Grade 4 · Operations & Algebraic Thinking · MAT-04-OA-003
Find Unknowns in Multiplication and Division Equations
Represent and solve find unknowns in multiplication and division equations problems using accurate mathematical models, notation, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve find unknowns in multiplication and division equations problems using accurate mathematical models, notation, and reasoning.
- Explain why a strategy for find unknowns in multiplication and division equations works and verify the result with an independent check.
Prerequisite check
Make sure the foundation is ready.
The new lesson builds directly on this prior concept, so the prerequisite should be usable rather than merely familiar.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Find Unknowns in Multiplication and Division Equations is a operations & algebraic thinking idea built from precise quantities, representations, and operations. The mathematical goal is to preserve the meaning of each quantity while choosing a representation that makes the relationship visible.
Represent the idea
The Math Box uses equation balance as the primary representation. Translate the situation into numbers or geometry first, identify what stays fixed, then change one quantity at a time and observe which relationships remain invariant.
Reason, calculate, and verify
A convincing solution names the quantities, shows the mathematical relationship, computes accurately, and includes a reasonableness check rather than relying on an unsupported answer.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- Build one example of find unknowns in multiplication and division equations in the Math Box.
- Change one input and predict the effect before moving the control.
- Explain which mathematical relationship stayed invariant after the change.
Worked examples
Twenty different ways to see the concept work.
Find the unknown: 3 × □ = 12.
12 ÷ 3 = 4; □ = 4Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 4 × □ = 20.
20 ÷ 4 = 5; □ = 5Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 5 × □ = 30.
30 ÷ 5 = 6; □ = 6Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 6 × □ = 42.
42 ÷ 6 = 7; □ = 7Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 7 × □ = 56.
56 ÷ 7 = 8; □ = 8Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 8 × □ = 72.
72 ÷ 8 = 9; □ = 9Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 9 × □ = 90.
90 ÷ 9 = 10; □ = 10Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 10 × □ = 40.
40 ÷ 10 = 4; □ = 4Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 3 × □ = 15.
15 ÷ 3 = 5; □ = 5Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 4 × □ = 24.
24 ÷ 4 = 6; □ = 6Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 5 × □ = 35.
35 ÷ 5 = 7; □ = 7Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 6 × □ = 48.
48 ÷ 6 = 8; □ = 8Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 7 × □ = 63.
63 ÷ 7 = 9; □ = 9Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 8 × □ = 80.
80 ÷ 8 = 10; □ = 10Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 9 × □ = 36.
36 ÷ 9 = 4; □ = 4Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 10 × □ = 50.
50 ÷ 10 = 5; □ = 5Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 3 × □ = 18.
18 ÷ 3 = 6; □ = 6Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 4 × □ = 28.
28 ÷ 4 = 7; □ = 7Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 5 × □ = 40.
40 ÷ 5 = 8; □ = 8Division reverses multiplication, so the quotient identifies the missing factor.
Find the unknown: 6 × □ = 54.
54 ÷ 6 = 9; □ = 9Division reverses multiplication, so the quotient identifies the missing factor.
Practice
Guided practice with hints
01Find the unknown: 7 × □ = 70.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 70 ÷ 7 = 10; □ = 10
02Find the unknown: 8 × □ = 32.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 32 ÷ 8 = 4; □ = 4
03Find the unknown: 9 × □ = 45.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 45 ÷ 9 = 5; □ = 5
04Find the unknown: 10 × □ = 60.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 60 ÷ 10 = 6; □ = 6
05Find the unknown: 3 × □ = 21.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 21 ÷ 3 = 7; □ = 7
06Find the unknown: 4 × □ = 32.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 32 ÷ 4 = 8; □ = 8
Practice
Independent practice
01Find the unknown: 5 × □ = 45.
Answer: 45 ÷ 5 = 9; □ = 9
02Find the unknown: 6 × □ = 60.
Answer: 60 ÷ 6 = 10; □ = 10
03Find the unknown: 7 × □ = 28.
Answer: 28 ÷ 7 = 4; □ = 4
04Find the unknown: 8 × □ = 40.
Answer: 40 ÷ 8 = 5; □ = 5
05Find the unknown: 3 × □ = 12.
Answer: 12 ÷ 3 = 4; □ = 4
06Find the unknown: 4 × □ = 20.
Answer: 20 ÷ 4 = 5; □ = 5
07Find the unknown: 5 × □ = 30.
Answer: 30 ÷ 5 = 6; □ = 6
08Find the unknown: 6 × □ = 42.
Answer: 42 ÷ 6 = 7; □ = 7
Common mistakes
Learn to catch the error, not just the answer.
Identify what each number represents and how the quantities are related before selecting an operation.
Estimate, use an inverse operation, or compare with a second representation.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use find unknowns in multiplication and division equations when solving multi-step problems in measurement, data, money, design, or everyday quantitative decisions.
- Explain the result with units and a second check so another person can audit the reasoning.
Challenge problems
Find the unknown: 7 × □ = 56.
56 ÷ 7 = 8; □ = 8
Find the unknown: 8 × □ = 72.
72 ÷ 8 = 9; □ = 9
Find the unknown: 9 × □ = 90.
90 ÷ 9 = 10; □ = 10
Find the unknown: 10 × □ = 40.
40 ÷ 10 = 4; □ = 4
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 4 Math Strategy Sprint
Solve and explain find unknowns in multiplication and division equations problems accurately, then check each result with a second method.
Practice
Mastery check
01Find the unknown: 3 × □ = 15.
Answer: 15 ÷ 3 = 5; □ = 5
02Find the unknown: 4 × □ = 24.
Answer: 24 ÷ 4 = 6; □ = 6
03Find the unknown: 5 × □ = 35.
Answer: 35 ÷ 5 = 7; □ = 7
04Find the unknown: 6 × □ = 48.
Answer: 48 ÷ 6 = 8; □ = 8
05Find the unknown: 7 × □ = 63.
Answer: 63 ÷ 7 = 9; □ = 9
06Find the unknown: 8 × □ = 80.
Answer: 80 ÷ 8 = 10; □ = 10
07Find the unknown: 9 × □ = 36.
Answer: 36 ÷ 9 = 4; □ = 4
08Find the unknown: 10 × □ = 50.
Answer: 50 ÷ 10 = 5; □ = 5
Terminology
Words to know
- representation
- A diagram, model, equation, table, graph, or other form showing a mathematical idea.
- strategy
- A purposeful method chosen to solve or analyze a problem.
- estimate
- A nearby value used to judge size or reasonableness.
- verify
- To confirm a result using a second calculation, representation, or logical check.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade-level instructional guidance, mathematical practices, and coherent progression.Common Core State Standards for Mathematics — Grades 4–5
Grade-level expectations for operations, number and fraction work, measurement, data, and geometry.Principles to Actions: Ensuring Mathematical Success for All
Evidence-based emphasis on reasoning, representations, discourse, and conceptual understanding.