One-to-one correspondence
When a child counts a pile of things, how do the counting words and the things fit together, and what does the last word tell you?
Where you are 23 in Counting and cardinality
Why this exists
Counting is matching: each object gets exactly one counting word, in order, with none skipped and none counted twice. When the matching is done, the last word said tells how many there are, and it does not matter which object you started with. These four days build that match with the hands first (one counter for each cup), then with the words, then with the last word, and finally with things scattered on a table.
A child only needs to be able to say the counting words in order to start. Keep every count at ten or below this week.
Day 1: One for each
You need: 5 small paper cups, a handful of counters (buttons or beans work).
Put 4 cups in a row. Hand the child some counters and say, “Put one counter in each cup. Just one.” Do not count anything yet. When every cup has one, ask, “Does every cup have a counter? Is any counter left over?” Then add a fifth cup and give no more counters.
Ask, “Are there more cups or more counters? How do you know?” The child can say “that cup has no counter” without saying any number. That is the point of the day: two groups can be matched one to one, and the leftover shows which group is bigger.
Practice.
- 5 cups and 5 counters, one in each cup. Answer: every cup gets one and no counter is left, so there are the same number of cups and counters.
- 5 cups and 3 counters. Answer: 2 cups stay empty, so there are more cups.
- 3 children and 4 crayons; give each child one crayon. Answer: 1 crayon is left over, so there are more crayons than children.
- Make a row of 4 blocks. Under it, build a second row with one block under each block. Answer: the new row has 4 blocks too, because every top block has exactly one partner.
Day 2: One word for each thing
You need: about 10 counters, a sheet of paper with a line drawn down the middle.
Put a row of counters on the left side of the line. Show how to slide one counter across the line while saying “one”, the next while saying “two”, and so on. The hand and the voice move together: one push, one word. A counter that has crossed the line has had its word, so it is never counted again, and every counter on the left still needs a word.
Ask, “Which counter got the word three? Point to it.” Ask, “What word did the last counter get?”
Practice.
- Push and count 4 counters. Answer: “one, two, three, four”, four pushes.
- Push and count 6 counters. Answer: the last push gets “six”.
- Push and count 8 counters. Answer: the last push gets “eight”.
- Touch each counter in the picture above and say a word for each. Answer: 5 touches, and the last counter gets “five”.
- Hold up one finger at a time while you count to 7. Answer: 7 fingers are up, one for each word.
Day 3: The last word is how many
You need: counters or small blocks.
Have the child count a row of 6 counters by touching each one. Right after, ask, “How many counters are there?” The answer you are waiting for is “six”, said at once, without counting again. The last word said names the whole group; that is the last word tells how many.
Then say, “Count them again, but start from the other end.” The words land on different counters, but the last word is still “six”.
Last, spread the same 6 counters out wide and ask, “Now how many?” If the child counts again, that is fine; the count comes out 6 again, because no counter was added or taken away.
Practice.
- Count 7 blocks, then answer “How many blocks?” Answer: 7, the last word said.
- Count the same 7 blocks starting from the other end. Answer: still 7.
- Make one row of 5 counters spread far apart and one row of 5 pushed close together. Which row has more? Answer: neither; both have 5. Matching them one to one, as on Day 1, leaves none over.
- Leo counts his toy cars: “one, two, three, four.” How many cars does he have? Answer: 4, because “four” was the last word.
Day 4: Counting things that are not in a line
You need: up to 10 counters, a cup, a plate.
Scatter counters on a plate. A row tells you where to go next, but a loose pile does not, so the child needs a way to keep track. Show two ways. Move each counter into the cup as you say its word, so the counted ones are out of the way. Or, first push the counters into a line and then count the line. On a picture, where nothing can move, touch each one and make a small mark on it as you say its word.
Ask, “How do you know you counted every one? How do you know you did not count one twice?”
Practice.
- Count the counters in the picture above. Answer: 9.
- Scatter 10 counters and count them by moving each one into a cup. Answer: 10.
- Drop 6 cubes into a cup, pour them out, and count them. Answer: 6, wherever they land.
- Scatter 8 counters and count them. Then line them up and count again. Answer: 8 both times.
Exercises
- How many counters are in the picture? Watch how the child keeps track of which ones are counted.
- Put 6 paper cups in a row and give the child 8 counters. Say, “Put one counter in each cup.” Then ask, “How many counters are left over? Are there more cups or more counters?”
Answers
- 7 counters. Each counter gets one word, from “one” to “seven”, and “seven” is the last word said. A child who has it touches or marks each counter once and answers “seven” without counting again.
- 2 counters are left over. Each of the 6 cups takes one counter, and 2 are still in the child’s hand, so there are more counters than cups.
Where this goes next
The matching of Day 1 is how a child will tell which group has more, and saying “the same” means every object found a partner. Counting one word per object grows into counting a set of objects up to twenty, into counting out a given number from a bigger pile, and into counting to one hundred. Writing one stroke for each thing is the idea behind tally marks. The words, the written numbers and the counters all naming one amount is the work of Number names, numerals and zero.