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Number lines with different scales

A number line from 0 to 1,000 with a mark for every number would never fit on a page. How can a line skip most of its marks and still say exactly where every number is?

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Number lines with different scales

When you reach for it

In Placing a number on a number line every mark on the line is one more than the mark to its left. That works well for a line from 80 to 90. It does not work for the minutes a class reads in a month, or for the posts along a trail numbered up to 1{,}000: a mark for every number would make the line far too long, and the marks too crowded to count.

So most number lines in books, on posters and on maps leave marks out. Their marks step by 2, by 5, by 10 or by 100. A child who can count by twos and fives, by tens and by hundreds already has the counting. This lesson turns that counting into reading a line. It covers four things:

  • reading a line when you know its step, and naming a mark that has no label;
  • finding the step when it is not given, from two labeled marks;
  • placing a number that has no mark of its own, between two marks;
  • the rule that makes any of this possible: equal spaces, each standing for the same amount.

Reach for it whenever a line shows only some of its numbers, which is nearly every number line a child will meet from now on. The next lesson, adding on an open number line, draws lines like these by hand to add and subtract within 100.

The procedure

The lesson takes three days, one part a day. Two settings run through it. The first is a reading poster on the classroom wall: a long number line where the class puts a sticker to show how many minutes each child read in a week. The second is a hiking trail with numbered posts set at equal spaces along it, and a trail map that shows them as marks on a line.

Day 1: lines that count by twos, fives, tens and hundreds

Draw a long line on the board with marks the same distance apart. Write 0 under the first mark and 10 under the second. Ask what the third mark should say. Most children will say 20, and the reason is worth saying out loud: the space from the first mark to the second stood for ten, and every space is the same size, so every space stands for ten. The third mark is ten more than 10.

That amount is the step of the line: how much more each mark is than the mark to its left. On the line in Placing a number on a number line the step is 1. On the reading poster it is 10. On a line that counts by twos it is 2. Two rules hold on every such line.

The spaces are equal. Each space is the same length as every other.

Each space stands for the step. Moving one mark to the right adds the step. Moving one mark to the left takes it away.

Together they say that walking along the marks is skip counting. On the poster, walking right from 0 reads 0, 10, 20, 30, the counting by tens a child already knows. Walking left from 100 reads 100, 90, 80.

A line like this usually has only a few labels. To name an unlabeled mark, start at the nearest labeled mark and skip count by the step, one count for each mark you pass, to the right to count up and to the left to count back. The nearest label matters: it keeps the count short.

Two more things carry over from the earlier lesson. A line that starts at 0 gets an arrow only on the right. A line that starts somewhere else is a piece cut from a longer line and gets an arrow on both ends. And the numbers still grow to the right: a mark further right is greater, whatever the step.

Hands first. A long strip of paper folded into equal parts makes a good line: unfold it and the creases are the marks. Write a label on the first crease and on one other, tell the child the step, and let them write in the rest. A hundred chart helps too. Reading down one column of the chart is counting by tens, and that column, laid on its side, is a line whose step is 10.

Day 2: finding the step from two labels

Often nobody says what the step is. The line has a few labels, and the step has to be read from them.

If two neighboring marks are labeled, the step is how much more the right label is than the left one. Marks that say 35 and 40 next to each other: count on from 35 to 40, five ones, so the step is 5.

If the two labels have marks between them, count the spaces between the labels first, not the marks. Then try a step. Start at the left label and skip count by that step, one count for each space. If the last count lands exactly on the right label, the step is right. If it stops short, the step is too small; if it goes past, the step is too big. Try the next one.

Why does the test work? Every space stands for the same amount, so going from one label to the other is the same jump made once for each space. A step that fits must reach the right label in exactly that many jumps. Only one step can do it: a bigger step reaches further in the same number of jumps, and a smaller one less far.

Here is why the spaces must be counted, not the marks. Between labels 0 and 20 with three bare marks there are four spaces. Counting the marks gives three, and three jumps of 5 reach only 15. The step lives in the spaces.

Once the step is known, the bare marks can be named as on Day 1.

For practice at the board, draw two lines with the same two end labels and different numbers of marks between them, and ask whether the middle marks say the same numbers. They do not, and seeing why is the heart of the day: the labels alone do not set the step; the labels and the number of spaces together do.

Day 3: numbers between marks, and honest lines

A line that counts by tens has no mark for 47. But 47 still has a place on it. The line is a picture of all the numbers, and the marks are only signposts. Since 47 is more than 40 and less than 50, its place is in the space between the marks for 40 and 50, further right than 40 and further left than 50.

To say more, ask which mark it is nearer. From 40 up to 47 is 7; from 47 up to 50 is 3. So 47 sits nearer the 50 mark. The middle of the space helps as a second landmark: halfway from 40 to 50 is 45, five from each end, and 47 is a little past that middle. That is as exact as a dot between two marks needs to be: between which two marks, and which one it is nearer. On a line by hundreds, the same thinking puts 250 exactly in the middle of the space from 200 to 300.

The second half of the day is about the first rule from Day 1. A number line is honest when its spaces are equal and each space stands for the same step. Only then does the line tell the truth about how far apart numbers are. Draw the reading poster with its marks bunched together on the left and spread out on the right, and a week of 20 minutes looks as long as a week of 50. The labels are all correct, and the picture still lies. To check a line, look at each space and ask how much it stands for, by finding the difference of the labels at its two ends. Every space must give the same answer.

Worked examples

Example 1

Stickers on the reading poster

This example belongs to Day 1. The reading poster runs from 0 to 100 and counts by tens. Only 0, 50 and 100 are labeled. Makeda’s sticker is at A and Luana’s at B. How many minutes did each of them read?

0 50 100 A B
Figure 1The reading poster counts by tens. Only three marks are labeled.
Solution

The poster counts by tens, so each mark is 10 more than the mark to its left.

Sticker A sits two marks to the right of the label 50, and 50 is the nearest label. Skip count by tens, one count for each mark: 60, 70. Makeda read 70 minutes.

Sticker B sits on the mark just before 100. One mark to the left takes away the step: 10 less than 100 is 90. Luana read 90 minutes.

Check B from the other label: counting by tens from 50 to the right, 60, 70, 80, 90, reaches B in four marks. Two counts from different labels landing on the same number is a good sign.

Example 2

A line that counts by twos

This example belongs to Day 1. The line counts by twos. Only 30 and 40 are labeled. Which numbers are at A and B? Which number is on the mark just left of 40?

30 40 A B
Figure 2A line that counts by twos, cut from the middle of a longer line.
Solution

The step is 2, so each mark is 2 more than the one to its left.

Dot A is three marks right of 30: 32, 34, 36. So A is 36.

Dot B is two marks right of 40: 42, 44. So B is 44.

The mark just left of 40 is 2 less: 38. Notice that the labels 30 and 40 are five spaces apart, 30, 32, 34, 36, 38, 40. On a line by ones they would have had ten spaces between them. A bigger step packs more numbers into each space, which is the whole reason for using one.

Example 3

Finding the step

This example belongs to Day 2. A line is labeled 0 at the left end and 20 at the right end, with three bare marks between them. What is the step? Name the bare marks.

0 20
Figure 3Four equal spaces get from 0 to 20. What does each space stand for?
Solution

Count the spaces between the two labels, not the marks: there are 4 spaces. Going from 0 to 20 is four equal jumps.

Try a step of 2: four jumps from 0 give 2, 4, 6, 8. That stops at 8, far short of 20, so 2 is too small.

Try a step of 5: four jumps give 5, 10, 15, 20. The last jump lands exactly on 20, so the step is 5.

The bare marks are the first three counts: 5, 10 and 15. A step of 10 would be too big, since four jumps of 10 go past 20 to 40.

Example 4

Posts on the trail map

This example belongs to Day 2. Chinonso’s trail map shows the posts along a hiking trail as marks on a line. The posts are evenly spaced, and only two are labeled: post 300 and post 700, with three posts between them and one more after 700. Number the four unlabeled posts.

300 700
Figure 4The trail map shows only two post numbers.
Solution

Between the labels there are 4 spaces. Try counting by hundreds from 300, one count for each space: 400, 500, 600, 700. The fourth count lands on 700, so the step is 100.

The three posts between the labels are 400, 500 and 600. The post after 700 is one more step: 800.

Counting by tens could not be the step: four jumps of 10 from 300 reach only 340. The two labels are far apart, so the step has to be big.

Example 5

The same labels, different steps

This example belongs to Day 2. Both lines in the figure are labeled 60 at the left end and 80 at the right end. What is the step on each line? What number is at the dot on the top line?

60 80 60 80
Figure 5The same two labels. Ten spaces on the top line, two on the bottom.
Solution

Start with the bottom line. It has 2 spaces between the labels. Try tens: 70, 80. Two jumps land on 80, so its step is 10, and its one bare mark is 70.

The top line has 10 spaces between the labels. Tens are far too big: ten jumps of 10 from 60 would reach 160. Fives are too big as well: 65, 70, 75, 80 reaches 80 after only four jumps, with six spaces still to go. Try twos: 62, 64, 66, 68, 70, 72, 74, 76, 78, 80. The tenth jump lands on 80, so the step is 2.

The dot is three marks right of 60, so it is 62, 64, 66: the dot is 66.

The two lines share their labels, and still the bare marks say different numbers. The step comes from the labels and the number of spaces together.

Example 6

Stickers between marks

This example belongs to Day 3. On the reading poster, which counts by tens with every mark labeled, Luana read 47 minutes one week and Chinonso read 83. Where do their stickers go?

Solution

Neither number is a count by tens, so neither has a mark. Each goes between two marks.

The number 47 is more than 40 and less than 50, so its sticker goes in the space between those two marks. From 40 up to 47 is 7, and from 47 up to 50 is 3, so the sticker sits nearer 50. The middle of the space is 45, and 47 is 2 past it: a little to the right of the middle.

The number 83 is between 80 and 90. From 80 up to 83 is 3, and from 83 up to 90 is 7, so this sticker sits nearer 80, to the left of the middle mark 85.

0 10 20 30 40 50 60 70 80 90 100 Luana Chinonso
Figure 6Neither 47 nor 83 has a mark of its own, but both have a place.
Example 7

An honest trail map

This example belongs to Day 3. The trail has a post at every ten, from 0 to 100. Chinonso’s first map showed only some of them. He drew five evenly spaced marks and labeled them 0, 10, 20, 50 and 100 (the top line in the figure). Is the map an honest number line? How should it be drawn?

0 10 20 50 100 0 10 20 30 40 50 60 70 80 90 100
Figure 7Top, the first trail map. Bottom, the honest one.
Solution

Check what each space stands for by finding how much more its right label is than its left one.

SpaceStands for
0 to 1010
10 to 2010
20 to 5030
50 to 10050

The spaces are equal, but they do not stand for the same amount, so the map is not honest. It makes post 50 look as near to post 20 as post 20 is to post 10, when the walk from 20 to 50 passes posts 30 and 40 on the way. The last space hides four posts.

The honest map gives every post its own mark, with the step 10 all the way: eleven marks, 0, 10, 20 and on to 100 (the bottom line). Now each space stands for 10, and a walk that looks twice as long on the map passes twice as many posts.

Boundaries

Equal spaces and one step, or the line cannot be read. Every move in this lesson, naming a bare mark, finding the step, placing a number between marks, depends on both. A line whose spaces are uneven, or whose spaces stand for different amounts, can still carry correct labels, but its bare marks cannot be named and its distances mean nothing.

Two labels, or one label and the step. One label alone does not fix a line: the bare marks could step by anything. A second label, or being told the step, settles it.

The test needs the spaces between the labels. If a picture hides some marks between the two labels, the step cannot be found from it.

A number between marks is placed, not pinned. On a line by tens, a dot for 47 can be shown between 40 and 50 and nearer 50; the picture does not show it more exactly than that. Splitting a space into equal smaller parts is later work, with tenths in Decimals on a number line and with the marks on a ruler or scale in Measuring tools and reading their scales.

Whole numbers, growing to the right. Every line here holds whole numbers, starts at 0 or later, and grows to the right. Lines that reach left of 0 come much later.

Where this goes next

The very next step is the open number line, a line drawn with only the marks a problem needs. A child adds 38 + 25 by jumping two tens to 58 and then five ones to 63, and the jumps are spaces of different sizes, each labeled with what it stands for. That is only possible for someone who knows that a space stands for an amount. Placing 47 between 40 and 50 and nearer 50 is the picture behind Rounding to the nearest ten or hundred. Equal spaces with a step bigger than one return in the scales of graphs, and once the spaces are split into equal parts, in Decimals on a number line.

Exercises

Day 1: lines that count by twos, fives, tens and hundreds

  1. A number line counts by fives. One mark says 15. What do the next three marks to the right say?
  2. A number line counts by twos. One mark says 50. What do the two marks to its left say?
  3. A number line counts by tens and is labeled only 0 at the left end and 100 at the right end. Which mark is 60?
  4. A number line counts by hundreds from 0 to 1{,}000. Which number is on the mark just before 1{,}000?
  5. A line counts by fives. Fill in the missing labels: 35, 40, \square, \square, 55, \square.
  6. A line counts by twos. Fill in the missing labels: \square, 74, 76, \square, \square, 82.
10 20 A 200 300 B 50 100 C
Figure 8Number lines for Exercises 7, 8 and 16.
  1. Line A counts by twos. Which number is at the dot?
  2. Line B counts by tens. Which number is at the dot?

Day 2: finding the step from two labels

  1. Two neighboring marks on a line say 46 and 48. What is the step?
  2. A line is labeled 0 and 10, with 5 spaces between the labels. What is the step? Name the bare marks.
  3. A line is labeled 200 and 500, with 3 spaces between the labels. What is the step? Name the bare marks.
  4. A line is labeled 40 and 80, with 4 spaces between the labels. What is the step? Name the bare marks.
  5. A line is labeled 15 and 25, with 2 spaces between the labels. What is the step? Name the bare mark.
  6. A line is labeled 90 and 100, with 5 spaces between the labels. What is the step? Name the bare marks.
  7. On a trail map, post 30 and post 60 are labeled, with 3 spaces between them. What numbers go on the two posts between them, and on the next post after 60?
  8. Line C in the figure does not say its step. Find the step, then the number at the dot.
  9. A line is labeled 0 at the left end and 30 at the right end. What is the step if there are 3 spaces between the labels? If there are 6? If there are 15?

Day 3: numbers between marks, and honest lines

  1. On a line that counts by tens from 0 to 100, between which two marks does 64 go? Which mark is it nearer?
  2. On a line that counts by fives, between which two marks does 38 go? Which mark is it nearer?
  3. On a line that counts by hundreds from 0 to 1{,}000, where does 550 go? Where does 920 go?
  4. On the reading poster, which counts by tens, Makeda read 60 minutes one week and 65 minutes the next. Which sticker sits on a mark? Where does the other one go?
  5. A line has five evenly spaced marks labeled 0, 2, 4, 6 and 10. Is it an honest number line? If not, how could it be fixed?

Reasoning

  1. The posts on a trail map are numbered by fives. Explain why the marks for the posts must be the same distance apart.
  2. Two lines both run from 0 to 100. One has 10 spaces and the other has 20. Does the mark in the middle say the same number on both lines? Explain.
  3. A line counts by tens and has no mark for 36. Does 36 have a place on it? Explain.
Answers
  1. 20, 25, 30. Each mark to the right is 5 more.
  2. 48 and 46. Each mark to the left is 2 less: 48, then 46.
  3. The sixth mark after 0: 10, 20, 30, 40, 50, 60. It is also the fourth mark before 100: 90, 80, 70, 60.
  4. 900. The mark to the left is one step less, and 100 less than 1{,}000 is 900.
  5. 35, 40, 45, 50, 55, 60. The missing labels are 45, 50 and 60.
  6. 72, 74, 76, 78, 80, 82. The missing labels are 72, 78 and 80.
  7. 16. The dot is three marks right of 10: 12, 14, 16. Counting back from 20 agrees: 18, 16.
  8. 270. The dot is seven marks right of 200: 210, 220, 230, 240, 250, 260, 270. From 300 it is three marks back: 290, 280, 270.
  9. The step is 2. Counting on from 46 to 48 takes two ones.
  10. The step is 2: five jumps of 2 from 0 give 2, 4, 6, 8, 10. The bare marks are 2, 4, 6 and 8.
  11. The step is 100: three jumps give 300, 400, 500. The bare marks are 300 and 400.
  12. The step is 10: four jumps give 50, 60, 70, 80. The bare marks are 50, 60 and 70.
  13. The step is 5: two jumps give 20, 25. The bare mark is 20.
  14. The step is 2: five jumps give 92, 94, 96, 98, 100. The bare marks are 92, 94, 96 and 98.
  15. The posts between are 40 and 50, and the next post after 60 is 70. Three jumps of 10 from 30 give 40, 50, 60, so the step is 10.
  16. The step is 10 and the dot is 80. There are 5 spaces from 50 to 100; five jumps of 10 give 60, 70, 80, 90, 100. The dot is three marks right of 50: 60, 70, 80.
  17. 10, 5 and 2. Three jumps of 10 reach 30; six jumps of 5 reach 30; fifteen jumps of 2 reach 30.
  18. Between 60 and 70, nearer 60. From 60 up to 64 is 4, and from 64 up to 70 is 6.
  19. Between 35 and 40, nearer 40. From 35 up to 38 is 3, and from 38 up to 40 is 2.
  20. 550 goes exactly in the middle of the space from 500 to 600, since it is 50 from each end. 920 goes between 900 and 1{,}000, nearer 900: it is 20 past 900 and 80 before 1{,}000.
  21. The 60 sticker sits on a mark. The 65 sticker goes in the middle of the space from 60 to 70, since it is 5 from each end.
  22. No. The first three spaces each stand for 2, but the last, from 6 to 10, stands for 4. Add a mark for 8 between 6 and 10, so that six evenly spaced marks read 0, 2, 4, 6, 8, 10 and every space stands for 2.
  23. Each space stands for the step, 5. If one space were longer than another, the map would show two neighboring posts as further apart than two other neighboring posts, though both pairs are 5 apart in their numbers. Equal spaces are what let the distance on the map say how far apart the numbers are.
  24. Yes, both say 50. On the line with 10 spaces the step is 10, and the middle mark is five marks in: 10, 20, 30, 40, 50. On the line with 20 spaces the step is 5, and the middle mark is ten marks in: 5, 10, 15 and on to 50. The middle of the line is halfway from 0 to 100 however many marks are drawn.
  25. Yes. Every whole number has a place on the line; the marks only label some of them. Since 36 is more than 30 and less than 40, its place is between those two marks, nearer 40 (6 past 30 and 4 before 40).
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0