Length on a number line
A ruler counts length from zero. If two things stand somewhere along a long line, how far apart are they, and how do lengths that are joined or cut show up on that line?
Where you are 7 in Length and units
Why this exists
Lehua and Saltan are walking on a boardwalk by the beach. Every meter along it there is a small marker post with a number on it, starting from 0 at the steps. Lehua stops by one marker and Saltan stops by another, farther along. Nobody has a tape measure. How far apart are they?
A child who can measure with a ruler knows how to read a length when one end sits at 0. Here neither end is at 0. The markers are a ruler laid along the ground, but the two children stand in the middle of it. The answer is in the two numbers: the distance between two points on a line is the difference of their numbers, the larger take away the smaller.
That one fact turns a The number line into a tool for length problems. Two pieces of string laid end to end become two stretches of the line, one after the other, and their total is a sum. A piece cut off the end leaves a stretch that ends at a difference. Two lengths compared side by side differ by the stretch between their ends. A story problem about length can be drawn on the line, and the drawing shows whether to add or to subtract.
The lesson runs over three days. Day 1 is the number line as a ruler, and the distance between two points (Examples 1 and 2). Day 2 is putting lengths together, cutting a piece off, and comparing (Examples 3 to 5). Day 3 is length story problems with a box for the unknown (Examples 6 and 7). Every length here is a whole number of units, and every number is 100 or less.
Definition
Day 1: the number line as a ruler
A number line is a straight line with marks spaced evenly along it, one mark for each whole number: 0 first, then 1, 2, 3 and so on to the right. Each gap between two neighboring marks is one unit long, and every unit on the line is the same length.
A whole number on the line names a point, and it also names a length: the number 7 is the point 7 units to the right of 0. Laying a ruler with its 0 at the start of the line makes the two the same thing.
The distance between two points on the line is the length of the stretch of line between them, counted in units. It is found by counting the spaces between the two points, not the marks.
Day 2: lengths together, apart and compared
A length can be drawn on the line as a shaded strip from one number to another. Three ways of working with two lengths are drawn this way.
- End to end: the second strip starts exactly where the first one stops, with no gap and no overlap. The two strips together make one longer strip.
- Cut off: a strip is taken off one end of a longer strip. The rest is a shorter strip.
- Compared: two strips start at the same point, one under the other. The part of the longer strip that sticks out past the shorter one is how much longer it is.
Day 3: a length story
A length story gives some lengths and asks for one more. The unknown length is written as a box, \square, in a number sentence such as \square + 38 = 91. The box can stand for the whole length, for one of the parts, or for where a stretch starts.
Intuition
Day 1: a ruler laid flat
Put a crayon on a ruler with one end at the 0 mark. The other end is at 7, so the crayon is 7 units long. Now draw a number line under the ruler with the same spacing. The crayon’s length is the stretch of the line from 0 to 7.
Nothing about the ruler is special except that it is stiff and has numbers on it. A number line is a ruler that goes on as far as you like, drawn on paper. That is why a number on the line can be read two ways: as a place (“the point 7”) and as a length (“7 units from 0”).
Now move the crayon so that one end sits at 3. The other end is at 10. The crayon has not changed, and its length is still the number of spaces it covers: from 3 to 10 that is 7 spaces. The numbers at its ends have both gone up by 3, and their difference is still 7. (Why a ruler can be read from any starting mark is the work of Measuring length; here the same fact is used on a long line that does not end.)
Count spaces, not marks. From 3 to 10 there are 8 marks, 3, 4, 5, 6, 7, 8, 9, 10, but only 7 spaces between them. A child who walks a finger along the line can say “one” as the finger leaves 3 and lands on 4, and so on: each step is one space, and the count stops when the finger lands on 10. The first mark is where the finger starts, so it is not counted.
Day 2: strips on the line
Children can act this out on a floor number line made from tape, or with paper strips cut to length and laid along a meter stick. Two strips end to end reach as far as their two lengths together. Cut a piece off the end of a strip, and the end of what remains moves back by the length of the piece. Put two strips under each other, both starting at the same mark, and the longer one sticks out past the shorter; the bit that sticks out is the difference.
Each picture matches a number sentence. End to end is adding. Cutting off is taking away. Comparing asks for the stretch between two ends, which is the distance from Day 1, so it is a difference too.
Day 3: drawing the story
A length story is easier to think about once it is drawn. Put the known lengths on the line as strips, put a box where the unknown is, and look at the drawing: if the unknown is a whole strip made of known parts, add; if it is a part, or the gap between two ends, subtract. The drawing does the deciding, so no key word in the story has to. This is the same thinking as bar models, with the strips sitting on a scale.
Results
Days 1 to 3
For whole numbers a and b with b the larger,
| Two lengths | On the line | Number sentence |
|---|---|---|
| end to end | a strip of a from 0, then a strip of b | ends at a + b |
| cut off | a strip of a, with b taken from its end | ends at a - b |
| compared | strips of a and b from the same point | longer by a - b |
The distance does not depend on the order of the two points: from a to b is the same stretch of line as from b to a.
Why it is true
Take two points, say 23 and 29. The stretch from 0 to 29 is made of two stretches laid end to end: from 0 to 23, which is 23 units, and from 23 to 29, which is the distance we want. So
The distance is the number that goes with 23 to make 29, and that is the difference 29 - 23 = 6. Nothing here used the numbers 23 and 29 themselves, so the same is true for any two points: the distance is the larger number take away the smaller one. This is subtraction as a missing part, drawn as length.
The rule about spaces comes from the same picture. Each space is one unit long, and the stretch is made of its spaces laid end to end, so its length is the number of spaces. The marks are the ends of the spaces. Going from the first point to the second, each space ends at a new mark, so there is one mark per space plus the mark you started on. That is why the marks always number one more than the spaces.
The three kinds of Day 2 come straight from this. End to end, the second strip starts at a and is b long, so it ends at the point b units past a, which is a + b. Cutting b off the end of a strip from 0 to a leaves a strip that ends b units before a, at a - b. Compared from the same start, the two strips end at a and b, and the bit that sticks out is the stretch between those two ends, whose length is their difference.
Worked examples
Day 1: Markers on the boardwalk
Saltan stands by the marker labeled 23 on the boardwalk. Lehua stands by the marker labeled 29. The markers are one meter apart. How far apart are they?
The markers are a number line in meters, so the distance is the number of one meter spaces between the two children. Walk a finger from 23 to 29 and count each space as the finger crosses it: 23 to 24 is one, 24 to 25 is two, and so on until 28 to 29, which is six. The strips in the picture are those six spaces. So they are 6 meters apart.
The difference gives the same answer at once:
Counting the marks instead would give 23, 24, 25, 26, 27, 28, 29, which is 7 marks. That is one too many, because the mark at 23 is where Saltan stands, not a space he has to cross.
Day 1: A race on the track
A running track has a mark every meter along its straight, starting from 0 at the start line. During a race, Yordanka is at the 28 meter mark and Saltan is at the 65 meter mark. How far ahead of Yordanka is Saltan?
Both runners are on the same number line, so the distance between them is the difference of their two numbers:
Saltan is 37 meters ahead. Counting 37 single spaces would be slow and easy to slip on, which is why the difference is the tool here. It can be found by any way the child knows: count up from 28 to 30 (two), from 30 to 60 (thirty), from 60 to 65 (five), and 2 + 30 + 5 = 37; or subtract by place value. Reading the stretch as a sum of easy pieces is taught in Adding on an open number line.
Check the other way: 28 + 37 = 65. Starting at Yordanka’s mark and going 37 meters lands on Saltan’s mark.
Day 2: Two jumps end to end
At a jump contest, each child does a standing jump from a line, then a second standing jump from where they landed. Lehua’s first jump is 44 inches and her second is 37 inches. How far from the starting line does she end up?
The starting line is 0. The first jump is a strip from 0 to 44. The second jump starts where the first one landed, so its strip starts at 44 and is 37 inches long. The two strips are end to end, and the question asks where the second one stops, which is the whole length:
Lehua ends up 81 inches from the starting line. In the picture the dashed strip on top runs from 0 to the end of the second jump, and it is exactly as long as the two shaded strips together.
Day 2: A piece cut off
The boardwalk is 85 meters long. The last 27 meters, at the far end, are closed for repairs. How much of the boardwalk is open? At which marker does the open part end?
The whole boardwalk is the strip from 0 to 85. The closed part is a piece 27 meters long cut off its far end. What is left is the open part, which starts at 0, so it ends at the point 27 meters before 85:
The open part is 58 meters long, and it ends at marker 58. Here the length and the marker have the same number because the open part starts at 0, the same reason a ruler read from its 0 mark gives the length directly.
Check by putting the pieces back: 58 + 27 = 85, the whole boardwalk.
Day 2: How much longer?
In the jump contest Yordanka’s single jump is 47 inches and Saltan’s is 39 inches. How much longer is Yordanka’s jump?
Both jumps start at the same line, so draw both strips from 0, one under the other. Yordanka’s ends at 47 and Saltan’s ends at 39. Up to 39 the two jumps cover the same ground; the part of Yordanka’s jump past 39 is the extra. It is the stretch from 39 to 47, which is the distance between those two points:
Yordanka’s jump is 8 inches longer. Counting the spaces from 39 to 47 gives 8 as well.
The same strip answers a second question: Saltan’s jump is 8 inches shorter. “Longer” and “shorter” are two ways of reading the same stretch, and both are found by subtracting.
Day 3: Where did the walk start?
Lehua walked 38 meters along the boardwalk, toward the far end, and stopped at marker 91. At which marker did she start?
The walk is a strip 38 meters long that ends at 91. The start is unknown, so put a box there. Starting at the box and going 38 meters lands on 91:
On the line, the strip runs backward from 91 by 38 units, so the start is at the difference:
Lehua started at marker 53. Check: from 53, going 38 meters lands on 53 + 38 = 91.
The sentence \square + 38 = 91 has an addition sign, but the drawing shows that the box is the start of a strip whose length and end are known, so the way to find it is to subtract.
Day 3: A jump compared, then two jumps together
In his two jumps end to end, Saltan’s second jump was 9 inches shorter than his first. His second jump was 34 inches. How long was his first jump? How far from the starting line did he end up?
SolutionDraw the two jumps as strips from the same point to compare them. The second jump is the shorter strip, 34 inches. The first is 9 inches longer, so its strip reaches 9 past the end of the second one. The word “shorter” in the story does not decide the sign; the drawing does, and it shows the first jump is the whole longer strip:
The first jump was 43 inches.
Now lay the jumps end to end, as they happened: a strip from 0 to 43, then a strip of 34 starting at 43. They end at
Saltan ended up 77 inches from the starting line. Check the first step: 43 - 34 = 9, so the second jump really was 9 inches shorter.
Boundaries
- Whole units only here. Every point in this lesson is at a whole number mark, so every distance is a whole number of spaces. A point between two marks has a length that is not a whole number of units, which is the work of Measuring tools and reading their scales and, later, fractions.
- A distance is never less than zero. Subtract the smaller number from the larger. The stretch from 65 to 28 is the same stretch as from 28 to 65, and its length is 37, not a “take away” that runs out. (Numbers to the left of 0 come much later, with integers.)
- One unit, the same all along. The rule works because every space is the same length. On a line drawn with uneven spaces, the drawing no longer shows lengths truly: a short gap can stand for the same one unit as a long gap. Two lengths can be added or compared only in the same unit: 44 inches and 37 centimeters cannot be added to get 81 of anything.
- End to end means no gap and no overlap. Two strips laid with a gap between them reach farther than the sum; laid so that they overlap, they reach less. The sum is the total only when the second strip starts exactly where the first one stops.
- A part cannot be longer than the whole. A piece cut off a strip is at most the whole strip. So “a 40 meter piece cut off a 25 meter board” has no answer.
- A straight line. The number line measures length along itself. A path that turns a corner is measured piece by piece and the pieces added, as when Measuring length lays string along a curve; the distance “as the crow flies” between the two ends is a different, shorter length.
Where this goes next
The number line is about to carry a great deal. Adding and subtracting on it, as Adding on an open number line shows, is the same jumping along a line, only with the units left out of the drawing. Marking a line by twos, fives or tens, and reading what each step is worth, is Number lines with different scales. Measured lengths collected from a class, plotted as dots over a number line, make a line plot.
Length story problems grow into Word problems with distance, time, volume, mass and money with every kind of measure, into two-step problems, and, through Customary length units and Metric length units, into problems that mix inches with feet or centimeters with meters. The number line is also how Elapsed time is pictured: a clock time is a point on a line of minutes, and how long something took is the distance between two points.
Much later the same rule, the distance is the larger number take away the smaller, reaches below zero. On a line with negative numbers the distance from -3 to 5 is still the number of spaces between them, which is the idea behind Distance on a number line and absolute value as distance.
Exercises
Day 1: the number line as a ruler
- Look at line A. How far apart are the two dots?
- On the boardwalk the markers are one meter apart. Lehua stands at marker 42 and Yordanka at marker 50. How far apart are they?
- How far apart are the points 17 and 60 on a number line?
- In a 100 meter race, Saltan is at the 35 meter mark. How far does he still have to run to reach the finish at 100?
- A point is 24 units to the right of 50. Which number is it at?
- On a number line, how many marks are there from 30 to 36, counting both ends? How many spaces? How far apart are 30 and 36?
Day 2: lengths together, apart and compared
- Look at line B. Where does the second strip end?
- In the jump contest, Yordanka jumps 26 inches, then jumps 33 inches more from where she landed. How far from the starting line does she end up?
- The boardwalk on another beach is 96 meters long. The first 40 meters are painted blue and the rest is not painted. How long is the part that is not painted?
- Saltan jumps 52 inches and Yordanka jumps 45 inches, both from the same line. How much longer is Saltan’s jump?
- Look at line C. How much longer is the top strip than the strip labeled 41?
- In a relay on the track, Lehua runs from the 0 meter mark to the 45 meter mark, and Saltan runs from there to the 100 meter mark. How far does Saltan run? Who runs farther, and by how much?
Day 3: length stories
- Saltan walked 29 meters along the boardwalk, toward the far end, and stopped at marker 83. At which marker did he start? Write a number sentence with a box.
- Yordanka’s jump was 12 inches longer than Lehua’s. Yordanka jumped 50 inches. How far did Lehua jump?
- Lehua’s jump was 15 inches shorter than Saltan’s. Lehua jumped 33 inches. How far did Saltan jump?
- In a 100 meter race, Saltan still has 18 meters to run. At which meter mark is he?
- On the boardwalk, Yordanka walks from marker 16 to marker 47, then turns around and walks 12 meters back. At which marker is she now? How far did she walk in all?
- The straight part of a track is 80 meters long. Lehua runs 25 meters from the start, then 30 meters more. How far is she from the end of the straight?
- Draw a number line for 63 - \square = 29 and find the number in the box.
Reasoning
- To find how far apart 54 and 59 are, why do we count spaces and not marks?
- Is the distance from 30 to 70 the same as the distance from 70 to 30? How far is it?
- A point is 10 units away from 45. Where could it be? Find both places.
- A strip from 15 to 35 and a strip from 0 to 20 are drawn on the same number line. Are they the same length? How do you know?
- Can two lengths laid end to end make a total that is shorter than one of them? Explain.
Answers
- 9 units. The dots are on 6 and 15, and 15 - 6 = 9. Counting the spaces from 6 to 15 also gives 9.
- 8 meters. Both are on the same line of markers, so the distance is 50 - 42 = 8.
- 43 units. 60 - 17 = 43. For example, count up 3 to 20 and 40 more to 60: 3 + 40 = 43.
- 65 meters. The distance from 35 to 100 is 100 - 35 = 65.
- 74. Going 24 units past 50 lands on 50 + 24 = 74.
- 7 marks and 6 spaces, so 30 and 36 are 6 apart: 36 - 30 = 6. The marks are one more than the spaces because the first mark is a starting place, not a space.
- At 77. The first strip ends at 52, and the second starts there and is 25 long, so it ends at 52 + 25 = 77.
- 59 inches. The jumps are end to end: 26 + 33 = 59.
- 56 meters. The painted part is cut off the start of the whole, so the rest is 96 - 40 = 56. Check: 40 + 56 = 96.
- 7 inches. Both strips start at the line, so the extra is the stretch from 45 to 52: 52 - 45 = 7.
- 27. Both strips start at 0, so the extra is the stretch from 41 to 68: 68 - 41 = 27.
- Saltan runs 100 - 45 = 55 meters. Lehua runs 45 meters, so Saltan runs farther, by 55 - 45 = 10 meters.
- Marker 54. \square + 29 = 83; the walk is a strip of 29 ending at 83, so it starts at 83 - 29 = 54. Check: 54 + 29 = 83.
- 38 inches. Lehua’s jump is the shorter strip, and Yordanka’s reaches 12 past it, so \square + 12 = 50 and Lehua jumped 50 - 12 = 38.
- 48 inches. Saltan’s jump is the longer strip, reaching 15 past Lehua’s, so 33 + 15 = 48.
- At the 82 meter mark. The stretch from where he is to 100 is 18, so he is at 100 - 18 = 82.
- At marker 35, and she walked 43 meters. The walk out is 47 - 16 = 31 meters. Walking back 12 from 47 lands at 47 - 12 = 35. In all she walked 31 + 12 = 43 meters.
- 25 meters. Her two runs are end to end, so she is at 25 + 30 = 55. The distance from 55 to 80 is 80 - 55 = 25.
- \square = 34. Draw a strip from 0 to 63 and cut a piece off its end so the rest ends at 29. The piece cut off is the stretch from 29 to 63, so \square = 63 - 29 = 34. Check: 63 - 34 = 29.
- The distance is how much line lies between the points, and the line is made of spaces, each one unit long. The marks are only the ends of spaces. Counting marks also counts the mark at 54, where you start, so it gives 6, one more than the 5 spaces: 59 - 54 = 5.
- Yes. It is the same stretch of line, walked in the other direction, so it has the same length: 70 - 30 = 40.
- At 35 or at 55. The point can be 10 to the left, 45 - 10 = 35, or 10 to the right, 45 + 10 = 55.
- Yes. Each is 20 units long: 35 - 15 = 20 and 20 - 0 = 20. A length is the number of spaces it covers, and sliding a strip along the line does not change how many spaces it covers.
- No. The second strip starts where the first one stops and adds more line, so the total reaches past the end of each one, unless a strip has length 0. Two strips end to end are always at least as long as each strip alone.