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Longer and shorter

Measurement · level 1, lesson 1 of 6

A question to start

Look at these two pencils. Which one is longer?

A B

Many people say "B", because the tip of B sticks out further to the right. But look at the left ends too. Pencil A starts a long way further left. So which is really longer?

Here is the surprise: A is longer. You cannot tell by looking at only one end. In this lesson you will learn the one simple trick that makes comparing lengths fair, and some clever ways to compare things that will not sit side by side.

The idea

Long, short, tall

The length of a thing is how far it goes from one end to the other end. A skipping rope has a long length. A crayon has a short length.

We use a few words for length:

  • Longer and shorter: a train is longer than a car. A crayon is shorter than a pencil.
  • Taller and shorter: we say "taller" when something stands up. A giraffe is taller than a horse.
  • The same length: two new crayons from one box are usually the same length.

"Taller" is just "longer" for things that stand up. If you lay your friend down on the floor (gently!), "taller" turns into "longer".

The fair way: line up the ends

To compare two lengths fairly, you need one rule:

Line up one end of each thing on the same starting line. Then look at the other end.

A B start line A is longer

Now it is easy. Both pencils start at the dashed line. A sticks out further at the other end, so A is longer.

Why does this work? When the starting ends are together, all of each pencil is on the same side of the line. The one that reaches further must have more length in it. When the starts are not together, one pencil gets a "head start", like a runner who starts a race halfway down the track. Reaching further then does not prove anything.

This is why, when you stand back to back with a friend to see who is taller, you both stand on the same floor. Your feet are the starting line. If your friend stood on a step, the test would not be fair.

Putting three things in order

Here are three ribbons, lined up at the start.

orange blue green

From longest to shortest: orange, blue, green. From shortest to longest: green, blue, orange. Putting things in order like this is the first step of measuring. The longest one is "the most", and the shortest one is "the least".

Here is a clever thing. Suppose you know two facts:

  • The orange ribbon is longer than the blue ribbon.
  • The blue ribbon is longer than the green ribbon.

Then you already know the orange ribbon is longer than the green one, without putting them side by side! Why? Orange reaches past blue, and blue reaches past green, so orange must reach past green too. The blue ribbon is in the middle, like a step on a staircase.

When things cannot be put side by side

Some things are too big or too stuck to move. Is the door wider than the table? You cannot carry the door to the table! Here are two ways to solve it.

Use a string. Hold a piece of string across the door, from one side to the other. Pinch the string where the door ends, or tie a knot there. Now carry the string to the table and hold it along the table. If the string is longer than the table, the door is wider than the table.

The string works like a copy of the door's width. A copy that you can carry around!

Use a stick or your arms. You can also use a stick, a strip of paper, or even your stretched-out arms, as long as they do not change while you carry them. (Arms are a bit wobbly, so a string is better.)

Wiggly things

Which is longer: the wiggly snake or the straight stick?

snake stick the snake's string, pulled straight

The snake looks short, because it is all bent up. But if you lay a string along every wiggle and then pull the string straight, you see how long the snake really is. In the picture, the snake's string is longer than the stick. A bendy thing hides its length in its bends.

Worked examples

Example 1. Who is taller? Ava and Ben stand back to back on the floor. A book laid flat on both heads tips down towards Ben. Who is taller?

  1. Both stand on the same floor, so their feet are lined up. This is fair.
  2. The book tips down towards Ben, so Ben's head is lower.
  3. So Ava is taller, and Ben is shorter.

Example 2. Is the bed longer than the rug? You cannot put the bed on the rug. You lay a string along the bed and cut it where the bed ends. Then you put the string on the rug. The string stops before the end of the rug.

  1. The string is exactly as long as the bed. It is a copy of the bed's length.
  2. The string is shorter than the rug.
  3. So the bed is shorter than the rug. The rug is longer.

Example 3. Order four friends. Here are facts about four friends.

  • Cam is taller than Dev.
  • Eli is shorter than Dev.
  • Fay is taller than Cam.

Put them in order from tallest to shortest.

  1. Start with Dev, who is in two facts. Cam is taller than Dev, and Eli is shorter. So far: Cam, Dev, Eli.
  2. Fay is taller than Cam, so Fay goes before Cam.
  3. Order: Fay, Cam, Dev, Eli. Fay is tallest and Eli is shortest.
  4. Check each fact: Cam before Dev, yes. Eli after Dev, yes. Fay before Cam, yes.

Try it

  1. Name something longer than your arm, and something shorter than your finger.
  2. Two straws lie on a table. Their left ends are lined up. The red straw's right end is further away than the blue straw's right end. Which straw is longer?
  3. Sam puts two sticks on the table, but does not line up the ends. He says, "My stick's end is further, so it is longer." Is Sam's test fair? What should he do?
  4. A crayon is shorter than a pencil. The pencil is shorter than a ruler. Is the crayon shorter or longer than the ruler?
  5. How could you find out if the window is wider than your bed, without moving either one?
  6. A long piece of rope is rolled up into a small ball. A short stick lies next to it. Mia says, "The stick is longer, because the ball is tiny." Is Mia right? How could you check?
  7. Order these from shortest to longest: a bus, an ant, a dog, a door.
Answers
  1. Many answers. Longer than your arm: a bed, a car, a skipping rope, a broom. Shorter than your finger: a grain of rice, a pea, an ant, a button.
  2. The red straw. The left ends are lined up, so the starts are fair, and the red one reaches further at the other end.
  3. No, it is not fair. One stick may have a head start. Sam should put the two sticks side by side with one pair of ends lined up on the same line, then look at the other ends.
  4. Shorter. The crayon is shorter than the pencil, and the pencil is shorter than the ruler, so the crayon is shorter than the ruler too. The pencil is in the middle.
  5. Use a string. Hold it across the window and pinch it (or tie a knot) where the window ends. Then hold the string along the bed. If the string is longer than the bed is wide, the window is wider.
  6. Probably not. A rope rolled up hides its length in the ball. Unroll the rope and lay it straight next to the stick, with the ends lined up. A long rope will reach much further than the stick.
  7. Ant, dog, door, bus. (A door is taller than most dogs are long, and a bus is longer than a door is tall.)

Common mistake

Looking at only one end. The thing whose end sticks out furthest is not always the longest. Always line up one pair of ends first. Then the other ends tell the truth.

Thinking a bent or rolled-up thing is short. A curled-up cat, a coiled hose and a wiggly worm are longer than they look. Straighten them out (or lay a string along them and straighten the string) before you compare.

Where it leads