Words to Equations Reteach Packet
7th Grade Math • Unit 3: Expressions, Equations & Inequalities Reteach Focus: NYS 7.EE.4a & 7.NS.3
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Student Reteach Packet
From Words to Equations: Understanding Context & Signs
SWBAT: Interpret real-world situations with positive and negative numbers, identify starting values, rates, and totals, write verbal equations, translate into two-step algebraic equations, and solve.
Lesson Frame: How can we break down the story in a word problem to find the starting value, rate of change, and ending value—and use their signs to write and solve an equation?
Key Takeaways: Anatomy of a Two-Step Problem
1. Starting Value (Constant): Where the situation begins before any time passes or actions happen.
• Positive (+): in the bank, above sea level, above zero temperature
• Negative (−): in debt, below sea level / depth, below zero temperature
2. Rate of Change (Coefficient): The amount added or subtracted each unit of time, item, or group (look for keywords "per," "each," "every").
• Positive (+): rising, climbing, depositing, earning, heating up
• Negative (−): descending, falling, withdrawing, spending, cooling down
3. Ending Value (Total): Where the story ends up after the changes occur.
The Golden 5-Step Protocol (Follow on Every Word Problem)
STEP 1
READ
Underline Start, circle Change, [box] End. Name unknown.
STEP 2
WRITE
Write a Verbal Equation using words from the story.
STEP 3
PLUG
Substitute signed numbers (+ / −) & your variable.
STEP 4
SOLVE
Use inverse operations (undo +/− first, then ×/÷).
STEP 5
CHECK
Write a complete sentence and check if it makes sense.
Everybody Writes!
Why is it helpful to write a verbal equation with words before writing an algebraic equation with numbers and variables?
STAMP (Core Rule of the Day)
Always identify the sign (+ or −) of the starting value and the rate of change BEFORE writing your equation:
Starting Value + (Rate of Change × Unknown) = Ending Value
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 1
Section 1
Activate Prior Knowledge: Signs in Real-World Contexts
8–10 Minutes
Directions: Before we write equations, we must master positive and negative values. Read each situation. Identify the integer or signed rate, explain what it represents, and state whether it is positive (+) or negative (−).
Part A: Identifying Signed Values & Meaning
| Real-World Scenario |
Signed Number |
What does it represent? |
| 1. The temperature outside is 8 degrees below zero. |
−8°F |
A temperature below 0° |
| 2. An anchor is dropped to a depth of 45 feet below sea level. |
|
________________________________ |
| 3. Marcus withdrew $25 from his savings account. |
|
________________________________ |
| 4. A submarine ascends at a rate of 12 feet every minute. |
|
________________________________ |
| 5. A hiker descends into a canyon at 18 feet per minute. |
|
________________________________ |
| 6. Jasmine has a bank balance showing a debt of $30. |
|
________________________________ |
Part B: Comparing Starting & Ending Values (Direction of Change)
Did the value increase (+) or decrease (−)? Find the direction and total amount of change.
7. Temperature Change
At 6:00 AM, the temperature was −5°F. At noon, the temperature was 12°F.
Did the temperature increase or decrease? ________
Is the change positive (+) or negative (−)? ________
What was the total change in degrees? ________
8. Bank Account Balance
On Monday, Elijah had $40. On Friday, his account was −$15 (in debt).
Did his balance increase or decrease? ________
Is the change positive (+) or negative (−)? ________
What was the total change in dollars? ________
9. Submarine Depth
A submarine starts at −150 feet and moves to −80 feet.
Did the submarine ascend or descend? ________
Is this vertical rate positive (+) or negative (−)? ________
What was the total change in feet? ________
10. Hot Air Balloon
A balloon floats at 850 feet and lands safely at 0 feet.
Did the balloon rise or fall? ________
Is this change positive (+) or negative (−)? ________
What was the total change in elevation? ________
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 2
Section 2
Explore / I DO: Teacher Models
Watch & Annotate Closely
Model 1: Positive Rate of Change (+) Temperature Warming Up
At 5:00 AM, the temperature outside was −14°F. As the sun came up, the temperature increased by 3°F each hour. After several hours, the temperature reached 10°F. How many hours did it take to reach 10°F?
Teacher Think-Aloud:
"The story starts cold at −14°. It increases, so the rate is +3. 'Each hour' tells me 3 multiplies by hours (h). The ending total is positive 10°."
Step 1: Identify Quantities
- Starting Temp: −14
- Rate of Change: +3 per hour
- Unknown: h = number of hours
- Ending Temp: 10
Step 2: Verbal Equation
Starting Temp + (Hourly Rate × Hours) = Ending Temp
Step 3 & 4: Algebraic Equation & Solve
−14 + 3h = 10
+14 +14 (Add 14 to both sides)
3h = 24
÷3 ÷3 (Divide by 3)
h = 8
Step 5: Check & Answer in Context
Check: −14 + 3(8) = −14 + 24 = 10 ✓
Answer: It took 8 hours for the temperature to reach 10°F.
Model 2: Negative Rate of Change (−) Submarine Descending
A submarine is cruising at a depth of 60 feet below sea level. It begins to descend at a rate of 15 feet per minute. It stops when it reaches 240 feet below sea level. How many minutes did the submarine descend?
Teacher Think-Aloud:
"Below sea level means negative! Start = −60. Descend means going deeper down, so rate = −15 per minute. Ending depth = −240. All values are negative!"
Step 1: Identify Quantities
- Starting Depth: −60
- Rate of Change: −15 per min (descends)
- Unknown: m = minutes descending
- Ending Depth: −240
Step 2: Verbal Equation
Starting Depth − (Descent Rate × Minutes) = Ending Depth
Step 3 & 4: Algebraic Equation & Solve
−60 + (−15m) = −240 or −60 − 15m = −240
+60 +60 (Add 60 to both sides)
−15m = −180
÷(−15) ÷(−15) (Divide by −15)
m = 12
Step 5: Check & Answer in Context
Check: −60 − 15(12) = −60 − 180 = −240 ✓
Answer: The submarine descended for 12 minutes.
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 3
Section 3
Guided Practice / WE DO: Deep Sea Submarine Lab
Scaffolded Practice (Problems 1–3)
Directions: We will practice using the exact same story context. Notice how the numbers and unknown change, but the verbal equation structure stays identical!
Problem 1: Finding Unknown Time (m) Descend (−)
A submarine starts at a depth of 80 feet below sea level. It then descends 12 feet every minute until it reaches a depth of 200 feet below sea level. How many minutes did it descend?
Start: −80
Rate: −12 per min
Unknown: m (mins)
End: −200
Verbal Equation:
Starting Depth + (Rate of Descent × Minutes) = Ending Depth
Algebraic Equation:
−80 + (−12m) = −200
Show Your Work (Inverse Operations):
Sentence: The submarine took ______ minutes to descend.
Problem 2: Finding Unknown Rate of Ascent (r) Ascend (+)
A submarine begins at a depth of 250 feet below sea level. It then ascends at a constant rate for 6 minutes until it is at 70 feet below sea level. What was its rate of ascent per minute?
Start: −250
Rate: +r per min
Time: 6 minutes
End: −70
Verbal Equation:
Starting Depth + (Rate of Ascent × 6) = Ending Depth
Algebraic Equation:
−250 + 6r = ______
Show Your Work:
Sentence: The submarine ascended at a rate of ______ ft/min.
Problem 3: Finding Unknown Starting Depth (d) Descend (−)
A research submarine begins at an unknown depth below sea level. It descends 20 feet each minute for 5 minutes and reaches a final depth of 280 feet below sea level. Where did it start?
Start: d
Rate: −20 per min
Time: 5 minutes
End: −280
Verbal Equation:
Start + (Rate × Time) = End
Algebraic Equation:
d + (−20)(5) = −280
Show Your Work:
Sentence: The submarine began at a depth of ______ feet.
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 4
Section 3 (Continued)
Guided Practice / WE DO: Temperature & Bank Accounts
Scaffolded Practice (Problems 4–6)
Problem 4: Cold Front Temperature Drop Falling Rate (−)
At sunset, the temperature in Buffalo was 14°F. A winter blizzard hit, causing the temperature to drop by 4°F each hour. After several hours, the temperature was −18°F. How many hours did the temperature drop?
1. Identify Signs & Variable:
Start: +14°
Rate: −4°/hr
Let h = hours
End: −18°
2. Verbal Equation:
Start + (Rate × Hours) = End
3. Algebraic Equation:
14 + (−4h) = −18
4. Solve (Show both steps):
5. Sentence: The temperature dropped for ______ hours.
Problem 5: Paying Off Debt with Equal Deposits Deposit (+)
Darius had an account balance showing a debt of $45. Over the next 5 weeks, he deposited the same amount of money each week into his account. His new balance was $80. How much money did Darius deposit each week?
1. Identify Signs & Variable:
Start: −$45 (debt)
Weeks: 5 weeks
Let d = deposit/wk
End: +$80
2. Verbal Equation:
Starting Debt + (Weeks × Deposit) = Ending Balance
3. Algebraic Equation:
−45 + 5d = 80
4. Solve (Show both steps):
5. Sentence: Darius deposited $______ each week.
Problem 6: Chemical Freezing Experiment Cooling Rate (−)
A beaker of chemical starts at 6°C. In a freezer, its temperature cools at a rate of 2.5°C per minute until it reaches −14°C. Write and solve an equation to find how many minutes (m) it took.
Verbal Equation:
Start Temp + (Cooling Rate × Minutes) = Final Temp
Algebraic Equation:
6 + (−2.5m) = −14
Solve & Sentence:
Sentence: It took ______ minutes to cool.
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 5
Section 4
Additional At-Bats: Connecting Words & Equations
Part 1: Structure Fluency
1. Match Each Story to its Correct Verbal Equation (Write letter A, B, or C):
Story 1: A hiker starts at 150 ft above sea level and descends 20 ft/min to sea level (0 ft). [ ___ ]
Story 2: A diver starts at 150 ft below sea level and ascends 20 ft/min to sea level. [ ___ ]
Story 3: A bank account has $20 and deposits $150 each month to reach a goal. [ ___ ]
A. Starting Depth + (Ascent Rate × Minutes) = Ending Depth
B. Starting Balance + (Monthly Deposit × Months) = Total
C. Starting Elevation − (Descent Rate × Minutes) = Ending Elevation
2. Match Each Verbal Equation to its Algebraic Equation:
Verbal I: A debt of $20 plus 4 weekly earnings equals $60. [ ___ ]
Verbal II: A temperature of 20°F drops 4° each hour to reach −60°F. [ ___ ]
Verbal III: A depth of 60 ft below sea level ascends 4 ft/min to −20 ft. [ ___ ]
X. 20 − 4h = −60
Y. −60 + 4m = −20
Z. −20 + 4w = 60
3. Sign Detective: Find and Correct the Sign Error
Scenario: Maya has a debt of $18. She deposits $6 each week for w weeks and now has a balance of $30.
Student writes: 18 + 6w = 30
What is the sign mistake?
Explain why 18 must be negative: _________________
Write the CORRECT equation:
___________________________
4. Fill in the Missing Information
A weather balloon starts at an altitude of 1,500 meters and descends at a constant rate of 45 meters per minute. After m minutes, its altitude is 600 meters.
Start: +1,500
Rate: [ _____ ]
Variable: m
End: +600
Write the equation: 1,500 + ( _____ ) = 600
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 6
Section 4 (Continued)
Additional At-Bats: Independent Equation Writing
Part 2: Verbal to Algebraic
Directions (Problems 5–8): For each problem, you must create the verbal equation yourself, write the algebraic equation, and solve.
5. Frozen Lake Temperature Integers
The temperature on a frozen lake is −8°F at 7:00 AM. The sun warms the air by 2.5°F each hour. After how many hours, h, will the temperature be 7°F?
Verbal Equation:
__________________________________________
Algebraic Equation:
__________________________
Solve & Check:
Answer: h = ______ hours
6. Underground Mine Elevator Negative Rate
An elevator starts at ground level (0 feet) and descends into a mine at 8 feet per second. How many seconds, s, does it take to reach a depth of 320 feet below ground level (−320 ft)?
Verbal Equation:
__________________________________________
Algebraic Equation:
__________________________
Solve & Check:
Answer: s = ______ seconds
7. Lunch Account Spending Decimals
Kayla has $28.50 in her school lunch account. Each day she buys a snack that costs $2.25. After d days, her account balance is $6.00. How many days did she buy snacks?
Verbal Equation:
__________________________________________
Algebraic Equation:
__________________________
Solve & Check:
Answer: d = ______ days
8. Multiple Choice Selection & Justification
A football team started on their own 20-yard line (+20). They received 3 penalties of equal yardage, p, moving them backward to their 5-yard line (+5). Which equation represents this situation?
a. 20 + 3p = 5
b. 20 − 3p = 5
c. −20 + 3p = 5
d. 3p − 20 = 5
Explain why choice b is correct using the words "starting," "backward/penalty," and "ending":
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 7
Section 5
Priority Practice: Independent Word Problems
Must Follow All 5 Steps!
Directions: For every problem, complete all 5 steps in the response box: Underline Start, circle Rate, [box] End. Write the verbal equation, write the algebraic equation, solve with inverse operations, and write your answer sentence.
1. Winter Sunrise Temperature 7.EE.4a / 7.NS.3
At 4:00 AM, the temperature was −12°F. Over the next several hours, the temperature rose by 4°F each hour. When the sun was fully up, the temperature was 8°F. How many hours did it take for the temperature to reach 8°F?
Step 2: Verbal Equation:
Start Temp + (Hourly Rate × Hours) = Ending Temp
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Inverse Operations):
Step 5: Sentence: It took ______ hours to reach 8°F.
2. Canyon Hike Descent Elevation Context
A park ranger begins at an elevation of 120 feet above sea level. She hikes down into a canyon, descending at a constant rate of 25 feet every minute. She stops when she is at a depth of 130 feet below sea level (−130 ft). How many minutes did she descend?
Step 2: Verbal Equation:
Start Elevation − (Descent Rate × Minutes) = Ending Depth
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Inverse Operations):
Step 5: Sentence: The ranger descended for ______ minutes.
3. Overdrawn Bank Account Financial Context
Anthony's bank account shows a debt of $35 (−$35). He makes 4 equal weekly deposits to his account. After the 4 deposits, his new account balance is $65. How much money was each weekly deposit?
Step 2: Verbal Equation:
Starting Debt + (4 × Weekly Deposit) = Final Balance
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Inverse Operations):
Step 5: Sentence: Each weekly deposit was $______.
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 8
Section 5 (Continued)
Priority Practice: Independent Word Problems
Problems 4–6
4. Hot Air Balloon Landing Negative Rate
A hot air balloon begins at an altitude of 950 feet. It descends at a constant rate of 40 feet per minute. After several minutes, the balloon is at an altitude of 150 feet. For how many minutes did the balloon descend?
Step 2: Verbal Equation:
__________________________________________
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Work):
Step 5: Sentence: The balloon descended for ______ minutes.
5. Laboratory Beaker Cooling Rational Numbers / Decimals
A liquid chemical starts at a temperature of 18°C. A scientist places it into a chiller where its temperature drops by 3.5°C every minute. How many minutes will it take for the liquid to reach −10°C?
Step 2: Verbal Equation:
__________________________________________
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Work):
Step 5: Sentence: It took ______ minutes to cool to −10°C.
6. Video Game Point Deficit Points Deficit Context
In a video game, Jordan's score is currently −160 points due to penalties. He completes 8 bonus stages, earning the exact same number of points, p, on each stage. His final score is 240 points. How many points did he earn per bonus stage?
Step 2: Verbal Equation:
__________________________________________
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Work):
Step 5: Sentence: Jordan earned ______ points per stage.
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 9
Section 5 (Continued)
Priority Practice: Independent Word Problems
Problems 7–9
7. Deep Sea Scuba Diver Negative to Negative
A diver is swimming at a depth of 95 feet below sea level (−95 ft). She ascends at a steady rate of 6 feet per minute. After m minutes, she reaches a coral reef that is 23 feet below sea level (−23 ft). How many minutes did she ascend?
Step 2: Verbal Equation:
__________________________________________
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Work):
Step 5: Sentence: She ascended for ______ minutes.
8. Spending a Gift Card Decimals / Money
Mr. Gomez had a gift card loaded with $50.00. He buys several smoothie bowls that cost $4.50 each. After buying his bowls, the remaining balance on the gift card is $14.00. How many smoothie bowls, b, did Mr. Gomez purchase?
Step 2: Verbal Equation:
__________________________________________
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Work):
Step 5: Sentence: He purchased ______ smoothie bowls.
9. Mountain Weather Station Negative Start & Negative Rate
On Mount Washington, the temperature at midnight was −5°F. Over the next several hours, the temperature dropped at a steady rate of 1.5°F per hour until it reached a low of −17°F. How many hours did the temperature continue dropping?
Step 2: Verbal Equation:
__________________________________________
Step 3: Algebraic Equation:
_________________________________
Step 4: Solve (Show Work):
Step 5: Sentence: It dropped for ______ hours.
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 10
Section 6
Error Analysis: Math Doctor
Diagnose & Fix 4 Errors
Directions: Each student below made a common mistake. Read the problem and the student's work. Identify the mistake, explain why it is wrong, and write the correct equation and solution.
Case 1: Sign Error on Starting Value Mistake in Sign
Problem: Nia's bank account was $25 in debt. She deposited $15 each week for w weeks until her balance reached $65.
Student Equation: 25 + 15w = 65
What is the mistake?
The student forgot that debt means...
Correct Equation & Solution:
Equation: __________________________
Correct w = ______ weeks
Case 2: Sign Error on Rate of Change Confusing Rising vs. Falling
Problem: An airplane flies at 2,800 feet. It descends at 120 feet per minute. How many minutes, m, until it is at 1,000 feet?
Student Equation: 2,800 + 120m = 1,000
What is the mistake?
The word "descends" means the rate must be...
Correct Equation & Solution:
Equation: __________________________
Correct m = ______ minutes
Case 3: Attaching the Variable to the Wrong Quantity Rate vs. Starting Amount
Problem: A gym charges a $40 sign-up fee plus $15 per month. Jayce has $145 to spend. How many months, m, can he join?
Student Equation: 40m + 15 = 145
What is the mistake?
The $40 is a one-time fee, so the variable belongs with...
Correct Equation & Solution:
Equation: __________________________
Correct m = ______ months
Case 4: Confusing Starting Amount and Total Mismatched Total
Problem: A temperature starts at −6°F and drops 3°F each hour for h hours until it reaches −21°F.
Student Equation: −21 − 3h = −6
What is the mistake?
The student switched the starting value and...
Correct Equation & Solution:
Equation: __________________________
Correct h = ______ hours
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 11
Section 7
Extra Practice: Fast Finisher / Additional Reps
Extension & Fluency
Directions: Write a verbal equation, translate into an algebraic equation, and solve. Check your final answer in context.
1. Submarine Sonar Test (Integer Rates)
A submarine is stationed at 115 feet below sea level (−115 ft). It ascends at a rate of 14 feet per minute. How many minutes will it take to reach a depth of 17 feet below sea level (−17 ft)?
Equation: _________________________________
Answer: m = ______ minutes
Work Space:
2. Camera Drone Elevation (Decimals)
A camera drone hovers at an altitude of 84.5 meters above the ground. Due to low battery, it descends at a steady rate of 4.2 meters per second until it reaches a safety landing pad at 13.1 meters. How many seconds did it descend?
Equation: _________________________________
Answer: s = ______ seconds
Work Space:
3. School Club Fundraising (Unknown Weekly Rate)
The robotics team begins with a balance of −$60 because they borrowed money for parts. Over the next 6 weeks, they hold bake sales and earn the exact same amount each week, x. At the end of 6 weeks, their club account has $150. How much did they earn each week?
Equation: _________________________________
Answer: x = $______ per week
Work Space:
4. Arctic Scientific Probe (Negative to Negative)
An ice probe is drilled into an ice sheet where the temperature is −4°C. As it drills deeper, the temperature drops by 1.8°C for every meter drilled. At what depth (in meters, d) will the temperature read −22°C?
Equation: _________________________________
Answer: d = ______ meters
Work Space:
5. Olympiad Extension Challenge Challenge
Tariq and Saniyah both start with bank account balances showing a debt. Tariq is $40 in debt (−$40) and saves $12 each week. Saniyah is $10 in debt (−$10) and saves $6 each week. In how many weeks, w, will their bank account balances be exactly equal?
Write an equation with variables on both sides and solve:
−40 + 12w = −10 + 6w ⟶ w = ______ weeks
7th Grade Math • L040 Reteach Packet "The only way to learn mathematics is to do mathematics." Page 12
7th Grade Math • Unit 3 Reteach EXIT TICKET • 10 POINTS
Two-Step Word Problems: Words to Equations
Score: / 10
Name:
Date:
Team:
Question 1: Interpreting Quantities & Signs [2 Points]
A diver is swimming at a depth of 40 feet below sea level. She then ascends at a rate of 5 feet per minute for 6 minutes.
Part A (1 pt): What signed number represents her starting depth?
Answer: ________ ft
Part B (1 pt): Is her rate of change positive (+) or negative (−)?
Answer: ________ (Explain: ascends)
Question 2: Guided Translation & Solution (Integers) [4 Points]
At midnight, the temperature in Fairbanks, Alaska was −8°F. A cold front caused the temperature to drop by 3°F each hour. After several hours, the temperature was −23°F. How many hours did the temperature drop?
Part A (1 pt): Write Verbal Equation:
Start Temp + (Hourly Drop × Hours) = Ending Temp
Part B (1 pt): Write Algebraic Equation:
_________________________________
Part C (2 pts): Solve & Write Sentence:
Sentence: The temperature dropped for ______ hours.
Question 3: Independent Context & Solution (Rational Numbers) [4 Points]
Malik has a bank account that is $32.50 in debt (−$32.50). Over the next 5 weeks, he makes equal deposits of d dollars each week. After 5 weeks, his bank balance is $67.50. How much did Malik deposit each week?
Part A (1 pt): Write Verbal Equation:
Starting Debt + (Weeks × Deposit) = Final Balance
Part B (1 pt): Write Algebraic Equation:
_________________________________
Part C (2 pts): Solve & Write Sentence:
Sentence: Malik deposited $______ each week.
7th Grade Math • L040 Reteach Packet • Exit Ticket "The only way to learn mathematics is to do mathematics." Page 13
