Division

Here you will learn about division, including dividing into equal groups, long division, dividing decimals, and dividing fractions.

Students will first learn about division as part of operations and algebraic thinking in 3 rd grade.

What is division?

Division is the process of splitting a number into equal parts or a group of objects into smaller, equal groups. Division is one of the four basic operations (arithmetic operations) and it is the inverse or opposite of multiplication.

To write a division equation, you write the number being divided, the dividend, then a division sign, and then the number that the dividend is being divided by, which is called the divisor. The answer to a division equation is called the quotient.

For example,

Annie wants to split her 28 marbles into equal groups of 4. How many marbles will be in each group?

You would write this as

To solve, you need to divide 28 into 4 equal groups.

There are 7 in each group, so 28 \div 4=7

You can also solve division facts by writing the equation as an unknown factor problem.

For example, you can solve 28 \div 4 by finding the number that makes 28 when multiplied by 4.

Long division

To divide larger numbers, you can use the standard algorithm for division, or long division.

Long division is a method of dividing a multi-digit number by another number by repeatedly subtracting multiples of the divisor from the dividend, determining the quotient digit by digit, and bringing down additional digits until the division is complete.

For example,

The quotient is shown at the top. There is no remainder, so

If a division problem ends in a remainder, the reminder can be written next to the quotient at the top. For example, if there was a remainder of 2, this could be written as R 2. Remainders can also be written as fractions or decimals.

Dividing fractions

To divide fractions, multiply the dividend (the first number in the equation) by the reciprocal of the divisor (second number in the equation). To find the reciprocal of a fraction, switch the numerator and denominator.

Note that a whole number written as a fraction would be the number over 1, so its reciprocal would be 1 over the number. Example: 5=\frac{5}{1} so its reciprocal is \frac{1}{5}.

To remember this process of dividing, think “Keep Change Flip.”

Dividing decimals

You can divide decimals using the standard algorithm, or long division. However, the divisor must be a whole number before you can start the long division process.

If the divisor is a decimal, you must multiply it by a power of 10 to shift the digits so there is no longer a decimal point. Whatever you multiply the divisor by, you must also multiply the dividend by.

For example,

a) 8.6 \div 2={?}

Since the divisor is a whole number, you can perform long division. Notice that the decimal point moves straight up to the same place in the quotient.

So, 8.6 \div 2=4.3

b) 24.66 \div 1.8={?}

In this example, the divisor, 1.8, is a decimal. Before you can begin the long division process, you will need to multiply 1.8 by a power of 10 so that it will be a whole number. Then you will need to multiply the dividend by that same power of 10.

To make 1.8 a whole number, you can multiply it by 10. Then you also multiply 24.66 by 10. So now, you would use long division to solve 246.6 \div 18.

What is division?

Common Core State Standards

How does this relate to 3 rd grade through 6 th grade math?

• 3rd Grade – Operations and Algebraic Thinking (3.OA.A.2)
  Interpret whole-number quotients of whole numbers, e.g., interpret 56 \div 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.
  
  For example, describe a context in which a number of shares or a number of groups can be expressed as 56 \div 8.

• 3rd Grade – Operations and Algebraic Thinking (3.OA.B.6)
  Understand division as an unknown-factor problem. For example, find 32 \div 8 by finding the number that makes 32 when multiplied by 8.

• 4th Grade – Number and Operations in Base Ten (4.NBT.B.6)
  Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

• 5th Grade – Number and Operations in Base Ten (5.NBT.B.6)
  Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

• 5th Grade – Number and Operations in Base Ten (5.NBT.B.7)
  Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

• 5th Grade – Number and Operations—Fractions (5.NF.B.7, 5.NF.B.7a, 5.NF.B.7b, 5.NF.B.7c)
  Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
  
  Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (\cfrac{1}{3}) \div 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (\cfrac{1}{3}) \div 4 = \cfrac{1}{12} because (\cfrac{1}{12}) × 4 = \cfrac{1}{3}.
  
  Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 \div (\cfrac{1}{5}), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 \div (\cfrac{1}{5}) = 20 because 20 \times (\cfrac{1}{5}) = 4
  
  Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share \cfrac{1}{2} lb of chocolate equally? How many \cfrac{1}{3} -cup servings are in 2 cups of raisins?

• 6th Grade – Number and Operations—Fractions (6.NS.A.1)
  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
  
  For example, create a story context for (\cfrac{2}{3}) \div (\cfrac{3}{4}) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (\cfrac{2}{3}) \div (\cfrac{3}{4}) = \cfrac{8}{9} because \cfrac{3}{4} of \cfrac{8}{9} is \cfrac{2}{3}. (In general, (\cfrac{a}{b}) \div (\cfrac{c}{d}) = \cfrac{ad}{bc}.)
  
  How much chocolate will each person get if 3 people share \cfrac{1}{2} lb of chocolate equally? How many \cfrac{3}{4} -cup servings are in \cfrac{2}{3} of a cup of yogurt? How wide is a rectangular strip of land with length \cfrac{3}{4} mi and area \cfrac{1}{2} square mi?

• 6th Grade – Number and Operations—Fractions (6.NS.B.2)
  Fluently divide multi-digit numbers using the standard algorithm.

• 6th Grade – Number and Operations—Fractions (6.NS.B.3)
  Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

How to divide

In order to divide objects into equal groups:

1. Identify the total number of objects.
2. Determine the number of groups.
3. Divide the objects into equal groups.

In order to divide using long division:

1. Set up the problem.
2. Divide, multiply and subtract.
3. Repeat step \bf{2} until the remainder is \bf{0}, or smaller than the divisor and finish to find the quotient.

In order to divide decimals:

1. If the divisor is a whole number, go to step \bf{3}. If not, decide the power of ten that will make the divisor a whole number.
2. Multiply both the divisor and the dividend by the same power of \bf{10} .
3. If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient. You may need to include zeros as place holders.
4. Perform long division.

In order to divide fractions:

1. Take the reciprocal (flip) of the divisor (second fraction).
2. Change the division sign to a multiplication sign.
3. Multiply the fractions together.
4. If possible, simplify or convert to a mixed number.

Division examples

Example 1: divide objects into equal groups

Marcus has 45 pieces of candy. He wants to divide his candy equally between himself and his 8 friends. How many pieces of candy will each person get?

1. Identify the total number of objects.

Marcus has a total of 45 pieces of candy.

2Determine the number of groups.

Marcus is going to divide his candy between himself and 8 friends, so the candy will be divided into 9 equal groups.

3Divide the objects into equal groups.

45 \div 9=5 in each group

So each person gets 5 pieces of candy.

Example 2: long division (no remainder)

Solve 3211 \div 13 using long division.

Set up the problem.

When using long division, the numbers are set up under a symbol that is commonly referred to as a “bus stop” or a “house.”

The divisor goes outside of the “house” on the left, and the dividend goes inside the “house.” The quotient will go on top.

Divide, multiply and subtract.

Repeat step \bf{2} until the remainder is \bf{0}, or smaller than the divisor and finish to find the quotient.

The remainder is zero, so the long division process is done.

That means 3,211 \div 13=247.

Example 3: long division (with remainder)

Solve 857 \div 5 using long division. Write the remainder as a fraction.

Set up the problem.

Divide, multiply and subtract.

Repeat step \bf{2} until the remainder is \bf{0}, or smaller than the divisor and finish to find the quotient.

There is a remainder of 2. The directions state to write the remainder as a fraction. To do this, you write the remainder as the numerator and the divisor as the denominator.

So the quotient is 171 \cfrac{2}{5}.

Example 4: dividing decimals (whole number divisor)

Solve 71.4 \div 12 using long division.

If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient. You may need to include zeros as place holders.

Do long division.

Notice that when getting a remainder when dividing decimals, you add a zero to the end of the dividend to bring down, then continue dividing.

Example 5: dividing decimals (decimal divisor)

Solve 5.13 \div 1.35 using long division.

Do long division.

So, 5.13 \div 1.35=3.8

Example 6: whole number divided by a fraction

Solve 6 \div \cfrac{2}{3}

Example 7: fraction divided by a whole number

Solve \cfrac{4}{5} \div 3 .

Example 8: fraction divided by a fraction

Solve \cfrac{3}{4} \div \cfrac{1}{2}

Teaching tips for division

• Review divisibility rules to assist students with long division. Divisibility rules help with long division by allowing students to more quickly identify potential divisors, reducing the number of attempts needed to find the correct one. For example, “A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. “

• Teach students different strategies for solving division problems, such as repeated subtraction, equal groups, or using multiplication facts. Encourage them to choose the strategy that works best for them.

• Offer plenty of opportunities for students to practice division through hands-on activities and games, not just worksheets.

• Knowing multiplication tables can help students divide, so provide practice for fluency as needed.

Easy mistakes to make

• Forgetting place value
  Students may forget to align digits correctly when performing long division, leading to errors in the quotient.

• Forgetting to check for remainders
  Students may forget to check for remainders or may incorrectly interpret them.

Related multiplication and division lessons

• Multiplication and division
• Multiplicative comparison
• Multiplying multi digit numbers
• Dividing multi digit numbers
• Multiplying and dividing integers
• Long division
• Negative times negative (coming soon)
• Negative numbers
• Multiplying and dividing rational numbers (coming soon)

The next lessons are

• Types of numbers
• Rounding numbers
• Factors and multiples

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