Dividing a number which is not a multiple of the divisor will leave a remainder.

Students should learn strategies to memorize the following sets of basic number facts:

addition of pairs of single-digit numbers, from 1 + 1 to 9 + 9

subtraction facts based on the inverses of basic addition facts, from 18 – 9 to 2 – 1

multiplication of single-digit numbers, from 1 x 1 to 9 x 9 or to 12 x 12

division facts based on the inverses of basic multiplication facts, from 144 / 12 or 100 / 10 to 1 / 1

Once the above sets of facts are memorized, another set of facts should be learned: division facts with remainders. The fact is, between the even multiples of each multiplier there are other numbers which can be divided. For example:

4 / 4 = 1

5 / 4 = ?

6 / 4 = ?

7 / 4 = ?

8 / 4 = 2

9 / 4 = ?

10 / 4 = ?

11 / 4 = ?

12 / 4 = 3

etc.

Note that here we are talking about mental processing, not carrying out a written algorithm or using a calculator. We want students to be able to recall the relevant multiple of the divisor and then how many “extras” are in the starting number. The steps to be mastered could be thought of as:

33 / 4 = ?

The multiple of 4 closest to, and less than, 33: 32

32 = 4 x 8

Remaining ones: 1

Answer: 33 / 4 = 8 rem. 1

Students could have practice handling materials and putting them into groups of 2 or 4, and recording the starting number, the number of groups, and the remaining “leftovers”. This can then be linked to questions about how to work out the answer to these “messy” division questions, based on knowledge of the even multiples.

Extending These Facts:

These number facts could be extended in a number of ways. Here are some suggestions:

Treating remainders as common fractions, by dividing the whole number remainder by the divisor (e.g., 23 / 8 = 2 rem. 7 = 2 7/8)

Write word problems based on division questions with remainders. Make them as realistic as you can.

Draw pictures of division operations where there are remainders. Try to tell a story with your picture.

Create PowerPoint slides to teach a younger student about division with remainder number facts.