Mathematics · 1st Grade · The Power of Ten and Place Value · Quarter 2

Representing Numbers with Place Value

Students represent two-digit numbers using base-ten blocks, drawings, and expanded form.

• Common Core State Standards
• CCSS.Math.Content.1.NBT.B.2a
• CCSS.Math.Content.1.NBT.B.2b

About This Topic

Mental math in first grade focuses on flexibility with numbers within 100. Students use place value strategies to add a two-digit number and a one-digit number, and a two-digit number and a multiple of ten. They also learn to mentally find ten more or ten less than a given number. This aligns with Common Core standards for mental computation and understanding properties of operations.

The goal is not rote memorization but the development of strategies like 'counting on by tens' or 'decomposing to make a ten.' For example, to solve 38 + 7, a student might think 38 + 2 = 40, and 40 + 5 = 45. Students grasp this concept faster through structured discussion and peer explanation where they can 'see' different ways to arrive at the same sum.

Key Questions

1. Explain how the position of a digit determines its value.
2. Differentiate between the digit '1' in the number 12 and the number 21.
3. Construct a number in multiple ways (e.g., 3 tens and 4 ones, or 34 ones).

Learning Objectives

• Represent two-digit numbers using base-ten blocks and drawings.
• Write two-digit numbers in expanded form.
• Explain how the position of a digit affects its value in a two-digit number.
• Compare the value of the same digit when it appears in different positions within a two-digit number.

Before You Start

Counting to 100

Why: Students need to be able to count reliably to 100 to understand the quantities represented by two-digit numbers.

Identifying Digits

Why: Students must be able to recognize and name individual digits (0-9) to understand their role in forming numbers.

Key Vocabulary

Tens	A group of ten ones. In a two-digit number, the tens digit tells us how many groups of ten we have.
Ones	Individual units. In a two-digit number, the ones digit tells us how many individual units we have left after making as many tens as possible.
Place Value	The value of a digit based on its position within a number. For example, in the number 23, the '2' has a value of 20 (tens) and the '3' has a value of 3 (ones).
Expanded Form	Writing a number to show the value of each digit. For example, 47 in expanded form is 40 + 7.

Watch Out for These Misconceptions

Common Misconception

Trying to use the standard algorithm (stacking) without understanding place value.

What to Teach Instead

Students often misalign columns or forget to 'carry.' Encouraging horizontal mental strategies and using base-ten blocks to model the 'composition' of a new ten prevents these procedural errors.

Common Misconception

Changing both digits when adding ten (e.g., 34 + 10 = 45).

What to Teach Instead

Students may think 'adding' means everything gets bigger. Using a 100-chart during a 'Gallery Walk' helps them see that moving down one row only changes the tens place.

Active Learning Ideas

Think-Pair-Share: Strategy Share-Out

Project a problem like 45 + 20. Give students 30 seconds of quiet 'think time' to solve it mentally. Then, they share their specific mental path with a partner (e.g., 'I just added 2 to the tens place').

15 min·Pairs

Inquiry Circle: The Ten-More Path

Groups are given a 100-chart and a set of 'Ten More/Ten Less' cards. They must move a game piece across the board based on the cards, explaining the pattern they see in the digits as they move vertically.

20 min·Small Groups

Role Play: The Human Number Rack

Ten students stand in a row. When adding 'ten more,' a whole new row of ten students joins them. This physical representation helps the class visualize how the tens digit increases while the ones stay the same.

20 min·Whole Class

Real-World Connections

• Cashiers at a grocery store use place value when counting out change. For instance, to give 37 cents back, they might count 3 dimes (tens) and 7 pennies (ones).
• Construction workers building with LEGO bricks might group bricks into tens to quickly estimate the total number of bricks needed for a large project, understanding that a stack of 10 bricks is different from 10 individual bricks.

Assessment Ideas

Exit Ticket

Give students a card with a two-digit number, like 52. Ask them to draw base-ten blocks to represent it, write it in expanded form (50 + 2), and then answer: 'What is the value of the digit 5 in this number?'

Quick Check

Display two numbers, such as 17 and 71. Ask students to hold up fingers to show how many tens and ones are in each number. Then, ask: 'Which number has more tens? Which number has more ones? Which digit has a greater value in 71?'

Discussion Prompt

Present the number 34. Ask students to explain in their own words how they know it means 3 tens and 4 ones. Prompt them to compare this to the number 43: 'How is 34 different from 43? What does the position of the digits tell us?'

Frequently Asked Questions

What does 'decomposing' a number mean?

Decomposing is breaking a number into smaller parts to make it easier to work with. For example, in 25 + 8, you might decompose 8 into 5 and 3, so you can add 25 + 5 to get 30, then add the remaining 3.

Why is 'ten more/ten less' so important?

It builds the foundation for mental addition and subtraction. It helps students understand the structure of the base-ten system and prepares them for adding and subtracting larger numbers in second grade.

How can I encourage students to use different strategies?

Value the process over the answer. During 'Number Talks,' highlight three different ways students solved the same problem. This shows the class that math is about flexible thinking, not just one 'right' way.

How can active learning help students understand mental addition?

Active learning turns mental math into a social and visual experience. When students explain their 'mental shortcuts' to a partner, they are forced to organize their thoughts. Using strategies like 'Station Rotations' with different manipulatives allows them to test their mental theories in a low-stakes, collaborative environment.