Saturday, December 25, 2010

Multiplying Numbers Using the Square of Binomial Formula: Lesson 2

Now let’s try to find the answer for the squares of these numbers:

24, 33, and 42

Example 1:

1-    24 = (20+4) and 24²= (20 + 4)²

The reason we want to change it to 20 + 4 is so that it is easier to find the square of 20 than the square of 24. The square of 20 = 400.

Let’s apply the following formula (a + b)² = a² + 2ab + b².  In this example
a = 20 and
b = 4
2-      a² = 20² = 20 × 20 =     400

3-      2ab = 2(20 × 4) = 2(80) = 160. Which is double the product of (a × b)

4-      b² = 4² = 4 × 4 =          16

Now let’s add all together using the Square of Binomial formula.

 a² + 2ab + b² = 400 + 160 + 16 the numbers we got from steps 2, 3 and 4.

(400 + 160) + 16 = 560 + 16 = 576
So 24 × 24 = 576

Let’s do another one.


Example 2:

1-    33 = (30 + 3) and 33²= (30 + 3)²

The reason we want to change it to 30 + 3 is so that it is easier to find the square of 30 than the square of 33. The square of 30 = 900

Let’s apply the following formula (a + b)² = a² + 2ab + b².  In this example
a = 30 and
b = 3
2-      a² = 30² = 30 × 30=      900

3-      2ab = 2(30 × 3) = 2(90) =180. Which is double the product of (a × b)

4-      b² = 2² = 3 × 3 =              9


Now let’s add all together using the Square of Binomial formula.

 a² + 2ab + b² = 900 + 180 + 9 the numbers we got from steps 2, 3 and 4.

(900 + 180) + 9 = 1080 + 9 = 1089
So 33² = 33 × 33 = 1089

For your own practice, try repeating the same process with the number 38. If it would help, try copying and pasting one of the examples and just replace the numbers with the new numbers that correspond to the number 42. You would use (40 + 2) to represent the number 38 in the formula. Since 4 × 4 = 16, 40 × 40 will equal to 1600.


Once you do several examples, you will start finding that it is much easier than you think. The more you practice, the better you will get at it. After you’ve done a few examples, try doing it mentally to see how fast you can come up with the answer. You can try applying this method to the numbers: 22, 23, and 24. However, for the numbers 26, 27, and 28, it would be easier to try the steps in lesson 1


Saturday, December 18, 2010

Multiplying Numbers Using the Square of Binomial Formula: Lesson 1

Now let’s try to find the answer for the squares of these numbers:
18, 27, and 38

Example 1:
1-    18 = (20-2) and 18²= (20 – 2)²
The reason we want to change it to 20 – 2 is so that it is easier to find the square of 20 than the square of 18. The square of 20 = 400.
Let’s apply the following formula (a - b)² = a² - 2ab + b².  In this example (a = 20) and (b = 2)
2-      a² = 20² = 400
3-      2ab = 2(20 × 2) = 2(40) = 80. Which is double the product of (a × b)
4-      b² = 2² = 2 × 2 = 4

Now let’s add all together using the Square of Binomial formula.
 a² - 2ab + b² = 400 – 80 + 4 the numbers we got from steps 2,3 and 4.
(400 – 80) + 4 = 320 + 4 = 324
So 18 × 18 = 324

Let’s do another one.

Example 2:
1-    27 = (30-3) and 27²= (30 – 3)²
The reason we want to change it to 30 – 3 is so that it is easier to find the square of 30 than the square of 27. The square of 30 = 900
Let’s apply the following formula (a - b)² = a² - 2ab + b².  In this example (a = 30) and (b = 3)
2-      a² = 30² = 900
3-      2ab = 2(30 × 3) = 2(90) =180. Which is double the product of (a × b)
4-      b² = 2² = 3 × 3 = 9

Now let’s add all together using the Square of Binomial formula.
 a² - 2ab + b² = 900 – 180 + 9 the numbers we got from steps 2,3 and 4.
(900 – 180) + 9 = 720 + 9 = 729
So 27 × 27 = 729

For your own practice, try repeating the same process with the number 38. If it would help, try copying and pasting one of the examples and just replace the numbers with the new numbers that correspond to the number 38. You would use (40 - 2) to represent the number 38 in the formula. Since 4 × 4 = 16, 40 × 40 will equal to 1600.

Once you do several examples, you will start finding that it is much easier than you think. The more you practice, the better you will get at it. After you’ve done a few examples, try doing it mentally to see how fast you can come up with the answer.

In the next lesson we will do examples using (a + b)² = a² + 2ab + b². This one should be a little easier since it is all addition, no subtraction involved. Lesson 2

Wednesday, December 15, 2010

Multiplication Utilizing The Square of Binomial

Now let’s take a look at the Square of Binomial, and how we can utilize this formula to ease multiplication of two digit numbers.

Here is the formula again (a ± b)² = a² ± 2ab + b².  
We want to first break the formula down so that it is easier to read.  Rather than the ± sign, we will break it down using the “+” sign once, and utilizing the “-“sign the other time.
(a + b)² = a² + 2ab + b².  This is what the formula looks like with the plus sign—the sum of two numbers raised to the power of 2. We will replace the ‘a’ and ‘b’ with digits and perform the multiplication as prescribed by the formula.

Let (a=30), for instance and we will let (b=1). So (a + b)² will look like this (30 1)².
We know that 30 + 1 = 31. So, what we are doing now is finding the square root of the number 31, or in other words 31 × 31.
The formula says that (a + b)² = a² + 2ab + b². or (30 + 1)² = 30² +( 2 ×30×1) + 1².  
This might look as if it makes the multiplication a bit more complicated, but when you get the hang of it, you will be able to do it mentally without using a pen and paper.
We will do it one segment at a time:

1- 30² or 30×30 = 900
2- (2 ×30×1) = 2 ×30 = 60. And 60 × 1 = 60. So the result of the middle part = 60.
3- The last part b², which is 1².  Or 1 × 1 = 1

Now let’s add 1,2 and 3 together as depicted in the formula:
900 + 60 + 1 = 961. This means that 31 × 31 = 961.
Now we will do the square of different between two numbers. (a - b)² = a² - 2ab + b².  

In this example we will be finding the square of the 29.
We will let a=30 and b=1. We want to stick with the number 30 to keep simple and relevant.
Now (30 – 1) is actually 29, which is the number we want to multiply. Let’s apply the formula now. Notice this time that the signs have changed a bit from the previous one. There is a ‘-‘ sign before the middle (2ab).
This is what it should look like with numbers instead of the letters ‘a’ and ‘b’.
(30 - 1)² = 30² -( 2 ×30×1) + 1² . Now let’s look at the sum of each segment.

1- 30² or 30×30 = 900. Same as above
2- (2 ×30×1) = 2 ×30 = 60. And 60 × 1 = 60. So the result of the middle part = 60.
3- The last part b², which is 1².  Or 1 × 1 = 1

The only difference here is, instead of adding the middle part; we will be subtracting it from the first sum, as shown here (a - b)² = a² - 2ab + b².  So here is the breakdown:

900 - 60 + 1 = 841. This means that 29 × 29 = 841.

If you’ve gotten the hang of it, good for you. If not, we will continue breaking it down in the upcoming post. Stay tuned, and feel free to post any questions that you might have.


Sunday, December 12, 2010

Fun Learning Way to Square a Number

There are several ways to perform multiplications. You can use a pen and paper, use a calculator, or memorize the results of particular numbers. Off course, trying to memorize the results of multiplications is not practical. You can only memorize so many. However, memorizing the multiplication table up to 10 or 12 might be a good idea to facilitate learning advanced multiplications. If you are able to memorize even up to 99, more power to you. But beyond that, I think is a waste of time and energy and takes the fun learning concept out of the equation. What a person needs to do instead is memorize and practice a method that will allow him/her to instantly process numbers in their brain to come up with the answer. I’m sure there are several methods that can practice with numbers. In this post, I will discuss only one fun and easy learning method that can be easily applied to a variety of two digit numbers.

If you know the basic multiplication table, you will be able to instantaneously answer the results of 10 × 10. 20 × 20. 30 × 30. and so on. I’m sure this should not be a problem for you. If so, here are the answers:
10 × 10 = 100. Basically this is the result of 1 × 1, and just adding the number of zeroes to the result 2 in this case. I hope I’m not insulting your intelligence or confusing you here.
20 × 20 = 400
30 × 30 = 900
If you can instantaneously know the answer for 8 × 8, for instance, which is 64, you would know
that 80 × 80 is equal to 6400.
How many seconds does it take you to do that? I hope it’s no more than 3 seconds.

Now we will only deal with multiplying like numbers—the square of a 2 digit number—so that the concept is easily understood, and then we can move on to other numbers. Let’s start with a different numbers than the, fun and easy one’s, that end with zero. How about (19²), (29²) and (39²), how can we multiply these numbers in our mind without using a pen and paper or calculator?

To do so, what we’re going to do is round them to the nearest number ending with a zero and take it from there. 
Let’s try the first number 19. The nearest zero number to it is 20. Why? Because 19 is actually (20 – 1). In other words, 19² is the same (20 – 1 )².  Realizing this much, we will first know that the answer will be close to 400, but not quite.

So the product of 19 × 19  is actually (400  - 19 – 20) = 361. 
And 29² is actually 30 × 30 = 900 – (30 – 29) = 900 – 59 = 841
The last one is 39² = (40)² - (40 – 39) = 1600 – 79 = 1521

If you didn't quite understand the concept here, don't worry, you'll get it later. In the next post, we will do similar multiplications utilizing a different approach, the Square of a Binomial. This approach should make things much easier than the previous method.
(a ± b)² = a² ± 2ab + b².  

Monday, December 6, 2010

Learn Math The Fun Way

Mathematics is the science of dealing with numbers, quantities, shapes, patterns and how they relate. It is not isolated skills and procedures. Mathematics involves many aspects of our daily lives. It relates to subjects like science, art, engineering, music and even cooking. In fact, it relates to many things we do on daily bases.


Although math involves many things in our lives, a lot of people dislike or hate math. Many students consider math to be their worst subject in school. It is the subject responsible for the big plunge in their grade point average. People hate math, most likely, because they had a terrible math teacher or teachers in grade school. Because they were not properly taught the basic concepts of mathematics, they find it hard and frustrating to understand and grasp later concepts. According to an official survey reveals that about 70% of the world’s student’s state that they hate math. They feel that math is daunting and intimidating. Students that begin to have trouble with math in early stages will dismiss the subject as something they are just not good at. It is true that math is not the easiest subject to learn; however, if approached properly, math could become fun to learn and very engaging. They must comprehend the “why” mathematical process is the way it is. Explain to them the reason, and they will learn how to reason.


To make it fun to learn for their kids, parents should promote reasoning by asking their kids questions and giving them the time to think it through. By doing so, they are encouraging their children to exercise critical thinking and reasoning. Finally, practice is the key to success in any endeavor. The three key elements to success is practice, practice and more practice.


There are many entertaining activities, games, and books that are geared towards enhancing math skills for different levels. You’ll be surprised at how much faster your kid will start learning when they are having fun and enjoying doing it. Even if the student is already doing well, he/she will start doing better. 

Sunday, November 28, 2010

Math Fluency

According to the Merriam Webster’s Dictionary, to be fluent is “to be capable of moving with ease and grace, effortlessly smooth and flowing, and showing mastery of a subject or skill.”


What is math fluency?

Math fluency means knowing the right method to correctly solve a mathematical problem with ease and in a relatively short period of time. It is also being comfortable with different approaches to solving a giving problem. The more fluent a person becomes in math, the better and smoother he/she becomes in accurately applying mathematical concepts to real life situations. Reaching good fluency in math requires proper, meaningful and consistent practice. Knowing how to apply a formula to solve a math problem does not necessarily mean one is good in math. It takes much more than applying a math formula to be considered good.

Reasoning and understanding of concept
A student is considered fluent if he/she is able to:
  •  Explain the solution process step by step and explain the reason for each step.
  •  Explain the formula and know how the formula came about.
  •  Use alternate methods to solve a problem, if one exists.
Building Blocks
It is very important to completely understand the basic building blocks of math in order to achieve mastery of the subject. One should have mastered the concepts of adding, subtracting, multiplication and division before moving on to higher math levels.
If you are able to establish a link between the math operations and build on them as you advance, you should be able to master the subject with ease. 

Tuesday, November 23, 2010

Good After School Recreational Activities

The words 'after school activities' are often misconceived. Some might feel that since these activities are after school, they are of little importance. Those who think so could not be more wrong. According to research, children gain some of their most important skills from after school programs.  It has been noticed that children who do not perform extracurricular activities are generally slow and less vibrant.

The after school activities must be implemented in an environment that is as disciplined and as efficient as that in school. Normally, these environments are where children acquire essential skill goal setting and time management. Time management is a crucial skill to learn. Children must experience the discipline in staying on task until it’s completed, to appreciate the gratitude of accomplishing the mission in the allotted time.  The best programs are those that are both recreational and educational, especially activities that are age-appropriate and integrate with what happens in school.

After school program s helps strengthen children’s interests.  They can also help them discover their natural abilities for some activities.  Children will need a good collection of skills in addition to their academic achievement in order to succeed in this dynamic world.  

In addition to the acquired educational and disciplinary benefits of these programs, they tend to also keep children off the streets and out of trouble while parents are still at work. According to Gottfredson, Gerstenblith, Soule, Womer, and Lu, "Children and adolescents who are not supervised by an adult for extended periods of time are at elevated risk for engaging in problem behavior".

Good after school programs keep children busy, active, healthier, and distracted of bad peer influence. 

Saturday, November 20, 2010

Make it Fun to Learn For Your Student

Often times student complain of boredom in school or in a certain subject.  This could be a signal for low or declining grades, and therefore the overall performance in school. Motivating students might not be an easy task. It could take time, skills and lots of patience. 

Some students are naturally, intrinsically motivated and enthusiastic about learning; however, many rely on outside forces, like parents and teachers, for continuous inspiration.  Unfortunately, there is not a magical formula for motivating students. Many factors affect the student’s motivation to effectively learn.  However, research has proved that students learn better when the subject is fun to learn and enjoyable. 

In many situations, the subject is not the problem; the problem is in the way the content is introduced to the student.  Feeding information to students is not proper teaching, and does not contribute to motivating students. Inspiration comes from the way the information is presented to the student so that it becomes interesting and fun to learn.  
According to (Lowman, 1984; Lucas, 1990; Weinert and Kluwe, 1987; Bligh, 1971), Instructors need to do the following to encourage self motivation:
  • Give frequent, early, positive feedback that supports students' beliefs that they can do well.
  • Ensure opportunities for students' success by assigning tasks that are neither too easy nor too difficult.
  • Help students find personal meaning and value in the material.
  • Create an atmosphere that is open and positive.
  • Help students feel that they are valued members of a learning community.

In order to make our students achieve we need to engage them in fun games and fun activities, relative to learning, and give them the opportunity to enjoy.  Unless students have a good interest in doing something, they will not approach it enthusiastically. This is where innovative methods must come in to aid the learning process and establish the fun to learn factor.  

Making it Fun to Learn

Anyone can learn almost anything they want as long as they have the proper tools and the time to learn it. Learning tools should be more than just instructional content. They should encompass diverse methods of presenting content so that it delivers excitement.  Look for reading and math games, for instance, that can make your student strongly interested and engaged in the learning activity. You will find many enjoyable, learning games for different subjects and different levels that could be of great help to you and your student. 


Go to Good After School Fun Activities