Here's some tips for kids dealing with multiplication tables.

There is an important rule shown in the multiplication tables. Any number multiplied by 0 is equal to 0, and is not included in the multiplication table. Any number multiplied by 1 is itself.

Here are the multiplication tables for 1 through 12

1 x 1 = 1    2 x 1 = 2   3 x 1 = 3   4 x 1 = 4    5 x 1 = 5    6 x 1 = 6   7 x 1 = 7

1 x 2 = 2   2 x 2 = 4   3 x 2 = 6   4 x 2 = 8   5 x 2 = 10    6 x 2 = 12   7 x 2 = 14

1 x 3 = 3    2 x 3 = 6   3 x 3 = 9   4 x 3 = 12    5 x 3 = 15   6 x 3 = 18   7 x 3 = 21

1 x 4 = 4    2 x 4 = 8   3 x 4 = 12   4 x 4 = 16    5 x 4 = 20   6 x 4 = 24   7 x 4 = 28

1 x 5 = 5   2 x 5 = 10    3 x 5 = 15   4 x 5 = 20   5 x 5 = 25   6 x 5 = 30   7 x 5 = 35

1 x 6 = 6   2 x 6 = 12   3 x 6 = 18   4 x 6 = 24   5 x 6 = 30   6 x 6 = 36   7 x 6 = 42

1 x 7 = 7   2 x 7 = 14   3 x 7 = 21   4 x 7 = 28   5 x 7 = 35   6 x 7 = 42   7 x 7 = 49

1 x 8 = 8   2 x 8 = 16    3 x 8 = 24   4 x 8 = 32   5 x 8 = 40   6 x 8 = 48   7 x 8 = 56

1 x 9 = 9   2 x 9 = 18    3 x 9 = 27   4 x 9 = 36   5 x 9 = 45   6 x 9 = 54   7 x 9 = 63

1 x 10 = 10   2 x 10 = 20   3 x 10 = 30   4 x 10 = 40   5 x 10 = 50   6 x 10 = 60   7 x 10 = 70

1 x 11 = 11   2 x 11 = 22   3 x 11 = 33    4 x 11 = 44   5 x 11 = 55   6 x 11 = 66   7 x 11 = 77

1 x 12 = 12   2 x 12 = 24   3 x 12 = 36   4 x 12 = 48   5 x 12 = 60   6 x 12 = 72   7 x 12 = 84

8 x 1 = 8   9 x 1 = 9   10 x 1 = 10   11 x 1 = 11   12 x 1 = 12

8 x 2 = 16    9 x 2 = 18   10 x 2 = 20   11 x 2 = 22   12 x 2 = 24

8 x 3 = 24   9 x 3 = 27   10 x 3 = 30   11 x 3 = 33   12 x 3 = 36

8 x 4 = 32   9 x 4 = 36   10 x 4 = 40   11 x 4 = 44   12 x 4 = 48

8 x 5 = 40   9 x 5 = 45   10 x 5 = 50   11 x 5 = 55   12 x 5 = 60

8 x 6 = 48   9 x 6 = 54   10 x 6 = 60   11 x 6 = 66   12 x 6 = 72

8 x 7 = 56   9 x 7 = 63   10 x 7 = 70   11 x 7 = 77   12 x 7 = 84

8 x 8 = 64    9 x 8 = 72    10 x 8 = 80    11 x 8 = 88    12 x 8 = 96

8 x 9 = 72   9 x 9 = 81   10 x 9 = 90   11 x 9 = 99   12 x 9 = 108

8 x 10 = 80   9 x 10 = 90   10 x 10 = 100   11 x 10 = 110   12 x 10 = 120

8 x 11 = 88   9 x 11 = 99   10 x 11 = 110   11 x 11 = 121    12 x 11 = 132

8 x 12 = 96   9 x 12 = 108   10 x 12 = 120    11 x 12 = 132   12 x 12 = 144

There are a few things to notice when looking at the multiplication tables. Staring with the 1's, each answer goes up by 1. We say that the multiples of 1 are 1,2,3,4,5,6, and so on.

For the 2's tables, each answer goes up by 2. The multiples of 2 are 2,4,6,8,10,12, and so on.

For the 3's tables, each answer goes up by 3. The multiples of 3 are 3,6,9,12,15,18, and so on.

The same pattern is true for all the multiplication tables.

The answer to a multiplication problem is also called the product.

Notice that any number multiplied by 10 ends in 0.

It may seem hard to learn the whole table, but notice that 1 x 2 is the same as 2 x 1, 3 x 2 is the same as 2 x 3, and so on

When dealing with confidence intervals for proportions, the formula is

p^ +/- Zcritical(standard deviation)

Where p^ = r/n,  r is the number of observations and n is the sample size

For a 95% confidence interval, Zcritical = 1.96

The standard deviation is square root[(p^)(1 - p^)/n]

The margin of error, denoted at E = Zcritical(standard deviation)

To find inflection points and concavity, take the second derivative and set equal to 0. Solve for x, then test a value on the left of the value for x and one on the right. If the second derivative of this value is less than zero, it's concave down on that interval, if the second derivative of this value is greater than zero, then it's concave up.

For example:

f(x) = 3x^3 + 2x^2 + 5x + 6

first derivative :   f ' (x) = 9x^2 + 4x + 5

second derivative :  f " (x) = 18x + 4

set the second derivative equal to 0 and solve for x

18x + 4 = 0

18x = -4

x = -4/18 = -2/9

Test a value to the left of -2/9,  I choose -1

f " (-1) = -14

Test value to the right of -2/9, I choose 0

f " (0) = 4

Since f " (-1) is negative, the concavity is downward from negative infinity to -2/9

Since f " (0) is positive, the concavity is upward from -2/9 to infinity.

During a course of algebra, teachers discuss functions and composition of functions. The topics can be confusing to many students, who also don't see any practical uses beyond the classroom. The next few paragraphs will clear any confusion you have on these topics.

Suppose we wish to represent the function g ( x ) by a token machine. The token machine yields one token for every quarter that is deposited into the machine. The tokens can be used to purchase prizes. We think of the quarters as the input x and the number of tokens as the output g ( x ). Suppose further that there is another machine that requires the use of tokens to obtain prizes. A certain number of tokens are needed to purchase each prize. We'll define the prize machine as f ( x ). The input is the number of tokens, which we defined as g ( x ). Notice that purchasing a prize out of the second machine is dependent on the number of tokens from the first machine. Such dependence can be interpreted in mathematical terms as composition of functions.
In the previous example, the domain x yields g ( x ), the number of tokens. Then g ( x ) becomes the input into f ( x ) to produce the output, which is the prize purchased. The end result is a composition function f º g , also noted as f ( g ( x )).

Next, notice the composition function f º g , also noted as f ( g ( x )), joining the two machines together as one machine which automatically deposits tokens into the prize generator, which ejects the appropriate prize corresponding to the number of tokens generated.

Here's a practical example using the composition of functions. A meteorologist predicts a low pressure area to move across the region over the next 36 hours. The current temperature of 60 degrees Fahrenheit is forecast to drop 1 degree every 3 hours. What is the composition function that expresses the Celsius temperature as a function of the number of hours from now? Note that expresses the Celsius temperature as a function of the number of hours from now? Note that C = (5/9)( F - 32).

To solve this we need to know the current temperature and the rate of change of the temperature. We know the current temperature is 60 and there is a 1/3 degree drop expected every hour. We will represent time in hours since the temperature is 60 degrees at t. The temperature in Fahrenheit at time t will be expressed by the function F (t ). The temperature at time t expressed in Celsius will be given by the composite function C (F ( t )).

The above example is just one application of composite functions in real life situations. The goal of the articles was to explain composition of functions as far as their structure is concerned and to show a real life application. I believe my explanation will clear questions one might have on these topics.

On the long running hit game show, "Let's Make a Deal," Monty Hall would ask the contestant which of three doors he or she wants. Behind each door is a prize, only one of which is valuable. After the contestant chooses a door, Hall opened one of the other doors and revealed a worthless prize. Then he would ask the contestant if he or she wanted to switch doors. To maximize chances of winning, one should always switch.

To analyze the problem, suppose I were to pick door number 1 and then was revealed door number 3 to be the worthless prize. Now I know the prize is either behind the door I chose or door number 2. It appears that the probability of winning is 50% and switching doesn't increase or decrease my chances.

Think of the problem this way. The probability that the prize is behind door number 1 given that the prize is not behind door number 3. The probability behind door number 1 equals 1/3 since there are 3 doors to choose from. The probability the prize is behind door number 1 or 2 knowing it is not behind door number 3 is 1/2.

The problem involves the use of conditional probability. Suppose you have events A and B, then the probability of A given B equals the probability of A and B divided by the probability of B. This is denoted as P(A/B) = P(A and B)/P(B).

Using the formula for conditional probability, you will find the probability of winning if switching is (1/3)/(1/2) = 2/3. Another way to think of this is that if you switch, the only way that you lose is if the prize was behind the door you initially picked, which has probability of winning 1/3. So the probability that you win is 1 - 1/3 = 2/3.

Although switching doesn't seem to give the best chance of winning, you can conduct an experiment. Play the game with someone assuming the role of Monty Hall and you'll find out as you increase the number of games you play, you'll win approximately 2 out 3 times.

To find the multiplicative inverse of a  2 X 2 matrix, there is a simple method.

But if you don't mind memorizing a formula, try the method shown below.

 



Example: Find the multiplicative inverse of

Over the course of mathematics, we learn how to raise a quantity to an exponent. We know, for example, that xⁿ means we multiply x by itself n times. But suppose n is zero. How do we multiply a number by itself zero times? We are taught that any number raised to the zero power equals one, but rarely does anyone explain why this is the case.

One way to explain the zero exponent phenomenon is to use rules for exponents. When multiplying like bases, exponents are added. For example, 3⁵ x 3⁰ = 3^{(5 + 0)}. Therefore 3⁵ x 3⁰ = 3⁵, so 3⁰ must equal 1.

When dividing like bases, exponents are subtracted. For example, (4⁶)/(4⁰) = 4^{(6 - 0)}. Therefore (4⁶)/(4⁰) = 4⁶, so 4⁰ must equal 1.

Another way to show that any base raised to the zero power is one is to examine some patterns. Notice the pattern in the following:

• 2⁴ = 16
• 2³ = 8
• 2² = 4
• 2¹ = 2
• 2⁰ = ?

Notice that as exponents decrease by 1, the result is divided by the base number 2. Continuing this pattern, 2⁰ = 1. This will hold true no matter what the base number is.

Finally, I can use the concept of limits to show that any number raised to the zero power is one. Suppose we take 5ⁿ, and we start with n = 1. We know that 5¹ = 1. Take the square root of 5, which is equivalent of 5^(1/2), which is approximately 2.24. Now take a smaller value for n, such as 1/3, which gives us 5^(1/3) = 1.71. Continue to take values for n smaller and smaller but not less than or equal to zero. You'll start to notice what is happening, 5^(1/10) = 1.17 , 5^(1/1000) = 1.002, 5^(1/100000) = 1.00002. Notice how the result is getting closer and closer to 1. We say that the limit as n approaches 0 of 5ⁿ = 1.

I just explained three methods that clarify why any number raised to the zero power equals one. Next time someone is puzzled by this fact, you can explain the reason behind this somewhat vague and often explained topic in mathematics.