038i - Plotting Linear Relations

Lesson 38 — Explicit Instruction: Plotting Linear Relations on the Cartesian Plane

Strand: Algebra | Descriptor: AC9M8A02 | Duration: 45 minutes

Learning Intentions

• To generate a table of values from a linear rule.
• To plot the resulting points on the Cartesian plane and recognise that they form a straight line.

Success Criteria

I can:

1. Identify the coordinates of a point in any of the four quadrants.
2. Complete a table of values for a given linear rule.
3. Plot points from a table of values accurately on the Cartesian plane.
4. Draw a straight line through plotted points, extending it in both directions.
5. Determine whether a given point lies on a linear relation, by substitution.

Warmup

(6 minutes — coordinate recall, mini whiteboards)

1. Plot and label , , , , .
2. Which quadrant does sit in?
3. What is special about any point on the -axis? On the -axis?
4. What are the coordinates of the origin?

Answers: 2. Quadrant II (top-left); 3. On the -axis, ; on the -axis, ; 4. .

Activities

Activity 1 — Explicit Instruction: Building a Table of Values (14 min)

The rule. A linear relation like is a rule that pairs every -value with exactly one -value.

I do: Build a table of values for , using .

Model the substitution explicitly for one value: at , .

I do — a rule with a coefficient. Build a table for , using .

We do: Complete a table for using .

You do: Complete a table of values, to , for:

(Answers: 1. ; 2. ; 3. ; 4. .)

Activity 2 — Explicit Instruction: Plotting and Drawing the Line (12 min)

I do. Plot the five pairs from : , , , , . Join them with a ruled straight line, extending past the first and last plotted points with arrows on both ends — this shows the relation continues infinitely in both directions.

Non-negotiable habits, modelled explicitly:

• Label both axes and choose an even scale before plotting a single point.
• Plot every point from the table, even if you’re confident — one point is not enough to confirm a line.
• Use a ruler. A “linear relation” is called linear precisely because it graphs as a straight line — a wobbly line suggests a plotting error.

We do: Plot from the table built in Activity 1, and draw the line.

You do: On grid paper, plot and rule the line for one of your Activity 1 tables (teacher’s choice or student’s choice), then answer:

1. Does your line slope upward or downward, left to right?
2. Where does your line cross the -axis?

(Answers depend on which relation was chosen — e.g. for : slopes upward; crosses the -axis at .)

Activity 3 — Inquiry: Does This Point Belong on the Line? (8 min)

Pairs, then whole-class share.

The rule produces a straight line.

1. Without plotting the whole line, decide whether each point lies on it: , , , .
2. Explain your method — did you need to draw the graph?
3. A classmate insists you must always plot the full line to check a point. Are they right?

Guiding questions for Q2–3:

Prompt	Purpose
Understand: what does “lies on the line” actually mean?	The point’s - and -coordinates must satisfy the rule — substituting into the rule must produce that exact .
Can you test this without drawing anything?	Yes — substitute the point’s -value into the rule and compare the result with the point’s -value.
Try it for .	— matches, so is on the line.
Try it for .	— does not match, so is not on the line.
Looking back — so was the classmate right?	No — substitution checks a point exactly and instantly; plotting is useful for seeing the whole relation, but is not required to test a single point.

Answers: ✓ on the line; ✓ on the line; ✗ not on the line ( should be ); ✗ not on the line ( should be ).

Checks for Understanding

(5 minutes — exit ticket, collected)

1. State the coordinates of the origin.
2. Complete a table of values for , using .
3. Plot the points from Q2 (on the grid provided) and rule a straight line through them.
4. Does the point lie on the line ? Show your working.
5. Reasoning. Explain why a single plotted point cannot confirm that a relation is linear.

Answers: 1. ; 2. ; 3. line through ; 4. — yes, it lies on the line; 5. a single point is consistent with infinitely many different curves and lines passing through it — at least two points are needed to identify a unique straight line, and a third is a useful check.

Common Misconceptions

Misconception	How to pre-empt it
Swapping the - and -coordinates when plotting, e.g. plotting as .	Drill “along the corridor, up the stairs” — first, then — on every point.
Joining plotted points with a freehand, wobbly line.	Insist on a ruler; a genuine linear relation is never curved.
Stopping the line exactly at the first and last plotted points, with no arrows.	Model extending the line beyond the table’s range with arrows, showing the relation continues.
Believing a table of values must start at .	Show tables starting at negative -values, and confirm the same line results regardless of which -values were chosen.
Assuming a point must be plotted to check whether it lies on a line.	Activity 3 shows substitution is faster and exact — reserve plotting for seeing the whole picture.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). The points and both lie on a linear relation passing through . If the rule is , verify both points satisfy it.

Answer

At : ✓. At : ✓. Both points lie on the line.

E2 (AMC Junior style). A point lies on the line . Find .

Answer

E3 (Challenge). Three points are given: , , . Determine, using substitution into , which points (if any) do not lie on this line.

Answer

At : ✓. At : ✓. At : ✓. All three lie on the line — meaning these three points alone would have been enough to strongly suggest the underlying rule.

E4 (Investigation). Every point on the line has the same -coordinate, whatever its -coordinate. Describe, in words, what this line looks like when plotted, and explain why.

Answer

It is a horizontal line, five units above the -axis, extending infinitely left and right — because the rule places no restriction at all on , only fixing for every possible .

Homework

1. Plot and label these points on a Cartesian plane: , , , , .
2. Complete a table of values, to , for: (a) (b) (c) .
3. Plot and rule the line for using your table from Q2(a).
4. Determine whether each point lies on : (a) (b) (c) (d) .
5. Reasoning. A student plots only and for the rule , then rules a line through them without checking a third point. Explain why checking a third point is good practice, even though two points are technically enough to define a line.
6. Challenge. A point lies on the line . Find .
7. Challenge. Find the value of so that the point lies on the line .

Answers: Q2 — (a) (b) (c) . Q4 — (a) ✓ (b) ✓ (c) , ✗ (d) , ✗. Q5 — a third point acts as a check against an arithmetic slip in the table or a plotting error — if all three don’t line up perfectly, a mistake has been made somewhere. Q6 — . Q7 — .