Study Guides
Teaching Multiples and Divisors — Why 1 and 12 Go Missing From "The Divisors of 12"
Say "write out the divisors of 12," and you might see your child write 2, 3, 4, 6 and stop there. 1 and 12 never make it onto the page. What's missing isn't calculation skill — it's a sense of how far to search for divisors.
Japan's Ministry of Education, in its commentary on the national curriculum guidelines for elementary mathematics (2017), states that the aim of this unit is "to make it possible to grasp the whole set of a given number's divisors, or the whole set of its multiples, as a single set." (Grade 5, Numbers and Calculation)
I had assumed the goal of this unit was being able to rattle off individual divisors quickly, the way times tables work. Looking into it, the goal wasn't the speed of individual recall at all — it was becoming able to survey "all" of a number's divisors as a single set. I didn't know the aim went this far.
The root cause: not seeing divisors and multiples as a set
Cause 1: leaving out 1 and the number itself when listing divisors
The commentary states: "a whole number that can divide 12 evenly is called a divisor of 12." 12 is divisible by 1, and it's divisible by 12 itself, so 1 and 12 are both bona fide divisors. Leaving these out is a sign that your child is searching for divisors only among numbers smaller than 12. Unless the search switches to finding every pair of numbers that multiply to make 12, 1 and 12 never come into view. A set of divisors missing 1 and 12 isn't the correct set.
Cause 2: confusing a common divisor problem with a common multiple problem
"Cut an 8 cm by 12 cm sheet of paper into same-size squares with nothing left over" calls for a common divisor. "A bus every 8 minutes and a bus every 12 minutes — when do they next leave at the same time?" calls for a common multiple. Mixing up these two kinds of problem is the most common mistake. The way to tell them apart is the size of the answer. A square's side has to be smaller than 8 and 12, so that's a common divisor; the time the buses line up has to come later than 8 or 12 minutes, so that's a common multiple. Deciding first which direction the answer should move from the original numbers keeps your child from having to guess whether to calculate a common divisor or a common multiple.
Cause 3: mixing 0 into the count of multiples
The commentary states: "a number made by multiplying 3 by a whole number is called a multiple of 3." Read just the phrase "multiplying by a whole number," and it looks like multiplying by 0 should be fine too, but schools have a rule that 0 isn't included among the multiples here. Multiples of 3 are counted starting from multiplying by 1 — 3, 6, 9, 12, and so on — and quite a few children get stuck on this point.
How to teach it at home
Step 1: Find every divisor by hunting for multiplication pairs
For 12, search in order from smallest for pairs that multiply to make 12: 1 × 12, 2 × 6, 3 × 4. Searching in pairs means 1 and 12 come up naturally as the partner in a pair, so they don't get left out. When a pair of matching numbers shows up, that's the signal the search is done (for 36, that's 6 × 6).
Step 2: Check "common divisor or common multiple" by asking how the answer compares to the original numbers
Before your child sets up an equation, have them say out loud: "should the answer be smaller than the original numbers, or bigger?" Once they can guess that a splitting-up problem should give a smaller answer and a lining-up problem should give a bigger one, they can choose for themselves whether to calculate the greatest common divisor (GCD) or the least common multiple (LCM).
Step 3: Write out multiples as the reverse of times tables, without 0
Ask "what do you get multiplying the 3 times table starting from 1?" and have your child build the multiples by multiplying from 1 up: 3 × 1, 3 × 2, 3 × 3, and so on. Since it's exactly the times-table sequence, this is light practice for a child who already has times tables down cold.
What to avoid
Having your child memorise the terms "greatest common divisor" and "least common multiple" first, then fit situations to them afterward, is an order worth avoiding. Before the terms, confirm the difference between a splitting-up situation and a lining-up situation using everyday words.
How to practise
- The Properties of Integers (Divisors and Multiples) unit runs in one continuous sequence: listing divisors and multiples, common divisors and the greatest common divisor, common multiples and the least common multiple
- Searching for divisors gets easier the faster your child can work Times Tables: 6 to 9 and 1 in reverse. If there's a times table your child still hesitates on, settling that first ends up being the shortcut
- Reducing and finding a common denominator for fractions are situations that use the common divisors and least common multiples practised here directly. Smoothing out the listing skill before moving on to Properties of Fractions makes things easier later
Frequently asked questions
Q. Is there a trick for finding every divisor of a large number?
A. Searching in order from 1 × the number for pairs that multiply to make it is the reliable method. Increase the smaller number in the pair from 1 upward — 1 and 12, 2 and 6, 3 and 4 — and the search is done once the two numbers in the pair cross over. For 36, work through 1 and 36, 2 and 18, 3 and 12, 4 and 9, up to 6 and 6 — since 6 × 6 is the same number twice, that's the signal it's fine to stop there. For numbers up to around 40, this method is enough to find them all.
Q. Should we teach prime numbers in this unit?
A. Prime numbers (numbers whose only divisors are 1 and themselves) are outside the scope of this unit. They're taken up properly starting in junior high. Noticing that "some numbers only have two divisors" is enough for now — there's no need to dig further.
Divisors and multiples aren't a unit about finding one answer — they're a unit about surveying a set. While the omissions keep happening, spend the time checking your child's search method rather than how well they've memorised it.
This article is written by a parent who builds diglabo. I am not a teacher or an education specialist. The general guidance here is based on how the topic is taught in Japanese primary schools, checked against Japan's national curriculum guidelines.
diglabo's worksheets follow the Japanese curriculum. What that means if you are outside Japan