Grade 8 · Functions · MAT-08-FN-010
Model Real-World Linear Relationships
Represent and solve model real-world linear relationships problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve model real-world linear relationships problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify model real-world linear relationships 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.
Meaning and structure
Model Real-World Linear Relationships belongs to functions. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent model real-world linear relationships 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 model real world linear relationships 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 quantity starts at 10 and increases by 2 each hour. Find it after 3 hours.
y=2x+10; y=16Constant rate plus fixed initial value gives a linear model.
A quantity starts at 11 and increases by 3 each hour. Find it after 4 hours.
y=3x+11; y=23Constant rate plus fixed initial value gives a linear model.
A quantity starts at 12 and increases by 4 each hour. Find it after 5 hours.
y=4x+12; y=32Constant rate plus fixed initial value gives a linear model.
A quantity starts at 13 and increases by 5 each hour. Find it after 6 hours.
y=5x+13; y=43Constant rate plus fixed initial value gives a linear model.
A quantity starts at 14 and increases by 6 each hour. Find it after 7 hours.
y=6x+14; y=56Constant rate plus fixed initial value gives a linear model.
A quantity starts at 15 and increases by 7 each hour. Find it after 8 hours.
y=7x+15; y=71Constant rate plus fixed initial value gives a linear model.
A quantity starts at 16 and increases by 8 each hour. Find it after 9 hours.
y=8x+16; y=88Constant rate plus fixed initial value gives a linear model.
A quantity starts at 17 and increases by 2 each hour. Find it after 10 hours.
y=2x+17; y=37Constant rate plus fixed initial value gives a linear model.
A quantity starts at 18 and increases by 3 each hour. Find it after 3 hours.
y=3x+18; y=27Constant rate plus fixed initial value gives a linear model.
A quantity starts at 10 and increases by 4 each hour. Find it after 4 hours.
y=4x+10; y=26Constant rate plus fixed initial value gives a linear model.
A quantity starts at 11 and increases by 5 each hour. Find it after 5 hours.
y=5x+11; y=36Constant rate plus fixed initial value gives a linear model.
A quantity starts at 12 and increases by 6 each hour. Find it after 6 hours.
y=6x+12; y=48Constant rate plus fixed initial value gives a linear model.
A quantity starts at 13 and increases by 7 each hour. Find it after 7 hours.
y=7x+13; y=62Constant rate plus fixed initial value gives a linear model.
A quantity starts at 14 and increases by 8 each hour. Find it after 8 hours.
y=8x+14; y=78Constant rate plus fixed initial value gives a linear model.
A quantity starts at 15 and increases by 2 each hour. Find it after 9 hours.
y=2x+15; y=33Constant rate plus fixed initial value gives a linear model.
A quantity starts at 16 and increases by 3 each hour. Find it after 10 hours.
y=3x+16; y=46Constant rate plus fixed initial value gives a linear model.
A quantity starts at 17 and increases by 4 each hour. Find it after 3 hours.
y=4x+17; y=29Constant rate plus fixed initial value gives a linear model.
A quantity starts at 18 and increases by 5 each hour. Find it after 4 hours.
y=5x+18; y=38Constant rate plus fixed initial value gives a linear model.
A quantity starts at 10 and increases by 6 each hour. Find it after 5 hours.
y=6x+10; y=40Constant rate plus fixed initial value gives a linear model.
A quantity starts at 11 and increases by 7 each hour. Find it after 6 hours.
y=7x+11; y=53Constant rate plus fixed initial value gives a linear model.
Practice
Guided practice with hints
01A quantity starts at 12 and increases by 8 each hour. Find it after 7 hours.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y=8x+12; y=68
02A quantity starts at 13 and increases by 2 each hour. Find it after 8 hours.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y=2x+13; y=29
03A quantity starts at 14 and increases by 3 each hour. Find it after 9 hours.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y=3x+14; y=41
04A quantity starts at 15 and increases by 4 each hour. Find it after 10 hours.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y=4x+15; y=55
05A quantity starts at 16 and increases by 5 each hour. Find it after 3 hours.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y=5x+16; y=31
06A quantity starts at 17 and increases by 6 each hour. Find it after 4 hours.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: y=6x+17; y=41
Practice
Independent practice
01A quantity starts at 18 and increases by 7 each hour. Find it after 5 hours.
Answer: y=7x+18; y=53
02A quantity starts at 10 and increases by 8 each hour. Find it after 6 hours.
Answer: y=8x+10; y=58
03A quantity starts at 11 and increases by 2 each hour. Find it after 7 hours.
Answer: y=2x+11; y=25
04A quantity starts at 12 and increases by 3 each hour. Find it after 8 hours.
Answer: y=3x+12; y=36
05A quantity starts at 13 and increases by 4 each hour. Find it after 9 hours.
Answer: y=4x+13; y=49
06A quantity starts at 14 and increases by 5 each hour. Find it after 10 hours.
Answer: y=5x+14; y=64
07A quantity starts at 15 and increases by 6 each hour. Find it after 3 hours.
Answer: y=6x+15; y=33
08A quantity starts at 16 and increases by 7 each hour. Find it after 4 hours.
Answer: y=7x+16; y=44
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
A quantity starts at 17 and increases by 8 each hour. Find it after 5 hours.
y=8x+17; y=57
A quantity starts at 18 and increases by 2 each hour. Find it after 6 hours.
y=2x+18; y=30
A quantity starts at 10 and increases by 3 each hour. Find it after 7 hours.
y=3x+10; y=31
A quantity starts at 11 and increases by 4 each hour. Find it after 8 hours.
y=4x+11; y=43
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve model real-world linear relationships problems and justify each result with a valid verification method.
Practice
Mastery check
01A quantity starts at 12 and increases by 5 each hour. Find it after 9 hours.
Answer: y=5x+12; y=57
02A quantity starts at 13 and increases by 6 each hour. Find it after 10 hours.
Answer: y=6x+13; y=73
03A quantity starts at 14 and increases by 7 each hour. Find it after 3 hours.
Answer: y=7x+14; y=35
04A quantity starts at 15 and increases by 8 each hour. Find it after 4 hours.
Answer: y=8x+15; y=47
05A quantity starts at 16 and increases by 2 each hour. Find it after 5 hours.
Answer: y=2x+16; y=26
06A quantity starts at 17 and increases by 3 each hour. Find it after 6 hours.
Answer: y=3x+17; y=35
07A quantity starts at 18 and increases by 4 each hour. Find it after 7 hours.
Answer: y=4x+18; y=46
08A quantity starts at 10 and increases by 5 each hour. Find it after 8 hours.
Answer: y=5x+10; y=50
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
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.