Chapter 3 · Lesson 6 of 6 · Grade 6 · Geometry · MAT-06-GEO-005
Find Distances Between Coordinate Points
Represent and solve find distances between coordinate points 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 6 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 find distances between coordinate points problems using precise mathematical notation, models, and reasoning.
- Explain why a method for find distances between coordinate points 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 6 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. This lesson closes the chapter's main sequence. Use it to connect the chapter ideas before the cumulative project and assessment: Create a coordinate map with elevations, temperatures, gains/losses, and route distances. Explain the meaning of each signed value.
Mathematical meaning
Find Distances Between Coordinate Points 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 find distances between coordinate points 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.
A(-5, 3) and B(-1, 3) are horizontal. Find AB.
|-1 − (-5)| = 4Horizontal distance is the absolute difference of x-coordinates.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A(-4, 3) and B(1, 3) are horizontal. Find AB.
|1 − (-4)| = 5The representation should show the same mathematical relationship as the calculation. Horizontal distance is the absolute difference of x-coordinates.
Explain why a valid method works, then solve: A(-3, 3) and B(3, 3) are horizontal. Find AB.
|3 − (-3)| = 6A complete explanation names the relationship or property being preserved. Horizontal distance is the absolute difference of x-coordinates.
Estimate or predict first, then calculate and decide whether the result is reasonable: A(-2, 3) and B(5, 3) are horizontal. Find AB.
|5 − (-2)| = 7The estimate is a reasonableness check, not a replacement for the exact result. Horizontal distance is the absolute difference of x-coordinates.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A(-1, 3) and B(7, 3) are horizontal. Find AB.
|7 − (-1)| = 8Verification should independently support the result. Horizontal distance is the absolute difference of x-coordinates.
Interpret the answer in context after solving. What does the result mean here? A(0, 3) and B(9, 3) are horizontal. Find AB.
|9 − (0)| = 9State the result with its meaning, units, direction, or comparison—not only a number. Horizontal distance is the absolute difference of x-coordinates.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A(1, 3) and B(11, 3) are horizontal. Find AB.
Label the figure and match each measure to the formula. Correct solution: |11 − (1)| = 10Error analysis requires identifying the broken idea, not merely replacing the final answer. Horizontal distance is the absolute difference of x-coordinates.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A(2, 3) and B(13, 3) are horizontal. Find AB. Case B: A(-5, 3) and B(6, 3) are horizontal. Find AB.
Case A: |13 − (2)| = 11 Case B: |6 − (-5)| = 11Compare the structure, representation, units, and result rather than only the numbers. Horizontal distance is the absolute difference of x-coordinates.
A(-5, 3) and B(7, 3) are horizontal. Find AB.
|7 − (-5)| = 12Horizontal distance is the absolute difference of x-coordinates.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A(-4, 3) and B(0, 3) are horizontal. Find AB.
|0 − (-4)| = 4The representation should show the same mathematical relationship as the calculation. Horizontal distance is the absolute difference of x-coordinates.
Explain why a valid method works, then solve: A(-3, 3) and B(2, 3) are horizontal. Find AB.
|2 − (-3)| = 5A complete explanation names the relationship or property being preserved. Horizontal distance is the absolute difference of x-coordinates.
Estimate or predict first, then calculate and decide whether the result is reasonable: A(-2, 3) and B(4, 3) are horizontal. Find AB.
|4 − (-2)| = 6The estimate is a reasonableness check, not a replacement for the exact result. Horizontal distance is the absolute difference of x-coordinates.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A(-1, 3) and B(6, 3) are horizontal. Find AB.
|6 − (-1)| = 7Verification should independently support the result. Horizontal distance is the absolute difference of x-coordinates.
Interpret the answer in context after solving. What does the result mean here? A(0, 3) and B(8, 3) are horizontal. Find AB.
|8 − (0)| = 8State the result with its meaning, units, direction, or comparison—not only a number. Horizontal distance is the absolute difference of x-coordinates.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A(1, 3) and B(10, 3) are horizontal. Find AB.
Label the figure and match each measure to the formula. Correct solution: |10 − (1)| = 9Error analysis requires identifying the broken idea, not merely replacing the final answer. Horizontal distance is the absolute difference of x-coordinates.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A(2, 3) and B(12, 3) are horizontal. Find AB. Case B: A(-5, 3) and B(5, 3) are horizontal. Find AB.
Case A: |12 − (2)| = 10 Case B: |5 − (-5)| = 10Compare the structure, representation, units, and result rather than only the numbers. Horizontal distance is the absolute difference of x-coordinates.
A(-5, 3) and B(6, 3) are horizontal. Find AB.
|6 − (-5)| = 11Horizontal distance is the absolute difference of x-coordinates.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A(-4, 3) and B(8, 3) are horizontal. Find AB.
|8 − (-4)| = 12The representation should show the same mathematical relationship as the calculation. Horizontal distance is the absolute difference of x-coordinates.
Explain why a valid method works, then solve: A(-3, 3) and B(1, 3) are horizontal. Find AB.
|1 − (-3)| = 4A complete explanation names the relationship or property being preserved. Horizontal distance is the absolute difference of x-coordinates.
Estimate or predict first, then calculate and decide whether the result is reasonable: A(-2, 3) and B(3, 3) are horizontal. Find AB.
|3 − (-2)| = 5The estimate is a reasonableness check, not a replacement for the exact result. Horizontal distance is the absolute difference of x-coordinates.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A(-1, 3) and B(5, 3) are horizontal. Find AB.
Hint: Choose a representation before computing.
Answer: |5 − (-1)| = 6
02Interpret the answer in context after solving. What does the result mean here? A(0, 3) and B(7, 3) are horizontal. Find AB.
Hint: Explain what relationship or property makes your method valid.
Answer: |7 − (0)| = 7
03Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A(1, 3) and B(9, 3) are horizontal. Find AB.
Hint: Choose a representation before computing.
Answer: Label the figure and match each measure to the formula. Correct solution: |9 − (1)| = 8
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A(2, 3) and B(11, 3) are horizontal. Find AB. Case B: A(-5, 3) and B(4, 3) are horizontal. Find AB.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: |11 − (2)| = 9 Case B: |4 − (-5)| = 9
05A(-5, 3) and B(5, 3) are horizontal. Find AB.
Hint: Choose a representation before computing.
Answer: |5 − (-5)| = 10
06Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A(-4, 3) and B(7, 3) are horizontal. Find AB.
Hint: Explain what relationship or property makes your method valid.
Answer: |7 − (-4)| = 11
Practice
Independent practice
01Explain why a valid method works, then solve: A(-3, 3) and B(9, 3) are horizontal. Find AB.
Answer: |9 − (-3)| = 12
02Estimate or predict first, then calculate and decide whether the result is reasonable: A(-2, 3) and B(2, 3) are horizontal. Find AB.
Answer: |2 − (-2)| = 4
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A(-1, 3) and B(4, 3) are horizontal. Find AB.
Answer: |4 − (-1)| = 5
04Interpret the answer in context after solving. What does the result mean here? A(0, 3) and B(6, 3) are horizontal. Find AB.
Answer: |6 − (0)| = 6
05Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A(1, 3) and B(8, 3) are horizontal. Find AB.
Answer: Label the figure and match each measure to the formula. Correct solution: |8 − (1)| = 7
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A(2, 3) and B(10, 3) are horizontal. Find AB. Case B: A(-5, 3) and B(3, 3) are horizontal. Find AB.
Answer: Case A: |10 − (2)| = 8 Case B: |3 − (-5)| = 8
07A(-5, 3) and B(4, 3) are horizontal. Find AB.
Answer: |4 − (-5)| = 9
08Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A(-4, 3) and B(6, 3) are horizontal. Find AB.
Answer: |6 − (-4)| = 10
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: A(-3, 3) and B(8, 3) are horizontal. Find AB.
|8 − (-3)| = 11
Estimate or predict first, then calculate and decide whether the result is reasonable: A(-2, 3) and B(10, 3) are horizontal. Find AB.
|10 − (-2)| = 12
Solve and verify the result with a second method, inverse operation, or equivalent representation: A(-1, 3) and B(3, 3) are horizontal. Find AB.
|3 − (-1)| = 4
Interpret the answer in context after solving. What does the result mean here? A(0, 3) and B(5, 3) are horizontal. Find AB.
|5 − (0)| = 5
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 3 Reasoning Lab
Solve find distances between coordinate points 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: A(1, 3) and B(7, 3) are horizontal. Find AB.
Answer: Label the figure and match each measure to the formula. Correct solution: |7 − (1)| = 6
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A(2, 3) and B(9, 3) are horizontal. Find AB. Case B: A(-5, 3) and B(2, 3) are horizontal. Find AB.
Answer: Case A: |9 − (2)| = 7 Case B: |2 − (-5)| = 7
03A(-5, 3) and B(3, 3) are horizontal. Find AB.
Answer: |3 − (-5)| = 8
04Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A(-4, 3) and B(5, 3) are horizontal. Find AB.
Answer: |5 − (-4)| = 9
05Explain why a valid method works, then solve: A(-3, 3) and B(7, 3) are horizontal. Find AB.
Answer: |7 − (-3)| = 10
06Estimate or predict first, then calculate and decide whether the result is reasonable: A(-2, 3) and B(9, 3) are horizontal. Find AB.
Answer: |9 − (-2)| = 11
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A(-1, 3) and B(11, 3) are horizontal. Find AB.
Answer: |11 − (-1)| = 12
08Interpret the answer in context after solving. What does the result mean here? A(0, 3) and B(4, 3) are horizontal. Find AB.
Answer: |4 − (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.