Multiplication from another perspective

Do you remember that old days when you were struggling to memorize the multiplication table? What if I told you that you didn’t need to memorize the whole table? 😀

While I was watching an online course on edX called “Effective Thinking through Mathematics” Professor Michael Starbird introduced an effective way to do multiplication operations. Suppose that you have a 5*5 grid as the following:

 

fivebyfive

Then I asked you to make that squared grid a 4*6 rectangular grid by simply removing the grid’s top row and concatenating it as a new column, which will result in the following:

 

fourbysix1

How many squares should we get rid off to obtain the correct 4*6 rectangular grid? Simply the one which is colored above!

Alright. Then I asked you again, starting from the 5*5 grid, to obtain a 3*7 rectangular grid from that squared grid by removing the top two rows and concatenating them as two new columns. The result will be:

threeby7

Now, how many squares should we get rid off to obtain the correct 3*7 grid? Again, the four colored squares.

Have you realized what’s going on? We are removing a^2 squares each time we’re converting the n-squared grid to a (n-a)*(n+a) rectangular grid.

12757724475_24ffc973fc_b

You are now probably wondering what’s the relation between those grids and multiplication. The answer is that if you want to do the following operation: x*y, where x<y, you can can calculate the number n as (x+y)/2 and the number a as (n-x) and do your operation by subtracting a^2 from n^2, which means that you only need to memorize the squares of the numbers not the whole multiplication table!

Will that method be valid if the sum of x and y is odd? YES! For example, if we are multiplying 6*7, then n equals 6.5 and a equals 0.5, by applying our method: (6.5)^2 – (0.5)^2 = (42.25)-(0.25) = 42 = 6*7!

I am feeling sorry for your suffer while you were a kid (and for myself too :D), but you can still save another kids from what we went through :D.

 

I’ll be happy to read your positive or negative feedback!

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How to install Ubuntu 14.04 alongside Windows 8.1

Hello, I am going to explain how to install Ubuntu 14.04 alongside Windows 8.1 step by step:

*NOTE: Please back up all your data before proceeding any step, I am not responsible for any loss of data! 😀

*This tutorial isn’t the best way to install Ubuntu 14.04 alongside Windows 8.1, but it’s a working one.

*This tutorial is for whom didn’t find the alongside option appearing in the following picture

1- Download Ubuntu 14.04 from the following link and follow the burning tutorial included:

http://www.ubuntu.com/download/desktop

2-Burn both windows 7 & windows 8.1 on DVDs.

3-Boot windows 7 by restarting your computer and pressing F12 on the computer start up to open the boot menu.

4-Choose the DVD drive where windows 7 disc is available.

5-Proceed until the partitions step. Make a partition for Ubuntu, Windows 7 and a partition for your data (optional).

6- Format the partitions on which Windows 7 and Ubuntu are going to be installed.

7-Choose the partition where you are going to install windows 7 on.

8-Proceed with Windows 7 installation.

9-Boot Ubuntu by Following the instructions in this tutorial until Step 3 ONLY.

http://www.ubuntu.com/download/desktop/install-ubuntu-desktop

10- Choose “Something Else” option.

11-Now you should find the three partitions you have made previously.

12-Select the partition where you want to install Ubuntu and press the minus sign “-” below.

13-Select the free space obtained from the previous step and press the plus sign “+” below.

14-Let the size be equal to your RAM size, choose “logical” at “Type for new partition”, choose “swap area” at “Use as”.

15-Select the free space again and press the plus sign.

16-Do not change the size, just choose “logical” and “/” at “mount point”.

17-Choose the partition made in the previous step to install Ubuntu on and proceed with the installation.

18-By reaching this step you should have two operating systems on your PC and most likely on the computer start up Ubuntu opens automatically, to solve this problem make the following step.

19-Open the terminal and write the following sudo grub-install /dev/sd1

20-Now you could choose either windows 7 or Ubuntu on start up from Ubuntu boot loader.

21-Insert Windows 8.1 disc and install it on the partition where Windows 7 is installed.

22-After finishing windows 8.1 installation you will find that the options on PC start up did’t change (either windows 7 or Ubuntu), don’t worry, just choose windows 7 if you want to open Windows 8.1 😉

23-Finally, un-tick the fast start up option from windows 8.1 control panel to access the data on the other partitions from Ubuntu.

24-Congrats! 😀