The Harbor Log

Common Core standards covered

• 6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
• 6.NS.C.6Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
• 6.NS.C.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., −(−3) = 3, and that 0 is its own opposite.
• 6.NS.C.6.cFind and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
• 6.NS.C.7Understand ordering and absolute value of rational numbers.
• 6.NS.C.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret −3 > −7 as a statement that −3 is located to the right of −7 on a number line oriented from left to right.
• 6.NS.C.7.bWrite, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write −3 °C > −7 °C to express the fact that −3 °C is warmer than −7 °C.
• 6.NS.C.7.cUnderstand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of −30 dollars, write |−30| = 30 to describe the size of the debt in dollars.
• 6.NS.C.7.dDistinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than −30 dollars represents a debt greater than 30 dollars.

© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

Teacher-facing overview

Big idea: Use positive and negative numbers to represent real-world quantities, position and order rational numbers on the number line, and interpret absolute value as magnitude — distinguishing it from statements about order.

Student target: “I can represent opposite-direction quantities with signed numbers, plot and order rationals on the number line, read inequalities as positions, and use absolute value as size — never confusing it with order.”

Success criteria: I can assign signed numbers with 0's meaning stated. · I can plot rationals, including fractions and decimals, on horizontal and vertical lines. · I can write and defend order statements like −3 °C > −7 °C. · I can compute |x| as distance from 0 and keep it separate from order.

Essential question

Below-zero numbers describe real depths, debts, and cold. What does −8 mean, why is −3 GREATER than −7 — and how can a smaller balance be a bigger debt?

Learning target: I can represent opposite-direction quantities with signed numbers, plot and order rationals on the number line, read inequalities as positions, and use absolute value as size — never confusing it with order.

Success criteria

• I can assign signed numbers with 0's meaning stated.
• I can plot rationals, including fractions and decimals, on horizontal and vertical lines.
• I can write and defend order statements like −3 °C > −7 °C.
• I can compute |x| as distance from 0 and keep it separate from order.

Why this matters

6.NS.C.5-7 is where the number line doubles in size, and the two ideas installed here — order is POSITION, absolute value is SIZE — are precisely the ones that collide in C.7.d's famous scenario: a balance below −30 is a debt above 30. Students who own that distinction sail through Grade 7's signed arithmetic (Subzero Expedition assumes it) and read every thermometer, elevation, and account statement correctly for life.

Students should already be able to

• The positive number line, including fractions and decimals.
• Comparing positive rationals.
• Real contexts: temperature, money, elevation.

Vocabulary to define explicitly

• Signed numbers (opposite directions)

  Positives and negatives describe opposite sides of a chosen zero.

  Example: Gull +12 m, reef −8 m — sea level is the 0 that gives both meaning.

  Not an example: '8 meters' without a sign or a zero — depth or height?

🔒 3 more terms with examples and non-examples — Opposite, Order as position, Absolute value (magnitude) — 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 zero sentences.
• Log columns during floor work.
• The two-rankings contrast.

Summative — the graded evidence: The exit ticket (assign, order, split), scored with the four-criterion rubric.

The task students work on

The harbormaster's log tracks everything against zero. HEIGHTS: gull +12 m, dock 0, buoy chain −3.5 m, reef −8 m, submarine −15 m — plot and order them. TEMPERATURES: the week read −3°, 2°, −7°, 0°, −1.5° — order the week and write the warmest-cold comparison as an inequality. THE LEDGER: accounts at −30, 25, and −45 dollars — order the balances, then rank the DEBTS, and explain why the two rankings differ. OPPOSITES: the crane lifts from −6 m to its opposite — where, and what is −(−6)?

Harbor log — position and size are separate columns.

Anticipated student strategies

Zero first (meaning), position second (order), magnitude third (size) — and the ledger forces the last two apart exactly where the standard wants the distinction drawn.

1. Zero-anchored assignment. The student names what 0 means (sea level, freezing, even account) before signing any number.

   C.5's exact demand — the zero sentence is the entry ticket to every context.

🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →

Lesson script

🔒 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

• “−7 > −3 because 7 is bigger than 3.” Plot them: −7 sits LEFT of (below) −3. Below zero, bigger digits mean farther down. −3 > −7 — the warmer day proves it.
• “|−30| = −30.” — the teacher response is in the full lesson.
• “A balance below −30 means a debt below $30.” — the teacher response is in the full lesson.
• “Negative fractions like −3.5 don't fit on the line.” — 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 4 independent.

1. I doPlot and order the week's temperatures (−3, 2, −7, 0, −1.5), then write the −3 vs −7 comparison as an inequality with its weather sentence.

🔒 5 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

1. A diver at −12 m, a kite at 9 m, a swimmer at 0. What does 0 mean, and who is farthest from it?

🔒 2 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

1. Step 1: Assign signed numbers to harbor heights, temps, and balances. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
2. Step 2: Plot and order them on vertical number lines. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
3. Step 3: Read inequalities as positions: −3 > −7. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
4. Step 4: Split order from size: absolute value as magnitude. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”

Differentiation and inclusion

Support: Vertical lines keep up/down physical; extend with fractions and decimals on the line and the debt-vs-balance paradox.

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 plotted-and-ordered set with an inequality chain, plus one order-vs-magnitude distinction in context.

Mastery: The student completes the task accurately and explains the strategy with limited prompting.

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