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Grade 1 · 1.NBT.A.1

Placing a number on a number line

Counting tells you how many. How do you see where a number sits among all the others, and which of two numbers comes later?

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Placing a number on a number line
Number lines with different scales

When you reach for it

A child who can count and read and write numbers to 120 knows the numbers as a list said out loud: one comes after another. A number line turns that list into a place. Each number gets its own spot on a straight line, and the order of the list becomes left to right. Once the numbers are laid out like that, a child can see things that counting only says: that 14 is a little past 10, that 85 sits halfway between 80 and 90, and that 17 is further along than 12.

A good way to think of it is a ruler for numbers. A ruler has marks the same distance apart, and each mark has a number. A number line is the same: the marks are evenly spaced, and each mark is one more than the mark to its left. This lesson is about the three things that follow from that:

  • a number is a point, one mark on the line, not a space between marks;
  • a line does not need every mark labeled, and it does not need to start at 0, because the missing labels can be found by counting marks from a label that is there;
  • a number further to the right is the greater number.

The number line comes back again and again: for adding, for measuring, for lines that count by twos, fives and tens, and much later for fractions. Every one of those uses rests on placing a single number correctly, which is what this lesson does.

Read it with the child, and build the lines on the floor, on paper strips and on a toy car track wherever you can. A line a child has walked along is understood in a way a printed one is not.

The procedure

The lesson takes three days, one part a day.

Part 1: a ruler for numbers

Make the first number line on the classroom floor with masking tape. Lay a long strip of tape in a straight line. Near the left end, put a short piece of tape across it and write 0 on it. Then take something that does not change size, a shoe, a strip of cardboard, a book, and lay it along the tape starting at the 0 mark. Where it ends, put the next mark and write 1. Move the strip so it starts at 1 and mark 2 where it ends. Keep going to 10, or to 20 if the room allows.

Building it this way shows the first rule without anyone stating it: the spaces are equal, because every space was made with the same strip. That is what makes it a ruler. If one space were long and the next short, the line would no longer tell the truth about how far apart the numbers are.

The second rule comes from the counting: each mark is one more than the mark to its left. Walking right, the child says the counting words in order, 0, 1, 2, 3, one per mark, exactly as in Counting on and counting back. Walking left, they count back: each mark is one less than the mark to its right.

The third rule is the one that matters most for everything that comes later: a number is a point. The number 6 is the mark with 6 on it. It is not the space after it or the space before it. The spaces are the steps between numbers. Have a child stand on 0 and step from mark to mark: the first step lands on 1, the second on 2, and the sixth step lands on 6. So the number on a mark also says how many equal steps it is from 0. That is why a number line can later be used to measure.

An arrow at the right end says the line keeps going: there are always more numbers than any line can show. The tape line starts at 0 and has no arrow on the left, because no whole number comes before 0.

To place a number on a line with every mark labeled, find the mark with that number on it. To place it on a line where only some marks are labeled, start at a labeled mark and count marks to the right, one more for each mark, until you reach the number. Example 2 shows that you may also start at a labeled mark further right and count back.

Part 2: lines with few labels, and lines that do not start at 0

A drawn line can only show a piece of the numbers. A line from 0 to 120 with a mark for every number would be far too long for a page, so a book shows the piece that is needed: from 80 to 90, say, or from 100 to 110. Think of it as a photograph of part of a very long tape. The marks to the left of 80 are still there, just outside the picture, so a line like this gets an arrow on both ends.

Such a line usually has only a few labels, often just its two ends. The rules from Part 1 still fill in the rest. If the left end says 80, the next mark is one more, 81, then 82, and so on. On a piece of line from 80 to 90 there are 11 marks: the two labeled ends and the nine numbers 81 to 89 between them.

Two things help a child here.

The tens stay put and the ones count up. Between 80 and 90 every number is “eighty something”, and the something is how many marks it is past 80. So 82 is two marks past 80 and 86 is six. This is the same pattern a child sees going across one row of a hundred chart. It holds past 100 too: between 100 and 110, the number 104 is four marks past 100.

The middle mark of a piece that runs from one ten to the next is the five. On a line from 80 to 90 it is 85: five marks from each end. Finding the middle first gives a second landmark, so a number like 87 can be found from 90 (three marks back) or from 85 (two marks on) as well as from 80.

The same counting works in the other direction: to say which number a dot names, count marks from the nearest label. A dot three marks right of 40 is 43.

Part 3: further right is greater

Lay a toy car race track on the floor and mark it like a number line, with a start line at 0 and a mark for every number up to 20. Let two cars roll and stop. The car that went further stops on the greater number.

The reason is the second rule from Part 1. Every step to the right is one more. To get from 12 to 17 you keep going right, counting 13, 14, 15, 16, 17, so 17 comes later in the count, and a number that comes later in the count is greater. Every number to the right of a mark is greater than it, and every number to the left is less. So to say which of two numbers is greater, place both on a line and look: the one further right is greater.

This turns ordering into looking. To put several numbers in order from least to greatest, place them all on one line and read the dots from left to right. Later the symbols > and < write the same facts down (Comparing numbers and the symbols >, < and =).

Worked examples

Example 1

Six steps on the tape

This example belongs to Part 1. Ebbe stands on the mark for 0 on the masking tape number line and steps from mark to mark, one step for each space, until he stands on 6. How many steps does he take? How many marks has he stood on, counting the one he started on?

Solution
0 1 2 3 4 5 6 7 8 9 10 6 steps
Figure 1The number 6 is a mark. Getting there from 0 takes 6 spaces.

Every step crosses exactly one space and lands on the next mark, and each mark is one more than the last. The first step lands on 1, the second on 2, and so on, so the step that lands on 6 is the sixth step. Ebbe takes 6 steps.

He has stood on the marks 0, 1, 2, 3, 4, 5, 6. That is 7 marks, one more than the steps, because he was already standing on 0 before he took any step. The number 6 counts the spaces, the steps from 0; it does not count the marks. This is what it means for 6 to be a point: it is the place 6 steps from 0.

Example 2

Fourteen on a line labeled only at 0, 10 and 20

This example belongs to Part 1. A number line runs from 0 to 20 with a mark for every number, but only 0, 10 and 20 are written. Where does 14 go?

Solution

The marks between the labels have no numbers on them, but each mark is one more than the mark to its left, so they can be counted. The labeled mark nearest 14 on the left is 10. Start there and count marks to the right: 11, 12, 13, 14. Fourteen is the fourth mark after 10.

0 10 20
Figure 2Only 0, 10 and 20 are written. The dot is 14.

Check it from the other side. Start at 20 and count back, one less for each mark: 19, 18, 17, 16, 15, 14. That is the sixth mark before 20. It is the same mark, because there are 10 spaces between 10 and 20, and 4 of them come before 14 while the other 6 come after it. Two counts that land on the same mark are a good sign the mark is right.

Example 3

Eighty-five on a line from 80 to 90

This example belongs to Part 2. A piece of number line has 11 evenly spaced marks. The left end is labeled 80 and the right end 90. Place 85.

Solution

The line does not start at 0; it is the piece of a long line between 80 and 90, so the rules still hold. Each mark after 80 is one more: 81, 82, 83, 84, 85. So 85 is the fifth mark after 80.

80 90
Figure 3A piece of the line from 80 to 90. The middle mark is 85.

Count from the other end too: 89, 88, 87, 86, 85 is five marks back from 90. Five marks from each end, with 10 spaces in all, puts 85 exactly in the middle of the line. That is worth remembering: on any piece of line from one ten to the next, the five sits in the middle.

Example 4

One hundred three on a line from 100 to 110

This example belongs to Part 2. A piece of number line has 11 evenly spaced marks, with 100 at the left end and 110 at the right end. Find 103.

Solution

Counting past 100 works just like counting past any other ten: 101, 102, 103. So 103 is the third mark after 100.

100 110
Figure 4Past 100 the line goes on the same way. The dot is 103.

Read the number aloud as the line says it: “one hundred three” is three more than one hundred. Thirteen belongs on a line near 10, far to the left of this piece, and 113 would be off the right end, three marks past 110.

Example 5

Reading two dots

This example belongs to Part 2. On the line in the figure, only 40 and 50 are labeled. Which numbers do the dots A and B name?

40 50 A B
Figure 5Which numbers do the dots name?
Solution

Count from the nearest label each time.

Dot A is near the left end. Counting marks to the right from 40: 41, 42, 43. It sits on the third mark, so A is 43.

Dot B is on the mark just before 50. A mark to the left is one less, and one less than 50 is 49. So B is 49. Counting all the way from 40 gives the same answer, nine marks to 49, but counting from the nearer label is shorter and leaves less room for a slip.

Example 6

Two cars on the race track

This example belongs to Part 3. Kjeld and Lisbet roll toy cars along the race track, which is marked like a number line from 0 to 20. Kjeld’s car stops at 12 and Lisbet’s at 17. Whose car went further, and which number is greater?

0 5 10 15 20 Kjeld Lisbet
Figure 6Two toy cars on the track. The one further right went further.
Solution

Both cars started at 0 and rolled to the right. Lisbet’s car stopped at 17, which is to the right of 12, so her car went further.

The track answers the second question too. Going right from 12, the marks are 13, 14, 15, 16, 17: each one more than the last. So 17 comes later in the count than 12, and 17 is the greater number. The car that went further is the car on the greater number, because on a number line further right always means more.

Example 7

Putting three numbers in order

This example belongs to Part 3. Put 110, 99 and 101 in order from least to greatest.

Solution

Place all three on one line. A piece of line from 95 to 115 holds them, with 100 and 110 labeled.

100 110 A B C
Figure 7Left to right is smallest to largest.

The number 99 is one less than 100, so it is the mark just to the left of 100 (dot A). The number 101 is one more than 100, the mark just to the right (dot B). The number 110 is the labeled mark (dot C). Reading the dots from left to right gives the order from least to greatest:

99, \quad 101, \quad 110

Notice that 99 and 101 are close together on the line, only two spaces apart, even though one has two digits and the other three. Crossing 100 is just one more step.

Boundaries

The spaces must be equal and each mark one more. Everything in this lesson, counting the unlabeled marks, finding the middle, reading further right as more, depends on it. A line drawn with uneven spaces, or with marks that jump by different amounts, cannot be read this way. Lines whose marks step by 2, 5 or 10 are still evenly spaced, but each mark is then more than one greater than the last; those are read in Number lines with different scales.

A line needs at least one label. If no mark has a number, nothing says where the counting starts. With one label and marks one apart, every other mark can be found (Exercise 13).

Whole numbers sit only on the marks. On a line whose marks are one apart, the points between two marks are not whole numbers. They are used later for fractions (Locating a fraction on the number line). Here a number goes on a mark, never between two.

Nothing comes before 0 here. A line that starts at 0 has no whole numbers to its left. A line that does extend left of 0 holds numbers below zero, which are a later idea (Placing and comparing integers on a number line).

A picture is only a piece. A drawn line shows the numbers it has room for. A number that is not on the piece, like 30 on a line from 80 to 90, still has a place, only off the edge.

Where this goes next

Placing one number is the first step to moving along the line. A child who can find 8 can hop on from it to add, and a child who can find 34 can jump on to ten more. Seeing that 85 is “eighty and five more” is the start of Tens and ones. Looking at which dot is further right becomes writing > and < and, with bigger numbers, Comparing and ordering whole numbers. The idea that a number is a point a certain number of equal steps from 0 is the idea behind every ruler, and so behind measuring length. Lines with marks every two, five or ten come next, in Number lines with different scales, and the spaces between the marks wait for fractions.

Exercises

Part 1: a ruler for numbers

  1. On the masking tape line, with every mark from 0 to 10 labeled, Kjeld stands on 0 and steps from mark to mark until he stands on 8. How many steps does he take?
  2. Lisbet makes a tape line from 0 to 20 with a mark for every number. How many marks does she put down? How many spaces are there between them?
  3. A line from 0 to 20 has a mark for every number, but only 0, 10 and 20 are labeled. Where does 7 go?
  4. On the same line, where does 18 go?
  5. On the same line, where does 11 go?
  6. On the same line, which number is on the mark exactly halfway between 10 and 20?

Part 2: lines with few labels, and lines that do not start at 0

  1. A line has 11 evenly spaced marks, labeled only 80 at the left end and 90 at the right end. Where does 83 go?
  2. On the same line, where does 88 go?
  3. A line has 11 evenly spaced marks, labeled only 100 and 110 at the ends. Where does 107 go?
  4. A line has 11 evenly spaced marks, labeled only 110 and 120 at the ends. Where does 119 go?
30 40 A 100 110 B 70 P Q C
Figure 8Number lines for Exercises 11 to 13.
  1. Which number does the dot on line A name?
  2. Which number does the dot on line B name?
  3. On line C, only 70 is labeled. Which numbers do the dots P and Q name?
  4. A line has every mark labeled, but some labels have rubbed off: 96, 97, \square, \square, 100, \square. Fill in the missing labels.
  5. On a line from 110 to 120 with only the ends labeled, which mark is 115?

Part 3: further right is greater

  1. Which is further right on a number line, 46 or 64?
  2. Which is further right, 98 or 102?
  3. Which is greater, 117 or 107? Say how a number line shows it.
  4. On the race track, Ebbe’s car stops at 15, Lisbet’s at 19 and Kjeld’s at 9. Put the cars in order from the one that went least far to the one that went furthest.
  5. Put 112, 102 and 120 in order from least to greatest.
  6. A number sits on a line from 80 to 90. It is further right than 87 and further left than 90. Which numbers could it be?

Reasoning

  1. Why must the spaces on a number line all be the same size? Use the masking tape line to explain.
  2. A number line starts at 80. Where is 0, and why can the line still be read?
  3. Is the number further right on a number line always the greater one? Explain why, using what each step to the right means.
  4. The numbers 102 and 120 use the same digits. Explain why they sit far apart on a number line.
Answers
  1. 8 steps. Each step lands on the next mark, so the eighth step lands on 8.
  2. 21 marks and 20 spaces. The marks are 0 through 20, which is 20 marks after the 0 mark, so 21 in all; there is one space between each pair of neighbors, so 20 spaces.
  3. The seventh mark after 0: count 1, 2, 3, 4, 5, 6, 7. It is also the third mark before 10 (9, 8, 7).
  4. The eighth mark after 10, which is also the second mark before 20 (19, 18).
  5. The first mark after 10, since 11 is one more than 10.
  6. 15. There are 10 spaces from 10 to 20, and the middle mark is five marks from each end: 11, 12, 13, 14, 15.
  7. The third mark after 80: 81, 82, 83.
  8. The second mark before 90: 89, 88. It is also three marks after the middle mark, 85.
  9. The seventh mark after 100, which is also the third mark before 110 (109, 108, 107).
  10. The mark just before 120, since 119 is one less than 120.
  11. 36. The dot is six marks after 30: 31, 32, 33, 34, 35, 36.
  12. 109. The dot is the mark just before 110, and one less than 110 is 109.
  13. P is 74 (four marks right of 70: 71, 72, 73, 74) and Q is 68 (two marks left of 70: 69, 68).
  14. 96, 97, 98, 99, 100, 101. Each mark is one more than the mark before it.
  15. The middle mark, five marks from each end: 111, 112, 113, 114, 115.
  16. 64. Counting goes past 46 long before it reaches 64, so 64 is further right.
  17. 102. The number 98 is two marks before 100 and 102 is two marks after it.
  18. 117 is greater. Both are “one hundred something”; 107 is seven marks past 100 and 117 is seventeen marks past 100, so 117 sits further right.
  19. Kjeld’s car (9), then Ebbe’s (15), then Lisbet’s (19). Reading the dots left to right on the track gives 9, 15, 19; Lisbet’s car went furthest.
  20. 102, 112, 120. On a line, 102 is two marks past 100, 112 is two marks past 110, and 120 is ten marks past 110.
  21. 88 or 89. The marks to the right of 87 and to the left of 90 are exactly those two.
  22. The tape line is built by laying the same strip again and again, so each space is one step of the same size. Then the number on a mark says how many equal steps it is from 0. If one space were longer, the mark for a number would sit further out than it should, and the line would show numbers as further apart or closer together than they are.
  23. The 0 is to the left, off the edge of the picture: the line is a piece cut from a long line, which is why it has an arrow at each end. It can still be read because every mark is one more than the mark to its left, so counting on from 80 names every mark.
  24. Yes. Each step to the right is one more, so moving right means counting on. A number further right is reached by counting on past the other one, so it comes later in the count and is greater.
  25. In 102 the 2 means two ones, so it is two marks past 100. In 120 the 2 means two tens, so it is twenty marks past 100. The same digits in different places name different amounts, and the line shows the difference as distance.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0