There are essentially two ways of thinking of division:

**Partition** division (also known as partitive, sharing and grouping division) is a way of understanding division in which you divide an amount into a given number of groups. If you are thinking about division this way, then 12 ÷ 3 means 12 things divided evenly among 3 groups, and we wish to know how many is in 1 (each) group.

**Measurement** division (also called repeated subtraction division), is a way of understanding division in which you divide an amount into groups of a given size. If you are thinking about division this way, then 12 ÷ 3 means 12 things divided evenly into groups of 3, and we wish to know how many groups we can make.

**Direct modeling:**These two ways of thinking of division correspond to two different ways to direct model to find the result of dividing.

Dividing 12 by 3 in a **partitive** way involves taking 12 objects and “dealing” them out evenly into three groups. Typically each counter is moved separately, and this takes 12 steps:

(On the Children's Mathematics Children's Strategies CD, you can watch a boy doing this. In the Multiplication & Division tab, watch Direct Modeling, Partitive Division–that version has the interesting detail that the boy uses a different colored Unifix cube to represent the sets)

Dividing 12 by 3 in a **measurement** way involves taking 12 objects and making groups of 3 from them. With a small number (3) in each group, this is typically a faster process because 3 counters are removed at a time, so this might take only 4 steps.

(On the Children's Mathematics Children's Strategies CD, you can watch a boy doing this. In the Multiplication & Division tab, watch Direct Modeling, Measurement Division)

While the answer you get in each case is the same, it represents different things. In the partitive interpretation, 4 is the number of objects in each group; in the measurement interpretation, 4 is the number of groups.

This makes division really tricky. One of the big things to learn about division is that both of these things are division, and they both give you the same numerical answers. This ties into multiplication being commutative.

The commutative law of multiplication says that 3 groups of 4 is the same amount as 4 groups of 3. You know the answer to the partitive question 12 divided into 3 groups if you know that 3 groups of 4 is 12.

You know the answer to the measurement question 12 divided into groups of 3 if you know that 4 groups of 3 is 12.

We can make the same array picture we did for multiplication to explain the two different versions of division:

Now, the two different direct modeling pictures we made for 12 ÷ 3 also show that 4 groups of 3 and 3 groups of 4 are the same amount (12), but when you are making your groups that way it seems really surprising that it should be true (Wow! You mean when you put 12 into 3 groups, there are 4 in each group, so 3 groups of 4, and when you put 12 into groups of 3, you get 4 groups, so that's 4 groups of 3? That's really cool how the numbers just switch around that way. I wonder if that works for any other numbers?). On the other hand, when you put them in an array, you can see that you just make the groups up and down for one grouping and side to side for the other grouping, so it's obvious that you should get the same number both ways (Well, duh, of course it's the same numbers when you switch it around–you're just counting in the opposite direction).

So–understanding the commutative law of multiplication, lets you start to understand that both ways of thinking of division will give you the same answers and you can use them interchangeably. That's really useful, because sometimes you can figure out the answer more quickly by one method than another.

On the Children's Mathematics Children's Strategies CD, you can watch how quickly a girl solves a measurement division by skip counting (In the Multiplication & Division tab, watch Counting, Measurement Division).

In the next interview (In the Multiplication & Division tab, watch Derived Facts, Partitive Division), the girl solves a partitive division problem with the same numbers.

She remembers from the previous problem that 7 groups of 3 is 21, but she doesn't have a commutative law perspective on multiplication yet, she doesn't make the connection that 3 groups of 7 is also 21, and so she goes through some pretty complicated mental arithmetic steps to convert those 7 groups of 3 into 3 groups of 7 (very cool thinking, by the way–you can tell that she has a really good grasp on numbers).

**Word problems **are almost always either measurement division or partitive division problems.

It's a useful thing for a teacher to be able to recognize which category different word problems fall into, and some of the ways children might go about solving the various problems.

We'll look at the 4 types of problems that are discussed in the Children's Mathematics book.

**Grouping** problems are the most basic. Usually they are problems about discrete*, physical objects that are grouped in a fairly familiar.

*discrete means indivisible. These are things you count by 1's rather than measuring in feet and inches (and half-inches, etc).

In these grouping examples, the objects are cookies, and they are grouped by putting them on plates.

Partitive example:

I have 24 cookies. I want to put them on 4 plates, so that I have the same number of cookies on each plate. How many cookies should I put on each plate?

You can tell this is a partitive problem because it tells how many plates (groups of cookies) there are, and asks how much is on 1 (each) plate.

A child could direct model this by acting out putting equal numbers of cookies on 4 plates until they are all gone. A child who was figuring this out by counting, would probably guess and check to find the solution (if I have 2 on each plate, how many is that? What if I have 5 on each plate? etc.)

Measurement example:

I have 24 cookies. I want to put 4 cookies on each plate. How many plates do I need to hold my cookies?

You can tell this is a measurement problem because it tells how many go on 1 (each) plate, and asks how many plates (groups) there are.

A child could direct model to figure this out by making groups of 4 from a set of 24 counters. A child could figure this out by skip counting up by 4's until they reached 24, and keeping track of the groups as they counted up, or they could figure this out (probably on paper) by subtracting 4's from 24 until all were gone.

**Price** or cost problems are often appropriate for solving by direct modeling if the numbers are small enough.

In price problems, the amount being grouped or shared is the money–the total cost, and it is grouped by the price per item.

So, if it is a problem about buying pencils, then the total cost will be grouped by the cost for each pencil, and a single group is the price of a single pencil.

Partitive example:

4 pencils cost 36 cents. How much does one pencil cost?

The cost of 36 cents must be divided among the 4 pencils. (Cost per pencil is one group)

Measurement example:

A pencil costs 8 cents. How many pencils can I buy for 24 cents?

The total cost is again divided among several pencils, but this time the price for one pencil is known, and the number of pencils (groups) in unknown.

**Rate** problems are a more general version of the sort of thinking involved in price problems. Whereas price problems involve a price per item (cost for 1 item), general rate problems can relate a wider variety of things.

Miles per hour is the most familar rate for most of us (relating distance–miles, and time–hours), but there are lots of others: words per minute (reading or typing), bushels per acre (corn or other crops), miles per gallon. Rate problems are usually appropriate for children at the age when they are familiar with the things being compared.

In these examples, if children had experience (perhaps in science) with measuring distances things moved, and elapsed time with stop watches, these problems would be appropriate.

Partitive example:

It takes a battery powered toy train 5 seconds to go 20 inches. How far does the train go in 1 second?

## Illustrative Mathematics

Alignments to Content Standards:

6.NS.A

Student View

**Part 1:** Match each of the three equations with one of the two diagrams below. Explain how the diagram represents the equation.

- $$frac{2}{3} imes 5 = ?$$
- $$frac{2}{3} imes ? = 5$$
- $$frac{3}{2} imes 5 = ?$$

**Part 2:** Explain how the diagram below represents both of the equations below.

$$frac{3}{5} imes ? = A$$

$$frac{5}{3} imes A = ?$$

Write the first equation as a division equation. Put it together with the second equation without a ? in it.

The purpose of this task is to help students understand that when we divide by a fraction, it is the same as multiplying by its reciprocal. By carrying out this work outside of a context, students can focus on numbers and operations. This task builds on the work of 5.NF.B, where students represented many different fractional multiplication and division contexts with diagrams.

## 5.4: Decimal Operations (Part 2)

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

To understand decimal division, let’s consider the multiplication problem

Remember, a multiplication problem can be rephrased as a division problem. So we can write 0.8 ÷ 4 = 0.2 We can think of this as “If we divide 8 tenths into four groups, how many are in each group?” Figure (PageIndex{1}) shows that there are four groups of two-tenths in eight-tenths. So 0.8 ÷ 4 = 0.2.

Using long division notation, we would write

Notice that the decimal point in the quotient is directly above the decimal point in the dividend.

To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.

HOW TO: DIVIDE A DECIMAL BY A WHOLE NUMBER

Step 1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.

Step 2. Divide as usual.

Example (PageIndex{9}):

Divide: 0.12 ÷ 3.

**Solution**

Write as long division, placing the decimal point in the quotient above the decimal point in the dividend. | |

Divide as usual. Since 3 does not go into 0 or 1 we use zeros as placeholders. |

0.12 ÷ 3 = 0.04

Exercise (PageIndex{17}):

Divide: 0.28 ÷ 4.

Answer

(0.07)

Exercise (PageIndex{18}):

Divide: 0.56 ÷ 7.

Answer

(0.08)

In everyday life, we divide whole numbers into decimals—money—to find the price of one item. For example, suppose a case of 24 water bottles cost $3.99. To find the price per water bottle, we would divide $3.99 by 24, and round the answer to the nearest cent (hundredth).

Example (PageIndex{10}):

Divide: $3.99 ÷ 24.

**Solution**

Place the decimal point in the quotient above the decimal point in the dividend. | |

Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place. | |

Round to the nearest cent. | $$ $0.166 approx $0.17$$ |

$3.99 ÷ 24 ≈ $0.17

This means the price per bottle is 17 cents.

Exercise (PageIndex{19}):

Divide: $6.99 ÷ 36.

Answer

($0.19)

Exercise (PageIndex{20}):

Divide: $4.99 ÷ 12.

Answer

($0.42)

So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let’s look at the same multiplication problem we looked at earlier, but in a different way.

$$(0.2)(4) = 0.8$$

Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, “Ho w many times does 0.2 go into 0.8?” Because (0.2)(4) = 0.8, we can say that 0.2 goes into 0.8 four times. This means that 0.8 divided by 0.2 is 4.

$$0.8 div 0.2 = 4$$

We would get the same answer, 4, if we divide 8 by 2, both whole numbers. Why is this so? Let’s think about the division problem as a fraction.

$$dfrac{0.8}{0.2}$$

$$dfrac{(0.8)10}{(0.2)10}$$

$$dfrac{8}{2}$$

$$4$$

We multiplied the numerator and denominator by 10 and ended up just dividing 8 by 2.

To divide decimals, we multiply both the numerator and denominator by the same power of 10 to make the denominator a whole number.

Because of the Equivalent Fractions Property, we haven’t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right.

We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign. It may help to review the vocabulary for division:

HOW TO: DIVIDE DECIMAL NUMBERS

Step 1. Determine the sign of the quotient.

Step 2. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.

Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.

Step 4. Write the quotient with the appropriate sign.

Example (PageIndex{11}):

Divide: −2.89 ÷ (3.4).

**Solution**

Determine the sign of the quotient. | The quotient will be negative. |

Make the divisor the whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places to the right. | |

Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero. | |

Write the quotient with the appropriate sign. | −2.89 ÷ (3.4) = −0.85 |

Exercise (PageIndex{21}):

Divide: −1.989 ÷ 5.1.

Answer

(-0.39)

Exercise (PageIndex{22}):

Divide: −2.04 ÷ 5.1.

Answer

(-0.4)

Example (PageIndex{12}):

Divide: −25.65 ÷ (−0.06).

**Solution**

The signs are the same. | The quotient is positive. |

Make the divisor a whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places. | |

Divide. Place the decimal point in the quotient above the decimal point in the dividend. | |

Write the quotient with the appropriate sign. | −25.65 ÷ (−0.06) = 427.5 |

Exercise (PageIndex{23}):

Divide: −23.492 ÷ (−0.04).

Answer

(587.3)

Exercise (PageIndex{24}):

Divide: −4.11 ÷ (−0.12).

Answer

(34.25)

Now we will divide a whole number by a decimal number.

Example (PageIndex{13})

Divide: 4 ÷ (0.05).

**Solution**

The signs are the same. | The quotient is positive. |

Make the divisor a whole number by 'moving' the decimal point all the way to the right. Move the decimal point in the dividend the same number of places, adding zeros as needed. | |

Divide. Place the decimal point in the quotient above the decimal point in the dividend. | |

Write the quotient with the appropriate sign. | 4 ÷ 0.05 = 80 |

We can relate this example to money. How many nickels are there in four dollars? Because 4 ÷ 0.05 = 80, there are 80 nickels in $4.

Exercise (PageIndex{25}):

Divide: 6 ÷ 0.03.

Answer

(200)

Exercise (PageIndex{26}):

Divide: 7 ÷ 0.02

Answer

(350)

We often apply decimals in real life, and most of the applications involving money. The Strategy for Applications we used in The Language of Algebra gives us a plan to follow to help find the answer. Take a moment to review that strategy now.

Strategy for Applications

- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Answer the question with a complete sentence.

Example (PageIndex{14}):

Paul received $50 for his birthday. He spent $31.64 on a video game. How much of Paul’s birthday money was left?

**Solution**

What are you asked to find? | How much did Paul have left? |

Write a phrase. | $50 less $31.64 |

Translate. | 50 − 31.64 |

Simplify. | 18.36 |

Write a sentence. | Paul has $18.36 left. |

Exercise (PageIndex{27}):

Nicole earned $35 for babysitting her cousins, then went to the bookstore and spent $18.48 on books and coffee. How much of her babysitting money was left?

Answer

($16.52)

Exercise (PageIndex{28}):

Amber bought a pair of shoes for $24.75 and a purse for $36.90. The sales tax was $4.32. How much did Amber spend?

Answer

($65.97)

Example (PageIndex{15}):

Jessie put 8 gallons of gas in her car. One gallon of gas costs $3.529. How much does Jessie owe for the gas? (Round the answer to the nearest cent.)

**Solution**

What are you asked to find? | How much did Jessie owe for all the gas? |

Write a phrase. | 8 times the cost of one gallon of gas |

Translate. | 8($3.529) |

Simplify. | $28.232 |

Round to the nearest cent. | $28.23 |

Write a sentence. | Jessie owes $28.23 for her gas purchase. |

Exercise (PageIndex{29}):

Hector put 13 gallons of gas into his car. One gallon of gas costs $3.175. How much did Hector owe for the gas? Round to the nearest cent.

Answer

($41.28)

Exercise (PageIndex{30}):

Christopher bought 5 pizzas for the team. Each pizza cost $9.75. How much did all the pizzas cost?

Answer

($48.75)

Example (PageIndex{16}):

Four friends went out for dinner. They shared a large pizza and a pitcher of soda. The total cost of their dinner was $31.76. If they divide the cost equally, how much should each friend pay?

**Solution**

What are you asked to find? | How much should each friend pay? |

Write a phrase. | $31.76 divided equally among the four friends. |

Translate to an expression. | $31.76 ÷ 4 |

Simplify. | $7.94 |

Write a sentence. | Each friend should pay $7.94 for his share of the dinner. |

Exercise (PageIndex{31}):

Six friends went out for dinner. The total cost of their dinner was $92.82. If they divide the bill equally, how much should each friend pay?

Answer

($15.47)

Exercise (PageIndex{32}):

Chad worked 40 hours last week and his paycheck was $570. How much does he earn per hour?

Answer

($14.25)

Be careful to follow the order of operations in the next example. Remember to multiply before you add.

Example (PageIndex{17}):

Marla buys 6 bananas that cost $0.22 each and 4 oranges that cost $0.49 each. How much is the total cost of the fruit?

**Solution**

What are you asked to find? | How much is the total cost of the fruit? |

Write a phrase. | 6 times the cost of each banana plus 4 times the cost of each orange |

Translate to an expression. | 6($0.22) + 4($0.49) |

Simplify. | $1.32 + $1.96 |

Add. | $3.28 |

Write a sentence. | Marla's total cost for the fruit is $3.28. |

Exercise (PageIndex{33}):

Suzanne buys 3 cans of beans that cost $0.75 each and 6 cans of corn that cost $0.62 each. How much is the total cost of these groceries?

Answer

($5.97)

Exercise (PageIndex{34}):

Lydia bought movie tickets for the family. She bought two adult tickets for $9.50 each and four children’s tickets for $6.00 each. How much did the tickets cost Lydia in all?

Answer

($43.00)

In the following exercises, add or subtract.

- 16.92 + 7.56
- 18.37 + 9.36
- 256.37 − 85.49
- 248.25 − 91.29
- 21.76 − 30.99
- 15.35 − 20.88
- 37.5 + 12.23
- 38.6 + 13.67
- −16.53 − 24.38
- −19.47 − 32.58
- −38.69 + 31.47
- −29.83 + 19.76
- −4.2 + (− 9.3)
- −8.6 + (− 8.6)
- 100 − 64.2
- 100 − 65.83
- 72.5 − 100
- 86.2 − 100
- 15 + 0.73
- 27 + 0.87
- 2.51 + 40
- 9.38 + 60
- 91.75 − (− 10.462)
- 94.69 − (− 12.678)
- 55.01 − 3.7
- 59.08 − 4.6
- 2.51 − 7.4
- 3.84 − 6.1

In the following exercises, multiply.

- (0.3)(0.4)
- (0.6)(0.7)
- (0.24)(0.6)
- (0.81)(0.3)
- (5.9)(7.12)
- (2.3)(9.41)
- (8.52)(3.14)
- (5.32)(4.86)
- (−4.3)(2.71)
- (− 8.5)(1.69)
- (−5.18)(− 65.23)
- (− 9.16)(− 68.34)
- (0.09)(24.78)
- (0.04)(36.89)
- (0.06)(21.75)
- (0.08)(52.45)
- (9.24)(10)
- (6.531)(10)
- (55.2)(1,000)
- (99.4)(1,000)

In the following exercises, divide.

- 0.15 ÷ 5
- 0.27 ÷ 3
- 4.75 ÷ 25
- 12.04 ÷ 43
- $8.49 ÷ 12
- $16.99 ÷ 9
- $117.25 ÷ 48
- $109.24 ÷ 36
- 0.6 ÷ 0.2
- 0.8 ÷ 0.4
- 1.44 ÷ (− 0.3)
- 1.25 ÷ (− 0.5)
- −1.75 ÷ (− 0.05)
- −1.15 ÷ (− 0.05)
- 5.2 ÷ 2.5
- 6.5 ÷ 3.25
- 12 ÷ 0.08
- 5 ÷ 0.04
- 11 ÷ 0.55
- 14 ÷ 0.35

In the following exercises, simplify.

- 6(12.4 − 9.2)
- 3(15.7 − 8.6)
- 24(0.5) + (0.3)2
- 35(0.2) + (0.9)2
- 1.15(26.83 + 1.61)

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