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Grade 2 · 2.NBT.A.2

Counting by twos, fives, threes and fours

Rain boots stand in pairs by the door and tulips grow in rows of five. Why say “two, four, six” instead of counting every boot, and how can you tell in advance which numbers such a count will land on?

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Counting by twos, fives, threes and fours

When you reach for it

Rain boots by a classroom door stand in pairs. A child can count them one boot at a time, “one, two, three, four...”, and get the right answer. But the boots come in twos, and the eye already sees the pairs. Counting by twos says only the number at the end of each pair: “two, four, six, eight”. The count is half as long, and there is half as much room for a slip.

The same move works for any collection made of equal groups: tulips planted in rows of five, stools with three legs each, chairs with four. Say the number at the end of each group, and each number said is the size of one group more than the one before. That is skip counting, the subject of Skip counting, and this lesson covers jumps of two, five, three and four. Jumps of ten and of a hundred are Counting by tens to one hundred and Counting by hundreds.

A child reaches for skip counting in four situations:

  • counting a collection that comes in equal groups, such as pairs of boots or rows of tulips;
  • counting on in equal jumps from a number other than zero, such as 13, 18, 23;
  • counting back in equal jumps, such as 40, 38, 36;
  • deciding whether a count will ever land on a given number, by looking at the ones digits.

The pictures are the The number line, where each count is a jump of the same length, and the hundred chart, the numbers 1 to 100 in rows of ten, where the squares a count lands on make a pattern. Counting itself, by ones and by tens, is Counting to 100 and beyond and Counting past one hundred; counting on and back by ones is Counting on and counting back.

The procedure

The lesson is taught over three days: twos on the first, fives on the second, threes and fours on the third. Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.

Part 1: Counting by twos

One count for each pair. Line up some pairs of boots. Point at the first pair and say “two”, because there are two boots so far. Point at the next pair and say “four”, because one more pair is two more boots. Each number said is two more than the one before, so it is the same as counting on by ones twice, saying only the second number aloud: “(one) two, (three) four, (five) six”. That is why the count is right: no boot is skipped, only the word for it is.

A child must point at exactly one pair for each number said. The pointing keeps the count tied to the boots, and it also gives a second fact for free. The number of counts said is the number of pairs. A child who counts “two, four, six, eight, ten” has said five numbers, so there are five pairs.

On a number line. Each count is a jump of two spaces to the right, and the count lands on every second mark. Example 1 draws it. The line makes plain that the jumps are all the same length, which is what “equal groups” means in a picture.

On a hundred chart. Shade every square a count by twos from 2 lands on.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Figure 1Counting by twos from 2 lands on every second square. The squares it lands on fill five whole columns.

Every second square is shaded, and the shaded squares fill five whole columns. Each row of the chart holds ten numbers, an even number of squares, so the pattern that starts a row starts the next row the same way. Read the ones digits: 2, 4, 6, 8, 0, then 2, 4, 6, 8, 0 again. Out loud the count has a rhythm: “twenty-two, twenty-four, twenty-six, twenty-eight, thirty”. The decade word holds for a few counts while the ending runs “two, four, six, eight”, and then the next decade starts.

The numbers a count by twos from 0 lands on are the even numbers, the ones a pile can make from pairs with nothing left over; Even and odd numbers works with them.

Starting somewhere else. A count by twos can start at any number. From 1 it goes 1, 3, 5, 7, 9, 11, 13. The ones digits are now 1, 3, 5, 7, 9, over and over. On the hundred chart this count walks along the white squares and never touches a shaded one. So the ones digit of the starting number decides which numbers the count can land on.

Counting back. Counting back by twos says two less each time: 20, 18, 16, 14. On the number line each jump goes two spaces to the left. The ones digits run the same pattern backward, 0, 8, 6, 4, 2. Counting back from 20 stops at 0. Counting back from an odd number such as 15 stops at 1, because one boot has no pair to take away with it.

Part 2: Counting by fives

Rows of five. Tulips planted in rows of five are counted one row at a time: “five, ten, fifteen, twenty”. The fingers on a hand are a group of five a child always has at hand, and the marks for minutes on a clock face come in fives too, which Telling time to the nearest five minutes uses.

The ones digit takes turns. Counting by fives from 5 gives 5, 10, 15, 20, 25, 30. The ones digits go 5, 0, 5, 0. Two fives make a ten, so every second count lands on a whole ten: 10, 20, 30. In between, the count lands halfway to the next ten.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Figure 2Counting by fives from 5 lands on two columns, the one ending in 5 and the one ending in 0.

On the hundred chart the count fills just two columns, the column ending in 5 and the column ending in 0. Twenty counts of five reach 100, because there are two counts in each row and ten rows.

Starting somewhere else. From 3 the count by fives goes 3, 8, 13, 18, 23, 28. The ones digits take turns again, but now between 3 and 8. Five more than a number ending in 3 ends in 8; five more than a number ending in 8 passes a ten and ends in 3. So this count lands on 48 and 53, but never on 50 or 55.

Counting back by fives. Five less each time: 50, 45, 40, 35, 30. The digits run the pattern backward. A count back from a number ending in 5 or 0 reaches 0 at the end; a count back from 23 goes 23, 18, 13, 8, 3 and stops at 3, because 3 has no five left to take.

Part 3: Counting by threes and fours

Threes. A stool with three legs is a group of three. Counting the legs of several stools goes 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. This count is harder to say from memory than twos or fives, because its ones digits do not settle into a short pattern. They go 3, 6, 9, 2, 5, 8, 1, 4, 7, 0: every digit once in the first ten counts. Only then does the pattern start over, at 33 and 36, because ten threes make 30, a whole number of tens.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Figure 3Counting by threes from 3 makes slanting stripes. In the first row it lands on 3, 6 and 9; in the next row on 12, 15 and 18.

On the hundred chart the squares make slanting stripes: 3, then 12 one row down and one square to the left, then 21; and 6, 15, 24, 33 along the next stripe. A child who loses the count can always rebuild the next number by counting on three ones: after 12 comes “thirteen, fourteen, fifteen”, so 15.

Fours. A chair has four legs. Counting chair legs goes 4, 8, 12, 16, 20, 24, 28. The ones digits run 4, 8, 2, 6, 0 and then repeat, because five fours make 20, a whole number of tens. Every number in this count ends in an even digit, and every one of them is also in the count by twos from 0: one jump of four is two jumps of two. The count by fours lands on every second number of the count by twos.

Other starts, and counting back. Threes from 1 go 1, 4, 7, 10, 13, 16. Fours from 2 go 2, 6, 10, 14, 18. Counting back by fours from 40 goes 40, 36, 32, 28, and counting back by threes from 30 goes 30, 27, 24, 21. Each one is the same jump, made the same way, from a different start or in the other direction.

Two kinds of group in one collection. A room with stools and chairs has two sizes of group. Count one kind first, then keep going from where that count stopped, now jumping by the other size. Example 7 does this. The order does not matter: the legs are the same legs whichever kind is counted first.

Worked examples

Each example can be built with real things or drawn: boots as pairs of rectangles, tulips as dots, stool and chair legs as short strokes. A number line or hundred chart on the table lets the child check each count.

Example 1

Rain boots by the door

Chidubem lines up the class’s rain boots by the door. There are 9 pairs. How many boots are there?

Solution

The boots come in pairs, so count by twos, pointing at one pair for each number:

2,\; 4,\; 6,\; 8,\; 10,\; 12,\; 14,\; 16,\; 18

Nine numbers were said, one for each pair, and the last one is 18. There are 18 boots.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 +2 +2 +2 +2 +2 +2 +2 +2 +2
Figure 4Nine pairs of boots, nine jumps of two. The count lands on 18.

On the number line the count is nine jumps of two, starting at 0. It lands on 18. To check, count the boots one at a time; the count by ones passes 2, 4, 6 and so on, and ends at 18 too. The count by twos simply did not say the numbers in between.

Example 2

Counting on and back from forty-seven

Chidubem starts at 47 and counts on by twos. What are the next five numbers he says? Will he ever say 60? Then he counts back by twos from 47 to 39. What does he say?

Solution

Each count is two more, so from 47 the count goes

47,\; 49,\; 51,\; 53,\; 55,\; 57

The step from 49 to 51 crosses into the next ten: two more than 49 is one to reach 50 and one more to reach 51.

He will never say 60. He started at 47, which ends in 7, and the ones digits of this count run 7, 9, 1, 3, 5 and back to 7. A number ending in 0 never comes up. On the hundred chart this count walks along the white squares of Figure 1, and 60 is shaded. He goes from 59 straight to 61.

Counting back is two less each time:

47,\; 45,\; 43,\; 41,\; 39

That is four counts back, so 39 is eight less than 47, two for each count.

Example 3

Rows of tulips

Brigida plants tulips in 7 rows with 5 tulips in each row. How many tulips did she plant?

Solution

Each row is a group of five, so count by fives, one number for each row:

5 10 15 20 25 30 35
Figure 5Seven rows of five tulips. The number at the end of each row is the count so far.
5,\; 10,\; 15,\; 20,\; 25,\; 30,\; 35

Seven rows, seven numbers, and the last is 35. Brigida planted 35 tulips.

The ones digits took turns, 5, 0, 5, 0, and every second row finished a ten: two rows are 10, four rows are 20, six rows are 30. That gives a check. Six rows are 30, and one more row of five is 35.

Example 4

Counting by fives from thirteen

Tamati starts at 13 and counts on by fives. Fill in the boxes:

13,\; 18,\; \square,\; \square,\; \square,\; 38

If he keeps going, will he say 53? Will he say 55?

Solution

Each number is five more than the one before. Five more than 18 passes 20: two to reach 20 and three more, so 23. Then 28, then 33:

13,\; 18,\; 23,\; 28,\; 33,\; 38

The last box is right, because five more than 33 is 38, the number printed.

The ones digits take turns between 3 and 8. So the count will say 53: after 38 come 43, 48 and 53. It will never say 55, which ends in 5; the count goes from 53 straight to 58.

Example 5

Legs on the stools

There are 6 three-legged stools in the art room. How many legs are there?

Solution

Each stool is a group of three legs, so count by threes, one number for each stool:

3,\; 6,\; 9,\; 12,\; 15,\; 18

There are 18 legs.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 +3 +3 +3 +3 +3 +3
Figure 6Six stools, six jumps of three. The count lands on 18.

The step from 9 to 12 is where children slow down: three more than 9 is “ten, eleven, twelve”. The number line shows six equal jumps of three from 0 to 18.

Nine pairs of boots in Example 1 also made 18. The same number can be counted out as different equal groups: 18 is nine twos and also six threes.

Example 6

Legs on the chairs

There are 7 chairs around a table, each with 4 legs. How many legs are there?

Solution

Count by fours, one number for each chair:

4,\; 8,\; 12,\; 16,\; 20,\; 24,\; 28

There are 28 legs. The hard step is from 8: four more is “nine, ten, eleven, twelve”. After that the ones digits run 2, 6, 0 and then 4, 8 again, the pattern of fours.

A check uses twos. Each chair’s legs are two pairs, so the 7 chairs hold 14 pairs of legs. Counting by twos to the fourteenth number says 2, 4, 6, \ldots, 26, 28 and reaches 28 as well.

Example 7

Stools and chairs together

Tamati’s classroom has 4 three-legged stools and 5 four-legged chairs. How many legs are there altogether?

Solution

There are two sizes of group, so count one kind and then go on with the other. The stools first, by threes:

3,\; 6,\; 9,\; 12

That is 12 legs so far. Now keep going from 12, by fours, one count for each chair:

16,\; 20,\; 24,\; 28,\; 32

There are 32 legs altogether.

Count the other way round to check. The chairs first, by fours: 4, 8, 12, 16, 20. Then the stools, going on by threes from 20: 23, 26, 29, 32. The total is 32 again. It has to be, because the same legs were counted, only in a different order.

Boundaries

  • The groups must all be the same size. Counting by twos counts pairs. If one boot has lost its partner, count the pairs by twos and then count on one: seven pairs and a single boot are 2, 4, 6, 8, 10, 12, 14 and then 15.
  • A count only lands on numbers with the right ones digits. Counting by twos from an odd number never lands on an even one; counting by fives from 13 lands only on numbers ending in 3 or 8. To reach any other number, some ones have to be counted on as well.
  • Counting back stops before it runs out. A count back by twos from 15 ends at 1, and a count back by fives from 23 ends at 3. There is no whole group left to take away.
  • The pattern in the ones digits depends on the jump. Twos repeat 2, 4, 6, 8, 0 every five counts and fives repeat every two counts. Fours repeat every five counts, but threes only repeat after ten counts, which is why counting by threes is the hardest of the four to say by heart.
  • The last number is the total only when the count starts at 0. Counting by fives from 13 gives no total of anything; it is a walk along the number line. Counting groups of objects starts from nothing.

Where this goes next

Skip counting turns into adding the same number again and again, and from there into multiplying. Arrays: rows and columns of objects lays equal rows out in a rectangle and counts them row by row, which is a count by twos, threes, fours or fives. What multiplication means gives the count a name and a sign, and Facts for two, five and ten collects the counts by twos, fives and tens as the first facts. Counting by fives runs the minute hand of a clock in Telling time to the nearest five minutes. Counting by tens from any number is in Ten more and ten less, and counting a large pile by making groups first is Counting a large pile by grouping.

Exercises

Part 1: twos

  1. Count by twos from 2 to 20.
  2. Chidubem lines up 11 pairs of rain boots. How many boots is that?
  3. Fill in the boxes: 58, 60, \square, \square, 66, \square
  4. Count on by twos from 73 to 85.
  5. Count back by twos from 40 to 28.
  6. Fill in the boxes: 91, 89, \square, \square, 83
  7. Brigida counts boots by twos and stops at 16. How many pairs of boots are there?
  8. Start at 35 and count by twos. Will you say 50? Will you say 51?

Part 2: fives

  1. Count by fives from 5 to 60.
  2. Tamati’s garden has 9 rows of 5 tulips. How many tulips are there?
  3. Fill in the boxes: 65, 70, \square, \square, 85
  4. Count on by fives from 4 to 39.
  5. Count back by fives from 100 to 70.
  6. Fill in the boxes: 63, 58, \square, \square, 43
  7. Brigida counts tulips by fives, one row at a time, and stops at 30. How many rows are there?

Part 3: threes and fours

  1. Count on by threes from 2 to 26.
  2. How many legs are there on 8 three-legged stools?
  3. Count by fours from 4 to 40.
  4. How many legs are there on 9 four-legged chairs?
  5. Count back by threes from 27 to 12.
20 25 30 35 40
Figure 7Jumps on a number line, for Exercise 21.
  1. What size is each jump on this number line? Which numbers does it land on?
  2. A room has 3 three-legged stools and 4 four-legged chairs. How many legs are there altogether?

Explain

  1. Why does every number in a count by fives from 0 end in 5 or 0?
  2. Chidubem counts by twos from 0 and Tamati counts by fours from 0. Does Chidubem say every number Tamati says? Does Tamati say every number Chidubem says? Explain.
  3. Can a count by threes from 0 land on a number ending in 7? Can a count by twos from 0? Give an example or a reason for each.
Answers
  1. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. Each number is two more than the one before; ten counts reach 20.
  2. 22 boots. Count one pair at a time: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, eleven numbers.
  3. 62, 64, 68. Each number is two more: 60, 62, 64, then 66 as printed, then 68.
  4. 73, 75, 77, 79, 81, 83, 85. The ones digits run 3, 5, 7, 9, 1; two more than 79 crosses into the eighties.
  5. 40, 38, 36, 34, 32, 30, 28. Each number is two less than the one before.
  6. 87, 85. Each number is two less: 89, 87, 85, then 83 as printed.
  7. 8 pairs. The count was 2, 4, 6, 8, 10, 12, 14, 16, eight numbers, one for each pair.
  8. No to 50, yes to 51. Starting at 35, the ones digits run 5, 7, 9, 1, 3, so the count never lands on a number ending in 0. It goes 35, 37, 39, 41, 43, 45, 47, 49, 51: from 49 straight to 51.
  9. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60. Twelve counts; the ones digits take turns, 5 and 0.
  10. 45 tulips. One count for each row: 5, 10, 15, 20, 25, 30, 35, 40, 45, nine numbers.
  11. 75, 80. Each number is five more: 70, 75, 80, then 85 as printed.
  12. 4, 9, 14, 19, 24, 29, 34, 39. The ones digits take turns between 4 and 9.
  13. 100, 95, 90, 85, 80, 75, 70. Each number is five less than the one before.
  14. 53, 48. Each number is five less: 58, 53, 48, then 43 as printed. The ones digits take turns between 3 and 8.
  15. 6 rows. The count was 5, 10, 15, 20, 25, 30, six numbers, one for each row.
  16. 2, 5, 8, 11, 14, 17, 20, 23, 26. Each number is three more than the one before; three more than 8 is “nine, ten, eleven”, so 11.
  17. 24 legs. One count for each stool: 3, 6, 9, 12, 15, 18, 21, 24, eight numbers.
  18. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40. The ones digits run 4, 8, 2, 6, 0 twice.
  19. 36 legs. One count for each chair: 4, 8, 12, 16, 20, 24, 28, 32, 36, nine numbers.
  20. 27, 24, 21, 18, 15, 12. Each number is three less than the one before.
  21. Jumps of 4, landing on 21, 25, 29, 33, 37. The first dot is one mark past 20, so it is 21; the next is 25, four marks along, which is the labeled mark. Each jump covers four marks, so the count goes on 29, 33, 37.
  22. 25 legs. The stools first, by threes: 3, 6, 9. Then on by fours from 9, one count for each chair: 13, 17, 21, 25.
  23. Two fives make a ten, so every second count lands on a whole ten, which ends in 0: 10, 20, 30. The counts in between are five more than a whole ten, so they end in 5: 5, 15, 25. The count only ever takes turns between the two.
  24. Chidubem says every number Tamati says, because one jump of four is two jumps of two: Tamati’s 4, 8, 12 are Chidubem’s second, fourth and sixth numbers. The other way round is not true. Chidubem says 2, 6 and 10, and Tamati jumps over them.
  25. Threes, yes: 3, 6, 9, 12, 15, 18, 21, 24, 27, and 27 ends in 7. Twos, no: a count by twos from 0 only ever ends in 0, 2, 4, 6 or 8, so it jumps from 16 to 18 and from 26 to 28, never landing on a number ending in 7.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0