Mission 17 · Unit 7

Rational Numbers

6.NS.C.6 · Unit 8
Today's objective: Place rational numbers on a number line and interpret their order.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

A marine biologist is studying ocean creatures at different depths. A dolphin swims at -3.5 meters (below sea level). A sea turtle dives to -12¾ meters. A flying fish jumps to +2.25 meters (above sea level). Your team must place these animals on a vertical number line, compare their positions, order them from deepest to highest, and find the distance between the dolphin and the sea turtle.

SEA LEVEL (0) +4 +2 0 -2 -3.5 -6 -8 -10 -12.75 🐟 Flying Fish: +2.25 m 🐬 Dolphin: -3.5 m 🐢 Turtle: -12.75 m Distance: 9.25 m

The Investigation

The Problem: Three ocean animals are at these positions relative to sea level: flying fish at +2.25 m, dolphin at -3.5 m, sea turtle at -12¾ m (which is -12.75 m). Place all three on a vertical number line. Order them from deepest to highest. Then calculate the distance between the dolphin and the sea turtle. Finally: which animal is closest to sea level?

Visual Model: Vertical Number Line

Ordering and Distance on a Vertical Number Line +4 +2.25 Flying Fish (+2.25) 0 Sea Level -3.5 Dolphin (-3.5) -12.75 Turtle (-12.75) 9.25 m Order (deepest to highest): 1. Turtle: -12.75 m (deepest) 2. Dolphin: -3.5 m 3. Flying Fish: +2.25 m (highest) Closest to sea level: Flying Fish Distance Calculation: |-3.5 - (-12.75)| = |-3.5 + 12.75| = |9.25| = 9.25 meters

Step-by-Step Investigation Guide

  1. Convert to the same form. Write -12¾ as a decimal: -12.75. Now all three positions are decimals. Guiding question: How do you convert ¾ to a decimal?
  2. Draw the vertical number line. Mark sea level as 0. Place positive numbers above and negative below. Plot all three creatures. Guiding question: Why does -12.75 go farther below 0 than -3.5?
  3. Compare positions. Which is deepest? Which is highest? Use < and > to compare each pair. Guiding question: Is -12.75 greater or less than -3.5? How do you know?
  4. Order from least to greatest. -12.75 < -3.5 < +2.25. Write this ordering. Guiding question: On a vertical number line, does "less than" mean higher or lower?
  5. Find the distance. Distance between dolphin and turtle: |-3.5 - (-12.75)| = |-3.5 + 12.75| = |9.25| = 9.25 m. Guiding question: Why do we use absolute value for distance?
  6. Identify closest to sea level. Compare |+2.25| = 2.25, |-3.5| = 3.5, |-12.75| = 12.75. Flying fish is closest (2.25 m from 0). Guiding question: What does absolute value tell us about position?

Language Support: Key Vocabulary

Rational number
Any number that can be written as a fraction: integers, decimals, fractions.
Negative number
A number less than zero. Written with a minus sign: -3.5.
Positive number
A number greater than zero: +2.25 or just 2.25.
Absolute value
The distance from zero. Always positive. |-5| = 5.
Opposite
Same distance from zero, other side. The opposite of -3 is +3.
Sea level
The surface of the ocean. We call this zero (0 meters).
Sentence Frames:

"_____ is deeper than _____ because it is farther below sea level."

"The absolute value of _____ is _____, which means it is _____ meters from sea level."

"I know -12.75 < -3.5 because on the number line, -12.75 is _____."

Multiple Representations

Approach 1: Vertical Number Line

Draw a vertical line with 0 in the middle. Plot each value. The lowest point is the least value. Read the order by going up the line.

Approach 2: Absolute Value Comparison

Find each distance from zero:
|+2.25| = 2.25
|-3.5| = 3.5
|-12.75| = 12.75
The smallest absolute value is closest to sea level.

Approach 3: Real-World Reasoning

Think of an elevator: floor -12 is much lower than floor -3. A positive floor is above ground. You would ride the elevator 9.25 floors from -3.5 to -12.75.

Team Roles

Facilitator Read the ocean scenario. Make sure everyone converts -12¾ to -12.75. Guide the team through each comparison.
Model Builder Draw the vertical number line with sea level at 0. Plot and label all three creatures at the correct positions with accurate spacing.
Precision Checker Verify each comparison (< and >). Check the distance calculation step by step. Confirm absolute values are correct.
Reporter Prepare the defense: state the order, the distance, and which animal is closest to sea level. Use complete sentences with math evidence.

Timed Lab Phases

Launch Phase
03:00

Read the marine biologist scenario. Assign roles. Convert -12¾ to a decimal.

  • What does "below sea level" mean in numbers?
  • What does a positive number mean here?
  • What is ¾ as a decimal?
Checkpoint: Everyone knows the three values and what positive/negative means in this context.

Draw the vertical number line. Place all three animals. Write comparison statements.

  • Where is 0 (sea level) on your number line?
  • Is -12.75 above or below -3.5?
  • Write: -12.75 ___ -3.5 (fill in < or >).
Checkpoint: Number line drawn with all 3 creatures plotted. Comparison statements written.

Order from least to greatest. Calculate the distance between dolphin and turtle. Find which animal is closest to sea level.

  • What is |-3.5 - (-12.75)|?
  • Which absolute value is smallest?
  • Write the complete ordering with < signs.
Checkpoint: Ordering complete. Distance calculated. Closest animal identified.

Reporter prepares the defense using the number line as visual evidence.

  • "The order from deepest to highest is _____ because _____."
  • "The distance is _____ meters because _____."
  • "The _____ is closest to sea level because its absolute value is _____."
Checkpoint: Defense uses the number line, comparisons, and absolute value calculations.

Challenge Extensions

Extension Problem: A submarine is at -85.5 meters. It rises 30.25 meters, then dives 15.75 meters. What is its final position? Plot the starting point, each move, and the final position on a number line.

What If?

  • What if the dolphin dives 8 more meters? What is its new position? Is it deeper than the turtle?
  • A whale is exactly halfway between the dolphin and the turtle. What depth is the whale at?
  • What temperature is the opposite of -3.5 degrees? Plot both on a horizontal number line.

Real-Life Connections

Negative numbers appear in temperatures below zero, bank account overdrafts, elevations below sea level (Death Valley is -86 m), and golf scores under par.

Defense Preparation

  1. Order the three animals from deepest to highest. Explain how you know.
  2. What is the distance between the dolphin and the sea turtle? Show your work.
  3. Which animal is closest to sea level? How does absolute value help you answer?
  4. Is -3.5 greater or less than -12.75? Explain using the number line.
Sentence Starters:
  • "The deepest animal is _____ at _____ meters because _____."
  • "I found the distance by calculating |_____ - (_____)| = _____."
  • "Absolute value tells us _____, which is useful because _____."

Rubric

Criteria Excellent (4) Proficient (3) Developing (2)
Number line Accurate vertical line with all 3 plotted and labeled All 3 plotted with minor spacing issues Some creatures plotted
Ordering Correct order with comparison symbols and explanation Correct order stated Partial ordering
Distance Correct calculation with absolute value shown Correct answer Attempted
Absolute value Used to compare distances from zero with explanation Absolute values calculated Concept mentioned

Exit Product

Your team submits: An Ocean Depth Report with:
  • A vertical number line with all three creatures plotted accurately
  • Ordering from least to greatest with < symbols: -12.75 < -3.5 < +2.25
  • Comparison statements for each pair (e.g., "-3.5 > -12.75 because...")
  • Distance calculation using absolute value: |-3.5 - (-12.75)| = 9.25 m
  • Identification of which creature is closest to sea level with justification

Self-Assessment Checklist