Chapter 3 · Lesson 9 of 10 · Grade 8 · Functions · MAT-08-FN-009
Write Linear Equations from Data and Graphs
Represent and solve write linear equations from data and graphs problems using precise mathematical notation, multiple representations, and justified reasoning.
Math V4 · Course chapter
Chapter 3: Functions and Linear Models
Functions
Essential question: How can functions connect inputs, outputs, rates of change, graphs, tables, and equations?
Where this lesson fits
Lesson 9 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Build and defend a multi-representation model using functions and linear models, including interpretation, verification, and error analysis.
Learning objectives
What you should be able to do
- Represent and solve write linear equations from data and graphs problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify write linear equations from data and graphs 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 3: Functions and Linear Models
Functions. Lesson 9 of 10 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
Write Linear Equations from Data and Graphs belongs to functions. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent write linear equations from data and graphs 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.
Linear Function Lab
- Model write linear equations from data and graphs 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.
A line has slope 1 and y-intercept 3. Write its equation.
y = 1x + 3Slope-intercept form records rate and starting value.
Represent the situation with a table, graph, diagram, number line, or equation before solving: A line has slope 2 and y-intercept 5. Write its equation.
y = 2x + 5A representation makes the mathematical structure visible before calculation. Slope-intercept form records rate and starting value.
Explain why the method works, not only what steps to perform: A line has slope 3 and y-intercept 7. Write its equation.
y = 3x + 7Justification connects each step to a definition, property, or relationship. Slope-intercept form records rate and starting value.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A line has slope 4 and y-intercept 9. Write its equation.
y = 4x + 9An estimate provides an independent reasonableness check. Slope-intercept form records rate and starting value.
Verify the result with substitution, an inverse operation, or a second representation: A line has slope 5 and y-intercept 11. Write its equation.
y = 5x + 11Independent verification should support the result without merely repeating the same procedure. Slope-intercept form records rate and starting value.
Interpret the result in context, including units, direction, rate, or geometric meaning: A line has slope 6 and y-intercept 13. Write its equation.
y = 6x + 13Interpretation connects a computed value to the mathematical situation. Slope-intercept form records rate and starting value.
Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: A line has slope 7 and y-intercept 15. Write its equation.
Outputs may repeat; one input cannot have two different outputs. Correct solution: y = 7x + 15Error analysis identifies the broken idea and replaces it with valid reasoning. Slope-intercept form records rate and starting value.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A line has slope 8 and y-intercept 17. Write its equation. Case B: A line has slope 19 and y-intercept 39. Write its equation.
Case A: y = 8x + 17 Case B: y = 19x + 39Comparison should address structure and representation, not only numerical answers. Slope-intercept form records rate and starting value.
A line has slope 9 and y-intercept 19. Write its equation.
y = 9x + 19Slope-intercept form records rate and starting value.
Represent the situation with a table, graph, diagram, number line, or equation before solving: A line has slope 10 and y-intercept 21. Write its equation.
y = 10x + 21A representation makes the mathematical structure visible before calculation. Slope-intercept form records rate and starting value.
Explain why the method works, not only what steps to perform: A line has slope 11 and y-intercept 23. Write its equation.
y = 11x + 23Justification connects each step to a definition, property, or relationship. Slope-intercept form records rate and starting value.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A line has slope 12 and y-intercept 25. Write its equation.
y = 12x + 25An estimate provides an independent reasonableness check. Slope-intercept form records rate and starting value.
Verify the result with substitution, an inverse operation, or a second representation: A line has slope 13 and y-intercept 27. Write its equation.
y = 13x + 27Independent verification should support the result without merely repeating the same procedure. Slope-intercept form records rate and starting value.
Interpret the result in context, including units, direction, rate, or geometric meaning: A line has slope 14 and y-intercept 29. Write its equation.
y = 14x + 29Interpretation connects a computed value to the mathematical situation. Slope-intercept form records rate and starting value.
Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: A line has slope 15 and y-intercept 31. Write its equation.
Outputs may repeat; one input cannot have two different outputs. Correct solution: y = 15x + 31Error analysis identifies the broken idea and replaces it with valid reasoning. Slope-intercept form records rate and starting value.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A line has slope 16 and y-intercept 33. Write its equation. Case B: A line has slope 27 and y-intercept 55. Write its equation.
Case A: y = 16x + 33 Case B: y = 27x + 55Comparison should address structure and representation, not only numerical answers. Slope-intercept form records rate and starting value.
A line has slope 17 and y-intercept 35. Write its equation.
y = 17x + 35Slope-intercept form records rate and starting value.
Represent the situation with a table, graph, diagram, number line, or equation before solving: A line has slope 18 and y-intercept 37. Write its equation.
y = 18x + 37A representation makes the mathematical structure visible before calculation. Slope-intercept form records rate and starting value.
Explain why the method works, not only what steps to perform: A line has slope 19 and y-intercept 39. Write its equation.
y = 19x + 39Justification connects each step to a definition, property, or relationship. Slope-intercept form records rate and starting value.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A line has slope 20 and y-intercept 41. Write its equation.
y = 20x + 41An estimate provides an independent reasonableness check. Slope-intercept form records rate and starting value.
Practice
Guided practice with hints
01Verify the result with substitution, an inverse operation, or a second representation: A line has slope 21 and y-intercept 43. Write its equation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y = 21x + 43
02Interpret the result in context, including units, direction, rate, or geometric meaning: A line has slope 22 and y-intercept 45. Write its equation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y = 22x + 45
03Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: A line has slope 23 and y-intercept 47. Write its equation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Outputs may repeat; one input cannot have two different outputs. Correct solution: y = 23x + 47
04Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A line has slope 24 and y-intercept 49. Write its equation. Case B: A line has slope 35 and y-intercept 71. Write its equation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Case A: y = 24x + 49 Case B: y = 35x + 71
05A line has slope 25 and y-intercept 51. Write its equation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y = 25x + 51
06Represent the situation with a table, graph, diagram, number line, or equation before solving: A line has slope 26 and y-intercept 53. Write its equation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y = 26x + 53
Practice
Independent practice
01Explain why the method works, not only what steps to perform: A line has slope 27 and y-intercept 55. Write its equation.
Answer: y = 27x + 55
02Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A line has slope 28 and y-intercept 57. Write its equation.
Answer: y = 28x + 57
03Verify the result with substitution, an inverse operation, or a second representation: A line has slope 29 and y-intercept 59. Write its equation.
Answer: y = 29x + 59
04Interpret the result in context, including units, direction, rate, or geometric meaning: A line has slope 30 and y-intercept 61. Write its equation.
Answer: y = 30x + 61
05Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: A line has slope 31 and y-intercept 63. Write its equation.
Answer: Outputs may repeat; one input cannot have two different outputs. Correct solution: y = 31x + 63
06Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A line has slope 32 and y-intercept 65. Write its equation. Case B: A line has slope 43 and y-intercept 87. Write its equation.
Answer: Case A: y = 32x + 65 Case B: y = 43x + 87
07A line has slope 33 and y-intercept 67. Write its equation.
Answer: y = 33x + 67
08Represent the situation with a table, graph, diagram, number line, or equation before solving: A line has slope 34 and y-intercept 69. Write its equation.
Answer: y = 34x + 69
Common mistakes
Learn to catch the error, not just the answer.
Outputs may repeat; one input cannot have two different outputs.
Slope is change per unit; intercept is output at x=0.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model constant rates and starting values with tables, graphs, and equations.
- Compare functions by rate of change and initial value.
Challenge problems
Explain why the method works, not only what steps to perform: A line has slope 35 and y-intercept 71. Write its equation.
y = 35x + 71
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A line has slope 36 and y-intercept 73. Write its equation.
y = 36x + 73
Verify the result with substitution, an inverse operation, or a second representation: A line has slope 37 and y-intercept 75. Write its equation.
y = 37x + 75
Interpret the result in context, including units, direction, rate, or geometric meaning: A line has slope 38 and y-intercept 77. Write its equation.
y = 38x + 77
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve write linear equations from data and graphs problems and justify each result with a valid verification method.
Practice
Mastery check
01Error analysis: a student says, "Rejecting a function because outputs repeat." Explain the error, then solve this related task correctly: A line has slope 39 and y-intercept 79. Write its equation.
Answer: Outputs may repeat; one input cannot have two different outputs. Correct solution: y = 39x + 79
02Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: A line has slope 40 and y-intercept 81. Write its equation. Case B: A line has slope 51 and y-intercept 103. Write its equation.
Answer: Case A: y = 40x + 81 Case B: y = 51x + 103
03A line has slope 41 and y-intercept 83. Write its equation.
Answer: y = 41x + 83
04Represent the situation with a table, graph, diagram, number line, or equation before solving: A line has slope 42 and y-intercept 85. Write its equation.
Answer: y = 42x + 85
05Explain why the method works, not only what steps to perform: A line has slope 43 and y-intercept 87. Write its equation.
Answer: y = 43x + 87
06Estimate or predict the result first, then calculate and judge whether the answer is reasonable: A line has slope 44 and y-intercept 89. Write its equation.
Answer: y = 44x + 89
07Verify the result with substitution, an inverse operation, or a second representation: A line has slope 45 and y-intercept 91. Write its equation.
Answer: y = 45x + 91
08Interpret the result in context, including units, direction, rate, or geometric meaning: A line has slope 46 and y-intercept 93. Write its equation.
Answer: y = 46x + 93
Terminology
Words to know
- function
- A relation assigning exactly one output to each input.
- slope
- Vertical change divided by horizontal change for a nonvertical line.
- y-intercept
- The output where a graph crosses the y-axis.
- rate of change
- Output change per unit input change.
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.