Quotients and Remainders

Contributor: Rachel Lewis. Lesson ID: 12133

Do you ever have leftovers? We don't mean from yesterday's dinner! Leftovers from division problems are called "remainders." Learn what to do when you can't divide evenly and a number is left behind!


Operations and Algebraic Thinking, Whole Numbers and Operations

learning style
Auditory, Visual
personality style
Beaver, Golden Retriever
Grade Level
Intermediate (3-5)
Lesson Type
Dig Deeper

Lesson Plan - Get It!


"The ants go marching two by two. Hoorah! Hoorah!" Wait a minute. There are nine ants. Can nine ants march in groups of two? There are two, four, six, eight ants. Then there is one left over. In this lesson, the leftover ant will be called a remainder of one!

There are eight ants marching in groups of two.

What is the quotient?

The answer to a division problem is called the quotient. 8 ÷ 2 is 4, so the quotient in our problem is 4.

Sometimes a number does not divide evenly into another number. The number that is left over after we find the quotient is called the remainder.

For example, we know 12 ÷ 3 = 4. If we make 12 tally marks, we can draw tallies in four equal groups of three:



There is none left over, so the remainder is 0.

What if we have 14 ÷ 3?



First, we can draw 14 tally marks. We have four equal groups of three, but we do not have enough tallies to make another group of three. So, we have four groups of three with two left over. 14 ÷ 3 will have a quotient of 4 with a remainder of 2.

For the next example, you will need 25 Unifix Cubes® or counters. If you don't have counters, you could use rocks, candies, or Lego® blocks.

Solve 21 ÷ 4. Count out 21 counters. Place your counters into groups of four. How many groups can you make?



Right! We can make five groups of four. So, the quotient is 5. Now, look at how many counters are left over from making your groups of four. What is the remainder?

Right again! There is one counter left over, so the remainder is 1.

21 ÷ 4 = 5 with a remainder of 1

Let’s look at another way to find the remainder. For this example, use long division to find 21 ÷ 4.

      5 < 20 ÷ 4 = 5
  4 2 1    
  - 2 0 < Subtract 20 from 21.
    0 1 < We get a remainder of 1.


Let’s look at another example. What is 73 ÷ 2?

Step 1:

    3   < 7 tens ÷ 2 = 3 with a remainder of 1                      
  2 7 3                          
  - 6   < Multiply 3 x 2.                      
    1   < Subtract 6 from 7.                      


Step 2:

    3 6 < Bring the 3 down from 73. 13 ÷ 2 = 6 with a remainder of 1.                      
  2 7 3                      
  - 6                            
    1 3                          
  - 1 2 < Multiply 6 x 2.                      
      1 < Subtract 12 from 13. We get a remainder of 1.


The answer to 73 ÷ 2 is 36 with a remainder of 1.

You can draw models, arrange counters, or use long division to find the quotient and remainder of a division problem. Practice your new strategies in the Got It? section.

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