Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage (link at left) to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!
Dr. Carol J.V. Fisher's Homepage
MULTIPLICATION
Jump right to the exercises!
See the best ALGEBRA PINBALL time for this exercise
The concepts for this exercise are summarized below; for a complete discussion,
read the text.
With the trend towards more and earlier calculator usage,
some people have lost a comfort with
basic arithmetic operations like 5·7 = 35 and 8+6 = 14 .
It is a waste of valuable time to use your calculator for problems such as these.
In this section, your basic multiplication skills are brought "up to speed" so you won't be wasting
mental energy on arithmetic and will be able to concentrate on higher-level ideas.
In mathematics, a vertically centered dot " · " is often used to denote multiplication.
Thus, 3 · 5 denotes " 3 times 5 ".
Consider the multiplication problem 3 · 5 .
One interpretation is 3 groups of 5 , so 3 · 5 = 5 + 5 + 5 = 15 .
Another interpretation is 5 groups of 3 , so 3 · 5 = 3 + 3 + 3 + 3 + 3 = 15 .
Thus, multiplication is a shorthand for repeated addition.
Algebra uses letters to represent numbers. (LOTS more on this later on!)
The expression 2x means " 2 times x " which is a shorthand for the addition problem x + x .
Similarly, the expression 3y means " 3 times y " which is a shorthand for the addition problem y + y + y , and so on.
Thus, for example, 5 · 3x means 3x+3x+3x+3x+3x which is 15x .
Of course, it's much easier to think in terms of multiplication rather than repeated addition, so
you immediately get 5 · 3x = 15x .
Similarly, 6 · 7t = 42t .
To commute means to change places.
The Commutative Property of Multiplication states that for all numbers x and y , x · y = y · x .
That is, you can change the places of the numbers in a multiplication problem, and this does not affect the result.
The expression x · y can more simply be written as xy .
That is, juxtaposition (placing two things side-by-side) can be used to denote multiplication.
Of course, this notation can't be used with numbers like 2 and 3, because "23" means the number "twenty-three," not 2 times 3!
So, we can restate the Commutative Property of Multiplication: For all numbers x and y , xy = yx .
If you're a sociable person, then you probably like being in groups;
i.e., you like to associate with other people.
In mathematics, associative laws have to do with grouping.
The Associative Property of Multiplication states that for all numbers x ,
y , and z , (x · y) · z = x · (y · z) .
Notice that the order in which the numbers are listed on both sides of the equation is exactly the same; only the grouping has changed.
The Associative Property of Multiplication states that in a multiplication problem, the grouping of
the numbers does not affect the result.
Thanks to the associative property, we can write things like 1 · 2 · 3 without ambiguity!
Think about this—if the grouping mattered, then (say) (1 · 2) · 3 and 1 · (2 · 3) would give different results,
so you'd always have to specify which way it should be done.
Multiplying a number by 1 does not change it.
In other words, multiplying by 1 preserves the identity of the original number.
That is, for all numbers x , x · 1 = 1 · x = x .
For this reason, the number 1 is called the multiplicative identity.
Multiplying a number by 0 always gives 0 .
That is, for all numbers x , x · 0 = 0 · x = 0 .
This is sometimes called the Multiplication Property of Zero.
In this exercise, you will practice multiplication problems of the form "x · y"
where x and y can be any of these numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 .
Follow the instructions below, and time yourself.
If you average less than 3 seconds per problem, then you'll get an online reward!
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.