The Salvage Dive
A salvage submarine descends at a steady signed rate — and running the clock backward makes negative-times-negative inevitable. Students extend multiplication and division to all rational numbers, then convert fraction depths for the digital gauge.
Common Core standards covered
- 7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
- 7.NS.A.2.aUnderstand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (−1)(−1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
- 7.NS.A.2.bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then −(p/q) = (−p)/q = p/(−q). Interpret quotients of rational numbers by describing real-world contexts.
- 7.NS.A.2.cApply properties of operations as strategies to multiply and divide rational numbers.
- 7.NS.A.2.dConvert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Multiply and divide rational numbers using the properties of operations, interpret signed products and quotients in context, and convert rational numbers to decimals that terminate or repeat.
Student target: “I can multiply and divide any rational numbers, justify the sign rules with patterns and contexts, and convert a fraction to its terminating or repeating decimal.”
Success criteria: I can interpret a signed product as rate × time in a context. · I can justify why negative × negative is positive, not just recite it. · I can divide signed numbers and use −(p/q) = (−p)/q = p/(−q). · I can convert a rational number to a decimal and say whether it terminates or repeats.
Essential question
Descending 15 m per minute is a NEGATIVE rate. What should multiplying by it mean — and what happens when the time is negative too?
Learning target: I can multiply and divide any rational numbers, justify the sign rules with patterns and contexts, and convert a fraction to its terminating or repeating decimal.
Success criteria
- I can interpret a signed product as rate × time in a context.
- I can justify why negative × negative is positive, not just recite it.
- I can divide signed numbers and use −(p/q) = (−p)/q = p/(−q).
- I can convert a rational number to a decimal and say whether it terminates or repeats.
Why this matters
The sign rules are where Grade 7 arithmetic most often turns into incantation — 'minus times minus is plus' recited without a why. The dive gives the why twice over: the pattern table (each step down in the multiplier adds +15) and the time-reversal story (one minute AGO, the sub was higher). Division inherits everything by undoing multiplication, and the decimal conversion closes the loop 7.NS.A.2.d asks for — a loop Grade 8 reopens when repeating decimals become the fingerprint of rationality itself.
Students should already be able to
- Adding and subtracting signed numbers on the number line — Subzero Expedition.
- Fraction multiplication and division (positive cases).
- Long division with decimals.
Vocabulary to define explicitly
- Signed rate
A rate with direction built in: negative for down or loss, positive for up or gain.
Example: Descending 15 m each minute is −15 m/min.
Not an example: '15 m/min downward' written as +15 loses the direction the arithmetic needs.
🔒 3 more terms with examples and non-examples — Product of signed numbers, Quotient of signed numbers, Terminating vs repeating decimal — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up repeated-addition-to-product compression.
- Dive-log rows during mission control — the −2 minute entry especially.
- Gauge-station labels and repeat bars.
Summative — the graded evidence: The exit ticket (two products, a quotient, two conversions), scored with the four-criterion rubric.
The task students work on
The salvage ship Peregrine lowers its sub to a wreck. The sub descends at a steady −15 m/min, passing the surface marker (depth 0) at time 0 — so negative times are BEFORE the marker, positive times after. Mission control needs the dive log completed: depths at +1 through +4 minutes, positions at −1 and −2 minutes (before the marker), the rate recovered from a −90 m change over 6 minutes, and the fraction depths on the paper chart — 3/8, 5/11, 7/12, −27/4 — converted for the digital gauge.
Dive log — products above the line, quotients and conversions below.
| Entry | Computation | Result | Meaning |
|---|---|---|---|
| Depth at 4 min | 4 × (−15) | ||
| Position at −2 min | (−2) × (−15) | ||
| Rate from the log | −90 ÷ 6 | ||
| Time to change −60 m | −60 ÷ (−15) | ||
| Chart depth −27/4 m | convert to decimal | ||
| Chart fraction 5/11 | convert to decimal |
Anticipated student strategies
Pattern first (it derives the rule), story second (it explains the pattern), distributive check last (it certifies any single computation).
- Pattern-table descent. The student builds 3×(−15), 2×(−15), 1×(−15), 0×(−15) and notices each step down in the multiplier ADDS 15 — so (−1)×(−15) must be +15.
The standard's own argument (properties preserved). Share first; the table converts the rule from decree to consequence.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
(−15) + (−15) + (−15) + (−15) = −60, compressed to 4 × (−15) = −60. Positive times negative: a positive amount of time at a negative rate. Then plant the hook: 'Where was the sub one minute BEFORE the marker?' Hold answers — the table will decide.
Listen for: 4 minutes × (−15 m/min) = −60 m of change. The signs carry direction; the units multiply like the numbers.
Board both sign rules as conclusions, not commandments. Then division inherits everything: −90 m over 6 minutes is −90 ÷ 6 = −15 m/min; and the triple identity −(90/6) = (−90)/6 = 90/(−6) — three costumes, one number. Distributive check ritual: −3 × 18 via −3 × (20 − 2) = −60 + 6 = −54.
- Same signs → positive product; different signs → negative. Derived, not decreed.
- Division is multiplication run backward, so the sign rules transfer whole.
Listen for: The +15-per-step pattern is forced by 3, 2, 1, 0 rows the skeptic already accepts — continuing it isn't a choice.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “Negative times negative is negative — two bad things make it worse.” Run the table: 1×(−15) = −15, 0×(−15) = 0… each step down ADDS 15, so (−1)×(−15) = +15. And the story agrees: before descending, the sub was higher. The 'worse' intuition is about addition, not multiplication.
- “−90 ÷ 6 and 90 ÷ (−6) are different problems with different answers.” — the teacher response is in the full lesson.
- “5/11 has no exact decimal, so fractions and decimals are different kinds of numbers.” — the teacher response is in the full lesson.
- “The sign rules are arbitrary conventions someone picked.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 5 independent.
- I doComplete the dive-log pair: the sub's depth after 4 minutes at −15 m/min, and its position 2 minutes BEFORE the marker.
🔒 6 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- A balloon drops 12 m per minute. Write and evaluate the product for its change over 5 minutes.
🔒 3 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Model descent as repeated signed addition, then multiplication. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Run the clock backward to discover the sign rules. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Divide to recover rates and times. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Convert fraction depths to decimals for the gauge. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: The pattern table carries the sign rules — keep it posted. Support with integer rates only; extend with fraction rates (−3/4 m per second) and the repeating-decimal hunt.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One completed dive log with signed products, quotients, and one fraction-to-decimal conversion.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.