Chapter 3 · Lesson 6 of 10 · Grade 7 · Expressions & Equations · MAT-07-EE-006
Write Equations from Real-World Problems
Represent and solve write equations from real-world problems problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 3: Expressions, Equations, and Inequalities
Expressions & Equations
Essential question: How can equivalent expressions and equations model constraints and support justified solutions?
Where this lesson fits
Lesson 6 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of expressions, equations, and inequalities, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve write equations from real-world problems problems using precise mathematical notation, models, and reasoning.
- Explain why a method for write equations from real-world problems works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 3: Expressions, Equations, and Inequalities
Expressions & Equations. Lesson 6 of 10 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 7 chapters, Grade 6 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Write Equations from Real-World Problems uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses equation balance 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 algebra by substituting the proposed solution into the original statement; equivalent expressions must agree for the same input.
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.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- Model write equations from real world problems in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
A service charges $5 plus $2 per unit. Find the cost for 3 units.
C = 5 + 2u; C = 11The fixed fee is constant and the per-unit charge multiplies units.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: A service charges $6 plus $3 per unit. Find the cost for 4 units.
C = 6 + 3u; C = 18The representation should show the same mathematical relationship as the calculation. The fixed fee is constant and the per-unit charge multiplies units.
Explain why a valid method works, then solve: A service charges $7 plus $4 per unit. Find the cost for 5 units.
C = 7 + 4u; C = 27A complete explanation names the relationship or property being preserved. The fixed fee is constant and the per-unit charge multiplies units.
Estimate or predict first, then calculate and decide whether the result is reasonable: A service charges $8 plus $5 per unit. Find the cost for 6 units.
C = 8 + 5u; C = 38The estimate is a reasonableness check, not a replacement for the exact result. The fixed fee is constant and the per-unit charge multiplies units.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A service charges $9 plus $6 per unit. Find the cost for 7 units.
C = 9 + 6u; C = 51Verification should independently support the result. The fixed fee is constant and the per-unit charge multiplies units.
Interpret the answer in context after solving. What does the result mean here? A service charges $10 plus $7 per unit. Find the cost for 8 units.
C = 10 + 7u; C = 66State the result with its meaning, units, direction, or comparison—not only a number. The fixed fee is constant and the per-unit charge multiplies units.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: A service charges $11 plus $2 per unit. Find the cost for 9 units.
Use equivalent operations on both sides. Correct solution: C = 11 + 2u; C = 29Error analysis requires identifying the broken idea, not merely replacing the final answer. The fixed fee is constant and the per-unit charge multiplies units.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A service charges $12 plus $3 per unit. Find the cost for 10 units. Case B: A service charges $5 plus $6 per unit. Find the cost for 10 units.
Case A: C = 12 + 3u; C = 42 Case B: C = 5 + 6u; C = 65Compare the structure, representation, units, and result rather than only the numbers. The fixed fee is constant and the per-unit charge multiplies units.
A service charges $5 plus $4 per unit. Find the cost for 11 units.
C = 5 + 4u; C = 49The fixed fee is constant and the per-unit charge multiplies units.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: A service charges $6 plus $5 per unit. Find the cost for 3 units.
C = 6 + 5u; C = 21The representation should show the same mathematical relationship as the calculation. The fixed fee is constant and the per-unit charge multiplies units.
Explain why a valid method works, then solve: A service charges $7 plus $6 per unit. Find the cost for 4 units.
C = 7 + 6u; C = 31A complete explanation names the relationship or property being preserved. The fixed fee is constant and the per-unit charge multiplies units.
Estimate or predict first, then calculate and decide whether the result is reasonable: A service charges $8 plus $7 per unit. Find the cost for 5 units.
C = 8 + 7u; C = 43The estimate is a reasonableness check, not a replacement for the exact result. The fixed fee is constant and the per-unit charge multiplies units.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A service charges $9 plus $2 per unit. Find the cost for 6 units.
C = 9 + 2u; C = 21Verification should independently support the result. The fixed fee is constant and the per-unit charge multiplies units.
Interpret the answer in context after solving. What does the result mean here? A service charges $10 plus $3 per unit. Find the cost for 7 units.
C = 10 + 3u; C = 31State the result with its meaning, units, direction, or comparison—not only a number. The fixed fee is constant and the per-unit charge multiplies units.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: A service charges $11 plus $4 per unit. Find the cost for 8 units.
Use equivalent operations on both sides. Correct solution: C = 11 + 4u; C = 43Error analysis requires identifying the broken idea, not merely replacing the final answer. The fixed fee is constant and the per-unit charge multiplies units.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A service charges $12 plus $5 per unit. Find the cost for 9 units. Case B: A service charges $5 plus $2 per unit. Find the cost for 9 units.
Case A: C = 12 + 5u; C = 57 Case B: C = 5 + 2u; C = 23Compare the structure, representation, units, and result rather than only the numbers. The fixed fee is constant and the per-unit charge multiplies units.
A service charges $5 plus $6 per unit. Find the cost for 10 units.
C = 5 + 6u; C = 65The fixed fee is constant and the per-unit charge multiplies units.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: A service charges $6 plus $7 per unit. Find the cost for 11 units.
C = 6 + 7u; C = 83The representation should show the same mathematical relationship as the calculation. The fixed fee is constant and the per-unit charge multiplies units.
Explain why a valid method works, then solve: A service charges $7 plus $2 per unit. Find the cost for 3 units.
C = 7 + 2u; C = 13A complete explanation names the relationship or property being preserved. The fixed fee is constant and the per-unit charge multiplies units.
Estimate or predict first, then calculate and decide whether the result is reasonable: A service charges $8 plus $3 per unit. Find the cost for 4 units.
C = 8 + 3u; C = 20The estimate is a reasonableness check, not a replacement for the exact result. The fixed fee is constant and the per-unit charge multiplies units.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A service charges $9 plus $4 per unit. Find the cost for 5 units.
Hint: Choose a representation before computing.
Answer: C = 9 + 4u; C = 29
02Interpret the answer in context after solving. What does the result mean here? A service charges $10 plus $5 per unit. Find the cost for 6 units.
Hint: Explain why the method works and name a verification strategy.
Answer: C = 10 + 5u; C = 40
03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: A service charges $11 plus $6 per unit. Find the cost for 7 units.
Hint: Choose a representation before computing.
Answer: Use equivalent operations on both sides. Correct solution: C = 11 + 6u; C = 53
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A service charges $12 plus $7 per unit. Find the cost for 8 units. Case B: A service charges $5 plus $4 per unit. Find the cost for 8 units.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: C = 12 + 7u; C = 68 Case B: C = 5 + 4u; C = 37
05A service charges $5 plus $2 per unit. Find the cost for 9 units.
Hint: Choose a representation before computing.
Answer: C = 5 + 2u; C = 23
06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: A service charges $6 plus $3 per unit. Find the cost for 10 units.
Hint: Explain why the method works and name a verification strategy.
Answer: C = 6 + 3u; C = 36
Practice
Independent practice
01Explain why a valid method works, then solve: A service charges $7 plus $4 per unit. Find the cost for 11 units.
Answer: C = 7 + 4u; C = 51
02Estimate or predict first, then calculate and decide whether the result is reasonable: A service charges $8 plus $5 per unit. Find the cost for 3 units.
Answer: C = 8 + 5u; C = 23
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A service charges $9 plus $6 per unit. Find the cost for 4 units.
Answer: C = 9 + 6u; C = 33
04Interpret the answer in context after solving. What does the result mean here? A service charges $10 plus $7 per unit. Find the cost for 5 units.
Answer: C = 10 + 7u; C = 45
05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: A service charges $11 plus $2 per unit. Find the cost for 6 units.
Answer: Use equivalent operations on both sides. Correct solution: C = 11 + 2u; C = 23
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A service charges $12 plus $3 per unit. Find the cost for 7 units. Case B: A service charges $5 plus $6 per unit. Find the cost for 7 units.
Answer: Case A: C = 12 + 3u; C = 33 Case B: C = 5 + 6u; C = 47
07A service charges $5 plus $4 per unit. Find the cost for 8 units.
Answer: C = 5 + 4u; C = 37
08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: A service charges $6 plus $5 per unit. Find the cost for 9 units.
Answer: C = 6 + 5u; C = 51
Common mistakes
Learn to catch the error, not just the answer.
Use equivalent operations on both sides.
Only terms with the same variable part are like terms.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Translate cost, distance, rate, and measurement situations into symbolic rules.
- Use equations and inequalities to determine unknown quantities, thresholds, and feasible ranges.
Challenge problems
Explain why a valid method works, then solve: A service charges $7 plus $6 per unit. Find the cost for 10 units.
C = 7 + 6u; C = 67
Estimate or predict first, then calculate and decide whether the result is reasonable: A service charges $8 plus $7 per unit. Find the cost for 11 units.
C = 8 + 7u; C = 85
Solve and verify the result with a second method, inverse operation, or equivalent representation: A service charges $9 plus $2 per unit. Find the cost for 3 units.
C = 9 + 2u; C = 15
Interpret the answer in context after solving. What does the result mean here? A service charges $10 plus $3 per unit. Find the cost for 4 units.
C = 10 + 3u; C = 22
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 3 Reasoning Lab
Solve write equations from real-world problems tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: A service charges $11 plus $4 per unit. Find the cost for 5 units.
Answer: Use equivalent operations on both sides. Correct solution: C = 11 + 4u; C = 31
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A service charges $12 plus $5 per unit. Find the cost for 6 units. Case B: A service charges $5 plus $2 per unit. Find the cost for 6 units.
Answer: Case A: C = 12 + 5u; C = 42 Case B: C = 5 + 2u; C = 17
03A service charges $5 plus $6 per unit. Find the cost for 7 units.
Answer: C = 5 + 6u; C = 47
04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: A service charges $6 plus $7 per unit. Find the cost for 8 units.
Answer: C = 6 + 7u; C = 62
05Explain why a valid method works, then solve: A service charges $7 plus $2 per unit. Find the cost for 9 units.
Answer: C = 7 + 2u; C = 25
06Estimate or predict first, then calculate and decide whether the result is reasonable: A service charges $8 plus $3 per unit. Find the cost for 10 units.
Answer: C = 8 + 3u; C = 38
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A service charges $9 plus $4 per unit. Find the cost for 11 units.
Answer: C = 9 + 4u; C = 53
08Interpret the answer in context after solving. What does the result mean here? A service charges $10 plus $5 per unit. Find the cost for 3 units.
Answer: C = 10 + 5u; C = 25
Terminology
Words to know
- variable
- A symbol representing a number that may vary or be unknown.
- coefficient
- A numerical factor multiplying a variable.
- expression
- A combination of numbers, variables, and operations.
- solution
- A value that makes an equation or inequality true.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.