Introduction to Algebraic Thinking: Missing Numbers

Active learning works for this topic because students need to see equations as balanced relationships, not just calculations. When they manipulate objects like balance scales or number lines, they build mental models of inverse operations that stick longer than abstract symbols alone.

• 3rd Year
• Mathematical Foundations and Real World Reasoning
• 4 activities
• 15 min–35 min

Learning Objectives

1. Calculate the missing addend or subtrahend in number sentences up to 100 using inverse operations.
2. Explain the relationship between addition and subtraction as inverse operations.
3. Design a strategy to solve for an unknown number in a given equation.
4. Justify the use of a balance scale model to represent the equality of an equation.
5. Identify the missing number in real-world scenarios involving simple quantities.

Inquiry Circle

⏱ 25 min·Pairs

Pairs Activity: Balance Scale Equations

Provide each pair with a balance scale and numbered weights. Present equations like 10 + ? = 20; students add weights to one side and find the missing one to balance. Pairs record their equation, solution, and explanation. Switch roles for a second equation.

Prepare & details

Explain how addition can help us find a missing number in a subtraction problem.

Facilitation Tip: During the Balance Scale Equations activity, circulate and ask pairs to explain why adding the difference to the missing subtrahend balances the scale.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 35 min·Small Groups

Small Groups: Number Line Challenges

Give groups dry-erase number lines and cards with equations like 25 - ? = 18. Students mark known numbers, jump forward with addition to find the missing subtrahend, then verify by subtraction. Groups share one strategy with the class.

Prepare & details

Design a strategy to find the unknown in a number sentence like '15 + ? = 23'.

Facilitation Tip: For Number Line Challenges, have students trace their jumps backward to reinforce the inverse relationship between addition and subtraction.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 20 min·Whole Class

Whole Class: Equation Bingo

Distribute bingo cards with missing number equations. Call out complete facts; students solve for the missing part to mark their card. First to complete a line explains their strategy using inverse operations.

Prepare & details

Justify why using a balance scale helps understand equations.

Facilitation Tip: In Equation Bingo, pause after each call to ask students how they knew which number to cover.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Inquiry Circle

⏱ 15 min·Individual

Individual: Real-World Story Problems

Students receive worksheets with scenarios like 'I had some apples, ate 7, now have 12. How many at start?' They write the equation, solve with inverse addition, and draw a bar model to justify.

Prepare & details

Explain how addition can help us find a missing number in a subtraction problem.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Teaching This Topic

Teachers should avoid rushing to abstract symbols. Concrete tools like balance scales and number lines help students visualize equality and inverse operations. Research shows that students who physically manipulate objects develop stronger mental models for solving equations. Emphasize student talk to make thinking visible, as verbalizing strategies helps correct misconceptions in real time.

What to Expect

Successful learning looks like students justifying their answers with clear, logical steps rather than relying on guesses. They should explain why addition undoes subtraction, or why counting back isn’t always efficient for larger numbers.

Watch Out for These Misconceptions

Common Misconception

During Balance Scale Equations, watch for students who count back from the minuend to find the missing subtrahend instead of using addition.

What to Teach Instead

Guide students to add the difference (the known side) to the missing subtrahend, then check if both sides match. Ask them to explain why adding works better than counting back for larger numbers.

Common Misconception

During Number Line Challenges, watch for students who guess numbers to place on the line rather than using systematic jumps.

What to Teach Instead

Have them start at the known number and jump backward to zero, then reverse the process to find the missing addend. Ask them to trace the jumps aloud to reinforce the inverse relationship.

Common Misconception

During Balance Scale Equations, watch for students who treat the equals sign as an operation symbol rather than a balance indicator.

What to Teach Instead

Ask them to place weights on both sides and adjust until the scale balances. Then have them write the equation to match, reinforcing that both sides must be equal.

Assessment Ideas

Quick Check

After the Number Line Challenges activity, present students with three number sentences: one with a missing addend, one with a missing subtrahend, and one inverse operation. Ask them to write the missing number for each and explain their strategy for one.

Discussion Prompt

During the Real-World Story Problems activity, pose the question: 'If you have some pencils, add 8 more, and now have 15, how can you use subtraction to find the original number?' Facilitate a class discussion where students share strategies and explain why subtraction works here.

Exit Ticket

After the Balance Scale Equations activity, give each student a card with a balance scale drawing. On one side, place the number 12. On the other side, place '9 + ?'. Ask students to write the number that makes the scale balance and one sentence explaining how they found it.

Extensions & Scaffolding

• Challenge: Provide a set of equations where the missing number is in any position (e.g., ? - 12 = 8, 15 + ? = 30, 25 - 5 = ?) and ask students to create their own balance scale problems for peers.
• Scaffolding: For students struggling with subtraction inverses, provide a number line with marked jumps to show how adding the difference returns to the minuend.
• Deeper exploration: Have students write their own real-world story problems with missing numbers, then swap and solve with a partner.

Key Vocabulary

Inverse Operations	Operations that undo each other, such as addition and subtraction.
Missing Addend	The unknown number that needs to be added to a known number to reach a specific sum.
Missing Subtrahend	The unknown number that needs to be subtracted from a known number to reach a specific difference.
Equation	A mathematical statement that shows two expressions are equal, often containing an equals sign and an unknown value.
Balance Scale	A tool used to visually represent equality, where both sides must hold the same weight or value to remain balanced.