Mathematics · 2nd Grade · Algebraic Thinking: Patterns and Equations · Weeks 19-27

Using Models for Subtraction within 100

Students use concrete models or drawings and strategies based on place value to subtract within 100, including decomposing a ten.

• Common Core State Standards
• CCSS.Math.Content.2.NBT.B.5

About This Topic

Subtraction within 100 using concrete models is a direct parallel to addition, but the cognitive demand is higher because decomposing a ten feels less intuitive than composing one. In the US second-grade curriculum, students use base-ten blocks to physically break apart a ten-rod into ten ones when the ones place of the minuend is too small. This makes the borrowing process visible and meaningful. Drawings allow students to show crossed-out or broken blocks, recording the same thinking on paper.

CCSS 2.NBT.B.5 expects fluency with subtraction within 100 using strategies based on place value. Fluency here includes both accuracy and flexible strategy use, not just speed. Concrete models are part of building that flexibility. Students who can draw a base-ten diagram for subtraction with decomposing, explain each step, and verify the result have met the spirit of the standard.

Active learning is especially important here because subtraction errors often trace back to procedural steps performed without understanding. When students compare models with a partner and explain their decomposition process, misconceptions surface and can be addressed before they calcify.

Key Questions

1. Compare the process of decomposing a ten in subtraction to composing a ten in addition.
2. Construct a drawing to illustrate how to subtract a two-digit number from another with borrowing.
3. Evaluate the benefits of using models before moving to abstract algorithms for subtraction.

Learning Objectives

• Demonstrate the process of decomposing a ten using base-ten blocks to solve subtraction problems within 100.
• Create a drawing that accurately represents the decomposition of a ten for subtraction with regrouping.
• Compare and contrast the steps involved in decomposing a ten for subtraction with composing a ten for addition.
• Explain the value of using concrete models or drawings before applying abstract subtraction algorithms.
• Calculate the difference between two-digit numbers within 100, showing work with place value strategies.

Before You Start

Addition with Composing Tens

Why: Students need to understand how to combine tens and ones, and how composing a ten works, to compare it with decomposing a ten.

Understanding Place Value to 100

Why: Students must know the value of digits in the tens and ones places to effectively use base-ten blocks and strategies for subtraction.

Key Vocabulary

Decompose	To break a number down into smaller parts. In subtraction, we decompose a ten into ten ones when we need more ones to subtract.
Regroup	To exchange a ten for ten ones, or a hundred for ten tens, to make it easier to subtract. This is also called borrowing.
Base-ten blocks	Manipulatives that represent ones, tens, and hundreds. We use rods for tens and units for ones to model subtraction.
Place value	The value of a digit based on its position in a number. We use place value to understand how to decompose and regroup numbers.

Watch Out for These Misconceptions

Common Misconception

Students may subtract the smaller digit from the larger regardless of which is the minuend or subtrahend (the 'flip the digits' error).

What to Teach Instead

Physically building the minuend with blocks and taking away the subtrahend's blocks makes the directionality concrete. Pair work where students narrate each removal step prevents this error from becoming automatic.

Common Misconception

Students may forget to adjust the tens column after decomposing, leaving the tens count unchanged.

What to Teach Instead

Use a two-column place value mat and require students to physically move the ten-rod before beginning to subtract ones. Partners check that the tens column decreased by one after any decomposition before proceeding.

Common Misconception

Students may attempt to decompose a ten even when the ones digit is large enough to subtract without it.

What to Teach Instead

Start each problem with the class asking 'do I have enough ones?' as a routine. Partner prediction before solving helps students develop the habit of checking before acting, rather than applying decomposition by default.

Active Learning Ideas

Think-Pair-Share: Break Apart or Borrow?

Teacher presents a subtraction problem like 63 - 28. Pairs decide whether decomposing a ten is needed, then each partner models it independently before comparing their drawings to identify any differences.

20 min·Pairs

Gallery Walk: Spot the Error

Post six worked subtraction problems around the room: two correct, two with decomposition errors, and two with correct answers reached by incorrect methods. Students identify and explain the errors in writing.

25 min·Small Groups

Inquiry Circle: Two Models, Same Problem

Groups solve one subtraction problem using base-ten blocks and then using a drawing. They write one sentence explaining how both models show the same decomposition and arrive at the same answer.

30 min·Small Groups

Stations Rotation: Subtraction Strategy Lab

Three stations feature physical blocks with decomposing, drawn models with circling and crossing out, and number line subtraction. Students solve assigned problems at each station and compare results with peers.

45 min·Small Groups

Real-World Connections

• When a cashier needs to give change, they might mentally decompose amounts. For example, to give 7 cents change from a dollar, they might think of the dollar as 9 dimes and 10 pennies to make the subtraction easier.
• Construction workers might estimate materials needed for a project. If they need 32 bricks and have 15, they might think about how many more tens and ones are needed, decomposing the total required amount to find the difference.

Assessment Ideas

Exit Ticket

Provide students with a subtraction problem, such as 42 - 17. Ask them to draw a picture using base-ten blocks or simple drawings to show how they solved it, including any decomposition of a ten. Write one sentence explaining their drawing.

Quick Check

Present a problem like 50 - 23. Ask students to hold up fingers to show how many tens they would need to decompose and how many ones they would get. Then, ask them to state the new number they have in the ones place.

Discussion Prompt

Ask students: 'When you add 25 + 37, you make a new ten. When you subtract 42 - 17, you break apart a ten. How are these actions similar, and how are they different?' Guide them to discuss the concept of exchanging value between place values.

Frequently Asked Questions

How does active learning help students understand subtraction with regrouping in 2nd grade?

Having students build the starting number with physical blocks and narrate each step aloud is highly effective. When pairs compare their drawn models after solving independently, discrepancies prompt rich mathematical discussion. Peer explanation of why a ten had to be decomposed deepens understanding more than repeated practice alone.

How do you teach borrowing in subtraction to second graders using models?

Present borrowing as decomposing: one ten-rod is broken into ten ones, giving more ones to work with. Students build the starting number with blocks, physically break apart a ten-rod when needed, then remove the subtrahend's blocks. This concrete experience makes the abstract notation of borrowing interpretable rather than arbitrary.

What is the difference between composing a ten in addition and decomposing a ten in subtraction?

Composing means grouping ten ones into one ten-rod (addition), while decomposing means breaking one ten-rod back into ten ones (subtraction). Both involve the same trade in opposite directions. Connecting the two actions explicitly helps students see subtraction as the inverse of addition.

Why should second graders use drawings for subtraction before learning the algorithm?

Drawings create a visual record of thinking that both students and teachers can review. When a student draws base-ten diagrams, crosses out what is subtracted, and circles what remains, the process is traceable. Errors are visible and correctable. The algorithm condenses this into symbols that can be applied without understanding.