Numbers with a trickGrades 5–6
29 pupils go on a trip, and each car takes 4 of them. How many cars are needed? In these tasks a child divides with a remainder and then decides what the remainder means: one more car, a sweet left over, a place in the next row. Rules of divisibility answer “does it divide?” without dividing at all.
●●● ●●● ●●● ●●● ●● 14 = 4 × 3 + 2
When dividing by 7, the remainder is 0, 1, 2 and so on up to 6. If it comes out 7 or more, one more group fits.
29 pupils in cars of 4 are 7 full cars and 1 pupil left over, so 8 cars are needed. The same division can be asked for 7, for 8 or for 1: the question decides.
A number divides by 2 when its last digit is even, by 5 when it ends in 0 or 5, and by 3 or by 9 when the sum of its digits does.
Every whole number can be written as full groups and what is left over. The question decides which of the two it wants.
| The question | The division | The answer |
|---|---|---|
| How many cars of 4 seats do 29 pupils need? | 29 = 7 × 4 + 1 | 8 cars |
| 29 sweets shared among 4 children: how many does each get? | 29 = 7 × 4 + 1 | 7 sweets each |
| How many of the 29 sweets are left over? | 29 = 7 × 4 + 1 | 1 sweet |
Write the number as full groups and what is left over. Then read the question again and decide what it wants: the groups, the remainder, or one group more. When it asks for the smallest number or the digit with some property, try them in order. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 5–6
150 pupils go on a school trip, and each bus has 40 seats. How many buses are needed?
[40] [40] [40] [30]
Answer 4 buses.
Example 2 · Grades 5–6
Which digit must stand in place of the star in 38*6 so that the number can be divided by 9 with nothing left over?
Answer The digit 1.
Example 3 · Grades 5–6
A hundred twos are multiplied together: 2 × 2 × 2 × … × 2. What is the last digit of the answer?
1 2 3 4 5 6 7 8 2 4 8 6 2 4 8 6
Answer 6.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The full groups and the remainder change places: the child gives the 30 pupils left over when the question asks for buses, or takes 25, the number of full rounds of the pattern, for the place in it.
What to say
“Is the question about the full groups or about what is left over?”
The answer is one group off: the bus for the last 30 pupils is forgotten, 3 instead of 4, or places in a pattern are counted from 0 instead of 1.
What to say
“Does what is left over need one more group?”
One case is found and the search stops: the star is 1, and the child does not check whether 4 and 7 work too when the question is about 3.
What to say
“Has every digit from 0 to 9 been tried?”
Share sweets or cards among the family and say the result in two parts: how many each one gets, and how many are left over.
Make up big numbers and test them for 3 and for 9 by adding up their digits, then divide on a calculator to see the rule work.
Look for patterns that go round: the days of the week, the colours of a necklace, the last digits of 3, 9, 27, 81. Ask what comes 50 steps on.
