A comprehensive 6th-grade math lesson introducing integer addition and subtraction through additive inverses, horizontal and vertical number line models, and real-world opposite scenarios like elevation and temperature.
4 Part 4: Distance Between Two Integers \( |p - q| \)
Standard 7.NS.A.1.C
Show that distance is the absolute value of the difference. Show your steps in the work box.
Problem 4A: Thermal Swings
In North Dakota, the high was +12°F and low was −9°F. Calculate the total temperature distance:
Distance = | 12 − (−9) | =
Total Distance: °F
Problem 4B: Reversing Order
Calculate the distance between −9°F and +12°F using the reverse subtraction | −9 − 12 |:
Distance = | −9 − 12 | =
Total Distance: °F
Part 5: Deep-Sea Ocean Trench Mission
Multi-Step
An oceanic research vessel deploys a weather drone flying at +125 meters above sea level and a robotic submersible operating at −85 meters below sea level.
Step 1: Vertical Distance
Write an absolute value subtraction expression for drone and sub distance:
Expression: = m
Step 2: Submersible Maneuver
Sub descends 40 m (\(-40\)), then ascends 60 m (\(+60\)). Find depth:
\( -85 + (-40) + 60 = \) m
Zero Zone • Student Practice Page 2 of 2
In \( (-2) + 5 \), students jump 5 from 0 rather than starting at −2. Emphasize: "Put your finger on \(p\) FIRST. The jump only begins once your feet are planted on \(p\)!"
Thinking distance can be negative.
When asked for distance between −3 and +5, students answer −8. Remind them: "Rulers have no negative numbers. Distance is always the absolute value \(
Zero Zone Facilitation Guide Page 1 of 2
P2
CLASSROOM SCRIPTS, ANCHORS & DIFFERENTIATION
Instructional Delivery
Scripted Turn-and-Talk Prompts
Prompt 1 (Concept of Additive Inverse):
"Imagine an elevator starts on floor 0, travels 7 floors down to the basement parking lot (−7), and then ascends 7 floors. Where does it end? Why must any integer and its opposite always sum to zero?"
Target response: They are equal distances from zero on opposite sides. Combining them returns the system to equilibrium.
Prompt 2 (Demystifying Subtracting Negatives):
"Look at \( 5 - (-3) \). If subtraction means 'taking away', how can taking away a negative value result in a BIGGER positive number (+8)?"
Target response: Taking away cold air makes a room warmer; taking away a bill makes you richer. Subtracting the opposite adds positive value.
Board Anchor Chart Setup
Positive Direction (Green Chalk/Marker)
Moves RIGHT on horizontal line
Moves UP on vertical thermometer/elevation
Represents: credits, climbing, warming, gains
Negative Direction (Red Chalk/Marker)
Moves LEFT on horizontal line
Moves DOWN on vertical line
Represents: debits, diving, cooling, losses
Tiered Differentiation Matrix
Tier 1: Targeted Support
Provide physical two-color integer chips (yellow = +1, red = −1). Allow students to physically pair up zero pairs and push them aside before moving to number lines.
Focus: Concrete Manipulatives
Tier 2: On-Level Scaffolding
Require students to speak the three steps aloud: 1) "I start at \(p\)." 2) "I face positive or negative based on sign." 3) "I jump \(|q|\) spaces." Use colored highlighters for vectors.
Focus: Procedural Fluency
Tier 3: Extension Challenge
Pose non-integer and variable challenges: "If \(p\) is negative and \(q\) is positive, under what condition is \(p + q > 0\)?" (Answer: when \(|q| > |p|\)). Connect to coordinate plane distances.
• 4 (Exemplary): Correctly rewrites subtraction with additive inverse, arrows drawn accurately with correct head direction, zero pair correctly identified.
• 2 (Developing): Computes correct numerical values but fails to demonstrate vector jumps on line or treats distance as negative.
Elevation change: −140 meters (descent of 140 m).
Equation: (+140) + (−140) = 0. Acceptable reasoning: It must descend the exact distance it climbed to return to starting elevation 0.
Q2: Vector on Line \( (-3) + (+5) \) 1 pt
Start at −3. Draw vector 5 units right.
Final Sum: +2. Scoring criteria: 0.5 pt for correct vector line drawing, 0.5 pt for accurate sum of +2.
Q3: Subtraction Rewrite \( -6 - (-9) \) 1 pt
Rewrite: \( -6 + (+9) \).
Final Sum: +3. Scoring criteria: Student must show addition of positive 9, not just the final numerical answer.
Q4: Distance Between −7°C & +8°C 1 pt
Expression: | 8 − (−7) | = | 8 + 7 | = 15.
Total Distance: 15°C. Common Error Alert: If student writes 1°C, they subtracted without accounting for the negative sign.
Remediation Threshold: Any student scoring ≤ 2/4 on the Exit Ticket should participate in a 10-minute small-group re-teaching station using physical two-color counters before moving to integer multiplication.
