Chapter 2 · Lesson 5 of 6 · Grade 6 · The Number System · MAT-06-NS-005
Find Greatest Common Factors
Represent and solve find greatest common factors problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 2: Division, Decimals, Factors, and Multiples
Extending arithmetic fluency with structure
Essential question: How can number structure make complicated calculations more understandable, efficient, and verifiable?
Where this lesson fits
Lesson 5 of 6. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Design and cost a batch-production plan that uses fractional quantities, decimal operations, common factors, and common multiples.
Learning objectives
What you should be able to do
- Represent and solve find greatest common factors problems using precise mathematical notation, models, and reasoning.
- Explain why a method for find greatest common factors works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Chapter 2: Division, Decimals, Factors, and Multiples
Extending arithmetic fluency with structure. This is Lesson 5 of 6 in Chapter 2. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes Chapter 1 ratio reasoning, fraction multiplication, place value, whole-number factors. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Find Greatest Common Factors extends arithmetic to signed, fractional, decimal, or coordinate quantities. Operations must preserve both magnitude and direction.
Represent the relationship
The Math Box uses multiplication grid to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
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.
Array Builder
Build equal rows and columns, then connect the grid to repeated addition.
- Model one example of find greatest common factors in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Find the greatest common factor of 4 and 14.
GCF(4, 14) = 2The greatest common factor is the largest positive integer dividing both numbers exactly.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Find the greatest common factor of 9 and 24.
GCF(9, 24) = 3The representation should show the same mathematical relationship as the calculation. The greatest common factor is the largest positive integer dividing both numbers exactly.
Explain why a valid method works, then solve: Find the greatest common factor of 16 and 36.
GCF(16, 36) = 4A complete explanation names the relationship or property being preserved. The greatest common factor is the largest positive integer dividing both numbers exactly.
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the greatest common factor of 25 and 50.
GCF(25, 50) = 25The estimate is a reasonableness check, not a replacement for the exact result. The greatest common factor is the largest positive integer dividing both numbers exactly.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the greatest common factor of 36 and 66.
GCF(36, 66) = 6Verification should independently support the result. The greatest common factor is the largest positive integer dividing both numbers exactly.
Interpret the answer in context after solving. What does the result mean here? Find the greatest common factor of 49 and 49.
GCF(49, 49) = 49State the result with its meaning, units, direction, or comparison—not only a number. The greatest common factor is the largest positive integer dividing both numbers exactly.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Find the greatest common factor of 16 and 64.
Use a number line, context, or inverse operation. Correct solution: GCF(16, 64) = 16Error analysis requires identifying the broken idea, not merely replacing the final answer. The greatest common factor is the largest positive integer dividing both numbers exactly.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the greatest common factor of 27 and 81. Case B: Find the greatest common factor of 54 and 72.
Case A: GCF(27, 81) = 27 Case B: GCF(54, 72) = 18Compare the structure, representation, units, and result rather than only the numbers. The greatest common factor is the largest positive integer dividing both numbers exactly.
Find the greatest common factor of 40 and 100.
GCF(40, 100) = 20The greatest common factor is the largest positive integer dividing both numbers exactly.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Find the greatest common factor of 10 and 22.
GCF(10, 22) = 2The representation should show the same mathematical relationship as the calculation. The greatest common factor is the largest positive integer dividing both numbers exactly.
Explain why a valid method works, then solve: Find the greatest common factor of 18 and 21.
GCF(18, 21) = 3A complete explanation names the relationship or property being preserved. The greatest common factor is the largest positive integer dividing both numbers exactly.
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the greatest common factor of 28 and 32.
GCF(28, 32) = 4The estimate is a reasonableness check, not a replacement for the exact result. The greatest common factor is the largest positive integer dividing both numbers exactly.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the greatest common factor of 10 and 45.
GCF(10, 45) = 5Verification should independently support the result. The greatest common factor is the largest positive integer dividing both numbers exactly.
Interpret the answer in context after solving. What does the result mean here? Find the greatest common factor of 18 and 60.
GCF(18, 60) = 6State the result with its meaning, units, direction, or comparison—not only a number. The greatest common factor is the largest positive integer dividing both numbers exactly.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Find the greatest common factor of 28 and 77.
Use a number line, context, or inverse operation. Correct solution: GCF(28, 77) = 7Error analysis requires identifying the broken idea, not merely replacing the final answer. The greatest common factor is the largest positive integer dividing both numbers exactly.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the greatest common factor of 40 and 56. Case B: Find the greatest common factor of 16 and 88.
Case A: GCF(40, 56) = 8 Case B: GCF(16, 88) = 8Compare the structure, representation, units, and result rather than only the numbers. The greatest common factor is the largest positive integer dividing both numbers exactly.
Find the greatest common factor of 54 and 72.
GCF(54, 72) = 18The greatest common factor is the largest positive integer dividing both numbers exactly.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Find the greatest common factor of 70 and 90.
GCF(70, 90) = 10The representation should show the same mathematical relationship as the calculation. The greatest common factor is the largest positive integer dividing both numbers exactly.
Explain why a valid method works, then solve: Find the greatest common factor of 4 and 20.
GCF(4, 20) = 4A complete explanation names the relationship or property being preserved. The greatest common factor is the largest positive integer dividing both numbers exactly.
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the greatest common factor of 9 and 33.
GCF(9, 33) = 3The estimate is a reasonableness check, not a replacement for the exact result. The greatest common factor is the largest positive integer dividing both numbers exactly.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the greatest common factor of 16 and 28.
Hint: Choose a representation before computing.
Answer: GCF(16, 28) = 4
02Interpret the answer in context after solving. What does the result mean here? Find the greatest common factor of 25 and 40.
Hint: Explain what relationship or property makes your method valid.
Answer: GCF(25, 40) = 5
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Find the greatest common factor of 36 and 54.
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: GCF(36, 54) = 18
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the greatest common factor of 49 and 70. Case B: Find the greatest common factor of 28 and 63.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: GCF(49, 70) = 7 Case B: GCF(28, 63) = 7
05Find the greatest common factor of 16 and 88.
Hint: Choose a representation before computing.
Answer: GCF(16, 88) = 8
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Find the greatest common factor of 27 and 63.
Hint: Explain what relationship or property makes your method valid.
Answer: GCF(27, 63) = 9
Practice
Independent practice
01Explain why a valid method works, then solve: Find the greatest common factor of 40 and 80.
Answer: GCF(40, 80) = 40
02Estimate or predict first, then calculate and decide whether the result is reasonable: Find the greatest common factor of 10 and 18.
Answer: GCF(10, 18) = 2
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the greatest common factor of 18 and 30.
Answer: GCF(18, 30) = 6
04Interpret the answer in context after solving. What does the result mean here? Find the greatest common factor of 28 and 44.
Answer: GCF(28, 44) = 4
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Find the greatest common factor of 10 and 35.
Answer: Use a number line, context, or inverse operation. Correct solution: GCF(10, 35) = 5
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the greatest common factor of 18 and 48. Case B: Find the greatest common factor of 36 and 42.
Answer: Case A: GCF(18, 48) = 6 Case B: GCF(36, 42) = 6
07Find the greatest common factor of 28 and 63.
Answer: GCF(28, 63) = 7
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Find the greatest common factor of 40 and 80.
Answer: GCF(40, 80) = 40
Common mistakes
Learn to catch the error, not just the answer.
Use a number line, context, or inverse operation.
For negatives, values closer to zero are greater.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model temperature, elevation, debt, gains and losses, coordinates, and measurements beyond whole numbers.
- Compute accurately with fractions, decimals, and signed quantities while checking magnitude and sign.
Challenge problems
Explain why a valid method works, then solve: Find the greatest common factor of 54 and 99.
GCF(54, 99) = 9
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the greatest common factor of 70 and 70.
GCF(70, 70) = 70
Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the greatest common factor of 4 and 16.
GCF(4, 16) = 4
Interpret the answer in context after solving. What does the result mean here? Find the greatest common factor of 9 and 27.
GCF(9, 27) = 9
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 2 Reasoning Lab
Solve find greatest common factors problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Find the greatest common factor of 16 and 40.
Answer: Use a number line, context, or inverse operation. Correct solution: GCF(16, 40) = 8
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the greatest common factor of 25 and 55. Case B: Find the greatest common factor of 10 and 50.
Answer: Case A: GCF(25, 55) = 5 Case B: GCF(10, 50) = 10
03Find the greatest common factor of 36 and 42.
Answer: GCF(36, 42) = 6
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Find the greatest common factor of 49 and 56.
Answer: GCF(49, 56) = 7
05Explain why a valid method works, then solve: Find the greatest common factor of 16 and 72.
Answer: GCF(16, 72) = 8
06Estimate or predict first, then calculate and decide whether the result is reasonable: Find the greatest common factor of 27 and 90.
Answer: GCF(27, 90) = 9
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the greatest common factor of 40 and 110.
Answer: GCF(40, 110) = 10
08Interpret the answer in context after solving. What does the result mean here? Find the greatest common factor of 10 and 14.
Answer: GCF(10, 14) = 2
Terminology
Words to know
- rational number
- A number expressible as a ratio of two integers with nonzero denominator.
- absolute value
- Distance from zero on a number line.
- opposite
- A number the same distance from zero on the other side.
- coordinate
- A number locating a point relative to an axis.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.