times tables and we will deal with
6, 7, 8, 9 & 12 soon. But before doing so we should re-visit the rectangle. The rectangle calculates unit square. It is exactly the same as calculating the area except for the dimension ( m2 ). There is no dimension involved.
Let’s do 4 x 5 using the rectangle


Answer : 4 +6 + 4 + 6 = 20
I split 4 into 2 + 2 and 5 into 2 + 3
The rest is obvious. This is a very useful tool and please encourage your kid to learn it. It will grow in usefulness as he/she grows in age.

Mathematics - Rules of Addition & Multiplication

Arithmetic Operations Addition Addition involves combining groups of 1 into a bigger group. Examples : 4 + 5 is 1111 + 11111 = 111111111 ( ...