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Grade 1 · 1.NBT.B.3 · 3.NF.A.3D

Comparing numbers and the symbols >, < and =

Two towers of cubes are too tall to match up one cube at a time. How can the two written numbers alone say which tower has more, and how do we write that answer down?

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Why this exists

Children already tell which of two small groups has more by matching them up or by counting (Comparing quantities, Comparing two numbers to 10). That works with seven cubes against five. It works badly with 45 cubes against 54. Matching that many cubes one to one is slow, and one slip spoils the answer.

The written numbers already hold the answer. A two-digit number says how many tens and how many ones it has (Tens and ones), and a ten is a lot more than a one. So we look at the tens first, and only look at the ones when the tens are the same. No cube has to be moved.

Once we know which is more, we want to write it down in a way anyone can read. Three signs do that: >, < and =. With them a whole comparison fits in a short sentence, 45 < 54, read aloud as “forty-five is less than fifty-four”.

The lesson runs over four days. Day 1 compares two-digit numbers by their tens. Day 2 handles numbers with the same tens, and numbers that are equal. Day 3 brings in the three signs. Day 4 reads each comparison both ways and carries the idea past 100, up to 120.

Definition

Take two whole numbers. The greater one names more things. The less one names fewer things. If both name the same amount, they are equal.

Three signs record which of these is true.

WriteSay
54 > 45fifty-four is greater than forty-five
45 < 54forty-five is less than fifty-four
54 = 54fifty-four is equal to fifty-four

The signs > and < are one shape turned around. Each has a wide, open side and a narrow point. The open side faces the greater number. The point faces the less number. A comparison is read from left to right, like any sentence, so the sign is read as whatever it says from the left-hand number: “is greater than” when the open side is on the left, “is less than” when the point is on the left.

The sign = says the two sides name the same amount; that meaning is the subject of The equal sign means the same amount.

Intuition

Day 1: tens first

The pictures in this lesson use base-ten blocks. A rod is ten small cubes snapped together in a column. A loose cube is a one. A rod plays the same part as a bundle of ten sticks.

4 tens 5 ones 45 5 tens 4 ones 54
Figure 1The same two digits, in different places. The picture on the right has one more rod.

Look at 45 and 54 in the picture. Both are written with a 4 and a 5, but the digits sit in different places, so they are worth different amounts. The left picture is 4 rods and 5 cubes. The right picture is 5 rods and 4 cubes.

Count the rods first. The right picture has one more rod. That rod is ten cubes. The left picture has only one more loose cube. Ten cubes against one cube: the right picture has more. So 54 is greater than 45.

Day 2: when the tens match

6 tens 8 ones 68 6 tens 1 one 61
Figure 2The rods match, six and six. The loose cubes decide.

Here both pictures have 6 rods. Pair the rods up, one against one, and none is left over. The tens cannot decide. The loose cubes can: 8 against 1. So 68 is greater than 61. Once the tens match, comparing two-digit numbers is just comparing the ones, which is comparing numbers to ten all over again.

Day 3: the signs

The open side of > or < is wide, and the two strokes meet at the point. The wide end goes next to the bigger amount, and the narrow end next to the smaller one. The sign has the same shape in both directions; only the way it faces changes.

Day 4: both ways

40 50 60 45 54
Figure 3Numbers grow to the right. 45 sits left of 54, so 45 < 54, and 54 sits right of 45, so 54 > 45.

On a The number line, numbers grow to the right. 45 is to the left of 54. Read from 45: “45 is less than 54”. Read from 54: “54 is greater than 45”. These are two sentences about one picture. In both, the point of the sign touches 45.

Results

Tens first. Of two numbers below 100, the one with more tens is greater, whatever their ones are. 54 > 45 because 5 tens is more than 4 tens.

Then ones. If the tens are the same, the one with more ones is greater. 68 > 61.

Equal. If the tens and the ones are both the same, the numbers are equal.

Exactly one. For any two numbers, exactly one of these is true: the first is less than the second, the two are equal, or the first is greater than the second.

Both ways. “First is less than second” and “second is greater than first” are the same fact. 45 < 54 and 54 > 45 say one thing.

Past one hundred. One hundred is ten tens. A number from 100 to 120 has ten, eleven or twelve tens, so it is greater than any number below 100, which has at most nine tens.

Why it is true

The ones place holds a digit from 0 to 9. So the most ones a number can show is 9. Then a number with 4 tens is at most 4 tens and 9 ones, which is 49. Add one more cube and the ten loose cubes snap into a fifth rod: 50. So every number with 4 tens is less than 50, and every number with 5 tens is 50 or more. The same holds for any two tens digits: a number with fewer tens always stops short of the next full ten. That is why the tens decide whenever they differ, no matter what the ones say.

When the tens are the same, both numbers are built from the same full rods. Match those rods one to one and they cancel out. All that is left to compare is the loose cubes.

Reading both ways is not a new fact. Lining up two amounts shows one of them falling short. Saying “the short one is less” or “the long one is greater” describes the same line-up from its two ends.

Worked examples

Example 1

Day 1: forty-five cubes or fifty-four?

Edvin’s block tower has 45 cubes. Alvar’s block tower has 54 cubes. Whose tower has more cubes? (The picture is the first figure in the lesson.)

Solution

45 is 4 tens and 5 ones. 54 is 5 tens and 4 ones. The two numbers use the same digits, but the digits are in different places, so they are worth different amounts.

Compare the tens first. Alvar has 5 tens and Edvin has 4, so Alvar has a whole extra ten cubes. Edvin has one extra loose cube (5 ones against 4). One cube cannot make up for ten.

So 54 is greater than 45. Alvar’s tower has more cubes.

Check: 45 is less than 50, since it has not reached 5 tens. 54 is more than 50. A number below 50 is less than a number above it.

Example 2

Day 1: lunchboxes on two carts

The lunch helpers stack lunchboxes in tens. The first cart has 3 stacks of ten and 8 more. The second cart has 4 stacks of ten and 2 more. Which cart holds more lunchboxes?

Solution

The first cart holds 3 tens and 8 ones, which is 38. The second holds 4 tens and 2 ones, which is 42.

Compare the tens first: 4 tens is more than 3 tens. The first cart has more loose lunchboxes, 8 against 2, but that cannot help it. Even with 9 loose lunchboxes, 3 tens and 9 ones is 39, still one short of 40, and the second cart has passed 40.

So 42 is greater than 38. The second cart holds more lunchboxes.

Example 3

Day 2: two tow trucks

Mathilde drives a tow truck. This year she towed 68 cars. Alvar drives another tow truck and towed 61 cars. Who towed more cars?

Solution

68 is 6 tens and 8 ones. 61 is 6 tens and 1 one. The tens are the same, so they do not decide. In the second picture of the lesson, the six rods on each side pair up with none left over.

Now compare the ones: 8 is more than 1. So 68 is greater than 61. Mathilde towed more cars.

Example 4

Day 2: two ways to build the same number

Mathilde has 4 rods and 13 loose cubes. Edvin has 5 rods and 3 loose cubes, which is 53. Who has more cubes?

Solution

It looks as if Edvin wins on tens, 5 against 4. But “tens first” works only when each person has at most 9 loose cubes, and Mathilde has 13. Ten of her loose cubes can be snapped into one more rod.

After snapping, Mathilde has 4 + 1 = 5 rods and 13 - 10 = 3 loose cubes. That is 5 tens and 3 ones, which is 53.

Now both have 5 tens and 3 ones. The tens match and the ones match, so the two amounts are equal. Neither has more: 53 = 53.

Example 5

Day 3: writing it down

Alvar’s tower has 39 cubes. Mathilde’s tower has 93 cubes. Compare the numbers, write the comparison with a sign, and read it aloud.

Solution

39 is 3 tens and 9 ones. 93 is 9 tens and 3 ones. Tens first: 3 tens is fewer than 9 tens, so 39 is the less number.

Write the two numbers in the order the question gives them, 39 then 93. The sign goes between them with its point at the less number, 39, and its open side at the greater number, 93:

39 < 93

Read it from the left: “thirty-nine is less than ninety-three”.

Notice that 39 shows nine ones, the most a number can show. It is still one short of 4 tens, and a long way short of 9 tens.

Example 6

Day 3: fill in the sign

Write >, < or = in each box.

(a) 72 \;\square\; 27 (b) 90 \;\square\; 9 \text{ tens} (c) 15 \;\square\; 51

Solution

(a) 72 has 7 tens and 27 has 2 tens. 72 is greater, so the open side faces 72: 72 > 27, “seventy-two is greater than twenty-seven”.

(b) 9 tens is 90, since nine rods of ten are ninety cubes. Both sides name the same amount: 90 = 9 \text{ tens}, “ninety is equal to nine tens”.

(c) 15 has 1 ten and 51 has 5 tens. 15 is less, so the point faces 15: 15 < 51, “fifteen is less than fifty-one”.

Example 7

Day 4: the same fact, read both ways

Write the comparison of Edvin’s 45 cubes and Alvar’s 54 cubes in two ways, once starting with 45 and once starting with 54.

Solution

Starting with 45: 45 is the less number, so

45 < 54

“forty-five is less than fifty-four”.

Starting with 54: 54 is the greater number, so

54 > 45

“fifty-four is greater than forty-five”.

The number line in the Intuition shows why both are right. 45 is to the left of 54, and numbers to the left are less. 54 is to the right of 45, and numbers to the right are greater. In both sentences the point of the sign touches 45, the less number, wherever 45 sits in the sentence.

Example 8

Day 4: past one hundred

10 tens 4 ones 104 9 tens 8 ones 98
Figure 4One hundred is ten tens, so 104 has ten rods and 98 has nine.

The school kitchen packed 104 lunchboxes on Monday and 98 on Tuesday. On which day did it pack more? Write the comparison both ways.

Solution

One hundred is ten tens, so 104 is 10 tens and 4 ones. 98 is 9 tens and 8 ones. Tens first: 10 tens is more than 9 tens. The kitchen packed more on Monday.

Both ways:

104 > 98 \qquad 98 < 104

Check by counting on from 98: 99, 100, 101, 102, 103, 104. The count reaches 104 after 98, and a number that comes later in the count is greater.

Boundaries

Tens first needs ones from 0 to 9. The tens decide only when no one has ten or more loose ones. With 4 tens and 13 ones, snap ten ones into a ten before comparing, as in the example with Mathilde’s cubes.

A comparison is about how many, not how big. 30 > 12 says thirty things are more things than twelve. It says nothing about which things are bigger: 12 tow trucks are much heavier than 30 toy cubes. Comparing two towers by their cube counts tells which is taller only when every cube is the same size.

Two numbers at a time. Each sign sits between two numbers. To put three or more numbers in order, compare them two at a time; Comparing and ordering whole numbers does this.

A sentence with a sign can be false. 45 > 54 is a sentence that can be read aloud, “forty-five is greater than fifty-four”, and it is false. Writing a sign does not make it true; the numbers do.

Whole numbers. Tens first, then ones, and “more tens means greater” all hold for whole numbers written in tens and ones. Numbers with a decimal point follow a different rule about length, taken up in Comparing and ordering decimals.

Where this goes next

Knowing which number is greater leads to the next question: greater by how much? That is Subtraction as the difference, where the gap between two amounts is found by subtracting.

The same “biggest place first” idea compares numbers with hundreds and thousands in Comparing and ordering whole numbers: look at the hundreds first, then the tens, then the ones.

The signs < and > come back in algebra, where one side holds an unknown and the question is which numbers make the sentence true. Inequalities on the number line draws those numbers on a number line.

Exercises

Day 1: tens first

A B C
Figure 5Pictures for Exercises 1 to 3.

Exercises 1 to 3 use the pictures. Write the number each picture shows, then say which number in the row is greater.

  1. Row A.
  2. Row B.
  3. Row C.
  4. Edvin’s block tower has 63 cubes. Alvar’s has 36 cubes. Whose tower has more cubes?
  5. Which number is less, 47 or 74?
  6. One shelf holds 29 lunchboxes. Another shelf holds 31 lunchboxes. Which shelf holds more?

Day 2: same tens, and equal numbers

  1. Which is greater, 84 or 87?
  2. Which is greater, 58 or 50?
  3. Which is greater, 9 or 19?
  4. Mathilde builds 6 rods and 2 loose cubes. Edvin counts out 62 cubes. Who has more?
  5. Alvar has 3 rods and 15 loose cubes. Mathilde has 44 cubes. Who has more?
  6. A tow truck towed 43 cars in the spring and 48 cars in the summer. In which season did it tow more cars?

Day 3: the signs

  1. Write >, < or =: 34 \;\square\; 43.
  2. Write >, < or =: 91 \;\square\; 89.
  3. Write >, < or =: 59 \;\square\; 61.
  4. Read aloud: 33 < 38.
  5. Write with a sign: “eighty-one is greater than eighteen”.
  6. Edvin’s lunchbox has 12 crackers. Mathilde’s has 21. Write a comparison of the two numbers with a sign, starting with Edvin’s number.

Day 4: both ways, and up to 120

  1. Write 56 > 49 the other way round, starting with 49.
  2. Compare 110 and 101. Write the comparison both ways.
  3. Which is greater, 99 or 102?
  4. Write >, < or =: 120 \;\square\; 119.
  5. List every number with 7 tens that is less than 73.

Explain

  1. Is a number with more ones always the greater number? Give an example to show your answer.
  2. Explain why any number with 8 tens is greater than any number with 7 tens.
Answers
  1. 36 and 29; 36 is greater. 3 tens against 2 tens; the 9 loose cubes in 29 cannot make up a whole ten.
  2. 52 and 25; 52 is greater. 5 tens against 2 tens.
  3. 80 and 46; 80 is greater. 8 tens against 4 tens; 80 has no loose cubes, and it does not need any.
  4. Edvin’s. 63 is 6 tens and 3 ones; 36 is 3 tens and 6 ones. 6 tens is more, so 63 is greater.
  5. 47. 4 tens is fewer than 7 tens.
  6. The shelf with 31. 3 tens beats 2 tens, even though 29 has more ones; 29 is one short of 30.
  7. 87. Both have 8 tens, so compare the ones: 7 is more than 4.
  8. 58. Both have 5 tens; 8 ones is more than 0 ones.
  9. 19. 9 has no tens and 19 has 1 ten; one ten is already more than 9 ones.
  10. Neither; they are equal. 6 rods and 2 cubes is 62, so 62 = 62.
  11. Alvar. Snap ten of his 15 loose cubes into a rod: 4 rods and 5 cubes, which is 45. 45 and 44 have the same tens, and 5 ones is more than 4, so 45 > 44.
  12. The summer. Both numbers have 4 tens; 8 ones is more than 3, so 48 is greater.
  13. 34 < 43. 3 tens is fewer than 4 tens, so the point faces 34.
  14. 91 > 89. 9 tens is more than 8 tens, so the open side faces 91.
  15. 59 < 61. 5 tens is fewer than 6 tens. 59 has more ones, but 5 tens and 9 ones is still short of 60.
  16. “Thirty-three is less than thirty-eight.” Both have 3 tens and 3 ones is fewer than 8, so the sentence is true.
  17. 81 > 18. The open side faces 81, the greater number.
  18. 12 < 21. 1 ten is fewer than 2 tens, so Edvin’s number is the less one, and the point faces it.
  19. 49 < 56. Same fact; the point still faces 49.
  20. 110 > 101 and 101 < 110. 110 is 11 tens; 101 is 10 tens and 1 one. 11 tens is more.
  21. 102. 102 has 10 tens; 99 has only 9 tens. Counting on from 99 gives 100, 101, 102, so 102 comes later.
  22. 120 > 119. 120 is 12 tens; 119 is 11 tens and 9 ones, one short of 12 tens.
  23. 70, 71 and 72. With 7 tens fixed, the ones must be fewer than 3: 0, 1 or 2.
  24. No. 29 has 9 ones and 32 has 2 ones, but 29 < 32, because 3 tens is more than 2 tens. More ones decides only when the tens are the same.
  25. A number with 7 tens has at most 9 ones, so it is at most 79. Ten more ones than 70 would make an eighth ten, so every number with 7 tens is less than 80. Every number with 8 tens is 80 or more. So any number with 8 tens is greater.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0