A three-day instructional unit mastering multi-digit addition and subtraction word problems with unknowns in any position. Students model situations using bar models, write algebraic equations with unknown symbols, and check solutions for reasonableness.
Officer Patel says: "In the problem \(s + 3,450 = 8,200\), I should add \(3,450 + 8,200\) to find \(s\) because it's an addition problem." Is Officer Patel correct? Explain why or why not, and show the correct way to solve for \(s\).
Equation Detectives • Day 1: Addition with Unknowns Page 2 of 2
Day 1 Worksheet Solutions
Case #101: The Wildlife Sanctuary (Change Unknown) Answer: 2,355 birds
Equation:
\(1,465 + b = 3,820\)
Inverse Work:
\(b = 3,820 - 1,465 = 2,355\)
Check:
\(1,465 + 2,355 = 3,820\) ✓
Case #102: The Community Food Drive (Start Unknown) Answer: 4,385 cans
Equation:
\(c + 2,745 = 7,130\)
Inverse Work:
\(c = 7,130 - 2,745 = 4,385\)
Check:
\(4,385 + 2,745 = 7,130\) ✓
Case #103: The School Fundraiser (Result Unknown) Answer: 8,323 pages
Correction & Explanation: Officer Patel is incorrect. Although the situation is modeled with addition (\(s + 3,450 = 8,200\)), the total is already known to be \(8,200\). Adding \(3,450 + 8,200\) yields \(11,650\), which cannot be a starting part because a part cannot be greater than the whole.
Correct Operation: To find the missing part \(s\), use the inverse operation: \(s = 8,200 - 3,450\). Therefore, \(s = 4,750\).
Equation Detectives • Day 1 Answer Key Page 2 of 2
Since money was spent from the original start:
m - 3,425 = 4,860
How Do We Solve for the Start?
To find what was there originally, combine the spent and remaining parts:
m = 4,860 + 3,425 = 8,285
Investigation Takeaway: When the starting amount is unknown in subtraction, add the two parts together!
Solving Matrix
How to Solve Subtraction Unknowns
Day 2 Strategy
Position 1
Start Unknown
? - 1,200 = 3,400
The whole is missing! Add the subtrahend and difference together.
Operation: Add
Position 2
Change Unknown
4,600 - ? = 1,800
The removed part is missing. Subtract the remaining from the total.
Operation: Subtract
Position 3
Difference Unknown
5,000 - 2,350 = ?
Both parts are known. Direct subtraction gives the difference.
Operation: Subtract
Verification Mandate: Every case today requires an addition check: Part + Part = Total!
Slide 5 of 5
Detective Defense Brief Explain & Justify
In Case #204, the problem used the words "1,690 fewer visitors". Why did you have to add rather than subtract to find the number of visitors at the Space Pavilion? Explain your mathematical reasoning using the idea of larger vs. smaller quantities.
Equation Detectives • Day 2: Subtraction & Comparison with Unknowns Page 2 of 2
Answer Key & Scoring Guide
Difference Detectives Complete Solutions
Day 2 Worksheet Solutions
Case #201: The Mountain Supply Depot (Change Unknown) Answer: 3,545 ration packs
Equation:
\(6,420 - r = 2,875\)
Calculation Work:
\(r = 6,420 - 2,875 = 3,545\)
Addition Check:
\(2,875 + 3,545 = 6,420\) ✓
Case #202: The Cargo Harbor (Comparison Unknown) Answer: 5,160 containers
Explanation: The word "fewer" describes the comparison between the two pavilions: Dinosaur had fewer than Space, which means the Space Pavilion had the larger quantity. To find a larger quantity when given the smaller quantity and the difference, you must combine (add) them together: \(3,415 + 1,690 = 5,105\).
If a student subtracted, they would get \(1,725\), which would incorrectly make the Space Pavilion have fewer visitors than the Dinosaur Pavilion.
Equation Detectives • Day 2 Answer Key Page 2 of 2
Estimate ≈ Actual
B
Inverse Operation
Did you subtract? Add the difference back to the subtrahend to see if you get the original total. Did you add? Subtract to verify.
Part + Part = Total
C
Meaningful Label
Numbers without units are unsolved clues! State exactly what your number represents in the context of the story.
e.g., "4,855 guests"
Investigation Standard: Every final answer in today's mission must include both an estimate check and a labeled unit!
Mission Ready
Master Detective Checklist
Day 3 Mission Protocol
1
Read the Case: Identify the hidden question and the final question.
2
Write Step 1 Equation: Use an unknown symbol (h) and calculate the intermediate value.
3
Write Step 2 Equation: Connect the intermediate answer to find the final unknown (x).
4
Explain & Check: Check by rounding, use inverse operations, and write the full unit label!
Capstone Assignment: Complete the Operation Mission Practice Packet!
Slide 5 of 5
Captain Davis started the month with a balance of frequent flyer miles. She earned 2,450 miles on Flight 1 and 3,780 miles on Flight 2. She then used 4,500 miles for an upgrade. She now has 8,230 miles. What was her starting balance (\(s\))?
Choose either Case #301 or Case #303. Explain why finding the hidden question was essential to solving the problem. Then show how you used estimation to confirm that your final calculated answer is completely reasonable.
Equation Detectives • Day 3: Two-Step Missions & Explain-and-Check Page 2 of 2
"What does your variable \(h\) represent in words?"
Equation Detectives • Day 3 Teacher Guide Page 1 of 2
Answer Key & Scoring Guide
Operation Mission Complete Solutions
Day 3 Worksheet Solutions
Case #301: The Wildlife Rescue Center Answer: 1,175 turtles
Exemplar (Case #303): "Finding the hidden question (how much money East Middle School received) was essential because the problem never stated East's funding directly—it only stated how much less East received than West. Without finding East's amount ($3,270), I could not combine both schools' awards.
I rounded West to $4,700 and the difference to $1,400. That gave an estimate for East of $3,300. Adding $4,700 + $3,300 gave a ballpark of $8,000. My exact answer of $7,920 is within $80 of my estimate, which proves my calculation is reasonable."
Unit Completion: Students who master all 3 days have fully achieved CCSS.4.OA.3 and 4.NBT.4 standard targets.
Equation Detectives • Day 3 Answer Key Page 2 of 2
