Mathematics · Grade 5 · The Power of Place: Large Numbers and Decimals · Term 1

Understanding Place Value to Millions

Students will investigate the structure of the base ten system for whole numbers up to millions, identifying the value of each digit.

• Ontario Curriculum Expectations
• 5.NBT.A.1

About This Topic

In Grade 5, students expand their understanding of the base ten system to include numbers up to 1,000,000 and decimals to hundredths. This topic focuses on the multiplicative relationship between places, where each position to the left is ten times greater and each position to the right is one tenth the value. This is a foundational shift from simply identifying place names to understanding the proportional reasoning that governs our number system. In Ontario, this aligns with the Number strand expectations for representing and comparing whole numbers and decimal tenths and hundredths.

Understanding the infinite scale of base ten allows students to see the symmetry around the ones place. They begin to realize that the decimal point is a fixed marker that separates whole units from fractional parts. This conceptual clarity is vital for later work with scientific notation and metric conversions. This topic comes alive when students can physically model the patterns using base ten blocks and interactive place value mats to see the physical growth and shrinkage of values.

Key Questions

1. Explain how the position of a digit determines its value in large numbers.
2. Compare the value of a digit in the thousands place versus the hundred thousands place.
3. Analyze how grouping by tens simplifies the representation of large quantities.

Learning Objectives

• Compare the value of a digit in the millions place to its value in the thousands place.
• Explain how the base ten system uses powers of ten to represent numbers up to one million.
• Identify the place value of any digit in a whole number up to one million.
• Analyze how regrouping ten units of one place value creates one unit of the next higher place value.
• Represent whole numbers up to one million using base ten blocks or place value charts.

Before You Start

Place Value to Thousands

Why: Students need a solid understanding of place value up to the thousands place before extending it to millions.

Representing Whole Numbers

Why: Prior experience with representing numbers using base ten blocks or expanded form is essential for understanding the structure of larger numbers.

Key Vocabulary

Place Value	The value of a digit in a number, determined by its position within the number. For example, in 345, the digit 4 has a value of 40 because it is in the tens place.
Base Ten System	A number system that uses ten digits (0-9) and groups quantities in sets of ten. Each place value is ten times greater than the place value to its right.
Millions	The place value representing one thousand thousands, or 1,000,000. It is the seventh digit from the right in a whole number.
Regrouping	The process of exchanging units from one place value for an equivalent number of units in an adjacent place value, such as exchanging ten ones for one ten.

Watch Out for These Misconceptions

Common Misconception

Thinking that the decimal point moves during multiplication or division by ten.

What to Teach Instead

Teach students that the decimal point is a fixed anchor. Use place value sliders to show that the digits shift positions while the decimal remains stationary. Peer discussion helps students articulate that it is the value of the digits changing, not the point itself.

Common Misconception

Believing that 'longer' decimals are always larger in value (e.g., 0.125 is greater than 0.5).

What to Teach Instead

Use base ten grids to compare values visually. Collaborative sorting activities where students justify their ranking of decimals help them focus on the value of the tenths place rather than the number of digits.

Active Learning Ideas

Stations Rotation: The Human Place Value Chart

Students move through stations where they act as digits in a large floor-sized place value chart. At one station, they must physically shift positions when the 'multiplier' calls out a power of ten, observing how their value changes. Other stations involve using digital tools to zoom into number lines between 0 and 1.

45 min·Small Groups

Inquiry Circle: The Million Dollar Walk

Groups use base ten blocks to model 1, 10, 100, and 1,000. They then work together to calculate and describe the physical size of a block representing 1,000,000. They present their findings using a gallery walk format to compare different visualization strategies.

60 min·Small Groups

Think-Pair-Share: Decimal Symmetry

Students examine a place value chart and discuss why there is no 'oneths' place. They work in pairs to find the 'mirror' of the tens place (tenths) and the hundreds place (hundredths). They share their theories with the class to build a collective understanding of the decimal point as an anchor.

20 min·Pairs

Real-World Connections

• City planners use large numbers up to millions to describe population statistics, budgets for infrastructure projects like new highways, and census data for urban development.
• Financial institutions track account balances, loan amounts, and stock market values that often extend into the millions, requiring precise understanding of place value for accuracy.
• Scientists recording data from astronomical observations might use numbers in the millions to represent distances to stars or the number of galaxies in a cluster.

Assessment Ideas

Exit Ticket

Provide students with the number 7,452,916. Ask them to: 1. Write the value of the digit 5. 2. Write the place value of the digit 4. 3. Explain in one sentence how the value of the digit 7 compares to the value of the digit 5.

Quick Check

Display a large number on the board, such as 3,805,124. Ask students to hold up fingers to indicate the place value of a specified digit (e.g., 'Show me the place value of the 8'). Then, ask them to write the value of that digit on a mini-whiteboard.

Discussion Prompt

Pose the question: 'Imagine you have 10,000 ones blocks. How many ten thousands blocks would you need to represent the same quantity? Explain your reasoning using the concept of place value.'

Frequently Asked Questions

How do I explain the relationship between places to Grade 5 students?

Focus on the 'ten times' rule. Show that moving one space left is multiplying by ten, and one space right is dividing by ten. Use concrete materials like base ten blocks so they can see that ten rods make one flat, and ten flats make one cube. This visual evidence makes the abstract concept of powers of ten much more accessible.

What is the best way to introduce decimals to hundredths?

Connect decimals to the Canadian dollar. Students already understand that 100 cents make one dollar. By framing hundredths as cents and tenths as dimes, you provide a familiar cultural context. Using play money in collaborative shopping simulations allows students to practice these exchanges in a low-stakes, practical environment.

How can active learning help students understand place value?

Active learning turns abstract numbers into physical relationships. Strategies like 'Human Place Value' require students to move their bodies to represent shifting values, which reinforces the directional nature of multiplication and division. Collaborative problem-solving encourages students to explain their reasoning out loud, which helps solidify their understanding of the proportional relationships between digits.

Why is the ones place the center of the number system?

The ones place is the starting point for both whole numbers and decimals. It represents a single whole unit. In Ontario's curriculum, we emphasize that the decimal point always sits to the right of the ones place. Helping students see the ones place as the 'balance point' helps them understand the symmetry of tens/tenths and hundreds/hundredths.