The City Grid
City hall sits at the origin and the town spreads into all four quadrants. Students plot landmarks by signed coordinates, discover that sign flips are reflections across the axes, and measure straight streets with absolute value — no ruler required.
Common Core standards covered
- 6.NS.C.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
- 6.NS.C.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Graph points in all four quadrants to solve problems, recognize that sign changes produce reflections across the axes, and use absolute value to find distances between points sharing a coordinate.
Student target: “I can plot and read points in all four quadrants, predict reflections from sign changes, and compute gridline distances as absolute-value differences.”
Success criteria: I can plot any ordered pair, rationals included, in the correct quadrant. · I can name a point's quadrant from its signs alone. · I can reflect a point across either axis (or both) by flipping signs. · I can find the distance between same-street points with |difference|.
Essential question
Two signed numbers pin any spot in town. What do the SIGNS say about where you are — and how far apart are two addresses on the same street?
Learning target: I can plot and read points in all four quadrants, predict reflections from sign changes, and compute gridline distances as absolute-value differences.
Success criteria
- I can plot any ordered pair, rationals included, in the correct quadrant.
- I can name a point's quadrant from its signs alone.
- I can reflect a point across either axis (or both) by flipping signs.
- I can find the distance between same-street points with |difference|.
Why this matters
6.NS.C.8 completes the coordinate plane the moment negatives exist, and its two ideas both echo forward: sign-flip reflections are the seed of Grade 8's transformation geometry (Coordinate Choreography runs on them), and absolute-value distance is the harbor's cable question gone 2-D — the direct ancestor of Grade 8's Pythagorean distances. Reading signs as location is also pure graph literacy for every scatter plot to come.
Students should already be able to
- The signed number line — The Harbor Log.
- Quadrant I plotting.
- Absolute value as distance.
Vocabulary to define explicitly
- Ordered pair (x, y)
East-west first, north-south second — order is the address system.
Example: (3, 4): three east, four north. (4, 3) is a different corner.
Not an example: '3 and 4' unordered — the pair's order IS the information.
🔒 3 more terms with examples and non-examples — Quadrants, Reflection by sign flip, Gridline distance — 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 mirror observation.
- Planner-sheet distances.
- The symmetry predictions.
Summative — the graded evidence: The exit ticket (locate, reflect, measure), scored with the four-criterion rubric.
The task students work on
City hall anchors the origin; avenues run east-west, streets north-south. PLOT: library (3, 4), pool (−3, 4), skating rink (−3, −4), market (3, −4), and the bus stop at (−2.5, 1.5). PATTERNS: what do the four landmarks' signs say about their quadrants — and what mirror moves connect them? MEASURE: the library→pool walk along 4th Street North, the library→market walk along 3rd Avenue East, and the full perimeter tour around all four landmarks. Every distance by coordinates — the ruler only checks.
City planner's sheet.
| Job | Work | Answer |
|---|---|---|
| Quadrants of the four landmarks | ||
| Pool as a reflection of the library | ||
| Rink as a reflection of the library | ||
| Library → pool distance | ||
| Library → market distance | ||
| Perimeter tour |
Anticipated student strategies
Walk to plot, read signs to locate, count-then-compute to measure — each skill grounds the next abstraction.
- Axis-walk plotting. The student walks x first (east/west), then y (north/south), narrating directions.
The order habit — share first; it makes (3, 4) vs (4, 3) impossible to confuse.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Three WEST, four north — same height, other side of the north-south axis. The negative x is a direction, not a defect. Plot both; the mirror is already visible.
Listen for: One sign flipped; the point jumped across one axis. Sign flips ARE mirror moves — today's headline.
Signs-to-quadrant chart posted; the three flips demonstrated on one point.
- Signs name the quadrant before any plotting.
- Which sign flips = which axis mirrors.
Listen for: (−5, −2) — x flips, y holds; the y-axis mirror only changes east-west. Prediction from structure, checked by plot.
🔒 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
- “(3, 4) and (4, 3) are the same corner.” Order is the address: three east four north vs four east three north — different intersections. Walk x first, always.
- “Distance from (3,4) to (−3,4) is 3 − 3 = 0.” — the teacher response is in the full lesson.
- “Reflecting across the y-axis flips y.” — the teacher response is in the full lesson.
- “Negative or fractional coordinates aren't real map points.” — 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.
- I doPlot the four landmarks, name each quadrant from its signs, and identify the pool and rink as reflections of the library.
🔒 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
- Plot (−5, 2) mentally: which quadrant?
🔒 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
- Step 1: Plot the landmarks, one per quadrant. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Read sign patterns as quadrant addresses. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Flip signs and name the reflections. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Measure the straight streets with absolute value. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Signs-to-quadrant chart scaffolds plotting; extend with rational coordinates and the perimeter tour.
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 four-quadrant plot with a reflection pair identified and one gridline distance computed by absolute value.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.