Chapter 3 · Lesson 5 of 6 · Grade 6 · Geometry · MAT-06-GEO-004
Graph Polygons in the Coordinate Plane
Represent and solve graph polygons in the coordinate plane problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 3: Rational Numbers and the Coordinate Plane
Direction, distance, order, and location
Essential question: How do signs, absolute value, and coordinates describe quantities and locations that extend beyond positive whole numbers?
Where this lesson fits
Lesson 5 of 6. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create a coordinate map with elevations, temperatures, gains/losses, and route distances. Explain the meaning of each signed value.
Learning objectives
What you should be able to do
- Represent and solve graph polygons in the coordinate plane problems using precise mathematical notation, models, and reasoning.
- Explain why a method for graph polygons in the coordinate plane 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 3: Rational Numbers and the Coordinate Plane
Direction, distance, order, and location. This is Lesson 5 of 6 in Chapter 3. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes number lines, fraction and decimal comparison, Chapter 2 arithmetic fluency. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Graph Polygons in the Coordinate Plane connects measurement to structure. Formulas are justified by decomposition, rearrangement, similarity, coordinate reasoning, or invariance.
Represent the relationship
The Math Box uses four quadrant plane 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.
Four-Quadrant Coordinate Lab
- Model one example of graph polygons in the coordinate plane 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.
Translate point (1, 1) left 4 and down 3.
(-3, -2)Horizontal movement changes x and vertical movement changes y.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Translate point (2, 3) left 5 and down 5.
(-3, -2)The representation should show the same mathematical relationship as the calculation. Horizontal movement changes x and vertical movement changes y.
Explain why a valid method works, then solve: Translate point (3, 5) left 6 and down 7.
(-3, -2)A complete explanation names the relationship or property being preserved. Horizontal movement changes x and vertical movement changes y.
Estimate or predict first, then calculate and decide whether the result is reasonable: Translate point (4, 1) left 7 and down 9.
(-3, -8)The estimate is a reasonableness check, not a replacement for the exact result. Horizontal movement changes x and vertical movement changes y.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Translate point (5, 3) left 8 and down 3.
(-3, 0)Verification should independently support the result. Horizontal movement changes x and vertical movement changes y.
Interpret the answer in context after solving. What does the result mean here? Translate point (6, 5) left 9 and down 5.
(-3, 0)State the result with its meaning, units, direction, or comparison—not only a number. Horizontal movement changes x and vertical movement changes y.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Translate point (1, 1) left 10 and down 7.
Label the figure and match each measure to the formula. Correct solution: (-9, -6)Error analysis requires identifying the broken idea, not merely replacing the final answer. Horizontal movement changes x and vertical movement changes y.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Translate point (2, 3) left 11 and down 9. Case B: Translate point (5, 3) left 11 and down 3.
Case A: (-9, -6) Case B: (-6, 0)Compare the structure, representation, units, and result rather than only the numbers. Horizontal movement changes x and vertical movement changes y.
Translate point (3, 5) left 12 and down 3.
(-9, 2)Horizontal movement changes x and vertical movement changes y.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Translate point (4, 1) left 4 and down 5.
(0, -4)The representation should show the same mathematical relationship as the calculation. Horizontal movement changes x and vertical movement changes y.
Explain why a valid method works, then solve: Translate point (5, 3) left 5 and down 7.
(0, -4)A complete explanation names the relationship or property being preserved. Horizontal movement changes x and vertical movement changes y.
Estimate or predict first, then calculate and decide whether the result is reasonable: Translate point (6, 5) left 6 and down 9.
(0, -4)The estimate is a reasonableness check, not a replacement for the exact result. Horizontal movement changes x and vertical movement changes y.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Translate point (1, 1) left 7 and down 3.
(-6, -2)Verification should independently support the result. Horizontal movement changes x and vertical movement changes y.
Interpret the answer in context after solving. What does the result mean here? Translate point (2, 3) left 8 and down 5.
(-6, -2)State the result with its meaning, units, direction, or comparison—not only a number. Horizontal movement changes x and vertical movement changes y.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Translate point (3, 5) left 9 and down 7.
Label the figure and match each measure to the formula. Correct solution: (-6, -2)Error analysis requires identifying the broken idea, not merely replacing the final answer. Horizontal movement changes x and vertical movement changes y.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Translate point (4, 1) left 10 and down 9. Case B: Translate point (1, 1) left 10 and down 3.
Case A: (-6, -8) Case B: (-9, -2)Compare the structure, representation, units, and result rather than only the numbers. Horizontal movement changes x and vertical movement changes y.
Translate point (5, 3) left 11 and down 3.
(-6, 0)Horizontal movement changes x and vertical movement changes y.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Translate point (6, 5) left 12 and down 5.
(-6, 0)The representation should show the same mathematical relationship as the calculation. Horizontal movement changes x and vertical movement changes y.
Explain why a valid method works, then solve: Translate point (1, 1) left 4 and down 7.
(-3, -6)A complete explanation names the relationship or property being preserved. Horizontal movement changes x and vertical movement changes y.
Estimate or predict first, then calculate and decide whether the result is reasonable: Translate point (2, 3) left 5 and down 9.
(-3, -6)The estimate is a reasonableness check, not a replacement for the exact result. Horizontal movement changes x and vertical movement changes y.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Translate point (3, 5) left 6 and down 3.
Hint: Choose a representation before computing.
Answer: (-3, 2)
02Interpret the answer in context after solving. What does the result mean here? Translate point (4, 1) left 7 and down 5.
Hint: Explain what relationship or property makes your method valid.
Answer: (-3, -4)
03Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Translate point (5, 3) left 8 and down 7.
Hint: Choose a representation before computing.
Answer: Label the figure and match each measure to the formula. Correct solution: (-3, -4)
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Translate point (6, 5) left 9 and down 9. Case B: Translate point (3, 5) left 9 and down 3.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: (-3, -4) Case B: (-6, 2)
05Translate point (1, 1) left 10 and down 3.
Hint: Choose a representation before computing.
Answer: (-9, -2)
06Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Translate point (2, 3) left 11 and down 5.
Hint: Explain what relationship or property makes your method valid.
Answer: (-9, -2)
Practice
Independent practice
01Explain why a valid method works, then solve: Translate point (3, 5) left 12 and down 7.
Answer: (-9, -2)
02Estimate or predict first, then calculate and decide whether the result is reasonable: Translate point (4, 1) left 4 and down 9.
Answer: (0, -8)
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Translate point (5, 3) left 5 and down 3.
Answer: (0, 0)
04Interpret the answer in context after solving. What does the result mean here? Translate point (6, 5) left 6 and down 5.
Answer: (0, 0)
05Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Translate point (1, 1) left 7 and down 7.
Answer: Label the figure and match each measure to the formula. Correct solution: (-6, -6)
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Translate point (2, 3) left 8 and down 9. Case B: Translate point (5, 3) left 8 and down 3.
Answer: Case A: (-6, -6) Case B: (-3, 0)
07Translate point (3, 5) left 9 and down 3.
Answer: (-6, 2)
08Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Translate point (4, 1) left 10 and down 5.
Answer: (-6, -4)
Common mistakes
Learn to catch the error, not just the answer.
Label the figure and match each measure to the formula.
Use square units for area and cubic units for volume.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Estimate material, floor space, packaging, distance, surface coverage, and capacity.
- Use diagrams and coordinates to justify measurements rather than relying on appearance.
Challenge problems
Explain why a valid method works, then solve: Translate point (5, 3) left 11 and down 7.
(-6, -4)
Estimate or predict first, then calculate and decide whether the result is reasonable: Translate point (6, 5) left 12 and down 9.
(-6, -4)
Solve and verify the result with a second method, inverse operation, or equivalent representation: Translate point (1, 1) left 4 and down 3.
(-3, -2)
Interpret the answer in context after solving. What does the result mean here? Translate point (2, 3) left 5 and down 5.
(-3, -2)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 3 Reasoning Lab
Solve graph polygons in the coordinate plane problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Translate point (3, 5) left 6 and down 7.
Answer: Label the figure and match each measure to the formula. Correct solution: (-3, -2)
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Translate point (4, 1) left 7 and down 9. Case B: Translate point (1, 1) left 7 and down 3.
Answer: Case A: (-3, -8) Case B: (-6, -2)
03Translate point (5, 3) left 8 and down 3.
Answer: (-3, 0)
04Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Translate point (6, 5) left 9 and down 5.
Answer: (-3, 0)
05Explain why a valid method works, then solve: Translate point (1, 1) left 10 and down 7.
Answer: (-9, -6)
06Estimate or predict first, then calculate and decide whether the result is reasonable: Translate point (2, 3) left 11 and down 9.
Answer: (-9, -6)
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Translate point (3, 5) left 12 and down 3.
Answer: (-9, 2)
08Interpret the answer in context after solving. What does the result mean here? Translate point (4, 1) left 4 and down 5.
Answer: (0, -4)
Terminology
Words to know
- dimension
- A measurable extent such as length, width, or height.
- area
- Two-dimensional measure in square units.
- surface area
- Total area of exterior faces or surfaces.
- volume
- Three-dimensional space measured in cubic units.
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.