Story Equations Reteach Packet
Uncommon Schools Mathematics | 7th Grade Reteach Unit: Expressions & Equations
From Words to Equations: Understanding the Story
Objective: SWBAT interpret real-world situations with signed numbers, construct a verbal equation, translate it into an algebraic equation, and solve.
Name: _________________
Date: _________
Homeroom: __________
Score: _____ / 100
The 5-Step Story-to-Equation Routine Master Framework: Starting Value + (Rate × Intervals) = Ending Value
1. READ What is happening? Identify start, rate, & end.
2. WRITE Describe the story relationship in words.
3. PLUG Substitute signed numbers & a variable.
4. SOLVE Use inverse operations to find the unknown.
5. CHECK Does the value make sense in context?
Section 1: Activate Prior Knowledge (Signed Numbers in Context)
8 Questions | 10 Minutes
Directions: For each scenario, write the signed number (positive or negative) that represents the change or position, then explain what that value means in context.
- An airplane descends 1,800 feet during landing.
Signed Value:
Meaning:
- A bank account is charged a $35 overdraft fee.
Signed Value:
Meaning:
- A ship anchor rests at 45 feet below sea level.
Signed Value:
Meaning:
- A hiker climbs 620 feet up a mountain ridge.
Signed Value:
Meaning:
- The temperature drops 14°F over three hours.
Signed Value:
Meaning:
- Marcus deposits $150 into his savings balance.
Signed Value:
Meaning:
- A research submarine operates at a depth of 280 feet.
Signed Value:
Meaning:
- Sarah owes her brother $40 for concert tickets.
Signed Value:
Meaning:
From Words to Equations: Understanding the Story Page 1 of 8
Section 2: Explore (Teacher Models) Annotating & Translating Stories
Key Teaching Principle: Before writing any numbers or variables, identify the starting state, the change per unit (rate), and the ending state. Write out the verbal equation first!
Teacher Model 1: Increasing Scenario (Temperature) Worked Example
Story: At 6:00 AM, the temperature on Mount Washington was −15°F. As the morning progressed, the temperature increased at a steady rate of 4°F each hour. After several hours, the thermometer read 9°F. How many hours did the temperature increase?
Step 1: Read & Annotate
- Starting Value: −15°F (below zero)
- Rate of Change: +4°F per hour (increase)
- Intervals: \(h\) hours (unknown)
- Ending Value: +9°F (above zero)
Step 2: Write Verbal Equation
Starting Temp + (Hourly Increase × Hours) = Ending Temp
Step 3 & 4: Plug & Solve
−15 + 4h = 9
−15 + 15 + 4h = 9 + 15 (Add 15 to both sides)
4h = 24
4h ÷ 4 = 24 ÷ 4 (Divide both sides by 4)
h = 6 hours
Step 5: Check Sense
−15 + 4(6) = −15 + 24 = 9°F. Matches ending temperature! The answer is reasonable.
Teacher Model 2: Decreasing Scenario (Bank Account / Debt) Worked Example
Story: Maya has $80 in her student lunch account. Each school day, she spends $6 on hot lunch. After a number of days, her account balance drops to −$16 (she owes the cafeteria money). For how many days did Maya purchase lunch?
Step 1: Read & Annotate
- Starting Value: +$80 (positive balance)
- Rate of Change: −$6 per day (spending/decrease)
- Intervals: \(d\) days (unknown)
- Ending Value: −$16 (negative debt balance)
Step 2: Write Verbal Equation
Starting Balance + (Daily Rate of Change × Days) = Ending Balance
Step 3 & 4: Plug & Solve
80 + (−6)d = −16
80 − 80 − 6d = −16 − 80 (Subtract 80 from both sides)
−6d = −96
−6d ÷ (−6) = −96 ÷ (−6) (Negative ÷ Negative = Positive)
d = 16 days
Step 5: Check Sense
80 + (−6)(16) = 80 − 96 = −16. The account is $16 in debt. The answer is reasonable.
From Words to Equations: Understanding the Story Page 2 of 8
Section 3: Guided Practice (Part 1) Structured Story Organizers
Directions: Complete the organizer table for each story. Fill in the verbal equation before substituting values. Solve the algebraic equation and check your work.
Problem 1: Cave Spelunker (Elevation) Heavy Scaffold
A spelunker is exploring a cave at an elevation of −120 feet relative to ground level. She begins climbing upward at a steady rate of 15 feet per minute until she reaches an underground ledge at −30 feet. How many minutes did she spend climbing?
| Starting Value |
Rate of Change (+ / −) |
Intervals (Variable) |
Ending Value |
|
|
\(m\) = minutes |
|
Step 2: Complete the Verbal Equation:
+ ( × Minutes ) =
Step 3 & 4: Algebraic Equation & Solve
Solution: \(m\) = _______ minutes
Step 5: Check Sense in Context
Sentence: She climbed for ______ minutes.
Problem 2: Checking Account Overdraft (Money) Moderate Scaffold
Jamal starts the week with $45 in his checking account. Over 5 consecutive days, he makes the exact same debit card purchase each day. At the end of the 5 days, his account balance is −$25. What was the daily change in his account balance?
| Starting Balance |
Daily Change (Rate) |
Number of Days |
Ending Balance |
|
\(x\) = daily change |
|
|
Step 2: Verbal Equation:
Step 3 & 4: Algebraic Equation & Solve
Solution: \(x\) = _______ dollars/day
Step 5: Check Sense in Context
Sentence: His balance changed by $______ each day.
From Words to Equations: Understanding the Story Page 3 of 8
Section 3: Guided Practice (Part 2) Fading Scaffolds
Directions: Construct the story organizer and write the verbal equation independently. Then translate into an equation and solve.
Problem 3: Laboratory Freezer (Temperature) Light Scaffold
A liquid compound in a science lab begins at a temperature of 8°C. A lab technician places it in an industrial cooling chamber that lowers the temperature by 3°C each minute. The experiment finishes when the temperature reaches −22°C. How many minutes was the compound inside the chamber?
| Starting Temp |
Rate of Change (+ / −) |
Minutes (\(t\)) |
Ending Temp |
|
|
\(t\) |
|
Step 2: Verbal Equation:
Step 3 & 4: Algebraic Equation & Solve
Solution: \(t\) = _______ minutes
Step 5: Check & Interpret
Sentence: It took ______ minutes to cool.
Problem 4: Research Drone Descent (Elevation) Minimal Scaffold
A surveillance drone starts hovering at 250 feet above a canyon rim. It descends into the canyon at a constant rate for 8 minutes until it reaches an altitude of −70 feet relative to the rim. What was the drone’s rate of elevation change per minute?
| Starting Altitude |
Rate of Change (\(r\)) |
Time (Minutes) |
Ending Altitude |
|
\(r\) |
|
|
Step 2: Verbal Equation:
Step 3 & 4: Algebraic Equation & Solve
Solution: \(r\) = _______ feet/minute
Step 5: Check & Interpret
Sentence: The drone descended at ______ ft/min.
From Words to Equations: Understanding the Story Page 4 of 8
Section 4: Additional At-Bats (Story Translation Focus) 8 Questions | DO NOT SOLVE
Instructional Focus: Your only task on this page is to understand and translate the story. Do not waste time solving the equations. Match, identify, and write the mathematical relationships.
Part A: Identifying Story Quantities
1. A debt of $180 is repaid with $15 weekly payments until $30 of debt remains.
Starting Value:
Rate (+ or −):
Intervals (\(w\)):
Ending Value:
2. Ice at −4°C is heated in an oven by 1.8°C every minute until it reaches 32°C.
Starting Value:
Rate (+ or −):
Intervals (\(m\)):
Ending Value:
Part B: Verbal Equation Matching & Completion
3. Match each story to its correct verbal equation by writing the letter (A, B, or C) in the blank:
A: Start − (Rate × Time) = End
B: −Start + (Rate × Time) = +End
C: −Start − (Rate × Time) = −End
An elevator at 100 feet descends 8 feet per second until it reaches ground level (0 feet).
A scuba diver at −20 feet swims down 5 feet per minute to a depth of −80 feet.
An overdrawn account at −$50 receives $20 weekly deposits and reaches +$90.
4. Fill in the missing words in the verbal equation for this scenario:
“A diver starts at 15 feet below sea level (−15) and ascends 3 feet per second until reaching the surface (0).”
Starting Depth (_______) + [ Rate of Climb (______) × Time in Seconds ] = Surface Elevation (_______)
5. Write the complete verbal equation for this situation (words only, no numbers):
“An account starts with money, incurs equal monthly service fees, and ends with a negative balance.”
6. Error Analysis: A student translated the story “A freezer at 10°F drops by 2°F each minute to −14°F” as \(10 + 2m = -14\). What error did the student make?
Part C: Select the Matching Algebraic Equation
7. A submarine at −150 feet descends 12 feet each minute (\(m\)) to −390 feet.
A. \(150 - 12m = 390\)
B. \(-150 - 12m = -390\)
C. \(-150 + 12m = -390\)
D. \(12m - 150 = 390\)
8. A bank account with −$60 has \(w\) equal weekly deposits of $25 added, ending at $140.
A. \(-60 + 25w = 140\)
B. \(60 - 25w = 140\)
C. \(-60 - 25w = -140\)
D. \(25w - 140 = 60\)
From Words to Equations: Understanding the Story Page 5 of 8
Section 5: Priority Practice (Independent Mastery) Problems 1 – 3 of 6 (Integers)
Directions: Execute the full 5-step routine independently. You must write the verbal equation before writing and solving the algebraic equation. Include units in your final answer.
Problem 1: Temperature Rise Standard: 7.EE.B.4a
At midnight, the temperature in Fairbanks, Alaska was −6°F. The temperature rose at a steady rate of 3°F each hour. After some hours, the temperature was 15°F. How many hours did it take to reach 15°F?
1. Story Values & 2. Verbal Equation:
Start: _____ | Rate: _____ | End: _____
3. Algebraic Equation:
4. Solve & 5. Check:
Solution: \(h\) = _____ Sentence: It took _____ hours.
Problem 2: Student Club Account Standard: 7.EE.B.4a
A middle school drama club account starts with $110. For a set number of school days, the club spends $14 per day on stage props and paint. At the end of the project, the account balance is −$30 (overdrawn). How many days did the club purchase supplies?
1. Story Values & 2. Verbal Equation:
Start: _____ | Rate: _____ | End: _____
3. Algebraic Equation:
4. Solve & 5. Check:
Solution: \(d\) = _____ Sentence: Purchased for _____ days.
Problem 3: Deep Sea Submersible Standard: 7.EE.B.4a
A submersible is conducting research at an elevation of −450 meters. It ascends toward the ocean surface at a rate of 25 meters per minute. It stops its ascent when it reaches a depth of −125 meters. How many minutes was the submersible ascending?
1. Story Values & 2. Verbal Equation:
Start: _____ | Rate: _____ | End: _____
3. Algebraic Equation:
4. Solve & 5. Check:
Solution: \(m\) = _____ Sentence: Ascended for _____ min.
From Words to Equations: Understanding the Story Page 6 of 8
Section 5: Priority Practice (Independent Mastery) Problems 4 – 6 of 6 (Rational Numbers / Decimals)
Directions: Apply the 5-step routine with positive and negative decimal values. Ensure you track the signs carefully during arithmetic.
Problem 4: Chemical Reaction Cooling Standard: 7.EE.B.4a
A beaker solution starts at 4.5°C. An endothermic reaction causes the temperature to drop by 1.5°C every hour. The reaction stabilizes when the solution temperature reaches −7.5°C. How many hours did the temperature decrease?
1. Story Values & 2. Verbal Equation:
Start: _____ | Rate: _____ | End: _____
3. Algebraic Equation:
4. Solve & 5. Check:
Solution: \(h\) = _____ Sentence: Decreased for _____ hours.
Problem 5: Resolving Debt Balance Standard: 7.EE.B.4a
Darius has a credit card balance of −$28.50 (he owes money). He decides to make equal weekly deposits of $12.50 into the card account to build a positive credit balance. After \(w\) weeks, his account balance is +$46.50. How many weeks did he make deposits?
1. Story Values & 2. Verbal Equation:
Start: _____ | Rate: _____ | End: _____
3. Algebraic Equation:
4. Solve & 5. Check:
Solution: \(w\) = _____ Sentence: Deposited for _____ weeks.
Problem 6: Water Reservoir Level Standard: 7.EE.B.4a
During a dry season, a municipal water reservoir is at −14.2 feet relative to its normal baseline level. During a rainy period, the water level rises steadily by 2.4 feet per day. After \(d\) days, the reservoir level reaches +2.6 feet above baseline. For how many days did it rain?
1. Story Values & 2. Verbal Equation:
Start: _____ | Rate: _____ | End: _____
3. Algebraic Equation:
4. Solve & 5. Check:
Solution: \(d\) = _____ Sentence: It rained for _____ days.
From Words to Equations: Understanding the Story Page 7 of 8
Section 6: Exit Ticket — Story to Equation Individual Assessment
Name: _________________
Date: _________
Homeroom: __________
Mastery: [ ] Yes [ ] No
Problem 1: Contextual Interpretation 3 Points
A deep-sea ocean trench sensor transmits the equation: \(\mathbf{-180 + 15m = -45}\).
a) What does −180 represent in the physical world?
b) What does +15 represent, and what does the positive sign indicate about direction?
c) What does −45 represent in this story?
Problem 2: Integer Story to Equation 4 Points
At 5:00 AM, the temperature was −18°F. Over the next 6 hours, the temperature rose by the exact same amount each hour. At 11:00 AM, the temperature was 12°F. What was the hourly change in temperature?
Verbal Equation (Words Only):
Algebraic Equation:
Solve & Check:
Final Answer with Units: __________________________
Problem 3: Rational Numbers Story to Equation 3 Points
A hiker starts at a lookout station at an elevation of +120.5 feet. Over 5 equal intervals, he descends into a deep ravine, ending at an elevation of −14.5 feet relative to the park station. What was his change in elevation per interval?
Verbal Equation (Words Only):
Algebraic Equation:
Solve & Check:
Final Answer with Units: __________________________
From Words to Equations: Understanding the Story Page 8 of 8
