048i - Conjectures About Gradient and Intercept

Lesson 48 — Making and Testing Conjectures About Gradient and Intercept

Strand: Algebra | Descriptor: AC9M8A04 | Duration: 45 minutes

Learning Intentions

• To make conjectures about the relationship between gradient, intercept and the visual features of a line.
• To test conjectures against multiple examples using a digital tool, and refine or reject them based on evidence.

Success Criteria

I can:

1. State a conjecture about gradient or intercept in clear mathematical language.
2. Test a conjecture against at least three examples, including a deliberately awkward one.
3. Refine or reject a conjecture based on evidence, including finding a counterexample.
4. Distinguish a conjecture (an untested claim) from a generalisation (a well-tested, justified pattern).

Warmup

(5 minutes — “always, sometimes, never,” pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. A line with a positive gradient rises from left to right.
2. A line with passes through the origin.
3. Steeper lines have a bigger -intercept.
4. Two lines with the same gradient are parallel.

Answers: 1. Always — this is the definition of positive gradient; 2. Always — substituting gives ; 3. Never — steepness depends only on , intercept is independent; e.g. is steep with a small intercept; 4. Sometimes — true unless they are the exact same line (coincident, not merely parallel).

Discussion: flag Q4 as today’s central idea — “sometimes” answers are exactly where careful conjecture-testing matters most.

Activities

Activity 1 — Modelling the Conjecture-testing Cycle (10 min)

Teacher models the cycle explicitly: State → Test → Refine (or Reject).

State: “A line with a negative gradient always slopes down from left to right.”

Test using a digital graphing tool: try , all with different intercepts.

In every case, as increases, decreases — the line falls left to right.

Refine: the conjecture holds for every negative tested, regardless of intercept. State the generalisation: “For any real , the line falls from left to right, for every value of .”

A second conjecture that needs refining: “A line with a larger gradient number is always steeper.”

Test: compare and . Is “larger” than ? No — but the line with is clearly steeper on the graph.

Refine the conjecture: steepness depends on the size of (its absolute value, ), not on whether itself is a bigger number. Restate: “A line with a larger is steeper.”

Activity 2 — Guided Pairs: Test the Claim Bank (14 min)

Pairs, at a device with a graphing tool. For each conjecture: test with at least three examples (including a tricky one), then classify as True, False, or Needs refining, with a one-sentence justification.

1. “Every line with a positive crosses the -axis at a positive value of .”
2. “Parallel lines always have the same gradient.”
3. “If a line passes through the origin, then .”
4. “Multiplying by reflects the line across the -axis.”
5. “A line with and passes through every point where .”

Socratic scaffolding for statement 1 (a genuinely false conjecture — good for testing negative ):

Prompt	Purpose
Understand: what is the claim asserting?	Any line crossing the -axis above zero also crosses the -axis at a positive .
Test with a positive-gradient example, e.g. .	-intercept: . Already a negative -intercept — the conjecture is failing!
Test with a negative-gradient example, e.g. .	-intercept: , positive — this one does fit the claim.
So is the conjecture always true?	No — it depends on the sign of , not just .
Look back — restate a corrected, narrower version.	”A line with positive and negative crosses the -axis at a positive -value.”

Answers: 1. False as stated — depends on the sign of too (see scaffolding); refine as above. 2. True — this is the definition used throughout Lesson 47’s Family 1. 3. True — substituting the origin forces . 4. True — reflecting across the -axis sends , which is exactly the effect of … (students should test carefully: reflecting across the -axis gives , so the conjecture as stated is only fully true when — a good “needs refining” example.) 5. True — by definition, passes through every point with equal coordinates.

Activity 3 — Write and Test Your Own Conjecture (10 min)

Pairs, then share two or three with the class.

Using the graphing tool, look for a pattern you haven’t already tested today. Write it as a conjecture starting with “If … then …” Test it against at least three examples, and report whether it survives, needs refining, or is rejected.

Sentence starters for pairs who need support:

• “If two lines have intercepts that are … then …”
• “If a gradient is a fraction rather than a whole number, then …”
• “If and are both negative, then …”

Whole-class share: collect two or three conjectures on the board; note which were confirmed, which needed refining, and which were rejected outright with a counterexample.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. State whether this is always, sometimes, or never true: “A line with has no -intercept.” Justify.
2. A student conjectures: “The line with the bigger value is always higher on the graph.” Find a counterexample or confirm it’s always true.
3. Test the conjecture “doubling doubles the steepness” using and . Does it hold?
4. Reasoning. Explain the difference between a conjecture and a generalisation, using an example from today’s lesson.

Answers: 1. Never — a horizontal line still crosses the -axis at ; 2. Counterexample needed only where lines cross — e.g. vs : at , the second is higher (), but at , the first is higher (); so the conjecture is only true at itself, not everywhere — it needs refining to specify “at the -axis”; 3. Yes — produces a visibly steeper line than , consistent with steepness scaling with ; 4. A conjecture is an untested or partly tested claim, like the class’s own attempts in Activity 3; a generalisation is a claim that has been tested across a wide range of cases (including tricky ones like negative or fractional ) and holds every time, such as “parallel lines share a gradient.”

Common Misconceptions

Misconception	How to pre-empt it
Testing a conjecture with only one example and declaring it proven.	Require at least three test cases, including one deliberately awkward case (negative, zero, or fractional).
Confusing “seems true in the examples I tried” with “true in general.”	Explicitly separate the words conjecture and generalisation throughout, as modelled in Activity 1.
Believing a single counterexample only weakens a conjecture rather than disproving it.	State plainly: one confirmed counterexample is enough to reject or force refinement of a general claim.
Comparing gradients by their signed value rather than their magnitude when discussing steepness.	Revisit Activity 1’s refined conjecture about $
Assuming every line comparison needs the same fixed to be meaningful (from CFU Q2).	Discuss that “higher on the graph” depends on where you look unless lines are parallel.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). True or false: “If two lines have the same -intercept, they must intersect only at that point.” Justify with an example or counterexample.

Answer

False — if they also share the same gradient, the lines are identical and “intersect” at every point, not just one. Example: and .

E2 (AMC Junior style). A line has gradient and intercept , both positive whole numbers less than . How many such lines pass through the point ?

Answer

Need with (both positive, less than , and isn’t required but must be a positive whole number so ): — 3 lines.

E3 (Challenge). Conjecture: “For any two lines and with , the -coordinate of their intersection is .” Test this with and , then explain why the conjecture needs .

Answer

. Formula: ✓. If , the denominator becomes zero (undefined) — matching the fact that parallel, non-identical lines never intersect.

E4 (Challenge). A family of lines is graphed for several values of . Conjecture what point every line in this family shares, and prove it algebraically.

Answer

Substitute : , regardless of . Every line passes through .

Homework

1. Test the conjecture “a line with crosses the -axis at a positive -value” using and . Is it always true?
2. State whether always, sometimes or never true, with justification: “Two lines with different gradients must intersect exactly once.”
3. A conjecture claims “the steeper line always has the bigger -intercept.” Find a counterexample.
4. Test whether passes through for every value of . Explain why or why not.
5. Reasoning. Explain why testing a conjecture with or is often a good strategy for finding counterexamples.
6. Challenge. A family of lines is given by . Find the point every member of this family shares, and prove your answer algebraically.

Answers: Q1 — : -intercept (positive) ✓; : -intercept (negative) ✗ — conjecture is false in general, it depends on the sign of too. Q2 — Always true (for straight lines in a plane) — different gradients guarantee exactly one intersection point. Q3 — e.g. (steep, small intercept) versus (flat, large intercept) — the flatter line has the bigger intercept. Q4 — yes, since gives regardless of ; every member of the family shares the fixed intercept . Q5 — these are boundary/extreme cases that often reveal hidden assumptions in a conjecture, such as division by zero, a “flat” special case, or a line through the origin, which general statements can overlook. Q6 — substituting : for all ; every line passes through .