Conjecture cycle
Observe → conjecture → test → refine → generalise → justify.
Learning goals: what you will learn
- explain the central idea: Explore functions and relations with digital tools, test conjectures against evidence and counterexamples, and generalise patterns carefully.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Prerequisite knowledge
You should be comfortable plotting coordinates, reading graphs, substituting into formulas, recognising linear, quadratic and exponential relations, and solving simple equations. You should also know that an intersection point satisfies both relations.
Concept teaching
Digital tools can reveal patterns quickly, but a screenshot is not a mathematical argument. A strong investigation varies one feature deliberately, records evidence, checks boundary/counterexamples and then generalises only as far as the evidence allows.
Intersection
At an intersection, both relations have the same ordered pair/output.
Bisection
If a continuous function changes sign over an interval, a root lies between; repeatedly halve the interval.
Digital-tool discipline
Change scale/zoom, test more than one case and record enough precision to justify the conclusion.
Worked examples
1. Zoom to an intersection
Graph y=x² and y=2ˣ. Find where their vertical order changes, then zoom and refine the x-interval. The x-coordinate of a crossing approximately solves x²=2ˣ.
2. Bisection for a root
f(x)=2x²−3x−7. f(2)=−5 and f(3)=2, so a root is between 2 and 3. Midpoint 2.5 gives f(2.5)=−2; root is now between 2.5 and 3. Continue.
3. Unit-circle stretch
x²+y²=1 is the unit circle. Replacing x by x/2 gives (x/2)²+y²=1, an ellipse stretched horizontally with x-intercepts ±2.
4. Line–quadratic intersections
For y=x+2 and y=x², solve x²=x+2 → x²−x−2=0 → x=−1 or 2. Points are (−1,1) and (2,4).
5. Line–circle intersections
For y=0 and x²+y²=9, x²=9 so x=±3. Intersections: (−3,0),(3,0).
6. Where a quadratic is positive
f(x)=(x−2)(x−5). Zeros at 2 and 5; upward-opening graph is positive for x<2 or x>5.
7. Exponential threshold
V=80(1.08)ᵗ. Use a table until V>100; t=2 gives 93.31, t=3 gives 100.78, so threshold is first crossed at t=3.
8. Test a conjecture about translation
Compare y=x², y=(x−3)² and y=x²+3. The first shifts right 3; the second shifts up 3. Testing points prevents confusing inside/outside translations.
9. Counterexample
Conjecture: “two graphs always intersect once.” y=x and y=x+2 never intersect. One counterexample is enough to disprove a universal claim.
10. Functions and machine learning
A model maps input features to an output. Changing parameters changes the function; optimisation searches for parameter values that reduce an error measure. At Year 10 level, focus on function input/output, parameter effects and evidence from graphs/tables.
Common misconceptions and corrections
- Trusting one graph window without changing the scale.
- Calling a visual intersection exact when the tool only gives an approximation.
- Generalising from one or two examples.
- Failing to test a boundary/counterexample.
- Changing several parameters at once and then not knowing which caused the effect.
Guided practice
Graph y=x², y=(x-2)² and y=x²+2. Before using a digital tool, predict each transformation. Then test, record one point that confirms each shift, and explain why changing the graph window can affect what you notice.
Teacher check: require a written method choice and one verification step before revealing the worked solution.
Independent practice
- Explain why a graph intersection solves f(x)=g(x).
- For f(x)=x²−5, use signs at x=2 and x=3 to trap a positive root.
- Describe the transformation from x²+y²=1 to (x/3)²+y²=1.
- Find intersections of y=2x and y=x².
- Find intersections of y=0 and x²+y²=16.
- Where is (x−1)(x−4)>0?
- When does 50(1.2)ᵗ first exceed 100?
- State a conjecture about y=(x−h)² and test h=−2.
- Give a counterexample to “all quadratics cross the x-axis twice”.
- Why should only one parameter be varied at a time in a digital experiment?
Check answers and explanations
- Because both outputs are equal at that ordered pair.
- f(2)=−1 and f(3)=4, so root lies in (2,3).
- Horizontal stretch by factor 3; x-intercepts ±3.
- x²=2x → x=0 or 2; points (0,0),(2,4).
- (−4,0),(4,0).
- x<1 or x>4.
- t=3: 86.4; t=4:103.68, so 4.
- y=(x−h)² shifts right by h; h=−2 gives y=(x+2)², shifted left 2.
- y=x²+1 has no real x-intercepts.
- So the effect of that parameter can be isolated.
Reasoning and problem-solving task
A student claims that increasing a in y=a(x-2)²+1 always moves the vertex upward. Design a digital investigation that tests the claim, include at least four values of a (positive, negative and zero), and write the strongest conclusion justified by your evidence.
Important questions and answers
- Explain why a graph intersection solves f(x)=g(x).
- Because both outputs are equal at that ordered pair.
- For f(x)=x²−5, use signs at x=2 and x=3 to trap a positive root.
- f(2)=−1 and f(3)=4, so root lies in (2,3).
- Describe the transformation from x²+y²=1 to (x/3)²+y²=1.
- Horizontal stretch by factor 3; x-intercepts ±3.
Assessment-style questions
A graphing tool suggests x²=2ˣ has more than one real solution. Describe a systematic zoom/bisection strategy for locating all visible intersections, explain why the answers are approximate, and state one check that guards against missing a solution. [6 marks]
Marking focus: method selection, mathematically correct working, interpretation and justification.
Review hints
- vary one parameter at a time
- test boundaries and counterexamples
- use ≈ for numerical solutions
- record the graph window or numerical interval used
Exit ticket: mastery check
- I can explain the concept without copying a formula sheet.
- I can solve a routine example and check the result.
- I can choose a method in an unfamiliar problem.
- I can explain a common error and correct it.
- I can connect the answer back to the context, including units or limitations.
Teacher and parent guidance
For teachers
Ask students to predict before using a graphing tool, then require a written evidence table and a counterexample search. Assess the quality of the generalisation, not the attractiveness of the graph.
For parents and carers
Ask your child to explain what they changed, what stayed fixed, and what evidence would prove their idea wrong. This turns digital graphing from button-pressing into mathematical reasoning.
Curriculum alignment
Australian Curriculum: AC9M10A05 — Year 10 Algebra. Explore functions and relations with digital tools, test conjectures against evidence and counterexamples, and generalise patterns carefully.
Victoria: VC2M10A11 and VC2M10A16 support the investigation; optional Level 10A VC2M10AA10 closely mirrors the digital-conjecture intent
NSW: Stage 5 Path — Functions and graphs; Working mathematically
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10A05 | VC2M10A11 and VC2M10A16 support the investigation; optional Level 10A VC2M10AA10 closely mirrors the digital-conjecture intent | Stage 5 Path — Functions and graphs; Working mathematically |
| Guided and independent practice | Builds fluency and application for AC9M10A05 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10A05: experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns.
- E1: use zoom/refined intervals to approximate function intersections such as x²=2ˣ. Example 1.
- E2: apply bisection to approximate quadratic intercepts such as f(x)=2x²−3x−7. Example 2.
- E3: transform x²+y²=1. Example 3.
- E4: find intersections of linear graphs with quadratics/circles. Examples 4–5.
- E5: identify intervals where quadratics are positive/negative. Example 6.
- E6: use tables to locate when exponential growth/decay crosses a threshold. Example 7.
- E7: investigate functions/relations as mathematical foundations of machine learning, including transformations, models and optimisation. Example 10.
Practice and teaching resources
Official curriculum references
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
Closest US CCSS alignment includes HSF-IF.C.7, HSF-BF.B.3 and HSA-REI.D.10–11; closest UK GCSE/NZ/Canadian/Victorian/NSW alignments cover functions, graphs, transformations, intersections and digital investigation.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Shifting functions
Khan Academy — Predicting how changing a function expression translates its graph.
As you watch: How can you check a predicted shift using a point on the original graph?
Load video player Loads YouTube in this lesson. See the video notice below.
Try it: Use a graphing tool to compare y = x², y = (x - 2)² and y = x² + 2; write and test a translation conjecture.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Experiment with functions and relations using digital tools, making and...
Mapped skill: experiment with functions and relations using digital tools, making and testing conjectures and generalising emerging patterns
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10A05 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10A11 + VC2M10A16 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-FNC-P-01 + MA5-NLI-P-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M10A05 — Functions, Relations and Digital Conjectures — AC9M10A05
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