Home, 391 Index

Definition whole number Division:                              Work in Progress

Division is defined as follows:  For real numbers a, b≠0, and c, a b=c if and only if there is a unique real number c for which b*c=a.  The number a is called the dividend, b is the divisor, and c is called the quotient.

Comments:

·       Division is defined in terms of multiplication in the same way that subtraction is numerically defined in terms of addition.

·       Division in the elementary grades is often first presented using the repeated subtraction model (see below).

·       Division is said to “undo” multiplication and is called the "inverse of multiplication."  Just as a technical discussion of subtraction as the "inverse of addition" requires the incorporation of the integers into the number system, a full discussion of division as the "inverse of multiplication" requires the group properties of the rational numbers and we are not ready to discuss those yet.

·       Division satisfies none of the associative or communicative etc. properties of multiplication and has no specific properties of its own.

·       Division is unique from the other three operations, +, -, and x, because division by zero is undefined.  It is not the case that division by zero is "impossible."  In 1950, it was impossible for man to walk on the moon.  This is not true today.  The word, "impossible" has come to mean that something is not possible now, but maybe it will be possible in the future.  Division by zero, however, is undefined today and it will remain undefined tomorrow.  This is discussed further in the missing multiplicand model given below.

The answers to the division question a b=c have two basic interpretations dependent upon on the context of the problem. We defined a b=c if and only if b*c=a. But multiplication is commutative, so b*c= c*b=a , which says that a c=b.  The array model for multiplication is usually presented as a row*column rectangle.  A 2 by 3 rectangular picture is shorter vertically than horizontally and cannot be viewed with the same orientation if it were hung in a 3 by 2 frame.  So while the area of both rectangles is the same, the perspective by which we see them is different. This difference is recognized in the two possible ways division questions that can be asked.

 For a c=b, the definition we gave says that our array is a b by c rectangle, so the answer c stands for the number of columns in the array.  For a c=b, then we focus on a c by b array where c represents the number of rows  or the height versus the length c of the rectangle.

If both b and c are known, it doesn’t matter whether we set up a word problem as a c=b or a c=b, because the product b*c=c*b is going to give us the same number of elements whether we have a bxc array or a cxb rectangle.  Most students would see a word problem with a as the unknown as a straight multiplication problem. But it should not be ruled out that some student could set it up as a division problem.

Couple these two interpretations of the division question with the possibility that any one of the three parameters could be the unknown, then it should not be so surprising that division is not the easiest subject for most students to master.  The possible division equations therefore are:

                  The Dividend a is unknown.                                       If and only if            .

                  The Divisor    b is unknown.                                       If and only if            .

                  The Quotient c is the unknown.                              If and only if            .   

Classroom Models for Division:

         i.)  The Repeated Subtraction or Measurement Model:   a b=c if and only if c is that unique number for which c*b=a is interpreted as asking the question, "how many times can b be subtracted from a before getting zero?"

The answer to this question is that you can subtract b c-many times from b before reaching zero.

Hundreds Charts are often used in second grade as a model to introduce division as repeated subtraction.  Suppose the student wants to know how many students will get suckers if there is a total of 28 suckers to distribute to the students in groups of 7. Have the students say the problem in their own words to show they understand what is being asked, for example, “ How many students will get suckers if all 28 of them are seven suckers to a bag? The division the equation is then 28  7 = c.  The question asked by this equation is, "How many groups of seven spaces can you count backward from 28 to end up at zero. Students circle the number 28 and cross off groups of seven consecutive spaces on their hundreds chart until they get to zero.  Then they count the number of groups circled and see that there are four circles each surrounding seven spaces.  Therefore, there are four students who can be the recipients of the suckers.

The number line is an oft-used tool to model repeated subtracted division problems after the students have ore of a physical mastery of working with pencil and paper.   Given the equation a b=c and a number line, we ask, "How many times must the b-cricket jump backward from a to get to zero".  It is understood, of course, that b-crickets cover b units in any one jump.  To find the answer to the equation 12 4=3 on a string number line, a student could hang a paper clip on the number twelve spot and add an additional clip to mark every four units subtracted. At the end the student would have four paper clips hanging on the numbers 0, 4, 8 and 12. They would then count three spaces each between then and conclude that 3*4=12.

Applying this model, our 12 4=3 number line says that the 4-cricket has to jump four steps backward for each of three separate jumps before landing on zero.  Another example, says that if you start the morning out with a dozen cookies, then you could eat four cookies for breakfast, lunch and dinner (i.e. three separate times) before having zero cookies left for bedtime.

Repeated subtraction is implicit in answering the measurement question, "What is the unit length of this room?"  or "Starting at one end of this room, how many unit steps must you take to end up with the toe of your shoe against the wall of the other end?"   Because of this, what some texts call the repeated subtraction model is called the measurement model in other texts.

         ii.)  The Partition Model: of the division statement a b=c if and only if  b*c=a says that partitioning (dividing, distributing) a-many things evenly among b-many boxes results in each box holding c-many items.  The sentence, 12 4=3 could ask, “How many pieces of candy will each student get if a total of 12 candies is distributed one by one among four students?” Or, the office aid went to the closet and picked up the remaining 12 boxes of chalk.  How many boxes of chalk will each classroom get if the boxes are distributed (partitioned) one by one to each of four classrooms.  In each case, it would be found that 4*3=12.  That is, start out with four rows and partition the room into row and columns (aisles) with 12 chairs.  “How many chairs will be in each aisle?”

The number line model for the partitive interpretation of the division equation a b=c asks, "How many times must the b-cricket jump forward from zero to get to a?". Applying this interpretation, 12 4=3 says that the 4-cricket has to start at zero and jump four steps at a time for three consecutive times before landing on 12.

iii.)        The Missing Factor Algorithm or the Definition of Division Model:  The missing factor is a way of checking one’s answer to a division question.  Another way to solve the division equation a b= is for the student to know that whatever is, it can be found by finding the missing factor to a multiplication equation whether the student chooses b* =a or *b=a.  The reason why some texts refer to this as a direct application of the definition of division should be very clear to you.  If not, reread the definition given above or in your textbook.

      iv.) Division Algorithm:  The Division Algorithm extends the concept of division to include a remainder.  Namely, for all whole numbers a, b, there are unique integers q and r such that a=bq+r, and r≤0<b.

The following number line problem says that the four-crickets jumps four steps at a time for three jumps and then walks three steps to get to 15.

Your concern as the teacher should not be rather the student interprets a problem as a measurement (repeated subtraction) partition problem.  You should not be evaluating your students on whether they start at zero and count four steps at a time to get to 12 or start at 12 and counts backward four steps at a time landing on zero.  Your attention should focus on two questions.  Did the student read the problem as a division problem?  And if so, did the student come up with the correct factor (What?) to put in the box whether the multiplication equation was b* or *b.

Discussion of the Zero Property of Division:

      Case  i.)  Dividing 0 by a nonzero number:  You can divide zero by any nonzero whole number because there exists an answer, namely zero itself, to the division sentence, 0 b=[?] and, the answer is unique.  That is there is one and only one answer to the question, “How many non-zero b’s can you add up to get the result of having zero-many of any quantity?”   Clearly, for whole numbers, a=0, b≠0 there exists c=0 such that 0 b=0 because 0*b=0=b*0.  Further, zero is the one and only such solution by the zero property of multiplication.

However, you cannot divide any whole number, zero or nonzero, by zero.

      Case ii.)  Division by 0:    

                  a.  Non-zero divided by zero:   You cannot divide a non-zero number by zero because there does not exist a number unique or otherwise to satisfy the definition of division equation.  For a≠0, a 0=[?]  iff  [?] *0=a by the zero property of multiplication.. No matter how many times you add zero to it, the sum can never be equal to a nonzero number.

                  b.  Zero divided by zero:  If a=0, then there is no unique answer to the expression 0 0=[?].  One can put any number x  in the box and x *0=0 by the zero property of multiplication.  So there does not exist a unique solution, hence the division equation  0 0=[?] is undefined.