Measurement Systems and Applications

Lesson: Iteration of Standard Units

When we measure length, we find out how long something is. To measure correctly, we use standard units. Standard units are measurement pieces that are always the same size, like an inch tile, a centimeter cube edge, or a ruler marked in equal parts.

Iteration of standard units means repeating the same-size unit again and again from one end of an object to the other. We place the units end-to-end to measure the whole length.

This is important because if the units are not the same size, or if they are not lined up correctly, the measurement will be wrong.

Why standard units matter

• Each unit is the same size.
• Everyone can measure in the same way.
• The answer is fair, clear, and easy to compare.

For example, if you use inch tiles, every tile is 1 inch long. If an object takes 5 inch tiles to cover its length, then the object is 5 inches long.

How to measure by iterating units

1. Start at the zero point.
2. Place one standard unit at the beginning of the object.
3. Put the next unit right next to it.
4. Keep placing units end-to-end.
5. Make sure there are no gaps and no overlaps.
6. Count the total number of units.

The zero point is very important. The zero point is where measurement begins. On a ruler, it is the mark labeled 0. If you do not start at 0, your answer may be too big or too small.

Think of the zero point as the starting line. Just like a race starts at the starting line, measuring starts at 0.

No gaps and no overlaps

When units are placed with gaps, the measured length looks longer than it really is. When units overlap, the measured length looks shorter than it really is.

So, to measure correctly, the units must:

• touch but not cover each other,
• be in a straight line,
• begin at 0,
• and all be the same size.

Worked Example 1: Measuring with inch tiles

A pencil is measured with inch tiles. The tiles are placed end-to-end with no gaps or overlaps. There are 6 tiles from one end of the pencil to the other.

Each tile is 1 inch long, so the pencil measures:

$$6 \text{ inches}$$

Answer: The pencil is 6 inches long.

Worked Example 2: Finding a mistake

A crayon is measured with 4 inch tiles, but there is a small gap between two of the tiles.

This is not correct measuring. The gap adds extra space that is not part of the crayon.

To fix it, move the tiles so they touch end-to-end. Then count again.

Important lesson: Units must have no gaps.

Worked Example 3: Why start at zero?

A book is lined up with a ruler, but the edge of the book starts at 2 instead of 0. The other edge ends at 9.

You should not just say the book is 9 units long. The measurement starts at 2, so we find the distance from 2 to 9:

$$9 - 2 = 7$$

Answer: The book is 7 units long.

This shows why the zero point matters. If you can start at 0, measuring is easier. If you start somewhere else, you must count only the units between the two ends.

Worked Example 4: Comparing correct and incorrect measuring

Two students measure the same marker.

• Student A places 5 centimeter tiles end-to-end, starting at 0, with no gaps or overlaps.
• Student B places 5 tiles, but two tiles overlap.

Student A measured correctly. Student B did not measure correctly because overlapping tiles make the marker seem shorter than it really is.

Correct answer: Trust Student A's measurement.

Things to remember when measuring length

• Use equal-sized standard units.
• Start at the zero point.
• Place units end-to-end.
• Make sure there are no gaps.
• Make sure there are no overlaps.
• Count the number of units carefully.

Try thinking about these questions

• If you measure with different-size pieces, can your answer be trusted?
• If your object starts at 1 on a ruler and ends at 8, is the length 8 or 7?
• If there is a gap between units, is the object really that long?

Brief Summary

Iteration of standard units means measuring by repeating the same unit over and over. The units must be placed end-to-end, starting at 0, with no gaps or overlaps. When we do this carefully, we can find the true length of an object.

Precision and Estimation in Length

We measure length to find out how long, tall, or wide something is. In real life, we measure many things, like a pencil, a book, a desk, or a doorway.

Sometimes we need an exact measurement. Other times, a good estimate is enough. In this lesson, you will learn how to choose the right measuring tool and how to read measurements to the nearest whole, half, or quarter unit.

1. What is precision?

Precision means measuring carefully and as closely as you can. When you are precise, you line up the object correctly and read the marks on the tool correctly.

For example, if a crayon is a little longer than 4 inches, you should not just say 4 inches if you can read the ruler more carefully. You might say it is 4 and a half inches, or 4 and a quarter inches, if that matches the marks.

2. What is estimation?

Estimation means making a smart guess that is close to the real measurement. Estimating helps when you do not need an exact answer right away.

For example, before measuring a table, you might estimate that it is about 3 feet long. Then you can measure it to check how close your estimate was.

3. Choosing the right tool

Different tools are better for different objects.

• Ruler: Good for small, straight objects like pencils, books, papers, and crayons.
• Measuring tape: Good for longer objects or objects that are hard to move, like a table, bed, wall, or doorway.

A ruler is usually short and stiff. A measuring tape is longer and can bend, so it is helpful for bigger spaces.

4. Standard units of length

We use standard units so everyone measures the same way. In 3rd Grade, common units are:

• Inches for smaller objects
• Feet for longer objects

Remember:

You do not always need to change units, but it is helpful to know that inches and feet are connected.

5. How to measure carefully

1. Choose the right tool.
2. Put the zero mark at the start of the object.
3. Make sure the object is lined up straight with the ruler or tape.
4. Look at where the end of the object lands.
5. Read the measurement to the nearest whole, half, or quarter unit.

Important: Start at the 0 mark, not at the edge of the ruler, unless the edge is exactly at 0.

6. Reading whole, half, and quarter units

On a ruler, some marks show whole numbers, and smaller marks show parts of a unit.

• Whole unit: 1, 2, 3, 4, ...
• Half unit: halfway between two whole numbers, like \(2\frac{1}{2}\)
• Quarter unit: one of four equal parts between whole numbers, like \(2\frac{1}{4}\) or \(2\frac{3}{4}\)

Between 2 and 3 inches, the quarter-inch marks are:

These marks help us be more precise than using only whole numbers.

7. When to use whole, half, or quarter units

Use the closest mark that the lesson or problem asks for.

• If it says nearest whole unit, choose the closest whole number.
• If it says nearest half unit, choose the closest whole or half.
• If it says nearest quarter unit, choose the closest whole, quarter, half, or three-quarter mark.

The more small marks you use, the more precise your measurement is.

8. Estimating before measuring

A good estimator thinks about the size of the object first.

You can ask yourself:

• Is this object shorter than a ruler?
• Does it look closer to 5 inches or 10 inches?
• Should I use inches or feet?
• Do I need a ruler or a measuring tape?

Estimating first can help you catch mistakes. If you estimate a book is about 10 inches long, but your measurement says 2 inches, something is probably wrong.

Worked Example 1: Choosing a tool

Question: What tool should you use to measure the length of a notebook? A ruler or a measuring tape?

Think: A notebook is a small, straight object.

Answer: Use a ruler.

Why? A ruler is best for smaller objects that can sit flat and straight.

Worked Example 2: Reading to the nearest whole unit

Question: A marker starts at 0 inches and ends a little past 6 inches, but not close to 7. What is its length to the nearest whole inch?

Think: The end is closest to 6.

Answer: The marker is about 6 inches long.

To the nearest whole unit, we pick the closest whole number.

Worked Example 3: Reading to the nearest half unit

Question: A pencil starts at 0 and ends exactly halfway between 4 inches and 5 inches. What is its length?

Think: Halfway between 4 and 5 is \(4\frac{1}{2}\).

Answer: The pencil is \(4\frac{1}{2}\) inches long.

This is more precise than saying just 4 inches or 5 inches.

Worked Example 4: Reading to the nearest quarter unit

Question: A crayon ends at the first quarter mark after 3 inches. What is its length?

Think: The first quarter mark after 3 is \(3\frac{1}{4}\).

Answer: The crayon is \(3\frac{1}{4}\) inches long.

If it ended at the second quarter mark, it would be \(3\frac{1}{2}\). If it ended at the third quarter mark, it would be \(3\frac{3}{4}\).

9. Comparing an estimate and an exact measurement

Suppose you estimate that a book is 8 inches long. Then you measure it and find it is \(7\frac{3}{4}\) inches long.

Your estimate was close. That means it was a good estimate.

Estimates do not have to be perfect. They just need to be reasonable and close.

10. Common mistakes to avoid

• Starting at the end of the ruler instead of the 0 mark
• Using a ruler for something too long, like a room
• Reading the wrong mark
• Forgetting whether the problem asks for whole, half, or quarter units
• Not lining up the object straight

11. Quick check ideas

After you measure, ask:

• Did I use the best tool?
• Did I start at 0?
• Does my answer make sense?
• Did I read to the nearest whole, half, or quarter unit like the problem asked?

Summary

Precision means measuring carefully and closely. Estimation means making a smart guess that is near the real length.

Use a ruler for small objects and a measuring tape for longer objects. Read measurements from the 0 mark and use the nearest whole, half, or quarter unit when needed.

When you estimate first and then measure carefully, you become a stronger and more confident mathematician.

Inverse Relationship of Unit Size

When we measure something, we use units. Units are the pieces we count to find how long, tall, or wide something is.

For example, we might measure with inches or centimeters. Both are units of length.

Here is the big idea for this lesson:

If the unit is smaller, you need more of them. If the unit is larger, you need fewer of them.

This is called the inverse relationship of unit size.

That name sounds big, but the idea is simple. Imagine covering a desk with books.

• If you use small books, you need more books.
• If you use big books, you need fewer books.

Measuring length works the same way.

If you measure the same object with small units like centimeters, the number will be bigger. If you measure the same object with larger units like inches, the number will be smaller.

So for one object:

• smaller unit  bigger number
• larger unit  smaller number

We are not changing the object. We are only changing the size of the unit we use to measure it.

For example, think about a ribbon. The ribbon stays the same length. But the number you say depends on the unit you use.

If the ribbon is measured with centimeters, it may be about 20 centimeters long. If the same ribbon is measured with inches, it may be about 8 inches long.

The ribbon did not shrink or grow. The unit size changed.

Why does this happen?

Measuring means asking, How many of this unit fit along the object?

If the unit is tiny, lots of them fit. If the unit is bigger, not as many fit.

You can think of it like steps across a room.

• With tiny steps, you take more steps.
• With giant steps, you take fewer steps.

The room is still the same size. Only the step size changes.

Important measurement idea: When we compare measurements, we should think about the size of the unit, not just the number.

A bigger number does not always mean a longer object.

For example, 12 centimeters and 12 inches are not the same length, because a centimeter and an inch are not the same size.

Worked Example 1

A pencil is measured two ways:

• 10 inches
• 25 centimeters

Which unit is smaller?

We look at the numbers for the same pencil. The pencil takes 25 centimeters but only 10 inches.

Because it takes more centimeters to measure the same object, centimeters are the smaller unit.

Answer: The centimeter is smaller than the inch.

Worked Example 2

A book is 30 centimeters long. Will the number of inches be more than 30 or less than 30?

Inches are larger than centimeters. Larger units mean fewer units are needed.

So the number of inches must be less than 30.

Answer: Less than 30 inches.

Worked Example 3

Two students measured the same table.

• Mia said the table is 6 hand spans long.
• Leo said the table is 8 hand spans long.

Who probably used the smaller hand span?

The same table took 8 hand spans for Leo and 6 hand spans for Mia.

More units means smaller units.

So Leo probably used the smaller hand span.

Answer: Leo used the smaller unit.

Worked Example 4

Fill in the blanks:

For the same object, if the unit gets smaller, the number of units gets ________.

If the unit gets larger, the number of units gets ________.

We use the main rule:

• smaller unit  more units
• larger unit  fewer units

Answer:

• smaller unit  larger number
• larger unit  smaller number

Let's look at it with a simple picture idea.

Imagine a line that is the same length every time.

If we use big units, only a few fit:

$$\text{same length} = 4\text{ big units}$$

If we use smaller units, more fit:

$$\text{same length} = 8\text{ small units}$$

The length did not change. Only the unit changed.

Things to remember

1. Always compare the same object.
2. Ask, Which unit is bigger?
3. If the unit is smaller, the measurement number is bigger.
4. If the unit is bigger, the measurement number is smaller.

Common mistake

Sometimes students think a bigger number means a bigger object. That is not always true.

Suppose one ribbon is 12 inches long. Another ribbon is 20 centimeters long.

You cannot decide which ribbon is longer by looking only at 12 and 20. The units are different sizes.

When units are different, think carefully about the size of each unit.

Try these thinking questions

• If you measure a marker with paper clips and then with tiny cubes, which measurement will probably be greater? The tiny cubes measurement, because tiny cubes are smaller.
• If a door is 200 centimeters tall, will the number of inches be more or less than 200? Less, because inches are larger than centimeters.
• If two people measure the same string and one gets 9 units while the other gets 5 units, who used the smaller unit? The person who got 9 units.

Summary

The inverse relationship of unit size means that the size of the unit and the number of units move in opposite ways.

When the unit gets smaller, the number gets larger. When the unit gets larger, the number gets smaller.

Remember: for the same object, more units means smaller units, and fewer units means larger units.

Perimeter as Linear Boundary

Have you ever walked all the way around a playground, a garden, or a table? The distance around the outside edge is called the perimeter.

Perimeter is the linear boundary of a shape. That means it measures the length all the way around a flat shape. It is like tracing the outside edge with your finger.

A shape can be big or small, but its perimeter is always found by adding the lengths of its outside sides.

What perimeter means

• Perimeter = the distance around a shape
• We only measure the outside boundary
• Perimeter is measured in length units, such as inches, feet, centimeters, or meters
• We add side lengths to find it

If a shape has side lengths of \(3\), \(4\), \(3\), and \(4\) units, then its perimeter is:

So the perimeter is 14 units.

Important idea: perimeter is about length around, not space inside. We are not counting the inside part of the shape. We are only following the edge.

How to find perimeter

1. Look at the shape.
2. Find the length of each outside side.
3. Add all the outside side lengths together.
4. Write the answer with a length unit.

You can think: around the shape, add the sides.

Worked Example 1: A simple square

A square has 4 sides. Each side is \(5\) inches long.

Add all 4 sides:

The perimeter is 20 inches.

Because all sides of a square are equal, you can also think of it as:

Worked Example 2: A rectangle

A rectangle has a length of \(7\) cm and a width of \(3\) cm.

A rectangle has 2 long sides and 2 short sides, so we add:

The perimeter is 20 cm.

You can also group equal sides:

Worked Example 3: A shape with different side lengths

A shape has side lengths of \(2\) m, \(6\) m, \(4\) m, and \(3\) m.

Add every outside side:

The perimeter is 15 meters.

Even when the sides are different lengths, the job is the same: add all the outside sides.

Worked Example 4: A missing side

A rectangle has one long side labeled \(8\) ft and one short side labeled \(2\) ft.

In a rectangle, opposite sides are equal. So the other long side is also \(8\) ft, and the other short side is also \(2\) ft.

Now add all the sides:

The perimeter is 20 ft.

Tips for success

• Make sure you use only the outside edge.
• Do not count a side twice unless it really appears twice on the shape.
• Check that you added all the sides.
• Always include the unit in your answer.

Common mistakes to avoid

• Adding only 2 sides instead of all the sides
• Forgetting a side on an odd-shaped figure
• Writing square units, like square inches, instead of length units
• Measuring the inside instead of the boundary

Let’s compare

If you put a fence around a yard, you need to know the perimeter because the fence goes around the outside.

If you wanted to cover the yard with grass, that would be about the inside space, not the perimeter.

So remember:

• Perimeter tells how far around
• It is a 1-dimensional length around a 2-dimensional shape

Practice thinking

Ask yourself these questions when finding perimeter:

• What are the outside side lengths?
• Did I add every outside side?
• Did I write the correct unit?

Summary

Perimeter is the total distance around the outside of a shape. To find it, add the lengths of all the outside sides. Perimeter is measured in length units like inches, feet, centimeters, and meters.

Area as a Measure of 2D Space

When we look at a flat shape, we can ask an important question: How much space is inside it? The answer to that question is called area.

Area is the amount of flat, 2D space inside a shape. A shape can be a rectangle, a square, or another closed shape. If the shape is flat like a page, a floor tile, or the top of a desk, we can talk about its area.

We measure area with square units. A square unit is a tiny square that covers part of the shape. We find the area by covering the inside of the shape completely with these same-size squares.

The squares must fit together with no gaps and no overlaps. That helps us measure fairly and correctly.

Here are some examples of square units:

• square inch
• square foot
• square centimeter

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

Think of area like tiling a floor. If you want to know how much floor space there is, you can imagine covering it with square tiles. The number of tiles tells the area.

Main Idea 1: Area is about the inside of a shape

Area measures the space inside the boundary of a shape. It does not tell how long the edges are. It tells how much flat space the shape covers.

For example, a notebook cover has area because it is a flat surface. A tabletop has area. A rug has area. These are all flat spaces that can be covered with square units.

Main Idea 2: Use equal-size square units

To measure area correctly, we use squares that are all the same size. If we used different-size squares, the measurement would not be fair.

Imagine covering one rectangle with big squares and another with tiny squares. The counts would not match unless the square units were the same size. That is why area is always measured with equal square units.

Main Idea 3: Count the square units

One way to find area is to count the number of square units inside a shape.

If a shape has rows of squares, you can count one by one, or you can count the rows to help you. For example, if there are 3 rows with 4 squares in each row, there are 12 square units in all.

That means the area is:

$$12\text{ square units}$$

Main Idea 4: Shapes can be tiled completely

To find area, the inside of the shape should be covered completely. Every part inside the shape must be included.

• No empty spaces
• No squares stacked on top of each other
• No counting squares outside the shape

This is called tiling the shape with square units.

Worked Example 1: Counting squares in a small rectangle

A rectangle is covered with 6 equal square units. What is its area?

Step 1: Count the squares.

There are 6 squares.

Step 2: Write the unit.

The area is:

$$6\text{ square units}$$

Answer: The rectangle has an area of 6 square units.

Worked Example 2: Using rows and columns

A rectangle has 3 rows of square units. Each row has 5 square units. What is the area?

Step 1: Count by rows.

There are 3 rows, and each row has 5 squares.

We can add:

$$5+5+5=15$$

Step 2: Write the area.

$$15\text{ square units}$$

Answer: The area is 15 square units.

Worked Example 3: A square shape

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

Step 1: Count the rows.

There are 4 rows of 4.

We can add:

$$4+4+4+4=16$$

Step 2: Write the area.

$$16\text{ square units}$$

Answer: The square has an area of 16 square units.

Worked Example 4: Finding the area of a larger rectangle

A rectangle is tiled with square units. It has 4 rows and 7 squares in each row. What is the area?

Step 1: Add the squares in all rows.

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

Step 2: Write the area with units.

$$28\text{ square units}$$

Answer: The area is 28 square units.

Tips for Finding Area

• Look only at the space inside the shape.
• Use equal-size square units.
• Make sure the shape is covered with no gaps and no overlaps.
• Count carefully.
• Always write square units after your answer.

Common Mistakes to Avoid

• Counting squares outside the shape
• Forgetting to count every square inside
• Using units like just “inches” instead of “square inches”
• Thinking area means the edge or border of the shape

Let’s Compare

If Shape A has 10 square units inside it, and Shape B has 14 square units inside it, then Shape B has the greater area because it covers more flat space.

If two shapes have the same number of square units inside, then they have the same area, even if they do not look exactly the same.

Real-Life Examples of Area

• How much space a rug covers on the floor
• How much paper is on the front of a poster
• How much space a garden bed covers
• How much tabletop space is inside the edges of a desk

Quick Check

1. A shape is covered by 9 square units. What is its area?
2. A rectangle has 2 rows of 6 square units. What is its area?
3. Why do we use square units to measure area?

Answers to Quick Check

1. $$9\text{ square units}$$
2. $$6+6=12\text{ square units}$$
3. Because area measures flat space inside a shape, and square units can cover that space completely with no gaps or overlaps.

Summary

Area tells us how much 2D flat space is inside a shape. We measure area by covering the shape with equal square units and counting them.

Remember: area is the space inside, not the edge. If you can tile the inside completely with same-size squares and count them, you can find the area.

Connecting Area to Multiplication

We use area to tell how much flat space a shape covers.

When we find the area of a rectangle, we can use multiplication to do it quickly.

This works because a rectangle can be filled with equal-size square units arranged in rows and columns, just like an array.

What is area?

Area is the number of square units needed to cover a shape with no gaps and no overlaps.

If a rectangle is covered by 12 little squares, then its area is 12 square units.

We write square units like this: \(12\text{ square units}\).

How does multiplication help?

Think about an array. An array has rows and columns.

• Rows go across.
• Columns go up and down.

A rectangle can be seen as an array of square units.

If we know how many rows and how many squares are in each row, we can multiply to find the total number of squares.

That total number of squares is the area.

The area rule for rectangles

For rectangles, we multiply the side lengths to find the area.

One side tells us how many squares fit across. The other side tells us how many squares fit down.

So we use:

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

If a rectangle is 4 units long and 3 units wide, then:

$$4 \times 3 = 12$$

The area is \(12\text{ square units}\).

Why square units?

We are counting little squares, not just lines.

That is why area is measured in square units.

For example:

• square inches
• square centimeters
• square feet
• square units

Picture it in your mind

Imagine a rectangle with 3 rows and 5 squares in each row.

You could count:

\(5 + 5 + 5 = 15\)

Or you could multiply:

$$3 \times 5 = 15$$

Both ways give the same area: \(15\text{ square units}\).

Multiplication is faster when there are many squares.

Worked Example 1

A rectangle has 2 rows of 4 squares.

Step 1: Count the rows and columns.

• Rows: 2
• Squares in each row: 4

Step 2: Multiply.

$$2 \times 4 = 8$$

Answer: The area is \(8\text{ square units}\).

Worked Example 2

A rectangle is 3 units long and 6 units wide.

This means it can be covered by 3 rows of 6 square units, or 6 rows of 3 square units.

Step 1: Write the multiplication sentence.

$$3 \times 6 = 18$$

Answer: The area is \(18\text{ square units}\).

Worked Example 3

A rectangle has side lengths 5 units and 5 units.

This is a special rectangle called a square, but we still find the area the same way.

Step 1: Multiply the side lengths.

$$5 \times 5 = 25$$

Answer: The area is \(25\text{ square units}\).

Worked Example 4

A classroom rug is 4 feet by 7 feet.

Step 1: Multiply the side lengths.

$$4 \times 7 = 28$$

Step 2: Use the correct unit.

Because the side lengths are in feet, the area is in square feet.

Answer: The area of the rug is \(28\text{ square feet}\).

Repeated addition and multiplication

Area connects to multiplication because multiplication is a quick way to add equal groups.

If a rectangle has 4 rows with 3 squares in each row, then:

\(3 + 3 + 3 + 3 = 12\)

This is the same as:

$$4 \times 3 = 12$$

So the area is \(12\text{ square units}\).

Important things to remember

• Area tells how much space is inside a flat shape.
• Rectangles can be covered with square units in rows and columns.
• To find the area of a rectangle, multiply the side lengths.
• Always write the answer in square units.

Watch out for these mistakes

• Do not just add the side lengths. For area, we multiply.
• Do not forget the unit. Area needs square units.
• Count rows and columns carefully. The total squares must match the multiplication sentence.

Try thinking about these

1. A rectangle has 3 rows and 4 columns. The area is \(3 \times 4 = 12\text{ square units}\).
2. A rectangle is 6 units by 2 units. The area is \(6 \times 2 = 12\text{ square units}\).
3. Both rectangles have the same area, even though they look different.

This shows that different rectangles can have the same area.

Summary

Area is the number of square units inside a shape.

A rectangle is like an array, with rows and columns of equal squares.

That is why we can multiply side lengths to find area:

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

When you see a rectangle, think: rows times columns or length times width.

Then write your answer with the correct square units.

Distinguishing Area and Perimeter

When we look at a shape, we can measure it in two different ways: by finding its perimeter or by finding its area.

These two ideas are not the same. Sometimes shapes can have the same perimeter but different areas. Sometimes shapes can have the same area but different perimeters.

Let’s learn how to tell them apart.

What is perimeter?

The perimeter is the distance around a shape. If you walked all the way around the edge of a shape, the total distance you walked would be the perimeter.

To find perimeter, we add the lengths of all the sides.

For example, if a rectangle has side lengths 5 units and 3 units, then its perimeter is:

$$5+3+5+3=16$$

So the perimeter is 16 units.

What is area?

The area is the amount of space inside a shape. Area tells how much surface a shape covers.

We often measure area by counting square units. A square unit is a little square that is 1 unit long and 1 unit wide.

For a rectangle, we can find area by multiplying the length and width:

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

If a rectangle has length 5 units and width 3 units, then its area is:

$$5\times 3=15$$

So the area is 15 square units.

Important clue:

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

A good way to remember is:

• Perimeter is like a fence around a yard.
• Area is like the grass covering the yard.

Worked Example 1: Find both perimeter and area

A rectangle is 4 units long and 2 units wide.

Step 1: Find the perimeter.

Add all the sides:

$$4+2+4+2=12$$

The perimeter is 12 units.

Step 2: Find the area.

Multiply length times width:

$$4\times 2=8$$

The area is 8 square units.

Notice that the perimeter and the area have different meanings and different units.

• Perimeter: 12 units
• Area: 8 square units

Worked Example 2: Same perimeter, different areas

Look at these two rectangles:

• Rectangle A: 5 units by 1 unit
• Rectangle B: 3 units by 3 units

Rectangle A perimeter:

$$5+1+5+1=12$$

Rectangle B perimeter:

$$3+3+3+3=12$$

Both rectangles have the same perimeter: 12 units.

Now find the areas.

Rectangle A area:

$$5\times 1=5$$

Rectangle B area:

$$3\times 3=9$$

The areas are different:

• Rectangle A: 5 square units
• Rectangle B: 9 square units

So this proves that two shapes can have the same perimeter but different areas.

Worked Example 3: Same area, different perimeters

Now look at these two rectangles:

• Rectangle C: 4 units by 2 units
• Rectangle D: 8 units by 1 unit

Rectangle C area:

$$4\times 2=8$$

Rectangle D area:

$$8\times 1=8$$

Both rectangles have the same area: 8 square units.

Now find the perimeters.

Rectangle C perimeter:

$$4+2+4+2=12$$

Rectangle D perimeter:

$$8+1+8+1=18$$

The perimeters are different:

• Rectangle C: 12 units
• Rectangle D: 18 units

So this proves that two shapes can have the same area but different perimeters.

Worked Example 4: How to decide whether to use area or perimeter

Read the problem carefully.

1. A gardener wants to put a border all the way around a flower bed.
2. A gardener wants to cover the flower bed with soil.

For problem 1, we use perimeter because the border goes around the shape.

For problem 2, we use area because the soil covers the inside of the shape.

Tips for telling them apart

• If the question asks for the distance around a shape, find perimeter.
• If the question asks how much space is inside a shape, find area.
• Perimeter is measured in units.
• Area is measured in square units.

Let’s compare what we learned

Why this matters

Shapes can look different and still share one measurement.

A long, skinny rectangle and a more square-shaped rectangle might have the same perimeter, but the space inside can be very different.

Also, two shapes can cover the same amount of space, but one shape may need more border around it than the other.

That is why we must always ask: Am I measuring around the shape, or inside the shape?

Brief Summary

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

Shapes can have the same perimeter but different areas, and they can also have the same area but different perimeters.

Remember: around = perimeter and inside = area.

Mass vs. Liquid Volume

We measure many things every day. We might measure how heavy something is, or we might measure how much liquid fits inside a container.

These are not the same kind of measurement. In this lesson, you will learn the difference between mass and liquid volume, the units we use, and how to tell which one to measure.

Mass tells how heavy an object is. We use mass for things like an apple, a book, or a rock.

Liquid volume tells how much liquid a container can hold. We use liquid volume for things like water, milk, or juice.

Main Idea: If you are measuring a solid object, think about mass. If you are measuring a liquid in a container, think about liquid volume.

Units for Mass

• grams (g) are used for lighter objects.
• kilograms (kg) are used for heavier objects.

Examples of mass:

• A paper clip might have a mass of a few grams.
• A watermelon might have a mass of a few kilograms.

Units for Liquid Volume

• milliliters (mL) are used for small amounts of liquid.
• liters (L) are used for larger amounts of liquid.

Examples of liquid volume:

• A small juice box might hold a few hundred milliliters.
• A large bottle of water might hold 1 liter or more.

How to Tell the Difference

1. Ask: Am I measuring how heavy something is?
   If yes, measure mass.
2. Ask: Am I measuring how much liquid fits in a container?
   If yes, measure liquid volume.

Look at these clues:

• Mass: apple, toy, backpack, dog food bag, rock
• Liquid volume: cup of milk, bottle of water, fish tank, soup pot, juice carton

Choosing the Right Unit

After you decide whether you need mass or liquid volume, choose the best unit.

• Use grams for light solid objects.
• Use kilograms for heavy solid objects.
• Use milliliters for small amounts of liquid.
• Use liters for large amounts of liquid.

Quick Think:

• A feather is very light, so grams make sense.
• A sack of potatoes is heavy, so kilograms make sense.
• A spoonful of cough syrup is a small amount of liquid, so milliliters make sense.
• A bucket of water holds a lot of liquid, so liters make sense.

Worked Example 1

Question: A banana is a solid object. Should you measure it in grams, kilograms, milliliters, or liters?

Step 1: A banana is not a liquid. It is a solid object, so we measure mass.

Step 2: A banana is not very heavy, so we use grams.

Answer: grams (g)

Worked Example 2

Question: A jug of lemonade is being filled. Should you measure it in grams, kilograms, milliliters, or liters?

Step 1: Lemonade is a liquid, so we measure liquid volume.

Step 2: A jug holds a lot more than a tiny amount, so liters is a good unit.

Answer: liters (L)

Worked Example 3

Question: Which is the better unit for a coin: grams or liters?

Step 1: A coin is a solid object, not a liquid.

Step 2: So we measure mass, not liquid volume.

Step 3: A coin is small and light, so grams is the better unit.

Answer: grams (g)

Worked Example 4

Question: A bottle has $$500\text{ mL}$$ of water. Is this mass or liquid volume?

Step 1: The unit is mL, which means milliliters.

Step 2: Milliliters measure liquid volume.

Answer: It is measuring liquid volume.

Helpful Comparisons

• Mass = how heavy
• Liquid volume = how much liquid fits inside
• g and kg = mass units
• mL and L = liquid volume units

Be Careful!

Sometimes a container is a solid object, but what you are measuring is the liquid inside it.

For example, a bottle is an object, but if you ask, “How much water does the bottle hold?” you are measuring liquid volume, not the mass of the bottle.

Also, do not mix the units:

• Do not measure milk in grams or kilograms when the question is asking how much liquid it holds.
• Do not measure a toy car in milliliters or liters.

Try Thinking About These

• A bag of rice: mass
• A cup of soup: liquid volume
• A puppy: mass
• A carton of juice: liquid volume

Summary

Mass measures how heavy something is. We use grams and kilograms for mass.

Liquid volume measures how much liquid a container can hold. We use milliliters and liters for liquid volume.

To choose the right measurement, ask yourself: Is it a solid object, or am I measuring liquid? That question will help you pick the correct kind of unit every time.

Reading Analog and Digital Time

We use clocks every day to know what time it is. Time helps us know when to wake up, go to school, eat lunch, and go to bed.

There are two common ways to show time:

• Analog clocks have a clock face with numbers and moving hands.
• Digital clocks show the time with numbers, like 3:15.

In this lesson, you will learn how to read both kinds of clocks and how to match an analog clock to a digital time.

Parts of an Analog Clock

An analog clock usually has numbers from 1 to 12 around a circle. It also has two main hands:

• The short hand is the hour hand. It tells the hour.
• The long hand is the minute hand. It tells the minutes.

Some clocks also have a very thin second hand, but for now we will focus on the hour hand and minute hand.

Reading the Hour Hand

The hour hand moves slowly from one number to the next. If the short hand is pointing right at 4, the hour is 4 o'clock.

If the hour hand is between two numbers, the hour is the smaller number it has already passed.

For example, if the short hand is between 7 and 8, the hour is still 7.

Reading the Minute Hand

The minute hand is the long hand. It moves faster around the clock.

When the minute hand points to numbers on the clock, we count by 5s:

• At 1 = 5 minutes
• At 2 = 10 minutes
• At 3 = 15 minutes
• At 4 = 20 minutes
• At 5 = 25 minutes
• At 6 = 30 minutes
• At 7 = 35 minutes
• At 8 = 40 minutes
• At 9 = 45 minutes
• At 10 = 50 minutes
• At 11 = 55 minutes
• At 12 = 0 minutes, or :00

You can think of the clock as having 60 minutes all the way around.

Each number is worth:

$$12 \times 5 = 60$$

How to Read an Analog Clock

1. Look at the short hand to find the hour.
2. Look at the long hand to find the minutes.
3. Put them together as hour:minute.

Be careful: if the minute hand is not at 12, the hour hand may be between two numbers. That is normal.

Reading a Digital Clock

A digital clock shows time with numbers, like 8:27.

• The number before the colon tells the hour.
• The number after the colon tells the minutes.

For example:

• 4:00 means 4 o'clock.
• 6:15 means 15 minutes after 6.
• 9:42 means 42 minutes after 9.

When the minutes are less than 10, a digital clock uses a zero first.

For example:

• 2:05 means 5 minutes after 2, not 2:5.
• 7:08 means 8 minutes after 7.

Matching Analog Time to Digital Time

To change an analog clock into digital time:

1. Read the hour hand.
2. Read the minute hand.
3. Write the time with a colon.

Example: If the hour hand is just past 3 and the minute hand points to 4, the minutes are 20. The digital time is 3:20.

To change digital time into an analog clock:

1. Put the minute hand on the correct minute mark.
2. Put the hour hand at the hour, or a little past it if minutes are more than 0.

Example: For 5:30, the minute hand points to 6 because that is 30 minutes. The hour hand goes halfway between 5 and 6.

Knowing AM and PM

A day has 24 hours, but many clocks use 12 hours two times each day.

• AM is from midnight to before noon.
• PM is from noon to before midnight.

You can remember:

• AM is usually when we sleep, wake up, eat breakfast, and go to school.
• PM is usually when we eat dinner, play after school, and go to bed in the evening.

Examples:

• 7:00 AM might be breakfast time.
• 1:00 PM might be after lunch.
• 8:00 PM might be bedtime for some children.

Special Minute Times

Some times show up often on clocks:

• :00 means exact hour
• :15 means 15 minutes after the hour
• :30 means 30 minutes after the hour
• :45 means 45 minutes after the hour

These are easy to spot on an analog clock because:

• :00 minute hand at 12
• :15 minute hand at 3
• :30 minute hand at 6
• :45 minute hand at 9

Worked Example 1

An analog clock has the short hand on 2 and the long hand on 12.

Step 1: The hour hand says 2.

Step 2: The minute hand at 12 means 00 minutes.

Answer: The time is 2:00.

Worked Example 2

An analog clock has the short hand a little past 4 and the long hand on 3.

Step 1: The hour is 4.

Step 2: The minute hand on 3 means 15 minutes.

Answer: The time is 4:15.

Worked Example 3

An analog clock has the short hand between 7 and 8 and the long hand on 8.

Step 1: The hour hand has passed 7, so the hour is 7.

Step 2: The minute hand on 8 means 40 minutes.

Answer: The time is 7:40.

Worked Example 4

A digital clock says 9:05 PM.

Step 1: The hour is 9.

Step 2: The minutes are 05, so the minute hand points to the 5-minute mark.

Step 3: The hour hand is just a little past 9.

Step 4: Because it says PM, it is at night, not in the morning.

Answer: The analog clock would show the minute hand on 1 and the hour hand just past 9.

Tips to Help You

• Read the hour hand first, then the minute hand.
• Count minutes by 5s around the clock.
• If the hour hand is between numbers, use the smaller number.
• On digital clocks, always read both numbers.
• Watch for AM and PM so you know morning or afternoon/evening.

Quick Check

• If the minute hand points to 6, the minutes are 30.
• If the minute hand points to 9, the minutes are 45.
• If a digital clock says 3:00, the minute hand on an analog clock points to 12.
• If a digital clock says 6:25, the minute hand points to the 5 because that is 25 minutes.

Summary

An analog clock uses a short hour hand and a long minute hand. A digital clock uses numbers written as hour:minute.

To read analog time, find the hour from the short hand and the minutes from the long hand. To read digital time, read the hour before the colon and the minutes after it.

Remember to count the minute marks by 5s and to notice whether the time is AM or PM.

Calculating Elapsed Time means finding out how much time passes from a start time to an end time.

We use elapsed time every day. We might ask:

• How long is school?
• How much time is left until dinner?
• How long did a movie last?

In this lesson, you will learn how to find elapsed time by using an open number line. An open number line helps us count forward from the start time to the end time in easy jumps.

What is elapsed time?

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

For example, if you start reading at 2:00 and stop at 3:00, the elapsed time is 1 hour.

How an open number line helps

An open number line for time is like a path. You begin at the start time and make jumps until you reach the end time.

You can make jumps by:

• hours
• half-hours
• groups of minutes, like 5 minutes, 10 minutes, or 15 minutes

After that, you add all the jumps together to find the total elapsed time.

Steps for finding elapsed time

1. Write the start time.
2. Write the end time.
3. Start at the start time on an open number line.
4. Jump forward by hours first, if that helps.
5. Then jump forward by groups of minutes.
6. Add all the jumps together.

Helpful idea: It is often easiest to jump to the next hour first. Then keep going until you reach the end time.

Worked Example 1

A game starts at 1:00 and ends at 3:00. How much time has passed?

Start at 1:00. Jump to 2:00. That is 1 hour.

Then jump from 2:00 to 3:00. That is 1 more hour.

Open number line jumps:

1:00 → 2:00 → 3:00

Add the jumps:

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

Answer: The game lasted 2 hours.

Worked Example 2

Art class starts at 9:15 and ends at 10:00. How much time has passed?

Start at 9:15. Jump to 10:00.

From 9:15 to 10:00 is 45 minutes.

You can think of it as:

• 9:15 to 9:30 = 15 minutes
• 9:30 to 10:00 = 30 minutes

Add the minutes:

$$15\text{ minutes} + 30\text{ minutes} = 45\text{ minutes}$$

Answer: The art class lasted 45 minutes.

Worked Example 3

Sam starts homework at 4:20 and finishes at 6:05. How much time has passed?

We will use an open number line and make friendly jumps.

Start at 4:20.

• Jump from 4:20 to 5:00 = 40 minutes
• Jump from 5:00 to 6:00 = 1 hour
• Jump from 6:00 to 6:05 = 5 minutes

Now add the jumps.

First add the minutes:

$$40\text{ minutes} + 5\text{ minutes} = 45\text{ minutes}$$

Then include the hour:

$$1\text{ hour } 45\text{ minutes}$$

Answer: Sam worked for 1 hour 45 minutes.

Worked Example 4

A movie begins at 2:35 and ends at 5:10. How much time has passed?

Start at 2:35.

• 2:35 to 3:00 = 25 minutes
• 3:00 to 4:00 = 1 hour
• 4:00 to 5:00 = 1 hour
• 5:00 to 5:10 = 10 minutes

Now add the hours and minutes.

Hours:

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

Minutes:

$$25\text{ minutes} + 10\text{ minutes} = 35\text{ minutes}$$

So the total elapsed time is:

$$2\text{ hours } 35\text{ minutes}$$

Answer: The movie lasted 2 hours 35 minutes.

Tips for success

• Start at the beginning time and count forward.
• Use easy jumps, like to the next hour.
• Keep hours and minutes organized.
• Add the hours together and the minutes together.
• Check that your ending jump lands exactly on the end time.

Things to watch out for

• Do not count backward unless your teacher asks you to.
• Do not forget to include all jumps.
• Be careful when the start time has minutes, like 4:20 or 2:35.

Try thinking about it this way

If the clock does not start right on the hour, first jump to the next full hour. Then count by hours. At the end, count any extra minutes.

That makes elapsed time easier to see.

Summary

Elapsed time tells how long something lasts. To find it, start at the beginning time and jump forward on an open number line.

Use jumps by hours and groups of minutes. Then add all the jumps together to find the total time passed.