Counting a large pile by grouping
There is a big pile of popsicle sticks on the table, far too many to count one at a time without losing your place. How can you count it quickly, be sure the count is right, and know roughly how many there are before you start?
Where you are 23 in Counting and cardinality
When you reach for it
A child who can count a set of objects touches each thing once while saying the next number word, and the last word said is how many. That works well for 8 crayons or 15 pom-poms. It starts to fail on a pile of 47 popsicle sticks or a bag of 63 pom-poms. The count is long, the things are all alike, and a single slip (a stick touched twice, a stick skipped, a lost place when someone talks) spoils the whole count with no way to tell where it went wrong.
The fix is to make equal groups as you count. Put the sticks in groups of ten, then count the groups, “ten, twenty, thirty, forty”, and count on through the few that are left over. Each small count only goes up to ten, so a slip costs one group, not the whole pile. The groups also stay on the table as a record of the counting already done.
Reach for it whenever a collection is too big to count comfortably in one go: roughly anything past twenty, and every collection up to 120 and beyond. Reach for its partner, estimating, before the counting starts: a sensible guess at how many, made by looking, which the count then confirms or corrects.
The lesson has four parts, one for each day. The first is why a long count goes wrong, and how groups of ten fix it. The second is groups of two, five and ten, and why ten is the handiest. The third is recording a count and checking it by grouping a different way. The fourth is estimating before counting, then comparing the estimate with the count.
Use real things in big numbers: popsicle sticks, pom-poms, crayons from a crayon tub, with small cups or paper plates to hold the groups. This lesson is about counting a collection. What each digit of the answer is worth, the 4 and the 7 in 47, is the next lesson, Tens and ones.
The procedure
Part 1: Why a long count goes wrong, and how groups help
What counting one at a time asks of a child. To count a pile one by one, a child has to do three things at once: say the number words in order, touch each thing exactly once (One-to-one correspondence), and keep the counted things apart from the ones not yet counted. With 10 things all three are easy. With 50 alike sticks, every one of them is a chance to slip, and the child is holding the whole count in their head. If they lose their place at “thirty-something”, nothing on the table says where they were, and the count starts again from one.
Move each thing as it is counted. The first help is to slide each stick to a new spot as it is counted. Then the counted sticks and the uncounted sticks are in two separate places, and no stick can be counted twice or missed. This is worth doing every time, but it does not solve the lost place: a heap of counted sticks still does not say how many it holds.
Make groups of ten as you count. Now count the sticks “one, two, ..., ten”, put those ten in a cup, and start again from one with the next stick. Each count only runs to ten, which a child can do reliably. When fewer than ten sticks are left, they cannot fill a cup, so they stay loose.
Count the groups, then count on. Every cup holds ten, so the cups are counted by tens: “ten, twenty, thirty, forty”. That is counting by tens, saying ten, twenty, thirty and on, one ten for each group. Then the loose sticks are counted on from there, one at a time: “forty-one, forty-two, ...” The last word said is how many sticks there are, exactly as when counting one by one (How counting tells how many). Putting sticks in cups did not add a stick or take one away, so the count comes out the same; it only comes out more surely.
A lost place is cheap. If a child is interrupted, the full cups are still there. Count them by tens, count on through any sticks already lined up beside them, and the count picks up where it stopped. At worst, the one group being filled has to be counted again. Example 1 shows this.
Every group must hold exactly ten. Counting cups by tens is only right if every cup really holds ten. A cup of nine counted as “ten” makes the whole count one too many. Children can check a cup by counting it, and it is worth doing when a count looks odd.
Part 2: Groups of two, five and ten
Any equal groups will do. Groups do not have to be tens. Pom-poms can be put in pairs and counted by twos, “two, four, six, eight”; sticks can be put in groups of five and counted by fives, “five, ten, fifteen, twenty”. This is Skip counting: counting in equal jumps rather than one at a time. The rule is the same for every size: all the groups are the same size, and the skip count matches that size. Pairs are counted by twos, fives by fives, tens by tens.
Leftovers are counted on by ones. When fewer things are left than a group needs, they stay loose. After the last skip count, they are counted on one at a time. Seventeen pom-poms in pairs make 8 pairs with 1 left over: “two, four, ..., sixteen, seventeen”. Whether a pile in pairs has one left over is what makes a number odd or even, in Even and odd numbers.
Bigger groups mean fewer counts. Thirty pom-poms make 15 pairs, or 6 fives, or 3 tens. Counting by twos takes fifteen skip counts; by fives, six; by tens, three. Each way ends on thirty.
Why ten is the handiest group. Ten wins for four reasons, and they reinforce each other.
- Fewer groups. A group of ten holds more than a group of two or five, so there are fewer groups to make and count.
- A small count inside each group. Filling a group means counting only to ten, which a child does without slips. A group of twenty would cut the number of groups again, but filling it is itself a long count, and counting by twenties is hard.
- The number words count tens. “Twenty”, “thirty”, “forty” and the rest are names for two tens, three tens, four tens. So counting groups of ten uses words the child already says when counting by ones, and a count like “forty, forty-one, forty-two” runs on smoothly. The names for fives and twos do not line up this way.
- Ten is everywhere. It is two hands of fingers and one full ten frame, and it is the size the written numbers are built on, which is why a collection counted in tens can be written down at once (the subject of Tens and ones).
Twos and fives still have their place. Pairs are quick to make and to check by eye, and fives match one hand. A child who counts a pile in fives and then in tens has two counts that should agree, which is the check in Part 3.
Part 3: Recording a count, and checking it
Recording what was counted. A count is worth writing down, so it is not lost when the cups are emptied. The record can say how the pile was grouped and what the count was:
| Groups of ten | Left over | How many |
|---|---|---|
| 4 | 7 | 47 |
A child might instead draw the groups as boxes with dots for the loose ones, or write the counting words, “10, 20, 30, 40, 41, 42, 43, 44, 45, 46, 47”. Any record that someone else can read, and that ends with the number, does the job. The record also lets the count be checked later without redoing it.
Counting past one hundred. Ten groups of ten is one hundred. The count goes on the same way: “one hundred, one hundred ten” for an eleventh group, “one hundred twenty” for a twelfth, and then on by ones, “one hundred eleven, one hundred twelve, ...” Reading and writing these numbers is Counting, reading and writing numbers to 120.
Checking a count by grouping another way. A count is a claim: “there are 47”. The strongest check is to count again in a different way, because a slip in one way is unlikely to be repeated exactly in another. Group the same pile in fives instead of tens, or in pairs, and count again. If the two counts agree, the count is very likely right. If they do not, at least one of them has a slip, and the pile is counted a third time, carefully.
Regrouping without starting over. Groups can be changed into other groups. Two fives make a ten, so a pile in fives can be turned into tens by pushing fives together in pairs. Untying a ten gives two fives or five pairs. Since no thing is added or removed by regrouping, the count after regrouping must be the same as before. Example 6 shows a count in fives checked by tens.
Part 4: Estimating before counting
What an estimate is. An estimate is a sensible guess at how many, made by looking, before or instead of counting. It is not a wild guess: it uses what the child can see and what they already know about the size of groups. It is not a count either, and it is not expected to match the count exactly.
Why estimate first. Guessing first gives the child a feel for how big the pile is, so the count has something to be compared with. If the guess was “about sixty” and the count comes out 16, something has gone wrong (perhaps a group was counted as one stick) and the count is worth checking. It also builds a feel for sizes of numbers: what thirty pom-poms looks like, and what a hundred looks like.
Saying an estimate as a range. A child can say an estimate as a single number, “about 60”, or as a range: “more than 50, fewer than 80”. A range is often easier to be sure of, and it is easy to check afterwards: the count either lands inside it or it does not. On a number line, a range is a stretch between two marks, and the count is one point that falls inside or outside the stretch.
Using one group to help. A good way to estimate is to count out one group of ten first and look at it. Then look at the rest of the pile and ask how many groups like that it would make. If the rest looks like about seven more groups, the whole pile is about eight groups, which is about 80. This uses a known amount to judge an unknown one, and it gets better with practice.
Comparing the estimate with the count. After the count, put the two side by side. Is the count more than the estimate, or fewer? Did it land inside the range? The aim is not to have guessed exactly; it is to notice how close the guess was and to make the next guess better. With groups of ten, comparing is easy: 63 is 6 groups of ten and some more, so it is more than 5 groups (50) and fewer than 8 groups (80). Comparing numbers in general is in Comparing quantities and Comparing numbers and the symbols >, < and =.
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4. Build each one with real things before reading the solution.
Picking up a lost count
Ljuba is counting the crayons in the crayon tub. Each time she counts ten, she puts them in a cup. She is called away partway through. When she comes back there are 3 full cups on the table and 6 crayons lined up next to them, and the rest of the crayons are still in the tub. How many crayons has she counted so far?
SolutionEvery cup holds ten, so the cups are counted by tens: “ten, twenty, thirty”. The 6 crayons in the line have been counted but not yet put in a cup, so they are counted on from thirty, one at a time: “thirty-one, thirty-two, thirty-three, thirty-four, thirty-five, thirty-six”.
She has counted 36 crayons so far. Her next crayon from the tub is “thirty-seven”, and when the line reaches ten crayons they go in a fourth cup. Had she been counting one by one into a single heap, nothing on the table would have told her where she had stopped, and she would have had to start again from one.
Forty-seven popsicle sticks
Jirina has a pile of popsicle sticks. She counts them into groups of ten, putting each group inside a rubber band, until fewer than ten are left. She makes 4 groups and has 7 sticks left. How many sticks are in the pile?
SolutionThe 7 loose sticks are fewer than ten, so they cannot make another group; that is why she stopped. Each group holds ten, so the groups are counted by tens: “ten, twenty, thirty, forty”. Then the loose sticks are counted on by ones: “forty-one, forty-two, forty-three, forty-four, forty-five, forty-six, forty-seven”.
There are 47 sticks. Figure 1 shows the count, with the counting word under each group and each loose stick. Every count she made while grouping went only up to ten, so a slip would have cost her one group, not the whole pile. Why the number is written with a 4 and a 7 is in Tens and ones.
Pom-poms in pairs
Stanko has 17 pom-poms. He puts them in pairs. How many pairs does he make, and how does he count the pom-poms from the pairs?
SolutionMake pairs until fewer than two pom-poms are left. Counting the pom-poms by twos as each pair is made: “two, four, six, eight, ten, twelve, fourteen, sixteen”. That is eight pairs, since each skip count was one pair. One pom-pom is left, and one cannot make a pair.
He counts the pom-poms by twos, one skip count for each pair, and then counts on by one for the pom-pom left over: “two, four, six, eight, ten, twelve, fourteen, sixteen, seventeen”.
So there are 8 pairs and 1 left over, which is 17 pom-poms, the number he started with.
Thirty pom-poms, three ways
Jirina has 30 pom-poms. Count them in groups of two, then in groups of five, then in groups of ten. How many groups does each way make? Which way is quickest to count?
SolutionIn pairs, count by twos: “two, four, six, ..., twenty-eight, thirty”. That is 15 skip counts, so 15 pairs.
In fives, count by fives: “five, ten, fifteen, twenty, twenty-five, thirty”. That is 6 skip counts, so 6 groups of five.
In tens, count by tens: “ten, twenty, thirty”. That is 3 skip counts, so 3 groups of ten.
| Group size | Number of groups | Counting words |
|---|---|---|
| 2 | 15 | two, four, ..., thirty |
| 5 | 6 | five, ten, ..., thirty |
| 10 | 3 | ten, twenty, thirty |
All three ways end on thirty, because the same pom-poms were counted each time; only the grouping changed. Tens is quickest: only three groups to count, and the words “ten, twenty, thirty” are the ones the child already knows. Figure 2 shows the three groupings side by side. The bigger the group, the fewer the groups, and ten is the biggest group that is still easy to fill and easy to count by.
More than one hundred pom-poms
Jirina counts a big box of pom-poms into cups of ten. She fills 11 cups and has 4 pom-poms left over. How many pom-poms are there? Record the count.
SolutionCount the cups by tens: “ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, one hundred”, which is ten cups, then “one hundred ten” for the eleventh. Then count on through the 4 left over: “one hundred eleven, one hundred twelve, one hundred thirteen, one hundred fourteen”.
There are 114 pom-poms. The record:
| Groups of ten | Left over | How many |
|---|---|---|
| 11 | 4 | 114 |
Ten cups of ten make one hundred, and the count simply keeps going past it in the same way. Anyone who reads the record can rebuild the pile, or check it, without asking Jirina how she counted.
Checking a count of fives with tens
Stanko counts a pile of popsicle sticks in groups of five. He makes 9 groups of five, with 3 sticks left over. How many sticks are there? Ljuba checks his count by pushing the fives together into tens. What does she find?
SolutionStanko’s count, by fives and then on by ones: “five, ten, fifteen, twenty, twenty-five, thirty, thirty-five, forty, forty-five”, then “forty-six, forty-seven, forty-eight”. There are 48 sticks.
Ljuba’s check. Two fives make a ten, so she pushes the fives together in pairs. Nine fives make 4 pairs of fives with 1 five left over, since 8 fives are 4 tens. That gives 4 groups of ten. The spare five and the 3 loose sticks are 8 loose sticks, too few for another ten. Counting by tens and on: “ten, twenty, thirty, forty, forty-one, ..., forty-eight”.
Both counts end on 48, and the two counts were made in different ways, so the count is very likely right. Regrouping did not add or remove a stick, which is why the two had to agree. If they had not, one of the counts had a slip, and the pile would be counted a third time.
Estimating a bag of pom-poms
Ljuba holds up a clear bag of pom-poms and asks: are there more than 50? Fewer than 80? The class guesses “more than 50, fewer than 80”. Then they count. They make 6 groups of ten with 3 left over. How many pom-poms are there, and was the guess right?
SolutionCount the groups by tens and then on by ones: “ten, twenty, thirty, forty, fifty, sixty”, then “sixty-one, sixty-two, sixty-three”.
There are 63 pom-poms. Now compare with the guess. Fifty is 5 groups of ten, and 63 is 6 groups of ten and 3 more, so 63 is more than 50. Eighty is 8 groups of ten, and 63 has only 6 full groups, so 63 is fewer than 80.
The count landed inside the range, so the estimate held. Figure 3 shows the range as a shaded stretch of the number line with the count marked inside it. The class can also notice that 63 is much nearer 60 than 80: next time, “more than 50, fewer than 70” would be a closer guess.
Estimating with one group as a guide
Ljuba wants to estimate the crayons in the crayon tub. She counts out ten crayons and lays them in a row. She looks at the rest and thinks they would make about 7 more rows like that. What is her estimate? Then she counts, and finds 7 groups of ten and 6 more. How close was the estimate?
SolutionHer row of ten is one group, and she thinks the rest make about 7 more groups. That is about 8 groups of ten in all: “ten, twenty, ..., eighty”. Her estimate is about 80 crayons.
The count: “ten, twenty, thirty, forty, fifty, sixty, seventy”, then “seventy-one, ..., seventy-six”.
There are 76 crayons, a few fewer than her estimate. Counting on from 76 to 80 is “seventy-seven, seventy-eight, seventy-nine, eighty”, four counts, so her estimate was 4 more than the count:
That is a close estimate. The rest of the tub held 7 groups of ten and 6 more, nearly the 7 more groups she guessed. Using one counted group to judge the rest is what made the estimate close.
Boundaries
- Every group must be the same size, and the skip count must match it. Counting by tens is right only if every group holds exactly ten. A group of nine counted as ten makes the count too big by one; groups of five counted by tens make it far too big.
- Leftovers are always counted on by ones. Whatever the group size, the things that do not fill a group are counted on one at a time after the last skip count. If more than a group’s worth is left loose, the count still comes out right as long as each is counted, but the record is clearest when the leftovers are fewer than one group.
- Grouping needs things that can be moved or marked. Sticks, pom-poms and crayons can be put in cups. Fish in a tank or birds on a wire cannot be grouped, and a count of them is hard to check; there an estimate may be the best that can be done.
- An estimate is not a count. An estimate that differs from the count is not wrong; it was never meant to be exact. But when the exact number matters, only a count will do.
- Past one hundred twenty. The method does not stop at 120. Ten groups of ten make one hundred, and bigger collections can be grouped in hundreds in the same way, in Reading and writing numbers to 1,000.
Where this goes next
A collection counted as groups of ten and some more is exactly what a two-digit number writes down. Tens and ones takes the next step: the 4 in 47 counts the groups of ten and the 7 counts the ones left over, and that is the start of Place value.
Groups of ten also make adding and taking away ten easy, since one more group adds exactly ten: that is Ten more and ten less. Counting on past one hundred is Counting, reading and writing numbers to 120 and then counting past one hundred. Estimating grows into rounding and checking whether an answer is sensible, in Rounding and Friendly numbers and deliberate over and under estimates.
Exercises
Part 1: Groups of ten
- A pile of popsicle sticks is counted into 3 groups of ten with 5 sticks left over. How many sticks are there?
- Stanko has 6 cups of ten pom-poms and 2 more pom-poms. How many pom-poms does he have?
- How many pom-poms are in row A?
- How many pom-poms are in row B?
- How many pom-poms are in row C?
Part 2: Twos, fives and tens
- Jirina makes 9 pairs of pom-poms. How many pom-poms are there?
- Ljuba makes 8 groups of five popsicle sticks. How many sticks are there?
- A pile is counted as 7 groups of five with 2 left over. How many are there?
- Stanko puts 24 crayons in pairs. How many pairs does he make?
- A pile of 50 popsicle sticks is put in groups of ten. How many groups are there? How many groups would there be in fives?
- A pile of 70 pom-poms could be counted in twos, fives or tens. How many groups would each way make? Which way is quickest?
Part 3: Recording and checking
- Jirina fills 11 cups of ten popsicle sticks and has 6 left over. How many sticks are there?
- A box of crayons makes exactly 10 groups of ten, with none left over. How many crayons are there?
- Stanko fills 12 cups of ten pom-poms, with none left over. How many pom-poms are there?
- A pile of crayons is counted as 7 groups of five with 4 left over. How many crayons are there? Pushed together into tens, how many groups of ten does it make, and how many are left over?
- A pile of popsicle sticks is counted as 4 groups of ten and 6 more. Ljuba checks by counting the same pile in pairs. How many pairs should she find?
Part 4: Estimating
- The class guesses there are about 40 pom-poms in a jar. The count is 6 groups of ten and 1 more. How many are there? Is that more or fewer than the guess?
- Stanko guesses a pile of popsicle sticks has more than 70 and fewer than 90. The count is 8 groups of ten and 5 more. Did the count land inside his range?
- Jirina guesses about 50 crayons. The count is 4 groups of ten and 9 more. How many crayons are there, and how far is the count from her guess?
- Ljuba fills one cup with ten pom-poms. The rest of the bag looks like it would fill about 8 more cups. What is a sensible estimate for the whole bag?
- A guess was “more than 80, fewer than 100”. The count is 10 groups of ten and 4 more. Did the count land inside the range?
Explain
- A pile is counted in tens and comes to 58. Counted again in fives, it comes to 57. What does that tell you, and what should happen next?
- Why is it easier to pick up a lost count when the things are in groups of ten than when they are in one heap?
- A pile is counted in tens and there are none left over. Could the same pile, counted in fives, have 3 left over? Explain.
- If the count is the real answer, why make an estimate first?
Answers
- 35 sticks. Count the groups by tens, “ten, twenty, thirty”, then on by ones through the 5: 10 + 10 + 10 + 5 = 35.
- 62 pom-poms. “Ten, twenty, ..., sixty” for the cups, then “sixty-one, sixty-two”: 60 + 2 = 62.
- 54. Row A has 5 groups of ten and 4 more: “ten, twenty, thirty, forty, fifty”, then on by ones to fifty-four, 50 + 4 = 54.
- 108. Row B has 10 groups of ten, which is one hundred, and 8 more: 100 + 8 = 108.
- 33. Row C has 2 groups of ten, “ten, twenty”, and then 13 loose pom-poms counted on by ones, “twenty-one, ..., thirty-three”: 20 + 13 = 33.
- 18 pom-poms. Count by twos, one count for each pair: “two, four, ..., eighteen”, 9 counts, ending on 18.
- 40 sticks. Count by fives, one count for each group: “five, ten, ..., forty”, 8 counts.
- 37. Count by fives to thirty-five for the 7 groups, then on by ones, “thirty-six, thirty-seven”: 35 + 2 = 37.
- 12 pairs. Count by twos, “two, four, ..., twenty-four”, and that takes 12 skip counts.
- 5 groups of ten, “ten, twenty, thirty, forty, fifty”. In fives it would be 10 groups, since counting by fives to fifty takes 10 counts (each ten is two fives).
- 35 pairs, 14 fives, or 7 tens. Tens is quickest: only 7 groups to make and count, “ten, twenty, ..., seventy”.
- 116 sticks. Ten cups are one hundred, the eleventh makes one hundred ten, and then on by ones through the 6: 100 + 10 + 6 = 116.
- 100 crayons. Count by tens ten times: “ten, twenty, ..., ninety, one hundred”.
- 120 pom-poms. Ten cups make one hundred, then “one hundred ten, one hundred twenty” for the eleventh and twelfth: 100 + 10 + 10 = 120.
- 39 crayons, which makes 3 groups of ten and 9 left over. By fives: “five, ..., thirty-five”, then “thirty-six, ..., thirty-nine”. Six of the fives pair up into 3 tens; the spare five and the 4 loose make 9, too few for another ten: 30 + 9 = 39.
- 23 pairs. The pile has 40 + 6 = 46 sticks. Counting by twos, “two, four, ..., forty-six”, takes 23 skip counts (each ten is 5 pairs, so 4 tens are 20 pairs, and 6 more sticks are 3 pairs).
- 61 pom-poms, which is more than the guess of 40: 61 has 6 full groups of ten and 40 is only 4 groups.
- Yes. The count is 80 + 5 = 85, which is more than 70 (seven groups) and fewer than 90 (nine groups): 70 < 85 < 90.
- 49 crayons, 1 fewer than her guess. The count is 40 + 9 = 49, and counting on from 49 to 50 takes one count: 49 + 1 = 50.
- About 90 pom-poms. One counted cup and about 8 more cups is about 9 groups of ten: “ten, twenty, ..., ninety”.
- No. The count is 100 + 4 = 104, which is more than 100, so it landed above the range. The estimate was not wrong to make; the count shows the collection was bigger than it looked.
- At least one of the two counts has a slip, because the same pile counted two ways must give the same number. Neither count can be trusted yet, so the pile should be counted a third time, carefully, and checking that every group really holds ten or five.
- In one heap, the counted things do not show how many there are, so a lost place means starting from one. Groups of ten stay on the table: count them by tens and on through any loose ones, and the count is back. At most one unfinished group has to be counted again.
- No. Every ten is exactly two fives, so a pile that is all tens with none left over is also all fives with none left over. A pile with 3 left over in fives would also have something left over in tens, so one of the two counts would have a slip.
- The estimate gives a feel for how big the pile is, so a count that comes out far from it (say 16 when the guess was about 60) is a signal to check the count. Comparing estimate and count afterwards also trains the eye to judge amounts, which makes the next estimate better.