Mathematics · Year 6 · Proportional Reasoning and Parts · Term 1

Mastering Decimal Multiplication and Division

Multiplying and dividing decimals by whole numbers and powers of ten.

• ACARA Content Descriptions
• AC9M6N06

About This Topic

Decimal multiplication and division involve multiplying and dividing decimals by whole numbers and powers of ten. This topic, aligned with AC9M6N06, focuses on the movement of digits within the place value system. Students learn that multiplying by 10, 100, or 1000 shifts digits to the left, while dividing shifts them to the right. This understanding is crucial for metric conversions and scientific calculations.

Students also explore how to multiply a decimal by a small whole number, often using estimation to ensure their answer is reasonable. In an Australian context, this might involve calculating the cost of multiple items at a grocery store or measuring lengths for a construction project. Students grasp this concept faster through structured discussion and peer explanation of place value shifts.

Key Questions

1. What happens to the value of a digit when we multiply a decimal by ten?
2. How can estimation help us check the placement of a decimal point in a product?
3. Why does dividing by a number less than one result in a larger quotient?

Learning Objectives

• Calculate the product when multiplying a decimal by a whole number, demonstrating understanding of place value shifts.
• Explain the effect of multiplying a decimal by powers of ten (10, 100, 1000) on the position of digits.
• Divide a decimal by a whole number, accurately placing the decimal point in the quotient.
• Compare the results of dividing a decimal by powers of ten (10, 100, 1000) and describe the pattern of digit movement.
• Estimate the product of a decimal and a whole number to verify the reasonableness of a calculated answer.

Before You Start

Understanding Place Value of Whole Numbers

Why: Students need a solid foundation in place value for whole numbers to extend this understanding to decimals.

Introduction to Decimals

Why: Students must be familiar with decimal notation and the relative values of digits in decimal places (tenths, hundredths, etc.).

Basic Multiplication and Division Facts

Why: Fluency with basic multiplication and division facts is necessary to perform the calculations accurately.

Key Vocabulary

Decimal point	A symbol used to separate the whole number part of a number from its fractional part. Its position indicates the value of the digits around it.
Place value	The value of a digit based on its position within a number. For example, in 3.45, the 4 is in the tenths place and the 5 is in the hundredths place.
Power of ten	A number that can be expressed as 10 multiplied by itself a certain number of times, such as 10 (10¹), 100 (10²), or 1000 (10³).
Quotient	The result obtained when one number is divided by another. In decimal division, correctly placing the decimal point in the quotient is essential.
Product	The result of multiplying two or more numbers. When multiplying decimals, understanding place value helps determine the correct position of the decimal point in the product.

Watch Out for These Misconceptions

Common Misconception

Multiplying always makes a number bigger.

What to Teach Instead

While true for whole numbers, multiplying by a decimal less than one (like 0.5) makes the number smaller. Use area models to show that 'half of a half' is a smaller piece.

Common Misconception

You just 'move the decimal point'.

What to Teach Instead

This shortcut often leads to errors. Teach that the *digits* move through place value columns while the decimal point remains fixed. Using place value mats helps clarify this movement.

Active Learning Ideas

Stations Rotation: Place Value Sliders

Students use physical 'sliders' to move digits across a place value chart as they multiply or divide by powers of ten. They record the 'before' and 'after' positions of the decimal point.

40 min·Small Groups

Think-Pair-Share: The Estimation Check

Before solving a decimal multiplication problem, students must estimate the answer. They share their estimation strategy with a partner (e.g., rounding 4.9 to 5) to predict where the decimal point should go.

15 min·Pairs

Inquiry Circle: Currency Converter

Groups are given 'exchange rates' for different Pacific currencies. They must multiply and divide decimals to convert Australian dollars into other currencies for a hypothetical travel trip.

45 min·Small Groups

Real-World Connections

• A baker calculating the total cost of ingredients for multiple batches of cookies. For example, if one batch requires 1.75 kg of flour and they need to make 4 batches, they must multiply 1.75 by 4.
• A sports statistician analyzing player performance data. They might divide a total score by the number of games played to find an average score per game, involving decimal division.
• Engineers converting measurements from meters to centimeters or millimeters for construction blueprints. This involves multiplying decimal values by powers of ten, like 100 or 1000.

Assessment Ideas

Quick Check

Present students with the calculation 3.45 x 10. Ask them to write the answer and explain in one sentence what happened to the digits. Then, ask them to write 12.68 ÷ 100 and explain the digit movement.

Exit Ticket

Give students two problems: 1. Calculate 4.8 x 3. Ask them to show their work and then estimate to check their answer. 2. Calculate 15.75 ÷ 5. Ask them to write the quotient and circle the digit in the tenths place.

Discussion Prompt

Pose the question: 'Why does multiplying 0.5 by 10 give you 5, but dividing 5 by 0.1 also gives you 50?' Facilitate a discussion where students explain the inverse relationship and the impact of multiplying/dividing by numbers greater or less than one, focusing on decimal place value.

Frequently Asked Questions

How can active learning help students understand decimal operations?

Active learning using place value mats and physical digit cards allows students to see the 'shift' that occurs during multiplication and division. Instead of memorising rules about moving dots, they experience the change in value. Peer-led 'estimation duels' also encourage students to think critically about the reasonableness of their answers before they even pick up a pencil.

What is the best way to teach division of decimals by 10?

Use a place value chart. Show that each digit moves one place to the right, making its value ten times smaller.

Why is estimation important with decimals?

It is the best way to catch errors in decimal point placement. If a student calculates 5.1 x 3 and gets 153, an estimate of 5 x 3 = 15 immediately shows the error.

How do decimals connect to the metric system?

The metric system is base-10, just like decimals. Converting centimeters to meters is simply a matter of dividing by 100, which is a key Year 6 skill.