Learn on PengiIllustrative Mathematics, Grade 5Chapter 6: Place Value Patterns and Decimal Operations

Lesson 2: Metric Conversion with Powers of Ten

In this Grade 5 Illustrative Mathematics lesson (Chapter 6, Lesson 2), students learn to convert metric length units — meters to centimeters, millimeters, and kilometers — by multiplying and dividing with powers of 10 (10, 100, and 1,000). Students explore place value patterns, such as how digits shift left when multiplying by a power of 10, connecting decimal multiplication to real-world measurement contexts like track and field. The lesson addresses standards 5.MD.A.1 and 5.NBT.A.2.

Section 1

Exponential Notation

Property

$a^n$ means multiply $a$ by itself, $n$ times.

\underbrace{a \cdot a \cdot a \cdots a}_{n\ \text{factors}}

The expression $a^n$ is read as $a$ to the $n$ th power. In this notation, $a$ is the base and $n$ is the exponent.

Examples

The expression $4^3$ means $4 \cdot 4 \cdot 4$ , which simplifies to $64$ . Here, 4 is the base and 3 is the exponent.
The product $y \cdot y \cdot y \cdot y \cdot y$ can be written in exponential notation as $y^5$ .
When evaluating $2^x$ for $x=4$ , we substitute to get $2^4$ , which means $2 \cdot 2 \cdot 2 \cdot 2 = 16$ .

Explanation

Exponential notation is a powerful shorthand for writing repeated multiplication. Instead of writing a long string of numbers, you can use a base and an exponent to represent the same value much more concisely.

Section 2

Powers of 10 and Exponents

Property

A power of 10 can be written in exponential form, $10^n$ , where the exponent $n$ indicates the number of zeros that follow the 1.
This is equivalent to multiplying 10 by itself $n$ times: $10^n = \underbrace{10 \times 10 \times \dots \times 10}_{n \text{ times}}$ .

Examples

Section 3

Metric Units of Length

Property

Metric Units of Length:

1 \text{ centimeter} = 10 \text{ millimeters}

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

1 \text{ kilometer} = 1000 \text{ meters}

The conversion factors in the metric system are all powers of 10.
This property makes the metric system easy to use, because we can convert between units simply by moving the decimal point.

Examples

A table is 2.5 meters long. To convert this to centimeters, you move the decimal two places to the right: $2.5 \text{ m} = 250 \text{ cm}$ .

A small insect measures 6 millimeters long. In centimeters, this is $6 \div 10 = 0.6$ centimeters.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Exponential Notation

Property

$a^n$ means multiply $a$ by itself, $n$ times.

\underbrace{a \cdot a \cdot a \cdots a}_{n\ \text{factors}}

The expression $a^n$ is read as $a$ to the $n$ th power. In this notation, $a$ is the base and $n$ is the exponent.

Examples

The expression $4^3$ means $4 \cdot 4 \cdot 4$ , which simplifies to $64$ . Here, 4 is the base and 3 is the exponent.
The product $y \cdot y \cdot y \cdot y \cdot y$ can be written in exponential notation as $y^5$ .
When evaluating $2^x$ for $x=4$ , we substitute to get $2^4$ , which means $2 \cdot 2 \cdot 2 \cdot 2 = 16$ .

Explanation

Section 2

Powers of 10 and Exponents

Property

Examples

Section 3

Metric Units of Length

Property

Metric Units of Length:

1 \text{ centimeter} = 10 \text{ millimeters}

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

1 \text{ kilometer} = 1000 \text{ meters}

The conversion factors in the metric system are all powers of 10.
This property makes the metric system easy to use, because we can convert between units simply by moving the decimal point.

Examples

A table is 2.5 meters long. To convert this to centimeters, you move the decimal two places to the right: $2.5 \text{ m} = 250 \text{ cm}$ .

A small insect measures 6 millimeters long. In centimeters, this is $6 \div 10 = 0.6$ centimeters.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Exponential Notation

Property

$a^n$ means multiply $a$ by itself, $n$ times.

\underbrace{a \cdot a \cdot a \cdots a}_{n\ \text{factors}}

The expression $a^n$ is read as $a$ to the $n$ th power. In this notation, $a$ is the base and $n$ is the exponent.

Examples

The expression $4^3$ means $4 \cdot 4 \cdot 4$ , which simplifies to $64$ . Here, 4 is the base and 3 is the exponent.
The product $y \cdot y \cdot y \cdot y \cdot y$ can be written in exponential notation as $y^5$ .
When evaluating $2^x$ for $x=4$ , we substitute to get $2^4$ , which means $2 \cdot 2 \cdot 2 \cdot 2 = 16$ .

Explanation

Section 2

Powers of 10 and Exponents

Property

Examples

Section 3

Metric Units of Length

Property

Metric Units of Length:

1 \text{ centimeter} = 10 \text{ millimeters}

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

1 \text{ kilometer} = 1000 \text{ meters}

The conversion factors in the metric system are all powers of 10.
This property makes the metric system easy to use, because we can convert between units simply by moving the decimal point.

Examples

A table is 2.5 meters long. To convert this to centimeters, you move the decimal two places to the right: $2.5 \text{ m} = 250 \text{ cm}$ .

A small insect measures 6 millimeters long. In centimeters, this is $6 \div 10 = 0.6$ centimeters.