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

Addition and Subtraction Strategies within 100

Students fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

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

About This Topic

Fluency with addition and subtraction within 100 is one of the most significant computational milestones in the US K-12 mathematics curriculum. CCSS 2.NBT.B.5 expects students to add and subtract fluently within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction. Fluency here means accurate, efficient, and flexible: students select the most efficient strategy for the specific numbers at hand, rather than applying a single procedure to every problem.

Key strategies include decomposing by place value (47 + 35 = 40+30 + 7+5 = 70+12 = 82), compensation (49 + 36: add 1 to 49 to get 50, then add 36, then subtract 1 = 85), and the open number line for subtraction by counting up. The commutative property helps with addition by allowing students to start with the larger number. The inverse relationship between addition and subtraction allows subtraction to be solved by asking what must be added.

Active learning is particularly powerful here because strategy comparison is the core intellectual work of the topic. When students solve the same problem with different strategies and then discuss which was most efficient and why, they develop the metacognitive flexibility that distinguishes fluent mathematicians from students who have memorized one algorithm. Partner strategy debates and structured comparison activities produce this kind of flexible thinking directly.

Key Questions

1. Compare different strategies for adding two-digit numbers, such as breaking apart and compensation.
2. Explain how the commutative property of addition can make problems easier to solve.
3. Assess the efficiency of various subtraction strategies for different types of problems.

Learning Objectives

• Compare the efficiency of at least two different strategies (e.g., breaking apart, compensation, open number line) for solving addition and subtraction problems within 100.
• Explain how the commutative property of addition can simplify the calculation of a two-digit addition problem.
• Calculate the sum or difference of two-digit numbers within 100 using at least three distinct strategies.
• Justify the choice of a specific strategy for solving a subtraction problem within 100, referencing the numbers involved.
• Demonstrate the relationship between addition and subtraction by solving a subtraction problem using an addition strategy.

Before You Start

Addition and Subtraction within 20

Why: Students need foundational fluency with basic addition and subtraction facts to build more complex strategies for larger numbers.

Understanding Place Value to 100

Why: Strategies for adding and subtracting within 100 rely heavily on understanding tens and ones.

Key Vocabulary

Place Value	The value of a digit based on its position in a number, such as the tens place or the ones place.
Decomposing	Breaking a number apart into smaller parts, often by place value (e.g., breaking 47 into 40 and 7).
Compensation	Adjusting one number in a problem to make it easier to solve, then adjusting the other number or the answer to account for the change.
Commutative Property of Addition	The property that states that the order of addends does not change the sum (e.g., 25 + 30 is the same as 30 + 25).
Open Number Line	A visual representation of a number line where students can make jumps to show addition or subtraction steps.

Watch Out for These Misconceptions

Common Misconception

Adding or subtracting the tens and ones separately but ignoring regrouping when the ones sum to more than 9.

What to Teach Instead

When decomposing by place value and the ones total more than 9, the extra ten must be added to the tens total. Students who forget this step produce systematic errors of exactly 10. Using a T-chart with Tens and Ones columns helps make the regrouping step explicit during decomposition.

Common Misconception

Thinking the commutative property applies to subtraction: 'I can start with the smaller number.'

What to Teach Instead

The commutative property holds for addition only. 73 - 28 is not the same as 28 - 73. When students try to subtract the larger digit from the smaller without regrouping, the commutative confusion is often the cause. Direct comparisons of 73-28 and 28-73 using blocks prove the distinction.

Common Misconception

Applying the compensation strategy incorrectly by adjusting the wrong number or adjusting in the wrong direction.

What to Teach Instead

In compensation for addition, adjusting one addend up requires adjusting the final sum down by the same amount. Students who adjust both addends or forget to compensate the answer produce errors. Walking through the logic step by step with a partner who checks each adjustment prevents this mistake from becoming a habit.

Active Learning Ideas

Think-Pair-Share: Strategy Showdown

Present one two-digit addition problem to the class (e.g., 38 + 47). Students solve it individually using any strategy. Partners compare strategies side by side, name each strategy, and decide which was more efficient for these specific numbers. Three pairs share to build a class list of strategy names and when each works best.

20 min·Pairs

Inquiry Circle: 100 Ways to Get There

Groups are given a target sum (e.g., 75). They must find at least four different addition equations that reach that target, using four different strategies. Groups post their equations and strategies on chart paper. The class identifies which strategy appeared most across groups.

35 min·Small Groups

Stations Rotation: Strategy Stations

Four stations each feature one strategy: place value decomposition, compensation, counting up on a number line, and using fact families. Students solve two problems at each station using that station's assigned strategy, then rate how efficient they found it for those specific numbers on a 1-3 scale.

45 min·Small Groups

Real-World Connections

• Grocery store cashiers use addition and subtraction strategies to quickly calculate change for customers. For example, they might use compensation to figure out how much change is due from a $20 bill for a $12.50 purchase.
• Construction workers often need to add or subtract measurements to ensure materials fit correctly. A carpenter might decompose lengths like 15 feet 6 inches and 8 feet 9 inches to determine the total length needed for a project.

Assessment Ideas

Exit Ticket

Present students with the problem 53 + 28. Ask them to solve it using two different strategies and write one sentence explaining which strategy they found more efficient and why.

Discussion Prompt

Pose the subtraction problem 72 - 35. Ask students to share how they would solve it. Facilitate a discussion comparing strategies like counting up on an open number line versus subtracting by place value. Ask: 'Which strategy feels easiest for these specific numbers?'

Quick Check

Write the problem 45 + 32 on the board. Ask students to show you their answer using a thumbs up if they used place value decomposition, thumbs sideways if they used compensation, and thumbs down if they used another strategy. Then, ask a few students to briefly explain their method.

Frequently Asked Questions

What are the best strategies for adding two-digit numbers in 2nd grade?

The three most reliable strategies are place value decomposition (split both numbers into tens and ones, add separately, combine), compensation (adjust one number to a friendly ten, then adjust the answer), and the open number line (start at the first number, make hops of ten then ones). Each has situations where it is most efficient, and students should be able to choose among them.

How does the commutative property of addition make problems easier?

The commutative property states that a + b = b + a. When students encounter 9 + 63, they can reorder it to 63 + 9 and count on just 9 steps from 63 instead of counting on 63 steps from 9. This makes counting-on strategies much faster and is a key reason why starting from the larger number is taught explicitly.

How does the compensation strategy work for two-digit addition?

Choose one addend and round it to the nearest ten. Add the rounded number instead, which is easy. Then adjust the answer to compensate for the rounding. For 38 + 47: round 38 up to 40 (+2), add 40+47=87, then subtract the 2 you added: 87-2=85. The adjustment keeps the sum accurate.

How does active learning improve fluency with two-digit addition and subtraction?

Strategy comparison discussions force students to evaluate efficiency, not just accuracy. When students justify why they chose compensation over decomposition for a specific pair of numbers, they develop the metacognitive awareness that defines genuine fluency. Collaborative strategy stations also expose students to methods their peers use, broadening their toolkit beyond what they discovered independently.