Reducing And Multiplication Practice

FRACTIONS	SHORT DIVISION
When you you click the fraction button, a fraction to reduce is displayed. The common divisors are displayed to the right of the division signs. To select a pair of Greatest Common Divisors (GCD), press the up/dn arrows until the GCD's you want to select are displayed.	You can use the short division application to divide the given numerator and denominator by a single digit GCD and find the reduced numerator and denominator.

After you select a pair of GCD's, divide and enter your new numerator and denominator in the textboxes to the right of the equal signs. Then, click the question button to see if your answers are correct. Choose a small fraction and below you can explore solving the fraction mechanically.
MECHANICAL REDUCING Click the frame below and follow the displayed instructions to generate a pattern of diamonds to represent the small fraction from above. Be sure to center your pattern of diamonds, then follow instructions below frame.

1.After you display the denominator and the numerator, to reduce the fraction, click the leftmost figure in the bottom and top rows. The diamonds will change into squares. 2.Cick a second diamond in the bottom row such that the square in the top row is equidistant from the two squares in the bottom row. 3.Continue clicking diamonds to the right alternately in the top and bottom rows such that any two squares in the bottom row with only diamonds between them are equidistant from a square in the top row. 4.When you cannot continue this pattern to the right, count repeatedly to the left along the bottom row a number of figures (diamonds and squares) equal the numerator. At the end of each count equal the numerator, click the diamond at that point in the bottom row. 5.If you reach the leftmost figure in the bottom row at the end of a count, reducing is complete. 6.If you do not reach the leftmost figure in the bottom row at the end of a count, starting where your count ended repeat steps '3, 4 and 5' 7.The GCD will equal 1 plus the number of diamonds remaining between any 2 squares. *Explore another way to find the GCD below.
REDUCING BY TAKING THE DIFFERRENCE Click frame and follow instructions on frame to enter numerator and denominator. To find the GCD of the pair, follow instructions below the frame.

1.In the denominator position of the rightmost pair enter the number in the numerator position of the previous pair. 2.In the numerator position of the rightmost pair enter the difference between the two numbers of the previous pair. 3.Create a new pair and repeat steps 1 and 2 until the two numbers of the rightmost pair are equal. *You have found the GCD of the original pair of numbers.
CLICK HERE TO SEE FRACTIONS ABOVE CLICK HERE TO SEE TIC-TAC-TIMES TABLES BELOW
PRIME FACTORING	For prime factoring, click the number button to enter a number between 0 and 100 in the textbox. Then, press the arrow keys to find a prime factor of the number in the textbox. The prime numbers are seen right below the textbox. Only prime number will be displayed. To select a prime factor click on the numbers across the vertical line directly to the right of the prime numbers. Click on these numbers in order from left to right to select prime factors. After you select a prime factor a new product will appear above the factor. For each product that appears, press the arrow keys to find another prime factor. Then, click the next number across the vertical line until the product '1' appears above the selected prime factor. By pressing the right arrow key, you can jump multiples of ten numbers at a time.

FINDING THE COMMON DIVISORS BY PRIME FACTORING	To find prime number common divisors of the numerator and denominator of a fraction, first place the cursor over the number above the numerator. Then, press the arrow keys to find a prime divisor of the numerator. To select the prime divisor, click on the numbers across the vertical line directly to the right of the numerator. To see if the number is a divisor of the denominator, enter the number below the denominator and to select click on the numbers across the vertical line directly to the right of the denominator. Each time you find a common divisor continue entering primes in order from left to right. When you can find no more common primes, multiply the found common divisors to get the Greatest Common Divisor (GCD).

In the table below, the number in the top left square (square 1) is the multiplicand or number you count by, and the order number of the square is the multiplier or number of times you count by the number in square 1 . Therefore, the content of each square is a multiple of the number in square 1. To find the order number of a square, start at square 1 and count from left to right zigzagging through the rows. To change the multiplicand, enter a number in the blue textbox and click square 1. To view the multiples, click the left gray button.
Tic-Tac Times Table Rules:
1) The top left small tic-tac table shows that if the sum of the order numbers of a pair of squares equals 10, then the ones digits of the multiples in those squares also sum to 10.
2) The middle small tic-tac table shows that the sequence of the one's digits, of the multiples, will make through the tic-tac table a horizontal zigzag, beginning or ending in square 1, or a vertical zigzag, beginning or ending in square 7. Read more below.

3) If the number in square 1 is odd, except in the case of 5, the ones digits sequence will include the counting numbers 1 through 9. If the number in square 1 is even, the ones digits sequence will include two sets of the even numbers 2 through 8 with a 0 in the center square.
4) Starting at square 1, if the one's digit of a next multiple in the sequence decreases in value, the ten's digit of the next multiple will increase in value by 1. Otherwise, the value of the ten's digit will not change.
If you are a little clever, the above rules will allow you to find the multiples of the 1's through the 9's tables without counting by the multiplicand. Each table has hidden patterns you may discover to make remembering the times tables easy. To test yourself, click the show/hide button to hide the ones digits or the tens digits of the multiples (products). Then, enter respectively the ones digits or the tens digits. If you enter the incorrect value, a blue 'X' will appear in the square, otherwise your correct entry will appear.