Measurement of Length

Length tells us how long something is from one end to the other end.

Distance tells us how far one place or object is from another.

When we measure length, we are finding the space from start to finish in a straight line. For example, the length of a pencil is the distance from one tip of the pencil to the other tip.

When we talk about distance, we can think about how far your chair is from the door, or how far your home is from school.

Length and distance both help us answer the question: How long? or How far?

Why learning length and distance is important

We use length and distance every day.

• We measure a book to see if it fits in a bag.
• We compare ribbons to see which one is longer.
• We think about how far we walk to the playground.
• We choose tools, like a ruler, to measure objects.

What length means

Length is the continuous distance from one end of an object to the other end.

An object has two ends. To find its length, we look at one end, then the other end, and measure all the way across without stopping.

Here are some things that have length:

• a crayon
• a pencil
• a book
• a table
• a jump rope

If one object reaches farther than another object, it is longer. If it does not reach as far, it is shorter.

What distance means

Distance is the space between two things or two places.

For example:

• The distance from your desk to the teacher's desk
• The distance from the door to the window
• The distance from one tree to another tree

If two things are very close, the distance is small. If they are far apart, the distance is large.

Comparing length

Before using a ruler, we can compare objects by looking at them or by lining them up carefully.

To compare two objects:

1. Put one end of both objects at the same starting point.
2. See which object goes farther.
3. The one that goes farther is longer.

This helps us compare fairly.

For example, if a marker and a crayon start at the same line, and the marker reaches farther, then the marker is longer than the crayon.

Non-standard units

Sometimes young learners measure with non-standard units. These are units that are not rulers or meter sticks.

Some examples are:

• paper clips
• cubes
• hand spans
• steps

If you measure a book with cubes, you place the cubes end to end with no gaps and no overlaps.

If the book is 5 cubes long, we can write:

\(\text{Length of book} = 5\text{ cubes}\)

When measuring distance with steps, one child may get a different answer from another child because their steps are different sizes. That is why standard tools are helpful too.

Standard tools for measuring length

A ruler is a standard tool for measuring small objects. It helps us measure carefully using equal spaces called units.

For 2nd Grade, you may use units like centimeters or inches.

• Use a ruler for a pencil, book, or eraser.
• Use a longer tool, like a measuring tape or meter stick, for a desk, door, or room.

When using a ruler:

1. Put the 0 mark at one end of the object.
2. Keep the ruler straight.
3. Look at where the other end stops.
4. Read the number.

If the end of the object stops at 8, then its length is \(8\) units.

Estimating length and distance

To estimate means to make a smart guess before measuring.

We can estimate by thinking about things we already know.

For example:

• A paper clip is short.
• A pencil is longer than a paper clip.
• A table is much longer than a pencil.

You might estimate that a book is about 6 cubes long, and then measure to check.

Estimating helps us think before we measure.

Choosing the right tool

Different objects need different tools.

• For a small object like a crayon, use a ruler.
• For a large object like a table, use a measuring tape or meter stick.
• For fun practice, you can use cubes or paper clips.

Choosing the right tool makes measuring easier and more correct.

Important measuring rules

• Start at the beginning, or 0.
• Measure from one end to the other end.
• Keep the measuring tool straight.
• Do not leave gaps.
• Do not let units overlap.
• Line up objects carefully when comparing.

Worked Example 1: Comparing two objects

A pencil and an eraser are placed with one end lined up together. The pencil reaches farther than the eraser.

Question: Which object is longer?

Answer: The pencil is longer.

Why? When both objects start at the same place, the pencil goes farther from end to end.

Worked Example 2: Measuring with cubes

A book is measured with cubes. The cubes are placed end to end with no gaps. There are 4 cubes.

Question: What is the length of the book?

Answer:

$$\text{Length of book} = 4\text{ cubes}$$

Why? The book covers 4 cubes from one end to the other end.

Worked Example 3: Measuring with a ruler

A crayon starts at 0 on a ruler and ends at 7.

Question: How long is the crayon?

Answer:

$$\text{Length of crayon} = 7\text{ units}$$

Why? The crayon begins at 0 and stops at 7, so its full length is 7 units.

Worked Example 4: Thinking about distance

The ball is near the box. The tree is far from the box.

Question: Which is a greater distance from the box: the ball or the tree?

Answer: The tree is a greater distance from the box.

Why? The tree is farther away from the box than the ball is.

Let's remember

• Length is how long an object is from end to end.
• Distance is how far apart two things are.
• We can compare objects to see which is longer or shorter.
• We can measure with non-standard units like cubes or paper clips.
• We can measure with standard tools like rulers.
• We should always start at 0 and measure carefully.

Brief summary

Length is the distance from one end of an object to the other end. Distance is the space between two objects or places. We can compare, estimate, and measure length and distance using tools like cubes, paper clips, and rulers. Careful measuring helps us find correct answers.

Measuring with Non-Standard Units

Sometimes we measure things without a ruler. We can use small objects that are all the same size, like paperclips, cubes, craft sticks, or erasers. These are called non-standard units.

When we measure with non-standard units, we find out how long something is by placing the units end-to-end. This means one unit goes right after the other in a line.

This lesson will help you learn how to measure carefully, count correctly, and compare lengths using non-standard units.

What is a non-standard unit?

A non-standard unit is a measuring object that is not a ruler, inch, or centimeter. It is something simple we can use again and again to measure length.

• paperclips
• connecting cubes
• small blocks
• craft sticks

If all the units are the same size, we can use them to measure fairly.

How to measure with non-standard units

1. Choose one kind of unit.
2. Make sure all the units are the same size.
3. Place the first unit at the end of the object.
4. Put the next unit right after the first one.
5. Do not leave gaps.
6. Do not let the units overlap.
7. Count the units to find the length.

When we measure, the units should line up like this:

correct: unit + unit + unit, all touching but not covering each other

not correct: units with spaces between them

not correct: units stacked on top of each other

Why no gaps or overlaps?

If there are gaps, the measurement can look longer than it really is. If units overlap, the measurement can look shorter than it really is.

We want a measurement that is careful and fair.

Use the same unit each time

If you measure a book with cubes, use cubes all the way across. If you start with cubes and then switch to paperclips, your answer may not be correct.

Also, if one paperclip is big and another paperclip is small, the measurement may not be fair. It is best to use units that match.

Worked Example 1

A crayon is measured with connecting cubes. There are 5 cubes placed end-to-end with no gaps or overlaps.

The crayon is 5 cubes long.

We can write:

length of crayon = \(5\) cubes

Worked Example 2

A pencil is measured with paperclips. A student places 6 paperclips, but there is a gap between two of them.

This is not a careful measurement.

To fix it, the student should slide the paperclips together so they touch end-to-end.

If the pencil is still covered by 6 touching paperclips, then the pencil is 6 paperclips long.

If the gap made the line too long, the correct answer might be less. That is why careful placement matters.

Worked Example 3

A book is measured two ways:

• It is \(4\) craft sticks long.
• It is \(8\) small cubes long.

Both can be true because craft sticks and cubes are different sizes.

Bigger units mean you may need fewer units.

Smaller units mean you may need more units.

So, \(4\) craft sticks and \(8\) cubes can both measure the same book.

Worked Example 4

Sam measures a marker with erasers. He lines up 3 erasers end-to-end.

Mia measures the same marker with small cubes. She lines up 7 cubes end-to-end.

Which unit is bigger?

The marker takes only \(3\) erasers but \(7\) cubes. That means the erasers are bigger than the cubes.

Fewer bigger units can measure the same object. More smaller units can measure the same object.

Estimating first

Before measuring, you can make a smart guess. This is called an estimate.

For example, before measuring a notebook with paperclips, you might guess it is about \(7\) paperclips long.

Then you measure to check your guess.

Estimating helps your brain think about size and length.

Comparing lengths

You can use non-standard units to compare objects.

If one ribbon is \(5\) cubes long and another ribbon is \(8\) cubes long, the ribbon that is \(8\) cubes long is longer.

Since \(8 > 5\), the second ribbon is longer.

If two objects are both \(6\) paperclips long, they are the same length using that unit.

Important measuring rules to remember

• Use units that are all the same size.
• Place units end-to-end.
• Make sure there are no gaps.
• Make sure there are no overlaps.
• Count carefully.
• Say the number and the unit, like 9 cubes long.

Let’s think together

If a toy car is \(6\) cubes long, and a stuffed animal is \(10\) cubes long, which is longer?

The stuffed animal is longer because \(10\) is more than \(6\).

If a table is \(12\) paperclips long and a book is \(5\) paperclips long, which is shorter?

The book is shorter because \(5\) is less than \(12\).

Common mistakes

• Leaving spaces between units
• Letting units overlap
• Starting away from the end of the object
• Using different-sized units together
• Forgetting to count one of the units

If you make one of these mistakes, that is okay. You can go back, line up the units carefully, and try again.

Summary

Measuring with non-standard units means using objects like cubes or paperclips to find length. Put the units end-to-end in a straight line. Do not leave gaps, and do not overlap the units.

Use the same size unit all the way across. Then count the units and name the length, like \(6\) cubes long. Bigger units need fewer pieces, and smaller units need more pieces.

Standard Units: Inches and Feet

We measure length to find out how long or tall something is. In the United States, two common standard units for measuring length are inches and feet.

A standard unit is a unit that everyone uses the same way. That means if two people measure the same pencil with a ruler, they should get the same answer.

In this lesson, you will learn:

• what inches and feet are,
• when to use inches or feet,
• how to measure with a ruler or measuring tape,
• and how inches and feet are related.

1. What is an inch?

An inch is a small unit of length. We use inches to measure smaller objects.

Here are some things you might measure in inches:

• a crayon
• a book
• a shoe
• a toy car

On a ruler, inches are marked by numbers. The space from one number to the next number is 1 inch.

2. What is a foot?

A foot is a larger unit of length. We use feet to measure longer or taller objects.

Here are some things you might measure in feet:

• a table
• a bed
• a door
• the length of a room

The short way to write foot is ft. So, 3 feet can be written as 3 ft.

3. How are inches and feet connected?

Inches and feet are connected in an important way:

$$12 \text{ inches} = 1 \text{ foot}$$

This means 1 foot is the same length as 12 inches.

If something is 2 feet long, that is the same as:

$$2 \times 12 = 24 \text{ inches}$$

You do not always need to change feet into inches, but it is helpful to know that 12 inches make 1 foot.

4. When should I use inches or feet?

Use inches for smaller things. Use feet for bigger things.

• If an object is short, use inches.
• If an object is long or tall, use feet.

Examples:

• A pencil is measured in inches.
• A classroom door is measured in feet.
• A notebook is measured in inches.
• A rug is measured in feet.

5. How to measure with a ruler

When you measure with a ruler, follow these steps:

1. Put the zero end of the ruler at the start of the object.
2. Make sure the ruler is straight.
3. Look at where the object ends.
4. Read the number at the end of the object.

If the object ends at 5, the object is 5 inches long.

Important: Start at 0, not at the edge of the ruler if the edge is before 0.

6. How to measure with a measuring tape

A measuring tape is good for measuring longer things, like a desk or a room.

You use it the same way:

1. Start at 0.
2. Stretch the tape straight along the object.
3. Read the number where the object ends.

If you are measuring something very long, feet may be the better unit to use.

7. Worked Examples

Example 1: Measuring a small object

A marker starts at 0 on a ruler and ends at 6. How long is it?

Step 1: Start at 0.

Step 2: Look where the marker ends. It ends at 6.

Answer: The marker is 6 inches long.

Example 2: Choosing the best unit

Would you use inches or feet to measure a desk?

A desk is bigger than a pencil or a book. It is a longer object.

Answer: We would use feet.

Example 3: Using what we know about 12 inches

A board is 1 foot long. How many inches long is it?

We know:

$$1 \text{ foot} = 12 \text{ inches}$$

Answer: The board is 12 inches long.

Example 4: Comparing two lengths

A ribbon is 8 inches long. A stick is 1 foot long. Which one is longer?

First, remember:

$$1 \text{ foot} = 12 \text{ inches}$$

Now compare 8 inches and 12 inches.

Since \(12 > 8\), the stick is longer.

Answer: The 1-foot stick is longer.

8. Tips for measuring correctly

• Always start at 0.
• Keep the ruler or tape straight.
• Use inches for small objects.
• Use feet for larger objects.
• Remember: 12 inches = 1 foot.

9. Let’s think

Which unit would you choose?

• Length of a spoon → inches
• Height of a door → feet
• Length of a paper clip → inches
• Length of a couch → feet

Summary

Inches and feet are standard units for measuring length. Inches are used for smaller objects, and feet are used for larger objects. A ruler can help measure inches, and a measuring tape can help measure feet. Remember: $$12 \text{ inches} = 1 \text{ foot}$$

Standard Units: Centimeters and Meters

We measure length to find out how long something is. In math, we use standard units so everyone measures the same way.

Two important metric units for length are centimeters and meters.

A centimeter is a small unit. We use centimeters to measure smaller objects, like a crayon, a pencil, or a book.

A meter is a bigger unit. We use meters to measure longer objects, like a door, a rug, or the length of a classroom.

We write centimeter as cm and meter as m.

Here is an important fact:

$$1 \text{ meter} = 100 \text{ centimeters}$$

That means a meter is made of 100 little centimeters.

When do we use centimeters?

• Use centimeters for small things.
• Use centimeters when a ruler is a good tool.
• Examples: coin, eraser, spoon, notebook.

When do we use meters?

• Use meters for big or long things.
• Use meters when a meter stick or measuring tape is a good tool.
• Examples: table, bed, door, room.

How to measure with centimeters

1. Put the end of the object at the 0 on the ruler.
2. Keep the ruler straight along the object.
3. Look at the number at the other end.
4. That number tells the length in centimeters.

If the object ends at 8 on the ruler, its length is \(8\) cm.

How to measure with meters

1. Use a meter stick or measuring tape.
2. Start at 0.
3. Measure to the end of the object.
4. Count how many meters long it is.

If something is as long as one meter stick, it is \(1\) m long.

Choosing the best unit

It is important to choose the right unit.

• A pencil is better measured in centimeters.
• A classroom is better measured in meters.

If we use a unit that is too big or too small, measuring can be hard. Good math thinkers choose the unit that makes the most sense.

Estimating length

Sometimes we make a smart guess before we measure. This is called an estimate.

For example:

• A marker might be about \(10\) cm long.
• A door might be about \(2\) m tall.

Estimating helps us decide whether to use centimeters or meters.

Worked Example 1: Measuring a small object

A crayon starts at 0 on a ruler and ends at 9.

The crayon is \(9\) centimeters long.

Answer: \(9\) cm

Worked Example 2: Choosing the correct unit

Should we measure a backpack in centimeters or meters?

A backpack is not as big as a room or as long as a hallway. It is a smaller object.

Answer: Use centimeters.

Worked Example 3: Measuring a larger object

A rug is the same length as 2 meter sticks placed end to end.

So the rug is \(2\) meters long.

Answer: \(2\) m

Worked Example 4: Comparing lengths

Sam measures a book. It is \(25\) cm long.

Lia measures a table. It is \(1\) m long.

Which object is better measured in centimeters? Which is better measured in meters?

• The book is better measured in centimeters.
• The table is better measured in meters.

Helpful reminders

• Start measuring at 0, not at the edge of the ruler if the edge is before 0.
• Keep the ruler or tape straight.
• Use cm for small lengths.
• Use m for big lengths.
• Remember: $$1 \text{ m} = 100 \text{ cm}$$

Let’s think!

• Would you use cm or m to measure your shoe?
• Would you use cm or m to measure your desk?
• Would you use cm or m to measure the classroom wall?

Summary

Centimeters and meters are standard units for measuring length. Centimeters are used for small objects, and meters are used for larger objects. When you measure, start at 0, use the right tool, and choose the unit that makes the most sense.

Selecting Measurement Tools

When we want to measure how long something is, we need to choose the right tool. Some objects are small, some are medium-sized, and some are very long. The tool we pick should match the size of the object.

In this lesson, you will learn how to choose between a ruler, a yardstick, and a measuring tape.

Why does the tool matter?

If you use a tool that is too short, it can be hard to measure the object. You may have to move the tool many times. If you use a tool that is too big for a tiny object, it may be hard to measure carefully. Choosing the right tool makes measuring easier and helps you get a better answer.

Meet the measurement tools

• Ruler: A ruler is good for measuring small objects. You can use it for things like a pencil, a crayon, a book, or a toy car.
• Yardstick: A yardstick is longer than a ruler. It is good for measuring medium or longer objects, such as a desk, a door, or the width of a table.
• Measuring tape: A measuring tape can be long and bendy. It is good for measuring long objects or objects that are not straight across, like a couch, a room, or around a box.

How to choose the right tool

1. Look at the object.
2. Ask, “Is it small, medium, or long?”
3. Think, “Is the object straight, or would a bendy tool help?”
4. Choose the tool that fits the object best.

A simple guide

• Use a ruler for small things you can hold in your hand.
• Use a yardstick for bigger things in the classroom or home.
• Use a measuring tape for very long things or around objects.

Think about size

A ruler is often about 12 inches long. A yardstick is much longer. We can think of a yardstick as about 3 rulers long.

$$3 \text{ rulers} \approx 1 \text{ yardstick}$$

You do not need to remember big number facts for this lesson. Just remember: ruler = smaller, yardstick = longer, and measuring tape = longest or bendy.

Worked Example 1

Object: a pencil

Question: Which tool should you use?

Think: A pencil is a small object.

Answer: Use a ruler.

Why? A ruler is just the right size for measuring a pencil. It helps you measure carefully.

Worked Example 2

Object: a classroom desk

Question: Which tool should you use?

Think: A desk is longer than a pencil or book. A ruler would be too short and would need to be moved many times.

Answer: Use a yardstick.

Why? A yardstick is better for medium or longer objects like desks and tables.

Worked Example 3

Object: the length of a room

Question: Which tool should you use?

Think: A room is very long. A ruler is too short. A measuring tape can stretch across a long distance.

Answer: Use a measuring tape.

Why? A measuring tape is good for long spaces.

Worked Example 4

Object: around a box

Question: Which tool should you use?

Think: You are measuring around the box, not just across it. A bendy tool will help.

Answer: Use a measuring tape.

Why? A measuring tape can wrap around an object.

Tips for choosing tools

• If the object is short, start with a ruler.
• If the object is longer than a ruler, think about a yardstick.
• If the object is very long or you need to measure around it, choose a measuring tape.
• Pick the tool that lets you measure with the fewest moves.

Common mistakes

• Using a ruler for everything: A ruler is helpful, but it is not the best choice for large objects.
• Using a yardstick for tiny objects: A yardstick is big and can make measuring small things harder.
• Forgetting about shape: If the object curves or you measure around it, a measuring tape is best.

Let's practice thinking

Ask yourself these questions:

• Can I hold this object in my hand easily?
• Is it bigger than a book or desk?
• Do I need a tool that bends?

If you answer those questions, you can usually choose the right measurement tool.

Quick check

• A spoon → ruler
• A door → yardstick
• A rug → measuring tape
• A notebook → ruler

Summary

Choosing the right measurement tool is an important skill. Small objects are best measured with a ruler. Longer objects are often best measured with a yardstick. Very long objects, or objects measured around the outside, are best measured with a measuring tape.

When you see an object, think about its size and shape. Then choose the tool that makes measuring easy and careful.

Proper Ruler Usage means using a ruler the right way so your measurement is correct.

When we measure length, we want to find out how long something is. A ruler helps us do that. But if we do not place the ruler correctly, our answer can be too big or too small.

The most important rule is this: start at the zero mark. The zero mark shows where the measuring begins.

Some rulers have a little empty space before the 0. That means the end of the ruler is not where you should start. You should start at the 0 mark, not at the edge of the ruler.

How to use a ruler correctly

1. Put one end of the object at the 0 mark on the ruler.

2. Make sure the object is lined up straight along the ruler.

3. Look at where the other end of the object stops.

4. Read the number at that end.

5. Say the length with the unit, like centimeters or inches.

Things to remember

• Do not start at the edge unless the edge is exactly at 0.

• Do not leave a gap between the object and the 0 mark.

• Do not let the object tilt or slant.

• Always check the number where the object ends.

Think of the ruler like a number line. You begin at 0 and count how far the object goes.

If an object starts at 0 and ends at 5, its length is:

$$5\text{ units}$$

If it starts at 0 and ends at 8, its length is:

$$8\text{ units}$$

Why starting at 0 matters

Let’s say a crayon is placed with its end at 1 instead of 0. If the other end is at 7, you might think the crayon is 7 units long. But that is not correct, because the crayon did not start at 0.

To find the real length, count how far it goes from 1 to 7:

$$7 - 1 = 6$$

So the crayon is really 6 units long.

This is why it is easier and safer to line the object up with 0.

Worked Example 1

A pencil starts at 0 on the ruler and ends at 4. How long is the pencil?

Step 1: Check the starting point. It is at 0.

Step 2: Look at the ending point. It is at 4.

Answer: The pencil is 4 units long.

$$4\text{ units}$$

Worked Example 2

An eraser is lined up with 0 and ends at 9. How long is the eraser?

Step 1: It starts at 0, so the ruler is being used correctly.

Step 2: The eraser ends at 9.

Answer: The eraser is 9 units long.

$$9\text{ units}$$

Worked Example 3

A marker does not start at 0. It starts at 2 and ends at 8. How long is the marker?

Step 1: Notice that it did not start at 0.

Step 2: Find the distance from 2 to 8.

$$8 - 2 = 6$$

Answer: The marker is 6 units long.

It would have been better to move the marker so one end touched 0.

Worked Example 4

Lena measures a book. She places the book at the very edge of the ruler, but the 0 mark is a little after the edge. The other end of the book reaches 7.

Is Lena using the ruler correctly?

No. She should start the book at the 0 mark, not at the edge.

If she starts at the edge when the 0 is not there, her measurement may be wrong.

Good ruler habits

• Check where 0 is before you measure.

• Line up one end of the object with 0.

• Keep the object straight.

• Read the number at the other end.

• Write the unit name.

Let’s think

If a strip of paper starts at 0 and ends at 6, it is 6 units long.

If the same strip starts at 3 and ends at 9, it is still 6 units long, because:

$$9 - 3 = 6$$

But to make measuring easy, we should place it at 0.

Summary

To use a ruler properly, always line up one end of the object with the 0 mark. Keep the object straight and read the number where the other end stops. Do not start at the edge unless the edge is exactly the same place as 0. Starting at 0 helps you measure correctly every time.

Inverse Relationship of Unit Size

Today we will learn an important idea about measuring length.

When we measure the same object with different-sized units, the number of units can change. If we use smaller units, we need more of them. If we use larger units, we need fewer of them.

This is called the inverse relationship of unit size. That is a big name, but the idea is simple:

• Smaller unit → bigger count
• Larger unit → smaller count

We are still measuring the same length. Only the size of the measuring unit changes.

For example, imagine walking across a room. If you take tiny steps, you need many steps. If you take giant steps, you need fewer steps. The room does not change. Your step size changes.

Measuring works the same way.

Main Idea 1: A unit is something we use to measure.

A unit is a piece we use again and again to measure length.

• A paper clip can be a unit.
• A cube can be a unit.
• An inch can be a unit.
• A foot can be a unit.

When we measure, we place the units end to end with no gaps and no overlaps.

Main Idea 2: Smaller units make larger numbers.

Let’s think about a pencil. If we measure it with tiny cubes, it might be 8 cubes long. If we measure the same pencil with longer paper clips, it might be 4 paper clips long.

The pencil did not get longer or shorter. The unit size changed.

Because cubes are smaller, we need more of them. Because paper clips are larger, we need fewer of them.

We can say it like this:

$$\text{same object} \qquad \text{smaller units} \to \text{more units}$$

$$\text{same object} \qquad \text{larger units} \to \text{fewer units}$$

Main Idea 3: The number alone does not tell the whole story.

If someone says, “The book is 6 long,” that does not make sense yet. We need to know 6 what.

Is it 6 cubes? 6 paper clips? 6 inches?

The unit matters. A larger number does not always mean a longer object. Sometimes it just means the unit is smaller.

Worked Example 1

A crayon is measured in two ways:

• It is 5 small cubes long.
• It is 3 large blocks long.

Why is the number 5 bigger than the number 3?

Step 1: Notice that the crayon is the same object both times.

Step 2: Compare the units. Small cubes are smaller than large blocks.

Step 3: Smaller units need a bigger count.

So the crayon is 5 small cubes long and 3 large blocks long because smaller units make larger numbers.

Worked Example 2

A desk is measured with footsteps.

• With small steps, the desk is 7 steps long.
• With big steps, the desk is 4 steps long.

Which steps are smaller?

Think: The same desk took more steps when it was measured with small steps.

Since 7 is more than 4, the 7-step measurement used smaller steps.

Answer: The 7 steps are the smaller steps.

Worked Example 3

A ribbon is measured with two units:

• 2 long sticks
• 8 tiny tiles

Which unit is larger: the stick or the tile?

Step 1: The same ribbon was measured both times.

Step 2: The measurement with fewer units uses the larger unit.

Step 3: Since 2 is fewer than 8, the stick is the larger unit.

Answer: The stick is larger than the tile.

Worked Example 4

Look at these two measurements for the same table:

• 10 paper squares long
• 5 markers long

What can we tell about the paper square and the marker?

Think: The same table needed 10 paper squares but only 5 markers.

If it takes more of one unit, that unit must be smaller.

So the paper square is smaller, and the marker is larger.

We can compare the counts like this: \(10 > 5\).

That means the unit used for 10 must be the smaller one.

How to Remember This Idea

• If the unit is small, the count is big.
• If the unit is big, the count is small.
• The object stays the same length.

You can also remember it with this little rule:

$$\text{small unit} \to \text{more} \qquad \text{big unit} \to \text{fewer}$$

What Good Measuring Looks Like

When measuring length, be careful to:

• Start at one end of the object.
• Put units in a straight line.
• Leave no gaps.
• Do not overlap the units.
• Count each unit one time.

If the units are not lined up correctly, the measurement may be wrong.

Try Thinking About These

1. A book is 9 cubes long and 3 erasers long. Which unit is smaller?
   Answer: The cube is smaller because it takes more cubes.
2. A rug is 4 broom lengths long and 12 shoe lengths long. Which unit is larger?
   Answer: The broom length is larger because it takes fewer broom lengths.
3. A shelf is 6 clips long. Can we know its exact length without knowing the size of the clips?
   Answer: No. We need to know the unit size.

Summary

When you measure the same object with different units, the size of the unit changes the number you get.

Smaller units need more pieces. Larger units need fewer pieces.

So if one measurement has a bigger number, that does not always mean the object is bigger. It may mean the measuring unit is smaller.

Estimating Length means making a smart guess about how long something is before you measure it.

We do not need to know the exact length right away. We can look at an object, think about things we already know, and make a good estimate.

Estimating helps us in real life. We estimate to decide if a book will fit in a backpack, if a ribbon is long enough, or which tool to use to measure something.

What is a benchmark?

A benchmark is something you already know the length of. You can use it to help make a good estimate.

Some helpful personal benchmarks are:

• the width of a finger
• the length of a hand
• the length of a foot
• the length of a small school item, like a crayon

For example, if you know your finger is about 1 centimeter wide, you can imagine finger-widths lined up to help estimate a small object.

If you know your foot is about 1 foot long, you can use that to estimate bigger objects on the floor.

How to estimate length

1. Look carefully at the object.
2. Choose a benchmark you know.
3. Think: How many of those benchmarks would fit along the object?
4. Make your best guess.
5. Measure to check how close your estimate was.

Choosing a good benchmark

Small objects need small benchmarks. Bigger objects need bigger benchmarks.

• Use a finger width for very small things.
• Use a hand length for medium things.
• Use a foot length for larger things.

If the benchmark is too big or too small, your estimate may be harder to make.

Think about the unit

When we estimate length, we also think about the unit we will use.

• Use centimeters for small objects.
• Use inches for small objects too, if that is the unit you use at school.
• Use feet for longer objects, like a rug or a table.

You are making an educated guess, not a wild guess. That means you use what you know to help you.

Worked Example 1

Estimate the length of an eraser.

Suppose the width of your finger is about 1 centimeter. When you look at the eraser, it seems about 4 finger-widths long.

So a good estimate is:

$$4\text{ centimeters}$$

After measuring, maybe the eraser is 5 centimeters long. That means your estimate was close.

Worked Example 2

Estimate the length of a pencil.

Suppose your hand is about 10 centimeters long. The pencil looks a little longer than one hand.

So you might estimate the pencil to be about:

$$12\text{ centimeters}$$

If you measure and find it is 13 centimeters, your estimate was a smart guess.

Worked Example 3

Estimate the length of a book.

Suppose one finger width is about 1 centimeter. The book looks about 18 finger-widths long.

Your estimate is:

$$18\text{ centimeters}$$

You could also think of the book as almost 2 hand lengths if one hand is about 10 centimeters. Then 2 hand lengths would be about \(20\) centimeters. Both estimates are reasonable.

Worked Example 4

Estimate the length of a rug.

Suppose your foot is about 1 foot long. The rug looks about 5 foot-lengths long.

So your estimate is:

$$5\text{ feet}$$

This is a better benchmark than using finger widths because the rug is much bigger.

Comparing estimates

Sometimes you may need to decide which object is longer just by looking and estimating.

Example: A marker looks about 5 inches long. A crayon looks about 3 inches long. Since \(5 > 3\), the marker is longer.

Estimating can help you compare objects quickly.

Tips for better estimating

• Use something familiar as your benchmark.
• Match the benchmark size to the object size.
• Look from one end to the other carefully.
• Practice often and then measure to check.

Things to avoid

• Do not pick a number without thinking.
• Do not use a tiny benchmark for a very large object unless you really need to.
• Do not worry if your estimate is not exact. An estimate is a close guess.

Let’s think together

If a glue stick looks about as long as 2 finger widths, would 20 centimeters be a good estimate?

No. That would be much too long. If one finger width is about 1 centimeter, then 2 finger widths would be about:

$$2\text{ centimeters}$$

A better estimate would be a small number, not a big one.

If a desk looks about 3 foot-lengths long, would 3 feet or 30 feet make more sense?

3 feet makes more sense. A classroom desk is not usually 30 feet long.

Why estimating matters

Estimating helps you become a stronger measurer. It helps you notice size, choose the right tool, and decide if an answer makes sense.

When you estimate first and measure second, you are training your brain to think carefully about length.

Summary

Estimating length means making a smart guess about how long something is. Use a benchmark you know, such as a finger width, hand length, or foot length. Pick a benchmark that matches the size of the object, make your estimate, and then measure to check how close you were.

Comparing Lengths and Finding the Difference

Sometimes we measure two objects and want to know which one is longer. We may also want to know how much longer one object is than another.

In this lesson, you will learn how to compare lengths and find the difference between them. This helps us answer questions like, “Which pencil is longer?” and “How many inches longer is it?”

What does compare mean?

To compare means to look at two things and decide which is longer, shorter, or if they are the same length.

• If one object measures more, it is longer.
• If one object measures less, it is shorter.
• If both objects measure the same, they are equal in length.

Step 1: Measure both objects

First, measure each object using the same unit. You might use inches, centimeters, cubes, paper clips, or another unit your teacher gives you.

It is important to use the same unit for both objects. If you use different units, your comparison will not be fair.

For example, if one ribbon is measured in inches and another ribbon is measured in centimeters, it is harder to compare them right away. Measure both in the same unit first.

Step 2: Compare the numbers

After measuring, look at the two numbers.

• The bigger number tells which object is longer.
• The smaller number tells which object is shorter.

For example, if one crayon is 8 inches long and another is 5 inches long, then 8 is greater than 5. So the 8-inch crayon is longer.

We can write:

\(8 > 5\)

Step 3: Find the difference

To find how much longer one object is than another, subtract the smaller length from the larger length.

Difference means how far apart the two lengths are.

Use this rule:

$$\text{longer length} - \text{shorter length} = \text{difference}$$

If one stick is 9 inches long and another stick is 6 inches long, then:

$$9 - 6 = 3$$

So the 9-inch stick is 3 inches longer.

You can also count on

Another way to find the difference is to count on from the smaller number to the larger number.

For example, compare 4 inches and 7 inches.

Start at 4 and count on: 5, 6, 7.

That is 3 counts, so:

$$7 - 4 = 3$$

So 7 inches is 3 inches longer than 4 inches.

Worked Example 1

A marker is 7 inches long. A crayon is 5 inches long. Which is longer, and by how much?

1. Measure or read the lengths: marker = 7 inches, crayon = 5 inches.
2. Compare: \(7 > 5\), so the marker is longer.
3. Subtract: $$7 - 5 = 2$$

Answer: The marker is 2 inches longer than the crayon.

Worked Example 2

A book is 12 centimeters long. A notebook is 12 centimeters long. Which is longer?

1. Compare the numbers: \(12 = 12\)
2. Both objects have the same length.
3. Subtract: $$12 - 12 = 0$$

Answer: Neither is longer. They are the same length. The difference is 0 centimeters.

Worked Example 3

A ribbon is 14 cubes long. Another ribbon is 9 cubes long. How much longer is the first ribbon?

1. Compare: \(14 > 9\), so the first ribbon is longer.
2. Subtract the smaller number from the larger number:

$$14 - 9 = 5$$

Answer: The first ribbon is 5 cubes longer.

Worked Example 4

A pencil is 11 centimeters long. A paintbrush is 8 centimeters long. How much shorter is the paintbrush?

When we ask how much shorter, we still find the difference.

1. Compare: \(11 > 8\), so the pencil is longer and the paintbrush is shorter.
2. Subtract: $$11 - 8 = 3$$

Answer: The paintbrush is 3 centimeters shorter than the pencil.

Important things to remember

• Measure both objects carefully.
• Use the same unit for both objects.
• Look for the bigger number to find the longer object.
• Subtract to find how much longer or shorter.
• If the numbers are the same, the objects are equal in length.

Watch out for these mistakes

• Using different units: Do not compare 8 inches to 8 cubes unless both were measured in the same way.
• Subtracting in the wrong order: Subtract the smaller length from the larger length when finding the difference.
• Forgetting the unit: Always say inches, centimeters, cubes, or whatever unit you used.

Try thinking like this

If you know the lengths are 10 cm and 6 cm, ask yourself:

• Which number is bigger?
• Which object is longer?
• What is \(10 - 6\)?

Then you can say: The 10 cm object is longer, and it is 4 cm longer.

Summary

To compare lengths, measure two objects using the same unit and look at the numbers. The bigger number shows the longer object. To find how much longer or shorter one object is, subtract the smaller length from the larger length.

Number Lines for Measurement Computation

When we measure length, we are finding out how long something is. A number line can help us think about length because a number line is also a line that shows distance from one number to another.

In this lesson, you will learn how to use a number line to solve measurement problems. You will see that moving along a number line is like measuring a line segment from one point to another.

What is a number line?

A number line is a straight line with numbers placed in order. The spaces between the numbers are equal. This is important because equal spaces help us measure correctly.

When we use a number line for measurement, we can think of:

• the start point as where the object begins,
• the end point as where the object stops,
• and the distance between them as the length.

If an object starts at 0 and ends at 5, its length is 5 units.

$$5 - 0 = 5$$

Why number lines help with measurement

A ruler is like a number line. It starts at 0 and goes up by equal steps. When we measure with a ruler, we look at where the object starts and where it ends. We can do the same thing on a number line.

Number lines help us:

• see length as a space, not just a number,
• find how much longer one line is than another,
• add lengths together,
• and find a missing length.

How to find length on a number line

To find the length of a line segment on a number line, follow these steps:

1. Look at the number where the segment starts.
2. Look at the number where the segment ends.
3. Count the spaces, or subtract the smaller number from the bigger number.

The rule is:

$$\text{length} = \text{end} - \text{start}$$

For example, if a segment starts at 2 and ends at 7, then:

$$7 - 2 = 5$$

So the segment is 5 units long.

Counting on with a number line

Sometimes subtraction can be thought of as counting on. If a segment starts at 3 and ends at 8, you can count from 3 up to 8:

3 to 4 is 1,
4 to 5 is 2,
5 to 6 is 3,
6 to 7 is 4,
7 to 8 is 5.

So the length is 5 units.

Adding lengths on a number line

You can also use a number line to put two lengths together. This means adding.

If one ribbon is 4 units long and another ribbon is 3 units long, put them together:

$$4 + 3 = 7$$

On a number line, start at 0, jump 4 units, then jump 3 more units. You land on 7. The total length is 7 units.

Finding a missing length

Sometimes you know the whole length and one part, but one part is missing. A number line can help you find the missing part.

For example, a rope is 9 units long. One piece is 4 units long. How long is the other piece?

$$9 - 4 = 5$$

The missing piece is 5 units long.

You can also think: “What do I add to 4 to get 9?”

$$4 + 5 = 9$$

Worked Example 1: Measuring from 0

A crayon is shown on a number line. It starts at 0 and ends at 6. How long is the crayon?

Step 1: Find the start point: 0

Step 2: Find the end point: 6

Step 3: Subtract:

$$6 - 0 = 6$$

Answer: The crayon is 6 units long.

Worked Example 2: Measuring when the start is not 0

A pencil starts at 2 and ends at 9 on a number line. How long is the pencil?

Step 1: Start point = 2

Step 2: End point = 9

Step 3: Find the difference:

$$9 - 2 = 7$$

Answer: The pencil is 7 units long.

You could also count the spaces from 2 to 9. There are 7 spaces.

Worked Example 3: Adding lengths

A string has two pieces. One piece is 5 units long. The other piece is 2 units long. How long are they together?

Step 1: Add the lengths:

$$5 + 2 = 7$$

Step 2: Think on a number line. Start at 0, jump to 5, then jump 2 more to 7.

Answer: Together, the string is 7 units long.

Worked Example 4: Finding a missing part

A line segment is 10 units long. One part is 6 units long. How long is the missing part?

Step 1: Whole length = 10

Step 2: Known part = 6

Step 3: Subtract:

$$10 - 6 = 4$$

Answer: The missing part is 4 units long.

Things to remember

• Length is the distance between two points.
• On a number line, count the spaces, not just the numbers.
• If the segment starts at 0, the end number tells the length.
• If the segment starts somewhere else, subtract: $$\text{end} - \text{start}$$
• You can use number lines to add lengths and find missing lengths.

Watch out for this common mistake

Some students count the numbers instead of the spaces. For example, from 2 to 5, the numbers are 2, 3, 4, 5. That is 4 numbers, but the length is only 3 spaces.

$$5 - 2 = 3$$

So always remember: measure the spaces between numbers.

Try thinking like this

• Where does the segment start?
• Where does it end?
• How many spaces are in between?
• Do I need to add or subtract?

Summary

A number line helps us measure length because it shows distance. We can find the length of a line segment by looking at its start and end points and finding the difference. We can also use number lines to add lengths together or find a missing part. Remember to count spaces, because spaces show the length.