Math 2008 Exam 4

Dividing by zero is undefined:

"by zero" is a prepositional phrase, this phrase should never be replaced with a pronoun. Doing so leads to an improper interpretation of the entire phrase. Why can't you divide by zero? "It's undefined." Is a bad answer!! What is undefined in the phrase? Many answer, "Zero!" This is incorrect - zero is very well defined!! What is undefined is the division. When is it undefined? When the divisor is zero. Prepositional phrases answer the question when. The prepositional phrase can be moved to the front or the back without altering the meaning, but is most commonly seen as originally given. By zero dividing is undefined. Dividing is undefined by zero.

Zero Property of Division

0/d=0; any whole number, except zero, when divided into 0 gives a quotient of zero

Sharing

Another way of thinking about division is like playing cards. The dividend is the deck and the divisor is the number of players. You deal out one card (one unit) to each player until the deck (the dividend) is exhausted. This model is employed with Base Ten Blocks (BTB) only.

Missing factor

Just like subtraction, we can model division using multiplication. The quotient is the missing factor

Estimation

One of the reasons division is the most complex operation performed with the whole numbers in elementary education is because you not only have to be good at addition, subtraction and multiplication to do division well. You must also be a good estimator. If you over or under estimate, then you have to start over. You have to estimate exactly the correct multiplier for the divisor to be placed in the quotient in the appropriate place. Later on when we want students to estimate approximations, they think they still have to get "THE" answer instead of something reasonable or helpful. It is part of the reason students struggle with estimations using approximation methods.

Measurement (repeated subtraction)

One way of thinking about division is to think of it as repeated subtraction of the divisor from the dividend. When the dividend is exhausted (equals zero), then the quotient is equal to the number of times the divisor was subtracted. This model is employed with Base Ten Blocks (BTB) and integer bars. Models with the integer bars are generally limited to working on division facts.

We can model division using multiplication and if I know multiplication facts, I get my division facts for free!

There are only 90 division facts because dividing by zero is underfund, eliminating ten potential facts

Distributive Property of Multiplication over addition

a whole number can be multiplied times both of the addends within parentheses: a(b+c)=ab+ac;(b+c)a=ba+ca NOTE: When a letter or number placed besides a parenthesis it implies multiplication by juxtaposition

Distributive Property of Multiplication over Subtraction

a whole number can be multiplied times minuend and the subtrahend within parentheses: a(b-c)=ab-ac; (b-c)a=ba-ca

Zero Property of Multiplication

any whole number multiplied by zero results in a product of zero: a*0=0*a=0

Model of Division

dividend/divisor=quotient

inverse

division is the inverse of multiplication, division undoes multiplication

Division identity property

f/1=f; any whole number divided by 1 is the same whole number

Multiplication Model

factor X factor = product

Model of Division

factor x missing factor=product

Dividend

is the number we are trying to divide into equal parts of a certain size

Quotient

is the result of dividing the dividend by the divisor. it is the number of equal parts the size of the divisor that were contained in the dividend

Associative Property of Multiplication

like addition, multiplication is a binary operation; in order to multiply three whole numbers or more together you must first get the product of two whole numbers and multiply that product by the next whole number. The order of the whole numbers being multiplied does not change, only the grouping of which whole numbers will be multiplied together first: a(bc)=(ab)c

Multiplicative Identity Property

multiplying any whole number by one does not change the value of the whole number. a*1=1*a=a

Repeated addition

one of the reasons you might have guessed the whole numbers would be closed for multiplication is because multiplication can be thought of as a repeated addition. Examples: 3x2=2+2+2=6(think to yourself three sets of two each) 2x5=5+5=10(think to yourself two sets of five each) 2x3=3+3=6(think to yourself two sets of three each) Note: It is important to remember that the order of the factors is important for a given model. The product will be the same, but the model will be different. Be careful to NOT commute the problems given on the test as you will not get credit for incorrect models.

Factor

one of two whole numbers to be multiplied together

Model of Division

product/factor=missing factor

Repeated Subtraction

repeated subtraction can be used to model division, just like repeated is used to model multiplication. The divisor is subtracted from the dividend until the result is zero. The number of times it was subtracted is the quotient

W

symbol for the whole number set; W= {0,1,2,...}

Commutative Property of Multiplication

the order of multiplication is not important; the same product is obtained regardless of the order: ab=ba

Product

the results of multiplying two or more factors

Closure property of multiplication

the set of whole numbers is closed for multiplication. Any whole number multiplied by an other number (even the same one) will result in a product that is also a whole number

Divisor

the size of the equal parts we are trying to divide the dividend into