The Relationship of Addition and Subtraction

Active learning works well for this topic because students need to physically and visually connect addition and subtraction. Moving objects, writing equations, and moving along a number line help them see the inverse relationship rather than just hear about it. These concrete experiences build the mental models that lead to fluency and confidence with fact families and commutative property.

• 2nd Year
• Foundations of Mathematical Thinking
• 4 activities
• 15 min–35 min

Learning Objectives

1. Calculate the missing number in a fact family by applying the inverse relationship between addition and subtraction.
2. Compare and contrast the results of adding numbers in different orders to demonstrate the commutative property.
3. Generate all four equations within a given fact family for three numbers.
4. Explain how knowing one addition fact can help solve related subtraction facts.

Collaborative Problem-Solving

⏱ 25 min·Pairs

Partner Game: Fact Family Match-Up

Pairs draw cards with numbers like 6, 7, 13 and create all four fact family sentences on mini-whiteboards. Switch roles after two minutes, checking work together. End with sharing one new insight per pair.

Prepare & details

If you know 6 + 7 = 13, how can that help you solve 13 − 7?

Facilitation Tip: During Partner Game: Fact Family Match-Up, circulate and ask pairs to explain how they knew their matches were correct, reinforcing the link between addition and subtraction.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 30 min·Small Groups

Small Groups: Domino Fact Families

Provide dominoes showing addends and sums. Groups write the four related sentences for each domino, then sort into fact family charts. Discuss commutative swaps as a group.

Prepare & details

What do you notice when you swap the numbers in 5 + 3 to make 3 + 5?

Facilitation Tip: For Small Groups: Domino Fact Families, remind students to turn the domino so they see both orientations, this helps them notice the commutative property in action.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 35 min·Whole Class

Whole Class: Human Number Line

Students line up to represent numbers, acting out addition by joining and subtraction by separating. Call out problems like 13 - 7; class adjusts positions to show the answer. Debrief patterns observed.

Prepare & details

What is 10 more than 24?

Facilitation Tip: When running Whole Class: Human Number Line, encourage students to step backward as well as forward to explicitly show subtraction as the reverse of addition.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 15 min·Individual

Individual: Ten-Frame Challenges

Students use ten-frames to model '10 more than 24,' then create subtraction facts. Record in journals and share one with a neighbor.

Prepare & details

If you know 6 + 7 = 13, how can that help you solve 13 − 7?

Facilitation Tip: With Individual: Ten-Frame Challenges, ask students to describe how filling the ten-frame relates to both the addition and subtraction equations they write.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Start with concrete tools like counters, dominoes, and number lines before moving to abstract equations. Ask students to verbalize their thinking as they build fact families. Avoid rushing to symbols; let students discover patterns through repeated hands-on practice. Research shows that students who manipulate objects and discuss relationships develop deeper understanding of inverse operations and number flexibility.

What to Expect

Successful learning looks like students confidently generating all four equations in a fact family and explaining how addition and subtraction relate. They should use the commutative property to switch addends without hesitation and use subtraction to solve for missing addends. Watch for flexible thinking when solving number puzzles and discussions that reference inverse operations naturally.

Watch Out for These Misconceptions

Common Misconception

During Partner Game: Fact Family Match-Up, watch for students who treat addition and subtraction facts as unrelated. Redirect by asking them to explain how the addition card helps them find the matching subtraction card using their counters.

What to Teach Instead

Prompt students to verbalize that knowing 6 + 7 = 13 means they also know 13 - 7 = 6 by physically removing 7 counters from a group of 13.

Common Misconception

During Small Groups: Domino Fact Families, watch for students who only write one addition equation per domino orientation. Redirect by asking them to write both addition equations and the corresponding subtraction equations to show the commutative property holds.

What to Teach Instead

Have students rotate the domino and write 5 + 3 = 8 and 3 + 5 = 8, then write 8 - 3 = 5 and 8 - 5 = 3 to see all four equations.

Common Misconception

During Whole Class: Human Number Line, watch for students who only move forward to add and backward to subtract. Redirect by asking them to explain how moving backward from 13 to 6 is the same as subtracting 7 from 13.

What to Teach Instead

Ask students to stand at 13, take 7 steps backward, and then explain the equation that matches their movement: 13 - 7 = 6.

Assessment Ideas

Quick Check

After Partner Game: Fact Family Match-Up, present students with a fact family like 4, 9, 13. Ask them to write all four equations on a mini-whiteboard and explain how one equation helped them find the others.

Exit Ticket

After Individual: Ten-Frame Challenges, give each student a card with an addition equation such as 6 + 5 = 11. Ask them to write the related subtraction equation and explain in one sentence how the addition fact helped them find the subtraction answer.

Discussion Prompt

During Whole Class: Human Number Line, pose the question: 'If you know that 14 - 5 = 9, what other number facts do you also know?' Facilitate a class discussion where students share multiple related facts and explain their reasoning.

Extensions & Scaffolding

• Challenge: Have students create their own fact family puzzles with three numbers, then trade with peers to solve them backwards using subtraction clues.
• Scaffolding: Provide partially completed fact family templates where students fill in missing equations or numbers to support students who struggle with generating all four equations.
• Deeper: Introduce missing addend problems in context, such as 'If 12 apples were picked and 7 are in the basket, how many are still on the tree?' to connect fact families to real-world situations.

Key Vocabulary

Fact Family	A set of related addition and subtraction equations that use the same three numbers. For example, 3, 4, and 7 form a fact family.
Inverse Operations	Operations that undo each other. Addition and subtraction are inverse operations because adding a number and then subtracting the same number returns you to the original value.
Commutative Property of Addition	The property that states that the order in which two numbers are added does not change the sum. For example, 5 + 2 is the same as 2 + 5.
Part-Whole Thinking	Understanding that a whole number can be composed of two or more smaller parts, and that these parts can be combined or separated to form the whole.