How to Teach Division with Remainders

In Japan, children learn division in grade 3. They start with problems that divide exactly, like 12 ÷ 3, and then move on to division with remainders, like 13 ÷ 4 = 3 remainder 1.

The commentary on Japan’s elementary school curriculum guidelines for mathematics (2017, in Japanese) lists this in the grade 3 content: understand the meaning of division and know where it is used, and also learn about remainders. In this post I read through what the commentary says and write about how to help at home. The ideas are the same wherever your child goes to school.

The commentary’s 13 ÷ 4

The commentary uses two situations: giving 13 cards out 4 at a time, and sharing 13 cards equally among 4 people. It explains 13 ÷ 4 as finding the whole number □ for which 4 × □ or □ × 4 is 13 or less and as close to 13 as possible, along with what is left over. It adds that the remainder must be smaller than the divisor.

So I think there are three things a child learns with remainders.

Use the times tables to find the largest product that does not go over the number being divided
Write what is left as the remainder
Check that the remainder is smaller than the divisor

At first, sharing real objects works best, I think. Get 13 counters or sweets and share them equally among 4 people.

Give one to each person and you have used 4. Once more, 8. Once more, 12. One is left. That is not enough to go round all 4 people, so the sharing stops there. Each person gets 3, and 1 is left over.

Try the other way too. Put the 13 counters into bags of 4. You make 3 bags, and 1 is left.

The commentary calls the first kind sharing division and the second kind grouping division. Both are 13 ÷ 4, and both answers are “3 remainder 1”. But the answers mean different things. The first is “3 each, with 1 left over”. The second is “3 bags, with 1 left over”. This difference matters later, when you deal with remainders in word problems.

Why the remainder is smaller than the divisor

A common mistake is 13 ÷ 4 = 2 remainder 5. Since 4 × 2 + 5 = 13, it looks right as a calculation.

With counters, though, it is clearly wrong. If 5 are left, you can give everyone one more. You could keep sharing, but you stopped.

I think showing this once with counters works better than explaining it in words. Say “if 4 or more are left, we can go round again” while moving the counters. Then “the remainder is smaller than the divisor” stops being a rule to memorize and becomes something obvious.

Finding the quotient

Without counters, use the times table of the divisor. For 13 ÷ 4, say the 4 times table.

4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16

16 is more than 13, so the last one you can use is 4 × 3 = 12. The quotient is 3. Take 12 from 13, and the remainder is 1.

The commentary describes 48 ÷ 6 and 13 ÷ 4 as calculations where the quotient can be found by using one times table fact once. I think many children who struggle with remainders are not really struggling with division. They cannot recall the times tables backwards quickly. If “in the 4 times table, what is closest to 13?” does not come quickly, I suggest going back to exact division and the times tables first. I wrote about learning them in How to Learn Times Tables.

Checking the answer

A division answer can be checked with multiplication and addition.

divisor × quotient + remainder = number divided

For 13 ÷ 4 = 3 remainder 1, 4 × 3 + 1 = 13, so it is right. According to the commentary, in grade 3 children learn to check division by seeing whether the quotient times the divisor gives the number divided. In grade 4 they learn the relationship as a formula: dividend = divisor × quotient + remainder. There too, the commentary says to pay attention to the remainder being smaller than the divisor.

So checking takes two steps. Does the formula give the number divided? And is the remainder smaller than the divisor? 13 ÷ 4 = 2 remainder 5 passes the first check but fails the second. A habit of doing both will help later with long division too, I think.

Remainders in word problems

The commentary has one more important example. 38 children sit on benches, 4 to a bench. 38 ÷ 4 = 9 remainder 2. But the answer is not 9 benches. The 2 children left over need a bench too, so the answer is 10.

On the other hand, if the question is “38 cards are given out 4 to each person. How many people get cards?”, the answer is 9 people, and 2 cards are left.

The same 38 ÷ 4 can mean dropping the remainder or adding 1 to the quotient, depending on the situation. The commentary says that sometimes the quotient cannot be used as the answer as it is, and children need to think about the best answer for the real situation.

When a child is unsure about a remainder in a word problem, I think it helps to suggest drawing a picture. Draw the benches and seat 4 children on each, and 2 are left standing. Once they see that, they know another bench is needed without being told.

How to write it

The answer is written like 13 ÷ 4 = 3 remainder 1, with the quotient, then the word “remainder”, then the number left over. Some schools write it as 3 r1 or 3 R1. Either way, it helps for a child to notice, even from the way it is written, that the answer to a division is not one number but a pair: a quotient and a remainder.

When the divisor gets bigger or the calculation moves to long division, the thinking does not change. Find the quotient, write what is left as the remainder, and check that the remainder is smaller than the divisor. Lining up place values, the basis of all column calculation, is in Teaching Column Addition and Subtraction at Home.

What MathCal can do

MathCal is an arithmetic drill app I make, where children write their answers by hand on the screen.

For the groundwork, use the Basic Calculation Drill. Turn off addition, subtraction and multiplication, and only division appears. Division in this drill always divides exactly, so it is practice for recalling the times tables backwards.

Problems with remainders are in the Column Arithmetic Drill (Division). You choose 1 to 3 digits for both the number divided and the divisor, and the child writes the answer by hand in the quotient and remainder boxes. Only those boxes are checked, and the working can be written freely on the grid. In my developer interview, when asked why the working is not marked, I answered, “Because it would take away the comfort.” The places for writing should be easy to write in, and only the answer is checked.

The lessons include Division Beads, where beads are shared out equally, and Long Division, which walks through the column method. In Long Division, when a problem has a remainder, the check is shown at the end: the divisor times the quotient, plus the remainder.

Today’s Drill, Today’s Challenge and Today’s Lesson on the home screen are free every day. Choosing any drill or lesson from the full lists requires a one-time purchase. There are no ads. For an overview of the arithmetic in each grade, see What Kids Learn in Arithmetic, Grade by Grade.