Counting and enumerationGrades 1–6
Six trees in a row: how many gaps between them? A log in five pieces: how many cuts? In these tasks a child counts either the things or the gaps between them, and the two numbers differ by one. That one is where most wrong answers come from, and a quick drawing of a small case is what puts it right.
1 2 3 4 |-----|-----|-----| 1 2 3
Gaps sit between things, so a row always has one gap fewer than things: six trees, five gaps; five pieces of a log, four cuts.
When the row closes into a ring, as bushes round a flowerbed do, the last one has a gap to the first: there are as many gaps as things.
From page 3 to page 7 is 7 − 3 = 4 steps, but 5 pages, because the first page is read too. Counting “from … to …” with both ends adds one.
Each question counts one of the two. A row, a ring and a range count differently, and a small picture shows how.
| The picture | What is counted | How many |
|---|---|---|
| 6 trees in a row | the gaps between them | 6 − 1 = 5 |
| a log in 5 pieces | the cuts | 5 − 1 = 4 |
| 8 bushes round a flowerbed | the gaps between them | 8 |
| pages 3 to 7 | the pages read | 7 − 3 + 1 = 5 |
Draw the things as dots or posts, first for a small number, two or three. Count the gaps on the drawing, see the rule, and only then use it for the numbers of the task. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 1–2
A bus route has 6 stops. The bus takes 3 minutes to get from one stop to the next. How many minutes does it take from the first stop to the last?
1 2 3 4 5 6 ●---●---●---●---●---● 3 3 3 3 3
Answer 15 minutes.
Example 2 · Grades 3–4
Lena lives on floor 9 and Masha on floor 3 of the same house, and both walk up from floor 1. All the flights of stairs are the same. How many times as many steps does Lena climb as Masha?
Answer 4 times as many.
Example 3 · Grades 5–6
Woodcutters sawed some logs, and every log was cut at least once. They made 9 cuts in all and got 14 pieces. How many logs did they saw?
Answer 5 logs.
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 child counts the stops instead of the gaps between them, or the gaps instead of the posts: 6 stops become 6 × 3 = 18 minutes. The answer is one off, so it looks almost right.
What to say
“Draw three stops. How many gaps are there?”
The child does something with the task's numbers that the picture does not ask for: adds them, 6 + 3, or doubles one of them, instead of counting the gaps and giving each its minutes.
What to say
“What does one gap take, and how many gaps are there?”
A number from the question becomes the answer: the 6 stops when the question asks for minutes, or floor 9 when it asks for flights of stairs.
What to say
“What does this number count: stops or gaps?”
Spread your fingers: five fingers, four gaps between them. It is the quickest picture of the whole topic, and it is always at hand.
On the stairs, count the floors and the flights together: from floor 1 to floor 4, how many flights? And from floor 2 to floor 4?
On a walk, count the posts of a fence and the spans between them, or the benches round a pond. In a row there is one gap fewer; in a ring, the same number.
