Above, Below & Across: Mapping a World of Integers
You are the navigator for the Mariana Deep-Dive submarine team. Sea level is zero. Wreck sites sit below it, drone buoys float above it, and every search point is a coordinate on the map. To find anything, you have to think in positives and negatives. Eres el navegante del equipo del submarino: el nivel del mar es cero.
1 Engage
Hook & essential question
The sub team logs four readings on the dive board. Each one is a distance from sea level (0). A drone buoy floats +18 m above the water; the wreck rests at −40 m; a diver hovers at −12 m; a gull is at +6 m.
Up is positive, down is negative. Zero is the surface.
Quick think (no calculator): Which object is closest to the surface? Which is farthest? Write your two answers and one reason in your notebook. You will prove it with absolute value in the Apply section.
Essential question: How do positive and negative numbers — and the coordinate plane — let me describe an exact location and the distance between any two points?
2 Explore
Investigate before the lesson
Explore each tool below for a few minutes. As you go, hunt for one big idea: a negative sign tells you the direction from zero, while absolute value tells you the distance from zero.
Direction vs. distance experiment
Look at the two divers below. One is at −12 m and one is at −40 m. The diver at −40 is a smaller number, but is farther from the surface. That is the whole trick of this unit.
Position: 12 m below 0.
Distance from 0: |−12| = 12 m
Position: 40 m below 0.
Distance from 0: |−40| = 40 m
Notice: On a number line, −40 sits to the left of −12, so −40 < −12. But its absolute value is larger, so it is deeper. Sign and size are two different questions.
3 Explain
The math, in plain language
A number line and a coordinate plane both use zero as their starting point. Moving one direction is positive; the opposite direction is negative. Two numbers the same distance from zero on opposite sides (like 5 and −5) are called opposites.
- integer
- a whole number: positive, negative, or zero · el número entero
- opposite
- same distance from 0, other side (5 and −5) · el opuesto
- absolute value |x|
- distance from 0, always 0 or positive · el valor absoluto
- rational number
- any number you can write as a fraction or decimal · el número racional
- coordinate plane
- two number lines that cross at the origin · el plano de coordenadas
- quadrant
- one of four regions, labeled I–IV · el cuadrante
Compare & order (6.NS.7)
On a number line, the number farther right is always greater. So −6 < −2 < 0 < 3.
A statement like −7 °C < −3 °C means −7° is the colder temperature.
Absolute value (6.NS.7c)
|−40| = 40 |+18| = 18
The bars strip the sign and report only the distance from 0. A bank balance of −$40 is a $40 debt — the size of what is owed.
Plotting points (6.NS.6c)
An ordered pair is (x, y). Start at the origin (0, 0). Move x left/right first, then y up/down.
Distance & reflection (6.NS.8)
When two points share an x- or y-value, the distance between them is the sum or difference of the absolute values of the coordinate that changes.
A reflection across an axis flips one sign: (4, 3) → (−4, 3) across the y-axis.
A → B is a reflection across the y-axis (x changes sign). B → C shares x = −4, so the distance is |3| + |−2| = 5 units.
Worked example — how deep is the wreck?
Back to Engage: the diver is at −12 m and the wreck is at −40 m.
Distance the diver must descend = |−40| − |−12| = 40 − 12 = 28 m.
The wreck is deeper because |−40| > |−12|, even though −40 < −12 as a number. Sign tells direction; absolute value tells distance.
Level 1 · support Step-by-step helper / Ayuda paso a paso
- Sign first. Up / right / above 0 = positive. Down / left / below 0 = negative.
- Compare: draw a number line. The number farther right is greater. El número a la derecha es mayor.
- Absolute value = how far from 0. Drop the sign: |−9| = 9.
- Plot (x, y): go across (x) before up/down (y). Primero x, luego y.
Level 2 · enrichment Push your thinking
For which values of n is |n| = −n a true statement? Test n = 5, n = 0, and n = −5, then write a rule describing exactly which numbers make it work, and why.
4 Apply
Show what you can do — auto-checked
Level 1 · support Sentence starters for explaining
"The number ____ is greater than ____ because it is farther to the ____ on the number line. Its absolute value is ____, which means it is ____ units from zero."
"El número ____ es mayor que ____ porque está más a la ____. Su valor absoluto es ____, así que está a ____ unidades del cero."
Level 2 · enrichment Reverse it
A point P is in Quadrant III and is exactly 6 units from each axis. Write its ordered pair. Then reflect P across the x-axis and name the quadrant the image lands in. Explain how you knew the signs.
Teacher Notes & Answer Key (not printed)
Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)
- Engage: dive depths motivate negative numbers and absolute value.
- Explore: place integers on a vertical number line and plot points.
- Explain: absolute value as distance; quadrants; reflections.
- Apply: answer key below.
- Reflect: students explain why |−40| = 40.
Apply — Answer Key
- Q1 — Distance |−40|: 40 m.
- Q2 — Compare depths (MC): answer c.
- Q3 — Distance −12 to −40: |−40 − (−12)| = 28 m.
- Q4 — Quadrant of (5, −3) (MC): answer b (Quadrant IV).
- Q5 — Reflect D(5, 2) across the y-axis: x changes sign → new x-coordinate −5 (new point (−5, 2)).
- Q6 — Distance A(−3,1) to B(4,1): 3 + 4 = 7.
Standard
CCSS 6.NS.C.5–8.
5 Reflect
Think about your thinking
A. In your own words, what is the difference between −40 < −12 and |−40| > |−12|? Use the words direction and distance.
B. Where in real life (temperature, money, elevation, a game map) would you use negative numbers or a coordinate plane? Describe one situation.
C. One thing that is now clear to me / Una cosa que ahora entiendo bien:
Your reflections save with this HyperDoc when you use Save as PDF or Save as DOC above.
6 Extend
Optional challenge project
Chart the dive site for the Mariana team
On grid paper, draw a four-quadrant coordinate plane and plot a real dive map with at least five labeled points, using at least one point in each quadrant. Then:
- Write the ordered pair for each point and name its quadrant (or axis).
- Pick two points that share an x- or y-value and find the exact distance between them using absolute value.
- Reflect one point across an axis, plot its image, and write a sentence explaining which coordinate changed sign and why.
Dibuja un plano de coordenadas con cinco puntos, calcula una distancia con valor absoluto y refleja un punto sobre un eje.