Unit 7 · Teacher Answer Key
Unit 7 Projects — Teacher Answer Key
Integers & Coordinate Plane · Grade 6 · Standards 6.NS.5, 6.NS.6, 6.NS.7, 6.NS.8
For teachers only — do not distribute to students.
Worked solutions below use each project's default input values.
Students who change the inputs will get different but mathematically similar answers;
the same reasoning steps apply. Use the "Sample Expert Response" paragraphs as
4-point anchor papers.
Phase 1 — Plot Your Attractions 6.NS.6 · Ordered Pairs & Quadrants
Default inputs:
Roller Coaster (x, y) = (−4, 6)
Water Slide (x, y) = (5, −3)
Haunted Mansion (x, y) = (−7, −5)
Roller Coaster (−4, 6): x = −4 (negative), y = 6 (positive) → sign pattern (−, +) →
Quadrant II
Water Slide (5, −3): x = 5 (positive), y = −3 (negative) → sign pattern (+, −) → Quadrant IV
Haunted Mansion (−7, −5): x = −7 (negative), y = −5 (negative) → sign pattern (−, −) → Quadrant III
Water Slide (5, −3): x = 5 (positive), y = −3 (negative) → sign pattern (+, −) → Quadrant IV
Haunted Mansion (−7, −5): x = −7 (negative), y = −5 (negative) → sign pattern (−, −) → Quadrant III
Key rule: Quadrant I (+,+) · Quadrant II (−,+) · Quadrant III (−,−) · Quadrant IV (+,−). Any point with a 0 coordinate lies on an axis, not in a quadrant.
Roller Coaster → QII | Water Slide → QIV | Haunted Mansion → QIII
Phase 2 — Mirror the Ride 6.NS.6b · Reflections Across Axes
Default inputs:
Original point (x, y) = (3, 4)
Reflect across x-axis
Rule — reflecting across the x-axis: keep x the same; negate y.
Original: (3, 4)
New x = 3 (unchanged)
New y = −4 (negate 4)
Reflected point: (3, −4)
Why y changes: The x-axis is horizontal. Reflecting flips the point to the opposite side vertically — the horizontal distance (x) stays the same, but the vertical position (y) reverses sign.
Compare to other axes:
Reflect (3, 4) across the y-axis: negate x, keep y → (−3, 4)
Reflect (3, 4) across both axes: negate both → (−3, −4)
Original: (3, 4)
New x = 3 (unchanged)
New y = −4 (negate 4)
Reflected point: (3, −4)
Why y changes: The x-axis is horizontal. Reflecting flips the point to the opposite side vertically — the horizontal distance (x) stays the same, but the vertical position (y) reverses sign.
Compare to other axes:
Reflect (3, 4) across the y-axis: negate x, keep y → (−3, 4)
Reflect (3, 4) across both axes: negate both → (−3, −4)
Reflected point: (3, −4)
Phase 3 — Measure Walking Distance 6.NS.8 · Distance on the Coordinate Plane
Default inputs:
Point A = (2, 3)
Point B = (2, −5)
Shared coordinate check: Both points have x = 2 (same) and y-values 3 and −5 (different).
Because they share x = 2, they lie on a vertical line; the distance is the absolute difference of their y-values.
Formula: distance = |y₂ − y₁| = |−5 − 3| = |−8| = 8 units
Why absolute value? Distance is always positive. Without absolute value we might subtract in the "wrong order" and get a negative result — |−8| = 8 eliminates that possibility.
Formula: distance = |y₂ − y₁| = |−5 − 3| = |−8| = 8 units
Why absolute value? Distance is always positive. Without absolute value we might subtract in the "wrong order" and get a negative result — |−8| = 8 eliminates that possibility.
Distance = 8 units
Phase 4 — Quick Checks 6.NS.6 · 6.NS.8
Quick Check 1: What is the distance between (2, 3) and (2, 9)?
Both points share x = 2 (vertical line).
Distance = |y₂ − y₁| = |9 − 3| = |6| = 6 units
Distance = |y₂ − y₁| = |9 − 3| = |6| = 6 units
Correct answer: 6
JavaScript check: if (v === 6)
Quick Check 2: What are the coordinates of (4, −5) reflected across the x-axis?
Reflect across x-axis: keep x the same, negate y.
New x = 4 (unchanged)
New y = −(−5) = 5
Reflected point: (4, 5)
New x = 4 (unchanged)
New y = −(−5) = 5
Reflected point: (4, 5)
Correct answer: x = 4, y = 5
JavaScript check: if (x === 4 && y === 5)
Sample Expert Response (4/4 Deliverable)
Sample Park Blueprint Reflection using default values:
"My theme park spans all four quadrants. The Roller Coaster is at (−4, 6) in Quadrant II because x is negative and y is positive. The Water Slide is at (5, −3) in Quadrant IV because x is positive and y is negative, and the Haunted Mansion is at (−7, −5) in Quadrant III because both coordinates are negative. I reflected the point (3, 4) across the x-axis to get (3, −4) — the x-coordinate stayed the same while the y-coordinate changed sign because reflecting across the x-axis flips the vertical position. The walking distance from A(2, 3) to B(2, −5) is 8 units because both points share x = 2 and |−5 − 3| = |−8| = 8."
"My theme park spans all four quadrants. The Roller Coaster is at (−4, 6) in Quadrant II because x is negative and y is positive. The Water Slide is at (5, −3) in Quadrant IV because x is positive and y is negative, and the Haunted Mansion is at (−7, −5) in Quadrant III because both coordinates are negative. I reflected the point (3, 4) across the x-axis to get (3, −4) — the x-coordinate stayed the same while the y-coordinate changed sign because reflecting across the x-axis flips the vertical position. The walking distance from A(2, 3) to B(2, −5) is 8 units because both points share x = 2 and |−5 − 3| = |−8| = 8."
Sample 4 / 3 / 2 Rubric Scoring Guide:
4 — Expert: All three quadrants correctly identified with sign reasoning explicitly stated ("x is negative, y is positive"); reflection explanation states which value changed and why; distance calculation shows the absolute-value work; blueprint uses real coordinates throughout.
3 — Proficient: All quadrants correct; correct reflection and distance; blueprint uses most coordinates but may skip reasoning steps.
2 — Developing: One or two quadrant errors; one coordinate of the reflection correct; distance attempted but has a computation error or skips absolute value.
4 — Expert: All three quadrants correctly identified with sign reasoning explicitly stated ("x is negative, y is positive"); reflection explanation states which value changed and why; distance calculation shows the absolute-value work; blueprint uses real coordinates throughout.
3 — Proficient: All quadrants correct; correct reflection and distance; blueprint uses most coordinates but may skip reasoning steps.
2 — Developing: One or two quadrant errors; one coordinate of the reflection correct; distance attempted but has a computation error or skips absolute value.
Phase 1 — Set Depth & Surface Height 6.NS.5 · Integers in Context
Default inputs:
Sub depth = −85 m
Cliff height = 34 m
Sub depth (−85 m): Negative because the sub is below sea level
(below the zero reference). −85 means 85 meters below sea level.
Cliff height (+34 m): Positive because the cliff top is above sea level. +34 means 34 meters above sea level.
Total vertical span from sub to cliff top:
|−85| + 34 = 85 + 34 = 119 meters
(Distance below sea level + distance above sea level = total span)
Cliff height (+34 m): Positive because the cliff top is above sea level. +34 means 34 meters above sea level.
Total vertical span from sub to cliff top:
|−85| + 34 = 85 + 34 = 119 meters
(Distance below sea level + distance above sea level = total span)
Sub: 85 m below sea level | Cliff: 34 m above | Total span: 119 m
Phase 2 — Distance from Sea Level (Absolute Value) 6.NS.7c · Absolute Value
Default input:
Depth reading = −120 m
Absolute value definition: |n| = distance from 0 on the number line; always ≥ 0.
|−120| = 120
The submarine is 120 meters below sea level.
Why absolute value? −120 is 120 units to the left of 0. The sign tells direction (below); absolute value gives only the magnitude (distance). Removing the negative sign gives 120.
|−120| = 120
The submarine is 120 meters below sea level.
Why absolute value? −120 is 120 units to the left of 0. The sign tells direction (below); absolute value gives only the magnitude (distance). Removing the negative sign gives 120.
|−120| = 120 meters from sea level
Phase 3 — Order the Sensor Readings 6.NS.7 · Compare & Order Integers
Default inputs (five temperature readings):
Reading 1 = −3 °C
Reading 2 = 0 °C
Reading 3 = −7 °C
Reading 4 = 2 °C
Reading 5 = −1 °C
Number-line reasoning: Numbers farther left are smaller (colder).
Place each value on a mental number line:
−7 < −3 < −1 < 0 < 2
Sorted from least to greatest (coldest to warmest):
−7, −3, −1, 0, 2 °C
Coldest: −7 °C | Warmest: 2 °C
−7 < −3 < −1 < 0 < 2
Sorted from least to greatest (coldest to warmest):
−7, −3, −1, 0, 2 °C
Coldest: −7 °C | Warmest: 2 °C
Ordered: −7, −3, −1, 0, 2 °C
Phase 4 — Safe-Depth Decision & Quick Checks 6.NS.5 · 6.NS.7
Default inputs:
Current depth = −175 m
Safety limit = 200 m from sea level
Step 1 — find distance from sea level: |−175| = 175 m
Step 2 — compare to limit: 175 ≤ 200
Decision: SAFE. The sub is 175 m from sea level, which is within the 200 m limit.
Step 2 — compare to limit: 175 ≤ 200
Decision: SAFE. The sub is 175 m from sea level, which is within the 200 m limit.
|−175| = 175 ≤ 200 → SAFE
Quick Check 1: What is |−45|?
−45 is 45 units to the left of 0 on the number line.
|−45| = 45. Absolute value removes the sign; distance is always positive.
|−45| = 45. Absolute value removes the sign; distance is always positive.
Correct answer: 45
JavaScript check: if (v === 45)
Quick Check 2: Order −3, 0, −7, 2 from least to greatest.
On the number line: −7 is leftmost (least), then −3, then 0, then 2 (rightmost = greatest).
−7 < −3 < 0 < 2
−7 < −3 < 0 < 2
Correct answer: −7, −3, 0, 2
JavaScript check: parses numbers and verifies [−7, −3, 0, 2] in that order.
Sample Expert Response (4/4 Deliverable)
Sample Mission Log Report using default values:
"Mission log: Nereid IV is currently at −85 meters, meaning it is 85 meters below sea level — a negative integer because the sub is below the zero reference point. The absolute value |−85| = 85 tells us the distance from sea level without the direction sign. Temperature sensor readings ordered from coldest to warmest are: −7, −3, −1, 0, 2 °C; the most negative value is coldest because it sits farthest left on the number line. At −175 m the sub is safe: |−175| = 175, and 175 ≤ 200, so it remains within the 200-meter safety limit."
"Mission log: Nereid IV is currently at −85 meters, meaning it is 85 meters below sea level — a negative integer because the sub is below the zero reference point. The absolute value |−85| = 85 tells us the distance from sea level without the direction sign. Temperature sensor readings ordered from coldest to warmest are: −7, −3, −1, 0, 2 °C; the most negative value is coldest because it sits farthest left on the number line. At −175 m the sub is safe: |−175| = 175, and 175 ≤ 200, so it remains within the 200-meter safety limit."
Sample 4 / 3 / 2 Rubric Scoring Guide:
4 — Expert: Depth correctly represented as negative with real-world explanation; |−85| = 85 stated and connected to "distance from zero"; all five temperatures ordered correctly with number-line reasoning; safe/unsafe decision justified by comparing |depth| to limit.
3 — Proficient: Correct signs, correct absolute value, correct ordering; safe-depth decision correct; log uses actual numbers but may skip some reasoning language.
2 — Developing: One or two values signed incorrectly; ordering has one transposition error; absolute value computed but not clearly connected to distance from zero.
4 — Expert: Depth correctly represented as negative with real-world explanation; |−85| = 85 stated and connected to "distance from zero"; all five temperatures ordered correctly with number-line reasoning; safe/unsafe decision justified by comparing |depth| to limit.
3 — Proficient: Correct signs, correct absolute value, correct ordering; safe-depth decision correct; log uses actual numbers but may skip some reasoning language.
2 — Developing: One or two values signed incorrectly; ordering has one transposition error; absolute value computed but not clearly connected to distance from zero.