LESSON PLAN

Cartesian Coordinates and Graphing

Apothem Team

Grade 7 · Geometry

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Project a treasure-map grid and call out "the treasure's at (3, 5)" — then have a volunteer place it. Half the room expects 3 across, 5 up; someone places 5 across, 3 up. Perfect: the ambiguity IS the lesson.

Convention settles it: first number horizontal (x), second vertical (y), always. Then the plot twist for Grade 7: the grid extends into NEGATIVE territory — four quadrants, and every point in the plane now has an address.

Explore

Four-quadrant fluency circuit: (1) Plot-and-join — a coordinate list that draws a picture when connected (crossing quadrants deliberately). (2) Sign-pattern census: pairs plot points and chart the quadrant sign-signatures — I: $(+,+)$ , II: $(-,+)$ , III: $(-,-)$ , IV: $(+,-)$ — plus the axes as homes of the zeros. (3) Distance hunts: how far from $(-3, 4)$ to $(5, 4)$ ? (Same y — pure horizontal: $5 - (-3) = 8$ .) From $(2, -1)$ to $(2, 6)$ ? (7, vertical.)

Station 3 quietly plants the integer-subtraction-as-distance idea from earlier in the year onto the grid.

Formalize

Formalize the plane's anatomy and the axis-aligned distance rule:

$(x, y): x \text{ first (horizontal)}, \; y \text{ second (vertical)} \qquad \text{same-}y \text{ distance} = |x_2 - x_1|$

The origin $(0,0)$ anchors everything; the quadrants are numbered counter-clockwise starting top-right (a convention with Roman numerals, worth one minute of why-not-clockwise shrugging — conventions need consistency, not justification). Points ON axes belong to no quadrant.

Practice

Practice: plot 12 points across all quadrants; name the quadrant (or axis) from signs alone WITHOUT plotting; three axis-aligned distances; one shape task (plot three vertices of a rectangle, deduce the fourth — sign reasoning beats counting).

Exit ticket: a point sits 4 left of the origin and 7 up. Coordinates? Quadrant? Distance to $(-4, -2)$ ? ( $(-4, 7)$ ; II; $9$ .)

Exit ticket

Exit ticket: a point sits 4 left of the origin and 7 up. Coordinates? Quadrant? Distance to $(-4, -2)$ ? ( $(-4, 7)$ ; II; $9$ .)

TIP "Run before you jump" (x before y) plus the sign-signature chart handles 90% of quadrant errors. The remaining 10% are axis points — drill "no quadrant" as a proud answer, not a failure.

WORKED EXAMPLES

Example 1 — The rectangle's missing corner

Given three vertices: $A(-2, 3)$ , $B(4, 3)$ , $C(4, -1)$ . Find $D$ .

Step 1: Read the structure: $A$ and $B$ share $y = 3$ (top edge, length $|4 - (-2)| = 6$ ). $B$ and $C$ share $x = 4$ (right edge, length $|3 - (-1)| = 4$ ).

Step 2: $D$ must close the loop: directly below $A$ and directly left of $C$ : $D = (-2, -1)$ .

Step 3: Verify by structure: $D$ shares $x = -2$ with $A$ ✓ and $y = -1$ with $C$ ✓; sides 6 and 4 opposite each other ✓.

Step 4: Bonus harvest: the rectangle's perimeter is $2(6+4) = 20$ ; its area $24$ — geometry formulas now running on coordinates, which is precisely where they'll live from Grade 8 onward.

Example 2 — The delivery route: taxicab distances on a city grid

The city: streets are grid lines, one block per unit. The courier starts at $(-3, 2)$ , delivers at $(4, 2)$ , then at $(4, -3)$ .

Step 1: Leg one — same $y$ : $|4 - (-3)| = 7$ blocks east.

Step 2: Leg two — same $x$ : $|2 - (-3)| = 5$ blocks south.

Step 3: Total ride: 12 blocks. (The crow flies $\sqrt{7^2 + 5^2} \approx 8.6$ — a Pythagorean teaser the couriers of Grade 8 will compute; today's courier is stuck on streets.)

Step 4: The subtraction-signs check worth pausing on: $4 - (-3) = 7$ used subtracting-a-negative from the integers unit — the grid gives that abstract rule a street address. Cross-topic connections like this one are what "the math is one subject" means.

Example 3 — Reflections by coordinates: the sign flip discovered

The task: triangle $P(1,2)$ , $Q(3,5)$ , $R(4,1)$ is reflected over the x-axis. Predict the images WITHOUT the grid, then verify with it.

Step 1: The hypothesis from one point: reflecting over the x-axis flips vertical position — $P(1,2) \to P'(1,-2)$ . Rule candidate: $(x, y) \to (x, -y)$ .

Step 2: Apply to all: $Q' = (3,-5)$ , $R' = (4,-1)$ .

Step 3: Verify by plotting both triangles: mirror-image across the x-axis ✓, each point equidistant above/below ✓.

Step 4: Complete the family and tabulate: over the y-axis: $(x,y) \to (-x, y)$ ; over BOTH (180° rotation about origin): $(x,y) \to (-x,-y)$ — the quadrant sign-signatures from the census are literally the images of quadrant I under these three moves. The plane's symmetry and the sign chart are the same fact.

MATERIALS

•Four-quadrant grids

•Plot-and-join picture lists

•Sign-signature charts

•Practice set (PDF)

WATCH FOR

!Coordinates reversed — the treasure-map opener makes the convention memorable by showing the cost of not having one.

!Quadrant signs memorized without the axes' zero-cases:

(0, -5)

assigned to a quadrant instead of the y-axis.

!Distances between negative coordinates computed by adding absolute values always — works across zero, fails on the same side (

-3

-8

is 5, not 11).