Dividing Whole Numbers and Applications Learning Objective(s) · Use three different ways to represent division. · Divide whole numbers. · Perform long division. · Divide whole numbers by a power of 10. · Recognize that division by 0 is not defined. · Solve application problems using division. Some people think about division as “fair sharing” because when you divide a number you are trying to create equal parts. Division is also the inverse operation of multiplication because it “undoes” multiplication. In multiplication, you combine equal sets to create a total. In division, you separate a whole group into sets that have the same amount. For example, you could use division to determine how to share 40 empanadas among 12 guests at a party. Division is splitting into equal parts or groups. For example, one might use division to determine how to share a plate of cookies evenly among a group. If there are 15 cookies to be shared among five people, you could divide 15 by 5 to find the “fair share” that each person would get. Consider the picture below.
15 cookies split evenly across 5 plates results in 3 cookies on each plate. You could represent this situation with the equation: 15 ÷ 5 = 3 You could also use a number line to model this division. Just as you can think of multiplication as repeated addition, you can think of division as repeated subtraction. Consider how many jumps you take by 5s as you move from 15 back to 0 on the number line.
Notice that there are 3 jumps that you make when you skip count by 5 from 15 back to 0 on the number line. This is like subtracting 5 from 15 three times. This repeated subtraction can be represented by the equation: 15 ÷ 5 = 3. Finally, consider how an area model can show this division. Ask yourself, if you were to make a rectangle that contained 15 squares with 5 squares in a row, how many rows would there be in the rectangle? You can start by making a row of 5: Then keep adding rows until you get to 15 small squares. The number of rows is 3. So, 15 divided by 5 is equal to 3.
Ways to Represent Division As with multiplication, division can be written using a few different symbols. We showed this division written as 15 ÷ 5 = 3, but it can also be written two other ways:
Each part of a division problem has a name. The number that is being divided up, that is the total, is called the dividend. In the work in this topic, this number will be the larger number, but that is not always true in mathematics. The number that is dividing the dividend is called the divisor. The answer to a division problem is called the quotient. The blue box below summarizes the terminology and common ways to represent division.
Once you understand how division is written, you are on your way to solving simple division problems. You will need your multiplication facts to perform division. If you do not have them memorized, you can guess and check or use a calculator. Consider the following problems: 10 ÷ 5 = ? 48 ÷ 2 = ? 30 ÷ 5 = ? In the first problem, 10 ÷ 5, you could ask yourself, “how many fives are there in ten?” You can probably answer this easily. Another way to think of this is to consider breaking up 10 into 5 groups and picturing how many would be in each group. 10 ÷ 5 = 2 To solve 48 ÷ 2, you might realize that dividing by 2 is like splitting into two groups or splitting the total in half. What number could you double to get 48? 48 ÷ 2 = 24 To figure out 30 ÷ 5, you could ask yourself, how many times do I have to skip count by 5 to get from 0 to 30? “5, 10, 15, 20, 25, 30. I have to skip count 6 times to get to 30.” 30 ÷ 5 = 6
Compute 35 ÷ 5.
Compute 32 ÷ 4. Sometimes when you are dividing, you cannot easily share the number equally. Think about the division problem 9 ÷ 2. You could think of this problem as 9 pieces of chocolate being split between 2 people. You could make two groups of 4 chocolates, and you would have one chocolate left over.
In mathematics, this left over part is called the remainder. It is the part that remains after performing the division. In the example above, the remainder is 1. We can write this as: 9 ÷ 2 = 4 R1 We read this equation: “Nine divided by two equals four with a remainder of 1.” You might be thinking you could split that extra piece of chocolate in parts to share it. This is great thinking! If you split the chocolate in half, you could give each person another half of a piece of chocolate. They would each get 4
Since multiplication is the inverse of division, you can check your answer to a division problem with multiplication. To check the answer 7 R3, first multiply 6 by 7 and then add 3. 6 • 7 = 42 42 + 3 = 45, so the quotient 7 R3 is correct.
Long division is a method that is helpful when you are performing division that you cannot do easily in your head, such as division involving larger numbers. Below is an example of a way to write out the division steps.
Dividing Whole Numbers by a Power of 10 Just as multiplication by powers of 10 results in a pattern, there is a pattern with division by powers of 10. Consider three quotients: 20 ÷ 10; 200 ÷ 10; and 2,000 ÷ 10. Think about 20 ÷ 10. There are 2 tens in twenty, so 20 ÷ 10 = 2. The computations for 200 ÷ 10 and 2,000 ÷ 10 are shown below.
Examine the results of these three problems to try to determine a pattern in division by 10.
Notice that the number of zeros in the quotient decreases when a dividend is divided by 10: 20 becomes 2; 200 becomes 20 and 2,000 become 200. In each of the examples above, you can see that there is one fewer 0 in the quotient than there was in the dividend. Continue another example of division by a power of 10.
Consider this set of examples of division by powers of 10. What pattern do you see?
Notice that when you divide a number by a power of 10, the quotient has fewer zeros. This is because division by a power of 10 has an effect on the place value. For example, when you perform the division 18,000 ÷ 100 = 180, the quotient, 180, has two fewer zeros than the dividend, 18,000. This is because the power of 10 divisor, 100, has two zeros.
You know what it means to divide by 2 or divide by 10, but what does it mean to divide a quantity by 0? Is this even possible? Can you divide 0 by a number? Consider the two problems written below.  We can read the first expression, “zero divided by eight” and the second expression, “eight divided by zero.” Since multiplication is the inverse of division, we could rewrite these as multiplication problems. 0 ÷ 8 = ? ? • 8 = 0 The quotient must be 0 because 0 • 8 = 0 Now let’s consider 8 ÷ 0 = ? ? • 0 = 8 This is not possible. There is no number that you could multiply by zero and get eight. Any number multiplied by zero is always zero. There is no quotient for Division by zero is an operation for which you cannot find an answer, so it is not allowed. We say that division by 0 is undefined. Using Division in Problem Solving Division is used in solving many types of problems. Below are three examples from real life that use division in their solutions.
Division is the inverse operation of multiplication, and can be used to determine how to evenly share a quantity among a group. Division can be written in three different ways, using a fraction bar, ÷, and |
What is the process of finding how many times a number is contained in another number or the process of separating a number into a given number of equal parts?
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