089i - Plotting on the Cartesian Plane

Lesson 89 — Plotting Relationships on the Cartesian Plane

Strand: Algebra | Descriptor: AC9M7A05 | Duration: 45 minutes

Learning Intentions

• To plot points and linear relationships in all four quadrants of the Cartesian plane.
• To see that a linear rule’s table plots as a straight line.

Success Criteria

I can:

1. Plot and read points in all four quadrants, in the order .
2. Plot a rule’s table and draw the line through the points.
3. Use the line to read off values between and beyond my table.
4. Connect the rule’s coefficient and constant to the line’s steepness and crossing point.

Warmup

(6 minutes — four-quadrant orientation, mini whiteboards)

On a printed plane from to on both axes:

1. Plot and label: , , , , , .
2. Which quadrant holds each of –? Where do and live?
3. What is special about the point ?

Answers: 2. : first, : second, : third, : fourth; on the -axis, on the -axis — axis points belong to no quadrant; 3. The origin — where the axes cross, the address .

The order rule, stated once, enforced always: along the hall, then up the stairs — first, second. and are different addresses.

Activities

Activity 1 — Explicit Instruction: from Table to Line (14 min)

I do — plot using yesterday’s table skills:

Plot all seven points, narrating the address of each. Then hold the ruler to them:

The headline observation — they are perfectly straight. Draw the line through and beyond the points, arrowheads both ends.

Why straight? Every step right by climbs by exactly — a constant step builds a constant slope. The table’s constant difference (Lesson 87) is the straightness (this is the whole connection the descriptor asks for).

Name the two anchors:

• The line crosses the -axis at — the constant term, exactly as the table’s entry.
• Steepness: up for every across — the coefficient.

We do — a falling line. with the class calling the table, then plotting: crosses the -axis at , falls per step. Where does it cross the -axis? At — read from the graph, confirmed by .

Reading between and beyond: from the drawn line of , read at (4 — between plotted points: the line interpolates), and at (9 — beyond the table: the line extrapolates). The line answers infinitely many questions the seven points never listed.

Activity 2 — Plotting Production (14 min)

Individually, two per grid sheet.

You do — for each rule: table ( to ), plot, draw the line, answer the reads.

1. — read: where does it cross each axis?
2. — read: at ; which gives ?
3. — read: describe its direction; where does it cross the -axis?
4. — read: compare its steepness with rule 1’s line.

(Answers: 1. and ; 2. ; ; 3. falls left-to-right; crosses at ; 4. flatter — half the climb per step.)

Craft checklist while circulating: even scales, points as neat crosses, ruler lines through the crosses, lines extended past the outer points with arrowheads, each line labelled with its rule.

The comparison moment (board, after most finish): stack all four rules. Coefficients order the steepness; negative means falling; constants are the -axis crossings. The class reads a rule’s graph personality from its two numbers.

Activity 3 — Inquiry: Two Points, One Line (11 min)

Pairs — closes the loop from Lesson 88’s guess-my-rule.

1. Plot the two points and . Draw the line through them.
2. From the drawing, read the -axis crossing and the steepness. Name the rule.
3. Verify your rule against both points by substitution.
4. Could a different straight line pass through both points? What about through just one of them?

Socratic scaffolding:

Prompt	Purpose
Steepness from the two points?	Up over across: per step.
Crossing point, read from the ruler line?	.
So the rule is…?	.
Verify.	✓, ✓.
A different line through both?	Impossible — the ruler shows one line only. Two points determine a line.
Through just one point?	Infinitely many — spin the ruler. Yesterday’s one-query problem, now visible.

Extension for fast pairs: the line through and — what happens? (Horizontal: rule ; steepness zero — the constant rules from L88 E3.)

Checks for Understanding

(6 minutes — exit ticket, on a small printed grid)

1. Plot , , and name each location (quadrant or axis).
2. Table and plot for to . Draw the line.
3. From your line: at ; the -axis crossing.
4. A line passes through and falls for every across. Name its rule.
5. Reasoning. Why do the points of a linear rule’s table always line up straight?

Answers: 1. Second quadrant; -axis; -axis; 2. points to ; 3. ; ; 4. ; 5. Equal steps in produce equal steps in (the constant difference), so every point continues the same direction — no bends possible.

Common Misconceptions

Misconception	How to pre-empt it
Swapping coordinates: plotting at .	The hall-then-stairs mantra; warmup addresses drilled first.
Dot-to-dot segments that stop at the outer points.	Lines extend with arrowheads — the rule doesn’t stop at the table’s edge.
Joining points of a linear rule with a wobbly freehand curve.	Ruler mandatory; the straightness is the content, not decoration.
Placing axis points “in” a quadrant.	Warmup Q2 names the convention.
Reading steepness from unequal axis scales as if equal.	Keep both axes at the same scale this lesson; name the hazard for later.
Believing a steeper-looking line always has the bigger coefficient.	True only with matched scales — connect to L83’s scale check.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which of these points lies on : , , ?

Answer

: ✓ and : ✓; : ✗. Two of the three.

E2 (AMC Junior style). A line passes through and . Find its rule and its -axis crossing.

Answer

Steepness : ; crossing where : , the point .

E3 (Challenge). The lines and are plotted on one grid. Find their crossing point by table/graph, confirm by equation, and describe the symmetry of the picture.

Answer

, : crossing . The lines have opposite slopes (, ), so the picture is a symmetric “X” — mirror images across the vertical line through the crossing.

E4 (Challenge). Three points , , are plotted. Show they are collinear, name the line, and find the next point in the pattern.

Answer

Steps: per , i.e. per , consistently: fits all three. Next: .

Homework

1. On a grid from to : plot and label , , , , , , and state each location.
2. For each rule: table to , plot, draw and label the line: (a) (b) (c) .
3. From your graph of (c): (a) both axis crossings (b) at (c) which gives ?
4. A line passes through climbing per step. Another passes through climbing per step. (a) Name both rules. (b) Plot both and read their crossing. (c) Verify the crossing by equation.
5. Do these points lie on one straight line: ? Decide by steps, then name the line if so.
6. Reasoning. Explain what the coefficient and constant of each control on the graph.
7. Reasoning. Why can you read at from a drawn line even though your table never used ?
8. Challenge. Plot the three lines , , on one grid. What do they share, what differs, and will any pair ever cross?

Answers: Q3 — (a) and (b) (c) . Q4 — (a) , (b)(c) , : crossing — off small grids; extending the lines or trusting the algebra is the point. Q5 — steps: over , then over — both per unit ✓: . Q6 — : steepness and direction; : the -axis crossing. Q7 — the line embodies the rule for every , not just tabulated ones; interpolation reads the rule, not the table. Q8 — equal slopes (): parallel lines, different crossings (); parallel lines never cross.