Measurement Systems and Applications

Attributes of Length and Standard Units

Length is a way to describe how long something is or how far apart two points are. When we measure length, we are measuring the distance from one end of an object to the other end, or the space between two places.

For example, you can measure the length of a pencil, the width of a desk, the height of a door, or the distance across a classroom. These are all examples of length.

To measure length correctly, we use standard units. Standard units are measurement units that everyone agrees on. This helps people get the same answer when measuring the same object.

If we did not use standard units, one person might say a table is 8 hand-lengths long, and another person might say it is 10 hand-lengths long. That happens because hands are different sizes. Standard units solve this problem.

Common standard units for length include:

• Inches  used for small objects
• Feet  used for medium-sized objects
• Yards  used for longer lengths
• Miles  used for long distances
• Centimeters  used for small objects
• Meters  used for longer objects or distances
• Kilometers  used for long distances

We choose a unit based on the size of the object or the distance we want to measure.

Here is a helpful way to think about it:

• Use small units for small lengths, like a crayon or a book.
• Use larger units for larger lengths, like a room or a playground.
• Use very large units for distances between places, like from one town to another.

Examples of choosing the best unit:

• The length of an eraser might be measured in inches or centimeters.
• The length of a bed might be measured in feet or meters.
• The length of a soccer field might be measured in yards or meters.
• The distance from one city to another might be measured in miles or kilometers.

Length has different attributes, or ways we can describe it. These include:

• Longer and shorter
• Taller and shorter
• Higher and lower
• Farther and nearer

Even though the words may change, they are all describing length or distance in some way.

For example:

• A tree can be taller than a bush.
• A ribbon can be longer than a shoelace.
• The park can be farther away than the school.

Important idea: Length must be measured from one point to another point. When using a ruler or measuring tape, one end should line up with 0. Then you read the number at the other end.

If you start measuring from the wrong place, your answer may be incorrect.

Worked Example 1: Measuring a small object

A marker is measured with a ruler. One end is lined up at 0, and the other end reaches 6 inches. What is the length of the marker?

Step 1: Check that the marker starts at 0.

Step 2: Read the number at the other end.

The marker is 6 inches long.

We can write this as \(6\) inches.

Worked Example 2: Choosing the best unit

What is the best unit to measure the length of a classroom: inches, feet, or miles?

Think:

• Inches are too small. It would take too many inches.
• Miles are too large. A classroom is much smaller than a mile.
• Feet are a good choice for something the size of a classroom.

Answer: The best unit is feet.

Worked Example 3: Comparing lengths

A jump rope is 8 feet long. A piece of string is 5 feet long. Which is longer, and how much longer is it?

Step 1: Compare the numbers.

Since \(8 > 5\), the jump rope is longer.

Step 2: Find how much longer.

$$8 - 5 = 3$$

Answer: The jump rope is 3 feet longer than the string.

Worked Example 4: Choosing between metric units

Which unit would be better for measuring the distance from your home to a library in another part of town: centimeters, meters, or kilometers?

Think:

• Centimeters are for very small objects.
• Meters are for objects or shorter distances.
• Kilometers are used for longer distances between places.

Answer: The best unit is kilometers.

Tips for measuring length well

• Start at 0 on the ruler or measuring tape.
• Make sure the object is straight while measuring.
• Use the correct unit for the size of the object.
• Check whether the question is asking about a small object, a large object, or a long distance.

How to choose a reasonable unit

1. Look at what you are measuring.
2. Decide if it is small, medium, or very large.
3. Pick a standard unit that matches the size.
4. Measure carefully from one end to the other.

Lets review with quick examples:

• A paper clip  centimeters or inches
• A table  feet or meters
• A football field  yards or meters
• A trip between cities  miles or kilometers

Summary

Length tells how long something is or how far apart two points are. We use standard units like inches, feet, yards, miles, centimeters, meters, and kilometers so measurements are clear and fair. Small objects need small units, and large distances need larger units. When measuring, always start at 0 and read to the other end carefully.

Measuring with Fractional Ruler Units

Sometimes a length is not a whole inch. A pencil might be longer than 4 inches but shorter than 5 inches. A ruler helps us measure these lengths using fractional units, such as halves and quarters of an inch.

In this lesson, you will learn how to use a ruler to measure to the nearest half inch and quarter inch. You will also learn how to line objects up correctly and read the marks between whole numbers.

1. Understanding the inch marks on a ruler

A ruler has numbers that show whole inches: 0, 1, 2, 3, and so on. The space from one whole number to the next is 1 inch.

That 1 inch can be split into equal smaller parts.

• If it is split into 2 equal parts, each part is one-half inch, or \(\frac{1}{2}\) inch.
• If it is split into 4 equal parts, each part is one-quarter inch, or \(\frac{1}{4}\) inch.

So between 0 and 1 inch, the marks can show:

The same pattern happens between every pair of whole numbers on the ruler.

2. What the fraction marks mean

Let us look at the inch between 2 and 3.

• The first quarter mark after 2 is \(2\frac{1}{4}\).
• The middle mark is \(2\frac{1}{2}\).
• The third quarter mark is \(2\frac{3}{4}\).
• Then comes 3.

This means a ruler can show lengths like:

• \(1\frac{1}{2}\) inches
• \(3\frac{1}{4}\) inches
• \(5\frac{3}{4}\) inches

3. Always start at zero

When measuring an object, place one end of the object at the 0 mark on the ruler. Then look at where the other end stops.

This is very important. If you start at the edge of the ruler instead of the 0 mark, your measurement may be wrong. Some rulers have a little space before 0, so always check where the 0 mark is.

• Line up one end of the object with 0.
• Keep the object straight along the ruler.
• Look at the mark where the other end lands.

4. How to read a measurement

Ask yourself these questions:

1. What whole inch is the object past?
2. Does it stop on a half-inch mark or a quarter-inch mark?
3. What mixed number names that length?

For example, if an object goes past 4 inches and stops at the middle mark between 4 and 5, the length is:

If it goes past 6 inches and stops at the first quarter mark, the length is:

5. Half inches and quarter inches

A half inch is bigger than a quarter inch because \(\frac{1}{2}\) is greater than \(\frac{1}{4}\).

Here are the quarter-inch steps from one whole number to the next:

Since \(\frac{2}{4} = \frac{1}{2}\), the middle mark can be read as either:

On a ruler, we usually say half inch instead of two-fourths inch.

6. Worked Examples

Example 1: Measuring a whole number length

A crayon starts at 0 and ends exactly at the 5-inch mark. What is its length?

Step 1: Check that the crayon starts at 0.

Step 2: Look where it ends. It ends at 5.

Answer: The crayon is 5 inches long.

Example 2: Measuring to the nearest half inch

A ribbon starts at 0 and ends at the middle mark between 2 and 3. What is its length?

Step 1: The ribbon is longer than 2 inches.

Step 2: It ends at the middle mark, which means \(\frac{1}{2}\) inch more.

Answer: The ribbon is $$2\frac{1}{2}\text{ inches}$$ long.

Example 3: Measuring to the nearest quarter inch

A paintbrush starts at 0 and ends at the first small mark after 3. What is its length?

Step 1: The paintbrush is longer than 3 inches.

Step 2: The first small mark after 3 is \(\frac{1}{4}\) inch.

Answer: The paintbrush is $$3\frac{1}{4}\text{ inches}$$ long.

Example 4: A harder quarter-inch measurement

A marker starts at 0 and ends at the third quarter mark after 7. What is its length?

Step 1: The marker is longer than 7 inches.

Step 2: The quarter marks after 7 are:

• first mark: \(7\frac{1}{4}\)
• middle mark: \(7\frac{1}{2}\)
• third mark: \(7\frac{3}{4}\)

Answer: The marker is $$7\frac{3}{4}\text{ inches}$$ long.

7. Tips for measuring correctly

• Start at 0, not at the edge of the ruler.
• Place the object straight along the ruler.
• Look carefully at the endpoint.
• Count the quarter marks if needed: \(\frac{1}{4}\), \(\frac{1}{2}\), \(\frac{3}{4}\), next whole inch.
• Write the unit: inches.

8. Common mistakes to avoid

• Starting at the edge instead of 0 — this gives the wrong length.
• Skipping the whole number — for example, saying \(\frac{1}{4}\) instead of \(4\frac{1}{4}\).
• Confusing \(\frac{1}{4}\) and \(\frac{3}{4}\) — count the quarter marks carefully.
• Forgetting the unit — say inches or write in.

9. How to think about quarter marks

Between any two whole numbers, there are 4 equal parts.

If you are between 5 and 6, the marks are:

This pattern works everywhere on the ruler. Once you know the pattern, you can read many measurements.

10. Try it in your head

• Middle mark between 1 and 2 = \(1\frac{1}{2}\)
• First quarter mark after 4 = \(4\frac{1}{4}\)
• Third quarter mark after 8 = \(8\frac{3}{4}\)
• Middle mark between 6 and 7 = \(6\frac{1}{2}\)

Summary

A ruler shows whole inches and smaller parts of inches. When an inch is split into 2 equal parts, each part is \(\frac{1}{2}\) inch. When an inch is split into 4 equal parts, each part is \(\frac{1}{4}\) inch.

To measure correctly, line the object up with the 0 mark, keep it straight, and read where the other end stops. Then name the length using a whole number and, if needed, a fraction such as \(\frac{1}{4}\), \(\frac{1}{2}\), or \(\frac{3}{4}\).

Customary Length Conversions

We measure length to find out how long, tall, or far something is. In 4th grade, you learn how to measure using customary units. Customary units are units we often use in the United States, like inches, feet, and yards.

Sometimes a length is given in one unit, but you need it in another unit. This is called a conversion. In this lesson, you will learn how to convert between feet and inches, and between yards and feet.

The two most important facts to remember are:

• 1 foot = 12 inches
• 1 yard = 3 feet

These facts help you change one unit into another.

When do we multiply?

You multiply when you change from a bigger unit to a smaller unit.

• Feet to inches: multiply by 12
• Yards to feet: multiply by 3

This works because one bigger unit is made of several smaller units. For example, 1 foot is made of 12 inches.

When do we divide?

You divide when you change from a smaller unit to a bigger unit.

• Inches to feet: divide by 12
• Feet to yards: divide by 3

This works because you are grouping the smaller units into larger units.

A helpful way to think about it

If you move from big to small, the number gets bigger. If you move from small to big, the number gets smaller.

• Bigger to smaller: more pieces, so multiply
• Smaller to bigger: fewer groups, so divide

For example, 2 feet is more than 2 inches, so when changing feet to inches, the number should become larger.

Conversion chart

• $$1 \text{ foot} = 12 \text{ inches}$$
• $$1 \text{ yard} = 3 \text{ feet}$$

You can use these facts in equations too:

• $$f \text{ feet} = 12f \text{ inches}$$
• $$y \text{ yards} = 3y \text{ feet}$$

That means if you know the number of feet, you can multiply by 12 to find inches. If you know the number of yards, you can multiply by 3 to find feet.

Worked Example 1: Convert feet to inches

How many inches are in 4 feet?

Since feet are bigger than inches, multiply by 12.

So, 4 feet = 48 inches.

Worked Example 2: Convert inches to feet

How many feet are in 36 inches?

Since inches are smaller than feet, divide by 12.

So, 36 inches = 3 feet.

Worked Example 3: Convert yards to feet

How many feet are in 5 yards?

Since yards are bigger than feet, multiply by 3.

So, 5 yards = 15 feet.

Worked Example 4: Solve a word problem

A jump rope is 3 yards long. How many feet long is it?

We know:

So multiply 3 yards by 3.

The jump rope is 9 feet long.

How to solve conversion problems step by step

1. Read the problem carefully.
2. Circle the unit you have.
3. Underline the unit you need.
4. Ask: Am I going from bigger to smaller, or smaller to bigger?
5. Use the correct fact:
   • $$1 \text{ foot} = 12 \text{ inches}$$
   • $$1 \text{ yard} = 3 \text{ feet}$$
6. Multiply or divide.
7. Write the correct unit in your answer.

Watch out for these common mistakes

• Do not mix up the facts. Remember:
  • 12 goes with feet and inches
  • 3 goes with yards and feet
• Do not forget the unit in your answer.
• Check if your answer makes sense.
  • If you changed feet to inches, your number should get bigger.
  • If you changed inches to feet, your number should get smaller.

Try thinking about these

• If a board is 2 feet long, then it is \(2 \times 12 = 24\) inches long.
• If a ribbon is 18 inches long, then it is \(18 \div 12 = 1\) foot and 6 inches long.
• If a sidewalk is 7 yards long, then it is \(7 \times 3 = 21\) feet long.

Sometimes a conversion does not make a whole number of larger units. For example, 18 inches is more than 1 foot because 1 foot is 12 inches. After using 12 inches, there are 6 inches left over. So 18 inches is 1 foot 6 inches.

Summary

Customary length conversions help you change one length unit into another. The two key facts are 1 foot = 12 inches and 1 yard = 3 feet.

Remember: multiply when changing from a bigger unit to a smaller unit, and divide when changing from a smaller unit to a bigger unit. Always label your answer with the correct unit and check that your answer makes sense.

Metric Length Conversions help us change one metric unit into another. In this lesson, you will learn how to convert between millimeters (mm), centimeters (cm), meters (m), and kilometers (km).

The metric system is helpful because it is based on powers of 10. That means we can convert by multiplying or dividing by 10, 100, or 1,000.

When we move from a bigger unit to a smaller unit, the number gets bigger. When we move from a smaller unit to a bigger unit, the number gets smaller.

Here are the metric length units we will use:

• 1 centimeter = 10 millimeters
• 1 meter = 100 centimeters
• 1 kilometer = 1,000 meters

You can think of the units in order from smallest to largest:

millimeter → centimeter → meter → kilometer

Each time you move to a smaller unit, you multiply. Each time you move to a larger unit, you divide.

Helpful conversion facts:

• To change cm to mm, multiply by 10.
• To change mm to cm, divide by 10.
• To change m to cm, multiply by 100.
• To change cm to m, divide by 100.
• To change km to m, multiply by 1,000.
• To change m to km, divide by 1,000.

Sometimes you need to move across more than one unit. For example, to change meters to millimeters, you can go:

• meters to centimeters: multiply by 100
• centimeters to millimeters: multiply by 10

So meters to millimeters means multiply by:

$$100 \times 10 = 1{,}000$$

That means:

$$1\text{ m} = 1{,}000\text{ mm}$$

In the same way:

$$1\text{ km} = 1{,}000\text{ m}$$

and

$$1\text{ m} = 100\text{ cm}$$

Let’s look at some worked examples.

Example 1: Convert 7 cm to mm

We are changing from centimeters to millimeters. Millimeters are smaller, so we multiply by 10.

$$7 \times 10 = 70$$

So, 7 cm = 70 mm.

Example 2: Convert 300 cm to m

We are changing from centimeters to meters. Meters are bigger, so we divide by 100.

$$300 \div 100 = 3$$

So, 300 cm = 3 m.

Example 3: Convert 5 m to cm

We are changing from meters to centimeters. Centimeters are smaller, so we multiply by 100.

$$5 \times 100 = 500$$

So, 5 m = 500 cm.

Example 4: Convert 2 km to m

We are changing from kilometers to meters. Meters are smaller, so we multiply by 1,000.

$$2 \times 1{,}000 = 2{,}000$$

So, 2 km = 2,000 m.

How to decide what to do

1. Look at the unit you have.
2. Look at the unit you want.
3. Ask: Am I moving to a smaller unit or a bigger unit?
4. If it is a smaller unit, multiply.
5. If it is a bigger unit, divide.

Quick check:

• Is 4 m more or less than 4 cm? It is more, because meters are bigger units.
• If you change 4 m to cm, should the number get bigger or smaller? It should get bigger.
• So, $$4 \times 100 = 400$$ and 4 m = 400 cm.

Common mistake to avoid: Do not multiply when you should divide, or divide when you should multiply.

For example, if you convert 900 mm to cm, do not multiply by 10. Since centimeters are bigger than millimeters, you must divide by 10:

$$900 \div 10 = 90$$

So, 900 mm = 90 cm.

Remember:

• Smaller unit means the number gets bigger.
• Bigger unit means the number gets smaller.
• The metric system uses 10, 100, and 1,000, which makes conversions easier.

Summary

Metric length conversions use millimeters, centimeters, meters, and kilometers. Use the basic facts you know: 1 cm = 10 mm, 1 m = 100 cm, and 1 km = 1,000 m. Multiply when changing to a smaller unit, and divide when changing to a bigger unit.

Perimeter is the total distance around the outside of a shape.

If you trace your finger all the way around the edge of a shape, the distance you travel is the perimeter.

Perimeter tells us how long the boundary is. We use it for shapes like squares, rectangles, triangles, and other closed shapes.

We measure perimeter in units of length, such as centimeters (cm), meters (m), inches (in), or feet (ft).

Important: Perimeter is about the outside edge, not the space inside. The space inside a shape is called area, and that is a different idea.

How to find perimeter

To find the perimeter of any closed shape, add the lengths of all its sides.

$$\text{Perimeter} = \text{sum of all side lengths}$$

You can think: Add around the shape.

Steps for finding perimeter

1. Look at the shape carefully.
2. Find the length of each side.
3. Add all the side lengths together.
4. Write the correct unit in your answer.

Perimeter of common shapes

Some shapes have sides that are the same length. This can help you add more quickly.

• Square: all 4 sides are equal, so perimeter is 4 times one side.
• Rectangle: opposite sides are equal, so add length + width + length + width.
• Triangle: add all 3 sides.
• Any polygon: add every side once.

For a square with side length \(s\):

$$P = 4 \times s$$

For a rectangle with length \(l\) and width \(w\):

$$P = l + w + l + w$$

This is the same as:

$$P = 2l + 2w$$

Worked Example 1: Find the perimeter of a square

A square has side length \(5\) cm. What is its perimeter?

All 4 sides are the same, so add \(5 + 5 + 5 + 5\).

$$P = 5 + 5 + 5 + 5 = 20$$

The perimeter is 20 cm.

Worked Example 2: Find the perimeter of a rectangle

A rectangle has length \(7\) m and width \(3\) m. What is its perimeter?

A rectangle has 2 long sides and 2 short sides.

$$P = 7 + 3 + 7 + 3$$

$$P = 20$$

The perimeter is 20 m.

You could also group equal sides:

$$P = 2 \times 7 + 2 \times 3 = 14 + 6 = 20$$

Worked Example 3: Find the perimeter of a triangle

A triangle has side lengths \(4\) in, \(6\) in, and \(5\) in. What is its perimeter?

Add all 3 sides:

$$P = 4 + 6 + 5 = 15$$

The perimeter is 15 in.

Worked Example 4: Find the perimeter of an irregular shape

An irregular shape has side lengths \(3\) cm, \(2\) cm, \(4\) cm, \(5\) cm, and \(6\) cm. What is its perimeter?

Even if the shape is not a square or rectangle, the rule is the same: add every outside side.

$$P = 3 + 2 + 4 + 5 + 6 = 20$$

The perimeter is 20 cm.

When some side lengths are missing

Sometimes a shape shows only some side lengths. You may need to use what you know about the shape.

For example, in a rectangle, opposite sides are equal. If one long side is \(8\) ft, the other long side is also \(8\) ft. If one short side is \(2\) ft, the other short side is also \(2\) ft.

Then the perimeter is:

$$P = 8 + 2 + 8 + 2 = 20\text{ ft}$$

Be careful about units

Always include the unit in your answer.

• Length might be in cm, m, in, or ft.
• Perimeter uses the same unit because it measures distance around the shape.

For example:

• Not just \(18\)
• Write 18 cm

Common mistakes to avoid

• Forgetting a side: Make sure you add every outside side once.
• Mixing up perimeter and area: Perimeter is around the outside.
• Leaving off the unit: Always write cm, m, in, ft, or another length unit.
• Adding inside lines: Only count the outer edge of the shape.

Helpful strategy

If a shape looks tricky, trace around it with your finger or pencil. Say each side length as you go, and then add them.

You can also make a list of the side lengths before adding.

For example, if the sides are \(6\), \(4\), \(6\), and \(4\), write:

$$6 + 4 + 6 + 4 = 20$$

Let’s practice thinking

• If a square has side length \(9\) cm, its perimeter is \(9 + 9 + 9 + 9 = 36\) cm.
• If a rectangle has length \(10\) m and width \(2\) m, its perimeter is \(10 + 2 + 10 + 2 = 24\) m.
• If a pentagon has sides \(3\) ft, \(3\) ft, \(4\) ft, \(2\) ft, and \(5\) ft, its perimeter is \(17\) ft.

Summary

Perimeter is the distance around the outside of a closed shape.

To find perimeter, add all the side lengths.

Squares, rectangles, triangles, and other polygons all follow the same big rule: add the outside edges.

Remember to include the correct length unit in your final answer.

Calculating Perimeter of Polygons

Have you ever wanted to know the distance all the way around a shape? That distance is called the perimeter.

A polygon is a closed shape made of straight sides. Some polygons are triangles, rectangles, squares, pentagons, and other many-sided shapes.

When we calculate perimeter, we add the lengths of all the outside sides of the polygon.

We can think of perimeter as the length of a fence around a yard or the border around a picture frame.

Main Idea:

To find perimeter, use this rule:

$$\text{Perimeter} = \text{sum of all side lengths}$$

If a shape has side lengths of 3 cm, 5 cm, 3 cm, and 5 cm, then its perimeter is:

$$3 + 5 + 3 + 5 = 16\text{ cm}$$

Important things to remember:

• Add every outside side one time.
• Make sure all side lengths use the same unit, like centimeters, inches, or meters.
• Write the unit in your answer.
• Perimeter tells us the distance around a shape, not the space inside it.

How to Find the Perimeter

1. Look at the polygon.
2. Find the length of each outside side.
3. Add the side lengths together.
4. Write the total with the correct unit.

Sometimes every side length is given. Sometimes one or more side lengths are not written, but you can figure them out from the shape.

For example, in a rectangle, opposite sides are equal. That means the top and bottom are the same length, and the left and right sides are the same length.

In a square, all 4 sides are equal. If one side is 6 cm, then all sides are 6 cm.

Worked Example 1: Perimeter of a Triangle

A triangle has side lengths 4 cm, 6 cm, and 5 cm. Find the perimeter.

Add all 3 sides:

$$4 + 6 + 5 = 15$$

So, the perimeter is 15 cm.

Worked Example 2: Perimeter of a Rectangle

A rectangle has a length of 8 in and a width of 3 in. Find the perimeter.

A rectangle has 2 long sides and 2 short sides.

So the sides are 8 in, 3 in, 8 in, and 3 in.

Add them:

$$8 + 3 + 8 + 3 = 22$$

So, the perimeter is 22 in.

You can also group equal sides:

$$8 + 8 + 3 + 3 = 16 + 6 = 22\text{ in}$$

Worked Example 3: Perimeter of a Square

A square has one side length of 7 m. Find the perimeter.

All 4 sides of a square are equal, so each side is 7 m.

Add all 4 sides:

$$7 + 7 + 7 + 7 = 28$$

So, the perimeter is 28 m.

You may also think of it as 4 groups of 7:

$$4 \times 7 = 28\text{ m}$$

Worked Example 4: Missing Side Lengths

A rectangle has a top side of 9 cm and a side of 4 cm. The bottom and the other side are not labeled. Find the perimeter.

Because it is a rectangle, opposite sides are equal.

• Top = 9 cm, so bottom = 9 cm
• One side = 4 cm, so the other side = 4 cm

Now add all sides:

$$9 + 4 + 9 + 4 = 26$$

So, the perimeter is 26 cm.

Tips for Success

• Trace your finger around the shape so you do not forget any side.
• If sides are equal, write the missing lengths before you add.
• Check that you added only the outside edges.
• Check your unit at the end.

Common Mistakes to Avoid

• Forgetting a side: Be sure to count every outside side.
• Using the wrong unit: If the sides are in centimeters, the answer should be in centimeters.
• Mixing up area and perimeter: Perimeter is around the shape, not inside it.
• Not using equal sides: In rectangles and squares, some sides have the same length.

Let's Review

• Perimeter means the distance around a polygon.
• To find perimeter, add all the outside side lengths.
• Rectangles have opposite sides equal.
• Squares have 4 equal sides.
• Always write the correct unit in your answer.

Brief Summary

Perimeter is the total distance around a polygon. You find it by adding the lengths of all the outside sides. If some sides are equal, like in rectangles and squares, use that to help find missing lengths before adding. Then write your answer with the correct unit.

Lesson: Understanding Area and Square Units

Have you ever looked at a floor covered with tiles or a wall covered with sticky notes? When we want to know how much flat space something covers, we measure its area.

Area is the amount of space inside a flat shape. It tells us how much surface a shape covers.

This is different from perimeter. Perimeter is the distance around the outside of a shape. Area is the space inside it.

To measure area, we use square units. A square unit is a square that has side lengths of 1 unit by 1 unit.

For example, a square that is 1 centimeter long and 1 centimeter wide is called 1 square centimeter, written as \(1\text{ cm}^2\). A square that is 1 inch by 1 inch is 1 square inch, written as \(1\text{ in}^2\).

Why do we use squares? Squares fit together neatly without gaps or overlaps. That makes them a great way to cover a flat surface and measure area fairly.

When we find area, we imagine covering the shape with equal-sized square units. The squares must:

• be the same size,
• not overlap, and
• not leave gaps.

If we count how many square units cover the shape, we know its area.

Main Idea 1: Area means covering the inside of a shape

Think of area like filling a rectangle with tiles. We are not measuring the border. We are measuring the inside part that the tiles cover.

If a shape is covered by 8 unit squares, then its area is 8 square units.

We can write that as:

$$\text{Area} = 8\text{ square units}$$

Main Idea 2: Square units are counted in rows and columns

Many shapes, especially rectangles, can be covered with square units arranged in rows and columns.

Instead of counting one square at a time, we can count how many rows there are and how many squares are in each row.

Then we multiply:

$$\text{Area} = \text{number of rows} \times \text{number in each row}$$

For rectangles, this is the same as:

$$\text{Area} = \text{length} \times \text{width}$$

This works because the rectangle is filled with equal square units in neat rows and columns.

Main Idea 3: Area and perimeter are not the same

It is important not to mix up area and perimeter.

• Perimeter = distance around the outside
• Area = space inside the shape

For example, a rectangle might have a perimeter of 14 units, but its area could be 12 square units. The numbers can be different because they measure different things.

Main Idea 4: Always include square units in your answer

When you measure area, your answer must use square units, not just units.

For example:

• correct: \(9\text{ square units}\)
• correct: \(9\text{ cm}^2\)
• not correct: \(9\text{ cm}\)

The little 2 in \(\text{cm}^2\) means "square centimeters." It reminds us that area is measured with squares.

Worked Example 1: Counting unit squares

A shape is covered by 6 equal square units. What is its area?

Step 1: Count the square units.

There are \(6\) squares.

Step 2: Write the answer with square units.

$$\text{Area} = 6\text{ square units}$$

Answer: The area is 6 square units.

Worked Example 2: Rectangle with rows and columns

A rectangle has 3 rows of square units, and each row has 4 square units. What is the area?

Step 1: Find the number of rows and columns.

• Rows: \(3\)
• Squares in each row: \(4\)

Step 2: Multiply.

$$3 \times 4 = 12$$

Step 3: Write square units.

$$\text{Area} = 12\text{ square units}$$

Answer: The area is 12 square units.

Worked Example 3: Using length and width

A rectangle is 5 units long and 2 units wide. Find its area.

Step 1: Use the area formula for rectangles.

$$\text{Area} = \text{length} \times \text{width}$$

Step 2: Substitute the numbers.

$$\text{Area} = 5 \times 2$$

Step 3: Multiply.

$$\text{Area} = 10$$

Step 4: Add the square units.

$$\text{Area} = 10\text{ square units}$$

Answer: The rectangle has an area of 10 square units.

Worked Example 4: Area or perimeter?

A student walks around the outside edge of a garden. Is the student finding area or perimeter?

Think: Walking around the outside means measuring the border of the shape.

Answer: The student is finding the perimeter, not the area.

If the student wanted to find area, they would need to know how much space is inside the garden.

Tips for Finding Area

• Look at the inside of the shape, not the outside edge.
• Count square units carefully.
• Make sure the squares do not overlap and do not leave gaps.
• For rectangles, multiply length by width.
• Always write square units in your answer.

Common Mistakes to Avoid

• Do not confuse area with perimeter.
• Do not write just "units" when the answer should be "square units."
• Do not count spaces outside the shape.
• Do not skip squares when counting rows and columns.

Let’s Compare

Imagine a rectangle that is 4 units long and 3 units wide.

Its area is:

$$4 \times 3 = 12\text{ square units}$$

That means 12 unit squares can cover the inside of the rectangle.

If you drew those squares, you would see 3 rows with 4 squares in each row, or 4 rows with 3 squares in each row. Either way, the total is 12 square units.

Quick Check for Understanding

1. What does area measure?
   Answer: The space inside a flat shape.
2. What unit is used to measure area?
   Answer: Square units.
3. A rectangle has 2 rows of 7 squares. What is its area?
   Answer: \(2 \times 7 = 14\text{ square units}\)
4. Does perimeter measure the inside or outside of a shape?
   Answer: The outside.

Summary

Area is the amount of space inside a flat shape. We measure area using square units, which are equal squares that cover a shape without gaps or overlaps.

To find the area of a rectangle, multiply the number of units in the length by the number of units in the width. Remember: area is inside, perimeter is around.

Area of Rectangles via Formula

Have you ever looked at a floor, a book cover, or a tabletop and wondered how much space it covers? That amount of covered space is called area.

In this lesson, you will learn how to find the area of a rectangle using a simple formula. You will also see how this formula matches what happens when we cover a shape with square units.

What is area?

Area is the amount of space inside a flat shape. We measure area using square units, such as square inches, square feet, or square centimeters.

A square unit is a square that measures 1 unit on each side. For example, a square inch is a square that is 1 inch long and 1 inch wide.

If we cover a rectangle with no gaps and no overlaps using square units, the total number of squares tells us the area.

From counting squares to using a formula

Imagine a rectangle made of rows and columns of square units.

• The length tells how many squares fit across.
• The width tells how many squares fit down.

If a rectangle has 5 squares across and 3 squares down, then it has 3 rows of 5 squares.

Instead of counting every square one by one, we can multiply:

We often write this as:

This means:

• A = area
• l = length
• w = width

Important: Area is always written in square units.

For example:

• square inches = \,inches\(^2\)
• square feet = \,feet\(^2\)
• square centimeters = \,cm\(^2\)

Why multiplying works

Suppose a rectangle is 4 units long and 3 units wide.

That means there are 4 square units in each row and 3 rows total.

So the total number of square units is:

So the area is 12 square units.

Multiplication helps us count all the equal rows quickly.

Steps for finding the area of a rectangle

1. Find the length of the rectangle.
2. Find the width of the rectangle.
3. Multiply length by width.
4. Write the answer using square units.

Worked Example 1

Find the area of a rectangle that is 6 units long and 2 units wide.

Use the formula:

Substitute the numbers:

The area is 12 square units.

Worked Example 2

A small rug is 7 feet long and 4 feet wide. What is its area?

Use the formula:

Substitute the measurements:

The area is 28 square feet.

Notice that we do not answer with just feet. We answer with square feet because area measures space inside the shape.

Worked Example 3

A rectangle is 9 centimeters long and 5 centimeters wide. What is the area?

Write the formula:

Substitute:

The area is 45 square centimeters.

Worked Example 4

A garden bed is 12 meters long and 8 meters wide. What is its area?

Use the formula:

Substitute:

The area is 96 square meters.

How area is different from perimeter

Sometimes students mix up area and perimeter.

• Area means the space inside a shape.
• Perimeter means the distance around a shape.

If you are asked for area, multiply length and width. If you are asked for perimeter, you add the side lengths around the shape.

Tips to remember

• Area is for the space inside a rectangle.
• Use the formula \(A = l \times w\).
• Multiply the two side lengths that meet at a corner.
• Always include square units in your answer.

Common mistakes to avoid

• Do not add length and width when finding area.
• Do not forget the units.
• Do not write just inches or feet. Write square inches or square feet.

Let’s check the idea one more time

If a rectangle has 10 squares in each row and 3 rows, then the total number of squares is:

So the area is 30 square units.

This is why the rectangle formula works: multiplying tells us the total number of equal square units that fit inside.

Summary

Area tells how much space is inside a rectangle. We can find it by counting square units, but multiplying is much faster.

For any rectangle, use the formula $$A = l \times w$$. Then write your answer in square units, such as square inches, square feet, or square centimeters.

Decomposing Rectilinear Figures for Area

Sometimes a shape is not just one rectangle. It may look like an L-shape or another shape made only of straight sides and right angles. These are called rectilinear figures.

To find the area of a rectilinear figure, we can decompose it. That means we break the larger shape into smaller rectangles. Then we find the area of each rectangle and add the areas together.

Remember: area tells how much space is inside a shape. We measure area in square units, such as square inches, square feet, or square centimeters.

The area of a rectangle is:

$$\text{Area} = \text{length} \times \text{width}$$

So if a rectangle is 5 units long and 3 units wide, its area is:

$$5 \times 3 = 15 \text{ square units}$$

How to decompose a rectilinear figure

1. Look at the shape carefully.

2. Split the shape into rectangles.

3. Find the length and width of each rectangle.

4. Find the area of each rectangle.

5. Add the areas together.

When you decompose a figure, the rectangles should fit together without overlapping. You are finding the area of all the parts exactly one time.

Important idea: Different students may split the same shape in different ways. That is okay. If the rectangles cover the whole shape and do not overlap, the total area will be the same.

Example 1: An L-shape split into 2 rectangles

Imagine an L-shaped figure made from:

• a rectangle that is 6 units by 2 units

• another rectangle that is 3 units by 4 units

Step 1: Find the area of the first rectangle.

$$6 \times 2 = 12$$

Step 2: Find the area of the second rectangle.

$$3 \times 4 = 12$$

Step 3: Add the areas.

$$12 + 12 = 24$$

Total area = 24 square units

This works because the whole L-shape is made from those 2 smaller rectangles.

Example 2: Finding a missing side before finding area

A rectilinear figure has a total height of 7 units. One part of the shape is 3 units tall. What is the missing height of the other part?

We subtract:

$$7 - 3 = 4$$

So the missing side length is 4 units.

Now suppose the figure is decomposed into these 2 rectangles:

• Rectangle A: 5 units by 3 units

• Rectangle B: 2 units by 4 units

Find each area:

Rectangle A:

$$5 \times 3 = 15$$

Rectangle B:

$$2 \times 4 = 8$$

Add them:

$$15 + 8 = 23$$

Total area = 23 square units

Sometimes you must find a missing side length first. Use what you know about the full side and the part of the side.

Example 3: One shape, two different ways

Suppose a rectilinear figure can be split in either of these ways.

Way 1:

• Rectangle A: 4 units by 6 units

• Rectangle B: 2 units by 3 units

Find the areas:

$$4 \times 6 = 24$$

$$2 \times 3 = 6$$

$$24 + 6 = 30$$

Way 2:

• Rectangle C: 6 units by 3 units

• Rectangle D: 2 units by 6 units

Find the areas:

$$6 \times 3 = 18$$

$$2 \times 6 = 12$$

$$18 + 12 = 30$$

Both ways give the same answer.

Total area = 30 square units

This shows that you can decompose the same figure in more than one correct way.

Example 4: A larger figure with 3 rectangles

Sometimes it helps to split a figure into 3 rectangles instead of 2.

Suppose a shape is decomposed into:

• Rectangle A: 3 units by 5 units

• Rectangle B: 4 units by 2 units

• Rectangle C: 2 units by 3 units

Find the area of each rectangle:

$$3 \times 5 = 15$$

$$4 \times 2 = 8$$

$$2 \times 3 = 6$$

Add all the parts:

$$15 + 8 + 6 = 29$$

Total area = 29 square units

Tips for success

• Draw lines to split the figure into rectangles.

• Label each side length clearly.

• If a side length is missing, use subtraction to find it.

• Use the area formula for each rectangle: \(l \times w\).

• Add all the rectangle areas at the end.

• Write the answer with square units.

Watch out for these mistakes

• Do not add side lengths when you should multiply to find area.

• Do not forget one of the smaller rectangles.

• Do not count the same part twice.

• Do not forget to include square units in your answer.

Let’s review

A rectilinear figure is a shape with straight sides and right angles. To find its area, decompose it into smaller rectangles. Find the area of each rectangle, then add the areas together.

If a side length is missing, you can often find it by subtracting one part from the whole. Take your time, organize your work, and check that all parts of the shape are included once.

Summary

To find the area of a rectilinear figure, break it into rectangles. Use $$\text{Area} = \text{length} \times \text{width}$$ for each rectangle. Then add the areas to find the total area of the whole shape.

Constant Area with Changing Perimeter

Sometimes two rectangles can have the same area but different perimeters. This means they cover the same amount of space inside, but the distance around the outside is not the same.

In this lesson, you will learn how area can stay constant while perimeter changes. You will also see how changing the side lengths of a rectangle can change the outside boundary, even when the inside space stays equal.

First, let’s remember what area and perimeter mean.

• Area is the amount of space inside a shape.
• Perimeter is the distance around a shape.

For rectangles:

• Area: \(\text{length} \times \text{width}\)
• Perimeter: \(2 \times (\text{length} + \text{width})\)

So for a rectangle with length \(l\) and width \(w\):

The big idea: If the product of the side lengths stays the same, then the area stays the same. But if the side lengths change, the perimeter can change too.

Let’s explore this with rectangles made from square units.

Example 1: Same area, different perimeter

Look at these two rectangles:

• Rectangle A: \(3\) units by \(4\) units
• Rectangle B: \(2\) units by \(6\) units

First, find the area.

Rectangle A:

Rectangle B:

Both rectangles have area \(12\) square units.

Now find the perimeter.

Rectangle A:

Rectangle B:

So both rectangles have the same area, but their perimeters are different.

What does this show? The inside space can stay the same, but the outside distance can change.

Why does this happen?

When a rectangle is shaped more like a square, the perimeter is often smaller.

When a rectangle gets longer and skinnier, the perimeter often gets larger, even if the area stays the same.

Let’s test that idea.

Example 2: Many rectangles with area 24

Here are some different rectangles that all have area \(24\) square units:

• \(1 \times 24\)
• \(2 \times 12\)
• \(3 \times 8\)
• \(4 \times 6\)

Check the area for each one:

• \(1 \times 24 = 24\)
• \(2 \times 12 = 24\)
• \(3 \times 8 = 24\)
• \(4 \times 6 = 24\)

Now find the perimeter of each rectangle.

• \(1 \times 24\): $$2(1+24)=2(25)=50$$
• \(2 \times 12\): $$2(2+12)=2(14)=28$$
• \(3 \times 8\): $$2(3+8)=2(11)=22$$
• \(4 \times 6\): $$2(4+6)=2(10)=20$$

All of these rectangles have area \(24\), but the perimeters are \(50\), \(28\), \(22\), and \(20\).

Important pattern: The rectangle that is closer to a square has the smaller perimeter. The long, skinny rectangle has the larger perimeter.

Let’s think about square units.

If you build rectangles with tiles, you can use the same number of tiles each time. That means the area stays the same. But when you arrange the tiles in different rectangle shapes, the outside edge changes. That changes the perimeter.

For example, \(12\) tiles can make:

• a \(3 \times 4\) rectangle
• a \(2 \times 6\) rectangle
• a \(1 \times 12\) rectangle

Each one uses \(12\) tiles, so each one has area \(12\). But the border around each rectangle is different.

Example 3: Which rectangle has the greater perimeter?

Rectangle C is \(5\) units by \(6\) units.

Rectangle D is \(3\) units by \(10\) units.

First, find the area.

Rectangle C:

Rectangle D:

The areas are the same.

Now find the perimeters.

Rectangle C:

Rectangle D:

Rectangle D has the greater perimeter.

Why? Rectangle D is more stretched out. Rectangle C is closer to a square.

How to solve these problems

1. Find the area of each rectangle by multiplying length and width.
2. Check if the areas are the same.
3. Find the perimeter by adding all sides, or using \(2(l+w)\).
4. Compare the perimeters.

Example 4: Make a new rectangle with the same area

A rectangle has side lengths \(4\) units and \(9\) units.

Its area is:

Can we make a different rectangle with area \(36\)? Yes.

Some possible rectangles are:

• \(1 \times 36\)
• \(2 \times 18\)
• \(3 \times 12\)
• \(4 \times 9\)
• \(6 \times 6\)

All of these have area \(36\) square units.

Now compare two of them:

For \(4 \times 9\):

For \(6 \times 6\):

Both have area \(36\), but the \(6 \times 6\) square has the smaller perimeter.

What should you remember?

• Rectangles can have the same area and different perimeters.
• Area tells how much space is inside.
• Perimeter tells the distance around the shape.
• Long, skinny rectangles often have larger perimeters.
• Rectangles closer to a square often have smaller perimeters.

Quick check questions

• Do \(2 \times 8\) and \(4 \times 4\) have the same area?
• Which has the larger perimeter: \(2 \times 8\) or \(4 \times 4\)?

Let’s check:

Area of \(2 \times 8\):

Area of \(4 \times 4\):

Yes, they have the same area.

Perimeter of \(2 \times 8\):

Perimeter of \(4 \times 4\):

The \(2 \times 8\) rectangle has the larger perimeter.

Summary

Area and perimeter are not the same thing. Two rectangles can cover the same amount of space but have different distances around them. When the rectangle becomes more stretched out, the perimeter usually gets bigger. When it is closer to a square, the perimeter is often smaller.

Attributes of Mass and Weight

When we measure how heavy something is, we often talk about mass or weight. In 4th grade, you can think of both as telling us how heavy or light an object feels.

We use different units of measurement to describe heavy and light objects. Some units are better for small, light things. Other units are better for large, heavy things.

In this lesson, you will learn how to:

• tell whether an object is lightweight or heavyweight,
• choose a good standard unit to measure it, and
• compare units in both the customary system and the metric system.

1. What do mass and weight tell us?

Mass and weight both help us describe how heavy something is. If an apple is lighter than a watermelon, then the apple has less mass and weight than the watermelon.

You do not always need to know the exact number first. Sometimes you can begin by asking:

• Is this object light or heavy?
• Should I use a small unit or a large unit?

2. Measurement systems for mass and weight

There are two common systems you may use in school.

• Customary system: ounces and pounds
• Metric system: grams and kilograms

Each system has a small unit and a larger unit.

Customary units

• ounce (oz): used for lighter objects
• pound (lb): used for heavier objects

Metric units

• gram (g): used for lighter objects
• kilogram (kg): used for heavier objects

3. Choosing the best unit

A good rule is:

• Use ounces or grams for things that are small and light.
• Use pounds or kilograms for things that are large and heavy.

For example:

• A paper clip is so light that grams would make sense.
• A loaf of bread might be measured in ounces or grams.
• A dog might be measured in pounds or kilograms.
• A backpack full of books is heavy enough for pounds or kilograms.

4. Understanding small units and large units

Small units are helpful when an object does not weigh very much. Large units are helpful when an object weighs a lot.

If you use a unit that is too small for a heavy object, the number can become very large. If you use a unit that is too large for a tiny object, the measurement may not make much sense for the job.

For instance, it is better to say a watermelon weighs several pounds than many, many ounces. It is better to say a coin weighs a few grams than part of a kilogram.

5. Helpful unit facts

You may use these facts when comparing units:

• In the customary system, $$1 \text{ pound} = 16 \text{ ounces}$$
• In the metric system, $$1 \text{ kilogram} = 1000 \text{ grams}$$

This means a pound is larger than an ounce, and a kilogram is larger than a gram.

6. Thinking about relative weight

You can compare objects even without measuring them exactly.

• A feather is lighter than a book.
• A book is lighter than a chair.
• A chair is lighter than a refrigerator.

This kind of thinking helps you choose the best unit. If an object seems very light, think about ounces or grams. If it seems much heavier, think about pounds or kilograms.

7. Worked Examples

Example 1: Choose the better customary unit

Which unit makes more sense for measuring an apple: ounces or pounds?

Step 1: Think about the object. An apple is fairly small and not very heavy.

Step 2: Small, light objects are often measured in ounces.

Answer: ounces is the better unit.

Example 2: Choose the better metric unit

Which unit makes more sense for measuring a bicycle: grams or kilograms?

Step 1: Think about the object. A bicycle is much heavier than a pencil or an eraser.

Step 2: Heavy objects are often measured in kilograms.

Answer: kilograms is the better unit.

Example 3: Compare two objects

A lunchbox weighs less than a suitcase. Which object would be more likely measured in pounds?

Step 1: A suitcase is usually heavier than a lunchbox.

Step 2: Heavier objects are often measured in pounds.

Answer: The suitcase would be more likely measured in pounds.

Example 4: Use a unit fact

A bag of rice weighs 2 pounds. How many ounces is that?

Use the fact $$1 \text{ lb} = 16 \text{ oz}$$

Step 1: There are 16 ounces in 1 pound.

Step 2: For 2 pounds, multiply:

$$2 \times 16 = 32$$

Answer: The bag of rice weighs 32 ounces.

8. Tips for choosing the right unit

• Ask yourself if the object is light or heavy.
• Use ounces or grams for lighter objects.
• Use pounds or kilograms for heavier objects.
• Think about objects you know. A coin is light. A backpack can be heavy.
• Choose the unit that keeps the measurement reasonable and easy to understand.

9. Common mistakes to avoid

• Do not choose kilograms for a tiny object like a button.
• Do not choose grams for a very heavy object like a person.
• Do not mix up the systems. Ounces and pounds go together. Grams and kilograms go together.

10. Quick check ideas

Try thinking about these on your own:

• Would a strawberry be measured better in grams or kilograms?
• Would a dog be measured better in ounces or pounds?
• Would a textbook be measured better in grams or kilograms?

If you answered grams, pounds, and kilograms, you are thinking carefully about object weight and unit size.

Summary

Mass and weight tell how heavy or light an object is. In the customary system, we use ounces for lighter objects and pounds for heavier objects. In the metric system, we use grams for lighter objects and kilograms for heavier objects.

When choosing a unit, think about the size and heaviness of the object. Small, light objects need smaller units. Large, heavy objects need larger units.

Customary and Metric Weight Conversions

We use weight to tell how heavy something is. In math, we often measure weight using two different systems:

• Customary system: ounces and pounds
• Metric system: grams and kilograms

When we convert, we change a measurement from one unit to another without changing how heavy the object is.

For example, a bag of flour might weigh the same amount whether we talk about it in pounds or ounces. The number changes, but the weight stays the same.

1. Customary Weight: Ounces and Pounds

In the customary system, the main fact to remember is:

$$1 \text{ pound} = 16 \text{ ounces}$$

This means one pound is made of 16 equal ounces.

To convert between pounds and ounces:

• Pounds to ounces: multiply by 16
• Ounces to pounds: divide by 16

You can think: if you are changing to a smaller unit, the number gets bigger. If you are changing to a larger unit, the number gets smaller.

2. Metric Weight: Grams and Kilograms

In the metric system, the main fact to remember is:

$$1 \text{ kilogram} = 1{,}000 \text{ grams}$$

This means one kilogram is made of 1,000 equal grams.

To convert between kilograms and grams:

• Kilograms to grams: multiply by 1,000
• Grams to kilograms: divide by 1,000

Just like before, changing to a smaller unit makes the number bigger, and changing to a larger unit makes the number smaller.

3. A Helpful Way to Think About It

Ask yourself: Am I changing to bigger units or smaller units?

• From pounds to ounces: smaller units, so multiply
• From ounces to pounds: bigger units, so divide
• From kilograms to grams: smaller units, so multiply
• From grams to kilograms: bigger units, so divide

This helps you decide what operation to use.

Worked Example 1: Convert pounds to ounces

A watermelon weighs 3 pounds. How many ounces is that?

We know:

$$1 \text{ pound} = 16 \text{ ounces}$$

Since we are changing pounds to ounces, we multiply:

$$3 \times 16 = 48$$

So, 3 pounds = 48 ounces.

Worked Example 2: Convert ounces to pounds

A loaf of bread weighs 32 ounces. How many pounds is that?

We know 16 ounces make 1 pound, so we divide by 16:

$$32 \div 16 = 2$$

So, 32 ounces = 2 pounds.

Worked Example 3: Convert kilograms to grams

A bag of rice weighs 4 kilograms. How many grams is that?

We know:

$$1 \text{ kilogram} = 1{,}000 \text{ grams}$$

Since we are changing kilograms to grams, we multiply:

$$4 \times 1{,}000 = 4{,}000$$

So, 4 kilograms = 4,000 grams.

Worked Example 4: Convert grams to kilograms

A bag of apples weighs 6,000 grams. How many kilograms is that?

We divide by 1,000:

$$6{,}000 \div 1{,}000 = 6$$

So, 6,000 grams = 6 kilograms.

4. Using What You Know to Check Your Answer

You can always do a quick check to see if your answer makes sense.

• If you changed to ounces or grams, your number should usually be bigger.
• If you changed to pounds or kilograms, your number should usually be smaller.

For example, if 2 pounds became 8 ounces, that would not make sense, because ounces are smaller than pounds, so the number should get bigger, not smaller.

5. Common Mistakes to Avoid

• Do not mix up the two facts:
  \(1\) pound = \(16\) ounces
  \(1\) kilogram = \(1{,}000\) grams
• Remember to multiply when changing to a smaller unit.
• Remember to divide when changing to a larger unit.
• Write the correct unit in your answer.

6. Quick Practice Thinking

Try asking yourself these questions:

1. Is the new unit bigger or smaller?
2. Should I multiply or divide?
3. Does my answer make sense?

Example: Convert 5 pounds to ounces.

• Ounces are smaller than pounds.
• So I multiply.
• $$5 \times 16 = 80$$

So, 5 pounds = 80 ounces.

Example: Convert 2,000 grams to kilograms.

• Kilograms are bigger than grams.
• So I divide.
• $$2{,}000 \div 1{,}000 = 2$$

So, 2,000 grams = 2 kilograms.

Summary

To convert customary weight, remember that 16 ounces = 1 pound. To convert metric weight, remember that 1,000 grams = 1 kilogram.

Multiply when changing to a smaller unit, and divide when changing to a larger unit. Always check if your answer makes sense by thinking about whether the number should get bigger or smaller.

Lesson: Attributes of Capacity and Volume

We use measurement every day. We measure how long something is, how heavy it is, and how much time has passed. We also measure how much a container can hold and how much space something takes up. These ideas are called capacity and volume.

In this lesson, you will learn what capacity and volume mean, how they are alike, how they are different, and which units are used to measure them in the customary system and the metric system.

What is capacity?

Capacity is the amount of liquid a container can hold. It tells us about containers such as cups, bottles, jugs, buckets, and tanks.

For example, a water bottle might hold 500 milliliters, and a milk jug might hold 1 gallon. These measurements describe the container’s capacity.

What is volume?

Volume is the amount of space that something takes up. A solid object has volume, and a box-shaped container also has volume because it takes up space.

If a box is large, it has a greater volume than a small box. If two containers are the same size inside, they can hold the same amount.

How are capacity and volume related?

• Capacity tells how much liquid a container can hold.
• Volume tells how much space an object or container takes up.
• They are related because a container with more inside space usually has a greater capacity.

Think about a juice box and a large pitcher. The pitcher has more inside space, so it can hold more liquid. That means it has a greater capacity.

Customary units for capacity

In the customary system, we often measure liquid with these units:

• cup (c)
• pint (pt)
• quart (qt)
• gallon (gal)

These units are often used in cooking, grocery shopping, and everyday life.

Helpful customary facts:

• $$2\text{ cups} = 1\text{ pint}$$
• $$2\text{ pints} = 1\text{ quart}$$
• $$4\text{ quarts} = 1\text{ gallon}$$

Metric units for capacity

In the metric system, we often measure liquid with these units:

• milliliter (mL)
• liter (L)

A milliliter is a small amount. A liter is a larger amount.

Helpful metric fact:

• $$1000\text{ mL} = 1\text{ L}$$

Choosing the right unit

It is important to choose a unit that makes sense for the object or container.

• Use milliliters for small amounts of liquid, like medicine or a juice box.
• Use liters for larger amounts of liquid, like a bottle of soda or a fish tank.
• Use cups, pints, quarts, and gallons for common customary measurements, such as milk, juice, and paint.

Examples of sensible units

• A spoonful of cough syrup: milliliters
• A carton of milk: quart or gallon
• A small water bottle: milliliters
• A large jug of juice: liters or gallons

Comparing capacities

We can compare containers by thinking about which one holds more or less.

• A bucket has more capacity than a cup.
• A gallon jug has more capacity than a pint container.
• A 2-liter bottle holds more than a 500-mL bottle.

Worked Example 1: Choosing the best unit

Question: Which unit makes more sense for a bottle of glue: liters or milliliters?

Step 1: Think about the size of the bottle. A school glue bottle is small.

Step 2: Small amounts of liquid are usually measured in milliliters.

Answer: milliliters is the better unit.

Worked Example 2: Using customary units

Question: A container holds 1 gallon of lemonade. Is that more or less than 1 quart?

Step 1: Remember the fact:

$$4\text{ quarts} = 1\text{ gallon}$$

Step 2: Compare 1 gallon to 1 quart. Since 1 gallon equals 4 quarts, it is greater than 1 quart.

Answer: 1 gallon is more than 1 quart.

Worked Example 3: Using metric units

Question: A pitcher holds 2 liters of water. A bottle holds 750 milliliters. Which holds more?

Step 1: It helps to compare using the same kind of unit.

We know:

$$1\text{ L} = 1000\text{ mL}$$

So:

$$2\text{ L} = 2000\text{ mL}$$

Step 2: Compare:

$$2000\text{ mL} > 750\text{ mL}$$

Answer: The pitcher holds more.

Worked Example 4: Capacity and volume thinking

Question: Two boxes are the same size inside. One is filled with juice cartons, and the other is empty. Do the boxes have the same capacity?

Step 1: Capacity depends on how much a container can hold, not whether it is empty or full right now.

Step 2: Since both boxes are the same size inside, they can hold the same amount.

Answer: Yes, they have the same capacity.

Tips to remember

• Capacity is about how much liquid a container can hold.
• Volume is about how much space something takes up.
• Small liquid amounts are often measured in mL or cups.
• Larger liquid amounts are often measured in L, quarts, or gallons.
• Always choose a unit that matches the size of the object or container.

Common mistakes to avoid

• Do not confuse capacity with weight. Capacity is about holding liquid, not how heavy something is.
• Do not choose a huge unit for a tiny container. For example, medicine is measured in milliliters, not gallons.
• Do not forget that capacity tells what a container can hold, even if it is not full.

Brief Summary

Capacity and volume both help us describe the physical world. Capacity tells how much liquid a container can hold, and volume tells how much space something takes up. We use units such as cups, pints, quarts, and gallons in the customary system, and milliliters and liters in the metric system. Choosing the correct unit helps us measure in a smart and useful way.

Customary and Metric Capacity Conversions

Capacity tells us how much liquid a container can hold. We use capacity when we measure water, juice, milk, soup, and other liquids.

In math, we use two common measurement systems for capacity:

• Customary units, such as cups, pints, quarts, and gallons
• Metric units, such as milliliters and liters

When we convert capacity, we change one unit into another unit without changing the amount of liquid.

For example, 1 gallon and 4 quarts can name the same amount of liquid.

Why is this useful?

• It helps us read recipes.
• It helps us compare containers.
• It helps us decide how much liquid we need.

Customary Capacity Units

Here are the customary capacity units you should know:

• 1 cup = 2 pints? No. Be careful! In 4th grade, we usually focus on these larger facts:
• 2 cups = 1 pint
• 2 pints = 1 quart
• 4 quarts = 1 gallon

You can also use these facts to make bigger conversions:

• 8 pints = 1 gallon because 2 pints make 1 quart and 4 quarts make 1 gallon
• 16 cups = 1 gallon

A simple way to remember some customary units is this order from smaller to larger:

cup  pint  quart  gallon

As the unit gets larger, the number used to name the same amount gets smaller.

For example, 1 gallon is the same as 4 quarts. Since quarts are smaller than gallons, it takes more of them.

Metric Capacity Units

In the metric system, the main capacity units for this lesson are:

• 1,000 milliliters = 1 liter

A milliliter is a small unit. A liter is a larger unit.

So if you change liters to milliliters, the number gets bigger. If you change milliliters to liters, the number gets smaller.

How to Convert Capacity

There are two main ideas to remember:

1. If you change from a larger unit to a smaller unit, you multiply.
2. If you change from a smaller unit to a larger unit, you divide.

For example:

• Gallons to quarts: multiply by 4
• Quarts to gallons: divide by 4
• Liters to milliliters: multiply by 1,000
• Milliliters to liters: divide by 1,000

Customary Conversion Facts to Know

• 2 cups = 1 pint
• 2 pints = 1 quart
• 4 quarts = 1 gallon

Metric Conversion Fact to Know

• 1,000 milliliters = 1 liter

Worked Example 1: Quarts to Gallons

Convert 12 quarts to gallons.

We know:

$$4 \text{ quarts} = 1 \text{ gallon}$$

We are changing from quarts to gallons. Quarts are smaller, so we divide by 4.

$$12 \div 4 = 3$$

So,

$$12 \text{ quarts} = 3 \text{ gallons}$$

Worked Example 2: Gallons to Quarts

Convert 5 gallons to quarts.

We know:

$$1 \text{ gallon} = 4 \text{ quarts}$$

We are changing from gallons to quarts. Gallons are larger, so we multiply by 4.

$$5 \times 4 = 20$$

So,

$$5 \text{ gallons} = 20 \text{ quarts}$$

Worked Example 3: Milliliters to Liters

Convert 3,000 milliliters to liters.

We know:

$$1{,}000 \text{ milliliters} = 1 \text{ liter}$$

We are changing from milliliters to liters. Milliliters are smaller, so we divide by 1,000.

$$3{,}000 \div 1{,}000 = 3$$

So,

$$3{,}000 \text{ milliliters} = 3 \text{ liters}$$

Worked Example 4: Liters to Milliliters

Convert 7 liters to milliliters.

We know:

$$1 \text{ liter} = 1{,}000 \text{ milliliters}$$

We are changing from liters to milliliters. Liters are larger, so we multiply by 1,000.

$$7 \times 1{,}000 = 7{,}000$$

So,

$$7 \text{ liters} = 7{,}000 \text{ milliliters}$$

How to Think About Conversion Problems

When solving a capacity conversion problem, ask yourself these questions:

1. What unit do I have now?
2. What unit do I need?
3. Is the new unit bigger or smaller?
4. Should I multiply or divide?

Example: Change 2 gallons to quarts.

• I have gallons.
• I need quarts.
• Quarts are smaller than gallons.
• So I multiply: \(2 \times 4 = 8\).

That means:

$$2 \text{ gallons} = 8 \text{ quarts}$$

Watch Out for These Mistakes

• Do not mix up multiply and divide. Larger to smaller means multiply. Smaller to larger means divide.
• Use the correct conversion fact. Quarts and gallons use 4. Liters and milliliters use 1,000.
• Keep the units in your answer. Do not write only the number.

Quick Practice Thinking

If you want, you can try these in your head:

• \(8\) quarts = ? gallons
• \(2\) liters = ? milliliters
• \(1\) gallon = ? quarts

The answers are:

• \(8 \div 4 = 2\), so 2 gallons
• \(2 \times 1{,}000 = 2{,}000\), so 2,000 milliliters
• 4 quarts

Summary

Capacity measures how much liquid a container holds. In customary units, important facts are 2 cups = 1 pint, 2 pints = 1 quart, and 4 quarts = 1 gallon. In metric units, 1,000 milliliters = 1 liter.

Remember: when you change from a larger unit to a smaller unit, multiply. When you change from a smaller unit to a larger unit, divide. Always check your units so your answer makes sense.

Elapsed Time on Number Lines means finding out how much time passes from a start time to an end time by showing the time in jumps.

A number line for time works like a path. You start at one time, make jumps forward, and stop at the ending time. Then you add the jumps to find the total elapsed time.

This is a helpful way to solve time problems, especially when the time goes past a new hour, like from 2:45 to 4:10.

What is elapsed time?

Elapsed time is the amount of time that goes by between two times.

For example, if a movie starts at 1:00 and ends at 2:30, the elapsed time is how long the movie lasted.

Why use a number line?

A number line helps you break the time into smaller, easier parts. You do not have to count every single minute. Instead, you can make smart jumps.

You can jump:

• to the next hour,
• by whole hours,
• and then by the extra minutes.

Steps for finding elapsed time on a number line

1. Write the start time on the left.
2. Write the end time on the right.
3. Make jumps from the start time to the end time.
4. Label each jump with the amount of time.
5. Add the jumps together.

Helpful idea: It is often easiest to jump to a friendly time first, such as the next hour.

For example, from 3:25, a friendly jump is to 4:00 because that is the next full hour.

Worked Example 1

Find the elapsed time from 2:15 to 3:00.

Start at 2:15. End at 3:00.

On the number line, make one jump from 2:15 to 3:00.

That jump is 45 minutes.

$$45\text{ minutes}$$

So, the elapsed time is 45 minutes.

Worked Example 2

Find the elapsed time from 1:20 to 3:20.

We can make easy jumps:

• 1:20 to 2:00 = 40 minutes
• 2:00 to 3:00 = 1 hour
• 3:00 to 3:20 = 20 minutes

Now add the jumps.

Minutes: \(40 + 20 = 60\) minutes

And \(60\) minutes = \(1\) hour

So the total is:

$$1\text{ hour} + 1\text{ hour} = 2\text{ hours}$$

The elapsed time is 2 hours.

Worked Example 3

Find the elapsed time from 9:45 to 12:10.

Use friendly jumps on the number line:

• 9:45 to 10:00 = 15 minutes
• 10:00 to 11:00 = 1 hour
• 11:00 to 12:00 = 1 hour
• 12:00 to 12:10 = 10 minutes

Add the hours and minutes.

Hours: \(1 + 1 = 2\) hours

Minutes: \(15 + 10 = 25\) minutes

So the elapsed time is:

$$2\text{ hours }25\text{ minutes}$$

Worked Example 4

Find the elapsed time from 4:58 to 6:12.

This one crosses more than one hour, so let us break it into parts.

• 4:58 to 5:00 = 2 minutes
• 5:00 to 6:00 = 1 hour
• 6:00 to 6:12 = 12 minutes

Add the parts:

Minutes: \(2 + 12 = 14\) minutes

Hours: \(1\) hour

So the elapsed time is:

$$1\text{ hour }14\text{ minutes}$$

Tips for success

• Always begin with the start time.
• Move forward to the end time.
• Jump to the next hour when it helps.
• Add hours to hours and minutes to minutes.
• Remember: \(60\) minutes = \(1\) hour.

What if the minutes add to 60 or more?

If your minutes make \(60\), trade them for 1 hour.

For example:

$$1\text{ hour } + 60\text{ minutes} = 2\text{ hours}$$

If the minutes are less than \(60\), keep them as minutes.

Check your thinking

• Does your answer make sense?
• Did you count forward, not backward?
• Did you include all the jumps?
• Did you add the hours and minutes correctly?

Let’s review

Elapsed time tells how much time passes between two events.

A number line helps by showing time in smaller jumps. You can jump to the next hour, then count full hours, then count extra minutes.

When you add all the jumps together, you get the total elapsed time.

Multi-Step Measurement Word Problems are problems where you have to do more than one step to find the answer.

Many of these problems ask you to measure, convert, and then solve. That means you may need to change one unit into another unit before you can add, subtract, multiply, or divide.

For example, you might need to change feet into inches, or hours into minutes, before solving the problem. If the units do not match, the math will not work correctly.

Big idea: In measurement word problems, always make sure the units are the same before doing the operation.

Common measurement facts to know

•  inches = 1 foot
• 3 feet = 1 yard
• 2 cups = 1 pint
• 2 pints = 1 quart
• 4 quarts = 1 gallon
• 60 seconds = 1 minute
• 60 minutes = 1 hour
• 7 days = 1 week
• 16 ounces = 1 pound

How to solve a multi-step measurement word problem

1. Read carefully. Ask yourself, “What is the problem asking me to find?”
2. Circle the numbers and units. Look for words like inches, feet, minutes, hours, cups, and pounds.
3. Make the units match. Convert if needed.
4. Choose the operation. Decide if you need to add, subtract, multiply, or divide.
5. Solve step by step.
6. Label your answer. Write the correct unit in your final answer.

Helpful question: Can I combine these numbers yet, or do I need to convert first?

Example 1: Length with addition

A ribbon is 2 feet long. Another ribbon is 8 inches long. How long are the ribbons altogether in inches?

Step 1: Find the units. One length is in feet. The other is in inches. We need the same unit.

Step 2: Convert feet to inches.

Since 1 foot = 12 inches:

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

Step 3: Add.

$$24 + 8 = 32$$

Answer: The ribbons are 32 inches long altogether.

Example 2: Time with subtraction

A movie lasted 1 hour and 25 minutes. If 40 minutes have already passed, how many minutes are left?

Step 1: Convert to one unit. Change 1 hour to minutes.

$$1 \text{ hour} = 60 \text{ minutes}$$

So the total movie time is:

$$60 + 25 = 85 \text{ minutes}$$

Step 2: Subtract the time that passed.

$$85 - 40 = 45$$

Answer: There are 45 minutes left.

Example 3: Weight with multiplication and conversion

A baker uses 2 pounds of flour each day. How many ounces of flour does the baker use in 3 days?

Step 1: Find the total pounds.

$$2 \times 3 = 6 \text{ pounds}$$

Step 2: Convert pounds to ounces.

Since 1 pound = 16 ounces:

$$6 \times 16 = 96 \text{ ounces}$$

Answer: The baker uses 96 ounces of flour in 3 days.

Example 4: Capacity with two conversions

Lena has 1 gallon of juice. She pours the juice equally into 8 pint-sized bottles. How many pints of juice go into each bottle?

Step 1: Convert gallons to quarts.

$$1 \text{ gallon} = 4 \text{ quarts}$$

Step 2: Convert quarts to pints.

Since 1 quart = 2 pints:

$$4 \times 2 = 8 \text{ pints}$$

Step 3: Divide equally into 8 bottles.

$$8 \div 8 = 1$$

Answer: Each bottle gets 1 pint of juice.

Tips for success

• Write the unit every time you write a number in your work.
• Convert first if the units are different.
• Work in order and do one step at a time.
• Check if your answer makes sense. If the problem is about a short ribbon, your answer should not be hundreds of feet.

Watch out for these mistakes

• Adding or subtracting measurements with different units without converting first.
• Forgetting a measurement fact, like  inches = 1 foot.
• Writing the wrong unit in the final answer.
• Only doing one step when the problem needs two or more steps.

Let's think through one more problem

A rope is 3 yards long. Tina cuts off 2 feet. How many feet of rope are left?

Step 1: Convert yards to feet.

Since 1 yard = 3 feet:

$$3 \text{ yards} = 3 \times 3 = 9 \text{ feet}$$

Step 2: Subtract the part cut off.

$$9 - 2 = 7$$

Answer: Tina has 7 feet of rope left.

Summary

Multi-step measurement word problems ask you to do more than one thing, such as convert units and then solve. The most important rule is to make sure the units match before you add, subtract, multiply, or divide.

If you read carefully, convert when needed, and solve one step at a time, you can solve these problems with confidence.