Vault Recovery: The Cryptographer's Shift
A power surge scrambled the smart-vault that protects the museum's rarest artifacts. The locks still work — but every dial now shows a mystery equation instead of a number, and the climate and weight safety zones must be reset. As tonight's junior cryptographer, you will solve one-step equations to recover each dial code, then write and graph the inequalities that keep the vault safe.
Your First Night on the Job
A vault dial is really just a hidden number waiting to be found. The surge replaced each code with a sentence like "a number plus 8 equals 21." A cryptographer's whole job is to undo what was done to that number until it stands alone. In this WebQuest you become that cryptographer — and the same "keep both sides balanced" thinking will reset the vault's safety zones, too.
The Driving Question
How can solving a one-step equation reveal an exact unknown, while an inequality describes a whole range of safe values — and how do you decide which one a real situation needs?
The Task
By the end of this WebQuest you will submit a Vault Recovery Log. A complete log must:
- Recover four dial codes by solving one-step equations (add, subtract, multiply, and divide).
- Show the inverse move you used on each dial and check your answer by substituting it back in.
- Write two inequalities from the vault's safety rules (a climate limit and a weight limit).
- Graph each inequality on a number line with the correct open or closed circle and arrow direction.
- Pass the Check Your Understanding self-check at the bottom to unlock the vault.
The Process
Work through the steps in order. Each step fills one section of your Vault Recovery Log.
Step 1 — Learn the four inverse moves
Every one-step equation hides the variable behind a single operation. Undo it with the inverse operation on both sides. Keep this reference open.
Step 2 — Recover the four dial codes
Solve each dial's equation and record the code in your log. Then check by putting your answer back into the original equation.
x + 8 = 21
m − 6 = 15
4n = 52
p ÷ 3 = 9
Worked example — Dial A (x + 8 = 21):
x + 8 = 21 → x + 8 − 8 = 21 − 8 → x = 13. Check: 13 + 8 = 21 ✓
Step 3 — Translate a safety rule into an inequality
The vault posts two safety rules. Each one describes a range of allowed values, so it becomes an inequality, not an equation. Match the words to the correct symbol:
- Climate rule: "The temperature t must stay below 68°F." → write the inequality.
- Weight rule: "A single shelf can hold at most 40 kg of artifacts (w)." → write the inequality.
Word-to-symbol guide:
"below / less than" → < · "more than / above" → > · "at most / no more than" → ≤ · "at least / no less than" → ≥
Step 4 — Graph each inequality on a number line
Translate your inequality into a picture. Use the rules below, then sketch both number lines in your log.
- Open circle ○ for < or > (the boundary value is not included).
- Closed circle ● for ≤ or ≥ (the boundary value is included).
- The arrow points toward every number that makes the inequality true (left for less-than, right for greater-than).
Worked example — graph of t < 68:
Open circle at 68 (68 is not allowed); the arrow shades every temperature below 68.
Step 5 — Assemble and verify the Recovery Log
Put it all together: four checked dial codes, two written inequalities, and two number-line graphs. Re-substitute each dial code one last time, then run the self-check below to confirm the vault will reopen.
Resources
Use these Neft Teacher tools as you work. They open in the same window — use your back button to return.
Key vocabulary · Vocabulario clave
- Equation / Ecuación — a math sentence with an equals sign and one exact solution.
- Inequality / Desigualdad — a sentence using <, >, ≤, or ≥ that allows a range of solutions.
- Variable / Variable — a letter that stands for an unknown number.
- Inverse operation / Operación inversa — the move that undoes another (subtract undoes add; divide undoes multiply).
- Solution / Solución — a value that makes the sentence true.
Evaluation
Your Vault Recovery Log will be scored on this rubric.
| Criteria | 4 · Lead Cryptographer | 3 · Analyst | 2 · Trainee | 1 · Getting started |
|---|---|---|---|---|
| Solving equations | All 4 dial codes correct with the right inverse move. | 3 of 4 codes correct. | 2 codes correct. | 0–1 codes correct. |
| Checking solutions | Every solution substituted back and verified. | Most solutions checked. | Some checks shown. | No checks shown. |
| Writing inequalities | Both inequalities use the correct symbol and variable. | One inequality fully correct. | Symbols attempted with errors. | No inequalities written. |
| Graphing on a number line | Both graphs: correct circle type and arrow direction. | One graph fully correct. | Circle or arrow error on each. | No usable graph. |
| Reasoning | Clear claim–evidence–reasoning explanation. | Explains most choices. | Brief or partial reasoning. | No explanation. |
Teacher Notes & Answer Key (not printed)
Vault Recovery · Recovery Log — pairs with the Evaluation rubric above.
Sample Answers — Vault Recovery Log
- Dial A: x + 8 = 21 → x = 21 − 8 = 13. Check: 13 + 8 = 21 ✓.
- Dial B: m − 6 = 15 → m = 15 + 6 = 21.
- Dial C: 4n = 52 → n = 52 ÷ 4 = 13.
- Dial D: p ÷ 3 = 9 → p = 9 × 3 = 27.
- Inequalities: "below 75°" → t < 75 (open circle, arrow left); "at most 400 lb" → w ≤ 400 (closed circle, arrow left).
Facilitation
- Name the inverse operation on each dial and substitute back to check.
- Open circle for < / >; closed circle for ≤ / ≥; arrow points toward the solutions.
Standard
CCSS 6.EE.B.5–8.
Check Your Understanding
Answer all six. Click Check My Answers when you are done. These match the dials and safety rules in your log.
Conclusion
You did what real cryptographers, engineers, and safety inspectors do: you reversed the operations hiding an unknown to recover an exact code (Dial A = 13, Dial C = 13, Dial D = 27), then used inequalities to describe a whole range of safe conditions. With the dials reset and the safety zones graphed, the vault reopens — the artifacts are secure. Mission complete!
Take it further · Level 2
Invent a fifth dial whose answer is also 13 but uses a different operation, plus a new shelf rule that needs ≥. Graph the new rule.
Reflect · Reflexiona
When does a situation need an equation (one answer) versus an inequality (a range)? Explain in 2–3 sentences. ¿Cuándo se necesita una ecuación y cuándo una desigualdad?
Real careers · Carreras reales
Cybersecurity analysts, structural engineers, and pharmacists all solve equations and set safe-range inequalities — exactly the skills you just used.