Chapter 4 · Lesson 4 of 12 · Grade 8 · Geometry · MAT-08-GEO-004
Dilate Figures by a Scale Factor
Represent and solve dilate figures by a scale factor problems using precise mathematical notation, multiple representations, and justified reasoning.
Math V4 · Course chapter
Chapter 4: Transformations, Similarity, Pythagorean Geometry, and Volume
Geometry
Essential question: How can transformations and geometric relationships explain shape, distance, similarity, and measurement?
Where this lesson fits
Lesson 4 of 12. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Build and defend a multi-representation model using transformations, similarity, pythagorean geometry, and volume, including interpretation, verification, and error analysis.
Learning objectives
What you should be able to do
- Represent and solve dilate figures by a scale factor problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify dilate figures by a scale factor results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Chapter 4: Transformations, Similarity, Pythagorean Geometry, and Volume
Geometry. Lesson 4 of 12 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 8 chapters, Grade 7 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier representation and explain the relationship.
Meaning and structure
Dilate Figures by a Scale Factor belongs to geometry. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent dilate figures by a scale factor numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
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.
Transformation Lab
- Model dilate figures by a scale factor in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Dilate (1,2) from the origin by factor 2.
(2,4)A dilation from the origin multiplies both coordinates by the scale factor.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Dilate (2,4) from the origin by factor 3.
(6,12)A representation makes the mathematical structure visible before calculation. A dilation from the origin multiplies both coordinates by the scale factor.
Explain why the method works, not only what steps to perform: Dilate (3,6) from the origin by factor 4.
(12,24)Justification connects each step to a definition, property, or relationship. A dilation from the origin multiplies both coordinates by the scale factor.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Dilate (4,2) from the origin by factor 5.
(20,10)An estimate provides an independent reasonableness check. A dilation from the origin multiplies both coordinates by the scale factor.
Verify the result with substitution, an inverse operation, or a second representation: Dilate (5,4) from the origin by factor 2.
(10,8)Independent verification should support the result without merely repeating the same procedure. A dilation from the origin multiplies both coordinates by the scale factor.
Interpret the result in context, including units, direction, rate, or geometric meaning: Dilate (6,6) from the origin by factor 3.
(18,18)Interpretation connects a computed value to the mathematical situation. A dilation from the origin multiplies both coordinates by the scale factor.
Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Dilate (7,2) from the origin by factor 4.
Rigid transformations preserve size; dilations generally do not. Correct solution: (28,8)Error analysis identifies the broken idea and replaces it with valid reasoning. A dilation from the origin multiplies both coordinates by the scale factor.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Dilate (1,4) from the origin by factor 5. Case B: Dilate (5,2) from the origin by factor 4.
Case A: (5,20) Case B: (20,8)Comparison should address structure and representation, not only numerical answers. A dilation from the origin multiplies both coordinates by the scale factor.
Dilate (2,6) from the origin by factor 2.
(4,12)A dilation from the origin multiplies both coordinates by the scale factor.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Dilate (3,2) from the origin by factor 3.
(9,6)A representation makes the mathematical structure visible before calculation. A dilation from the origin multiplies both coordinates by the scale factor.
Explain why the method works, not only what steps to perform: Dilate (4,4) from the origin by factor 4.
(16,16)Justification connects each step to a definition, property, or relationship. A dilation from the origin multiplies both coordinates by the scale factor.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Dilate (5,6) from the origin by factor 5.
(25,30)An estimate provides an independent reasonableness check. A dilation from the origin multiplies both coordinates by the scale factor.
Verify the result with substitution, an inverse operation, or a second representation: Dilate (6,2) from the origin by factor 2.
(12,4)Independent verification should support the result without merely repeating the same procedure. A dilation from the origin multiplies both coordinates by the scale factor.
Interpret the result in context, including units, direction, rate, or geometric meaning: Dilate (7,4) from the origin by factor 3.
(21,12)Interpretation connects a computed value to the mathematical situation. A dilation from the origin multiplies both coordinates by the scale factor.
Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Dilate (1,6) from the origin by factor 4.
Rigid transformations preserve size; dilations generally do not. Correct solution: (4,24)Error analysis identifies the broken idea and replaces it with valid reasoning. A dilation from the origin multiplies both coordinates by the scale factor.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Dilate (2,2) from the origin by factor 5. Case B: Dilate (6,6) from the origin by factor 4.
Case A: (10,10) Case B: (24,24)Comparison should address structure and representation, not only numerical answers. A dilation from the origin multiplies both coordinates by the scale factor.
Dilate (3,4) from the origin by factor 2.
(6,8)A dilation from the origin multiplies both coordinates by the scale factor.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Dilate (4,6) from the origin by factor 3.
(12,18)A representation makes the mathematical structure visible before calculation. A dilation from the origin multiplies both coordinates by the scale factor.
Explain why the method works, not only what steps to perform: Dilate (5,2) from the origin by factor 4.
(20,8)Justification connects each step to a definition, property, or relationship. A dilation from the origin multiplies both coordinates by the scale factor.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Dilate (6,4) from the origin by factor 5.
(30,20)An estimate provides an independent reasonableness check. A dilation from the origin multiplies both coordinates by the scale factor.
Practice
Guided practice with hints
01Verify the result with substitution, an inverse operation, or a second representation: Dilate (7,6) from the origin by factor 2.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (14,12)
02Interpret the result in context, including units, direction, rate, or geometric meaning: Dilate (1,2) from the origin by factor 3.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (3,6)
03Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Dilate (2,4) from the origin by factor 4.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: (8,16)
04Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Dilate (3,6) from the origin by factor 5. Case B: Dilate (7,4) from the origin by factor 4.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Case A: (15,30) Case B: (28,16)
05Dilate (4,2) from the origin by factor 2.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (8,4)
06Represent the situation with a table, graph, diagram, number line, or equation before solving: Dilate (5,4) from the origin by factor 3.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: (15,12)
Practice
Independent practice
01Explain why the method works, not only what steps to perform: Dilate (6,6) from the origin by factor 4.
Answer: (24,24)
02Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Dilate (7,2) from the origin by factor 5.
Answer: (35,10)
03Verify the result with substitution, an inverse operation, or a second representation: Dilate (1,4) from the origin by factor 2.
Answer: (2,8)
04Interpret the result in context, including units, direction, rate, or geometric meaning: Dilate (2,6) from the origin by factor 3.
Answer: (6,18)
05Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Dilate (3,2) from the origin by factor 4.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: (12,8)
06Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Dilate (4,4) from the origin by factor 5. Case B: Dilate (1,2) from the origin by factor 4.
Answer: Case A: (20,20) Case B: (4,8)
07Dilate (5,6) from the origin by factor 2.
Answer: (10,12)
08Represent the situation with a table, graph, diagram, number line, or equation before solving: Dilate (6,2) from the origin by factor 3.
Answer: (18,6)
Common mistakes
Learn to catch the error, not just the answer.
Rigid transformations preserve size; dilations generally do not.
a²+b²=c² applies to right triangles.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use transformations in design and coordinate geometry and Pythagorean reasoning for distance.
- Model capacities of cylindrical, conical, and spherical objects.
Challenge problems
Explain why the method works, not only what steps to perform: Dilate (7,4) from the origin by factor 4.
(28,16)
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Dilate (1,6) from the origin by factor 5.
(5,30)
Verify the result with substitution, an inverse operation, or a second representation: Dilate (2,2) from the origin by factor 2.
(4,4)
Interpret the result in context, including units, direction, rate, or geometric meaning: Dilate (3,4) from the origin by factor 3.
(9,12)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve dilate figures by a scale factor problems and justify each result with a valid verification method.
Practice
Mastery check
01Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Dilate (4,6) from the origin by factor 4.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: (16,24)
02Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Dilate (5,2) from the origin by factor 5. Case B: Dilate (2,6) from the origin by factor 4.
Answer: Case A: (25,10) Case B: (8,24)
03Dilate (6,4) from the origin by factor 2.
Answer: (12,8)
04Represent the situation with a table, graph, diagram, number line, or equation before solving: Dilate (7,6) from the origin by factor 3.
Answer: (21,18)
05Explain why the method works, not only what steps to perform: Dilate (1,2) from the origin by factor 4.
Answer: (4,8)
06Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Dilate (2,4) from the origin by factor 5.
Answer: (10,20)
07Verify the result with substitution, an inverse operation, or a second representation: Dilate (3,6) from the origin by factor 2.
Answer: (6,12)
08Interpret the result in context, including units, direction, rate, or geometric meaning: Dilate (4,2) from the origin by factor 3.
Answer: (12,6)
Terminology
Words to know
- transformation
- A rule mapping a figure to an image.
- congruent
- Having the same size and shape.
- similar
- Having equal corresponding angles and proportional corresponding lengths.
- Pythagorean theorem
- For a right triangle, a²+b²=c².
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 8 scope, sequence, and public task structure
Coherent sequencing, representations, proof-oriented reasoning, and tasks.Eureka Math² Grade 8 program structure
Module coherence, cumulative practice, and algebraic fluency.enVision Mathematics Grade 8 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 8
Grade-level content and mathematical-practice expectations.