Patterns & testing conjectures in linear functions
Mathematics · Year 8
Name: ______________________Date: ____________
A conjecture is an educated guess or hypothesis about a pattern, made before you've proven it's always true. In maths, you can test a conjecture by trying several examples — if it holds every time, that's good evidence (though not full proof); if you find one example where it fails, the conjecture is disproven. With linear functions, this might mean noticing a pattern in a table of values (like "y always increases by 3 when x increases by 1") and testing whether that pattern holds by generating more values or checking it algebraically.
Example
Given the table x=1,y=4; x=2,y=7; x=3,y=10, you might conjecture "y increases by 3 each time x increases by 1." Testing x=4 predicts y=13. If the underlying rule is y = 3x + 1, checking x=4 gives y = 3(4)+1 = 13 — the conjecture holds, and you've found the rule.
Key terms
Conjecture:
An educated guess about a pattern, not yet proven true in every case.
Counterexample:
A single example that disproves a conjecture.
Questions
1. A conjecture is:
An educated guess about a pattern, not yet proven
A proven mathematical law
A random guess with no reasoning
Always false by definition
2. A counterexample is:
A single example that disproves a conjecture
An example that always proves a conjecture true
A type of graph
A type of equation only
3. If a conjecture fails for even one example, it is:
Disproven
Still definitely true
Automatically a law
Unrelated to the example
4. Testing several examples that all support a conjecture provides:
Good evidence, though not full proof
A complete, formal proof
No evidence at all
Disproof of the conjecture
5. In a table where y increases by 3 each time x increases by 1, this describes a:
Linear pattern
Random, unpredictable pattern
Pattern with no rule
Non-numeric pattern
6. A pattern in a table of values can be tested by:
Generating more values and checking if it still holds
Ignoring the table entirely
Assuming it is true with no checking
Changing the table until it matches
7. For y = 3x + 1, when x = 4, y equals:
13
12
10
15
8. Given x=1,y=5; x=2,y=8; x=3,y=11, what is the likely rule?
y = 3x + 2
y = 2x + 5
y = 5x
y = x + 4
9. Using y = 3x + 2, predict y when x = 5:
17
15
13
19
10. A student conjectures "doubling x always doubles y" for y = 2x + 3. Testing x=2 (y=7) and x=4 (y=11), is the conjecture true?
No — y did not double (7 to 11, not 14), disproving the conjecture
Yes, the conjecture is proven true
The test is inconclusive either way
This cannot be tested with these numbers
11. Why is finding one counterexample enough to disprove a conjecture, even if it held true many times before?
A conjecture must hold true in every single case to be considered a proven rule
One failure does not matter if it held true most of the time
Counterexamples only apply to shapes, not numbers
A conjecture can be true and false at the same time
12. Given a pattern 2, 5, 8, 11, the next term (following the linear pattern) is:
14
13
15
12
13. Why might testing a conjecture with several different examples, not just one, be important?
A conjecture could appear true for a single example by coincidence
A single example is always sufficient to prove any conjecture
Testing multiple examples is unnecessary once you have one that works
More examples always disprove a conjecture
14. Given x=0,y=4; x=1,y=7; x=2,y=10, the rule connecting x and y is:
y = 3x + 4
y = 4x + 3
y = 7x
y = x + 4
15. A student conjectures "for y = x², doubling x always quadruples y." Testing x=2 (y=4) and x=4 (y=16), is this conjecture supported?
Yes, this example supports it, but more testing (or algebraic proof) would be needed to be fully certain
No, this immediately disproves it
This test is completely irrelevant to the conjecture
One example is always sufficient proof
16. Why is generating an algebraic rule (like y = 3x + 1) more powerful than just testing individual number examples?
An algebraic rule can predict every value at once, rather than checking cases one at a time
Algebraic rules are never as reliable as testing examples
Testing individual examples always proves a rule for certain
There is no difference in power between the two approaches
17. A mathematician tests a conjecture on 100 examples and it holds every time. Why is this still not a complete proof?
There could still be an untested case where the conjecture fails
100 examples is always sufficient for complete mathematical proof
Conjectures cannot be tested with numerical examples at all
A conjecture becomes a proven law once tested twice
18. Given the pattern 3, 7, 11, 15, find the rule and predict the 10th term:
Rule: 4n − 1; 10th term = 39
Rule: 4n + 1; 10th term = 41
Rule: 3n + 4; 10th term = 34
Rule: n + 3; 10th term = 13
19. Why might using digital tools (like a spreadsheet) to generate many values of a linear function help test a conjecture more thoroughly?
It allows quickly checking far more cases than would be practical by hand
Digital tools cannot be used to test mathematical conjectures
Generating more values never adds confidence to a conjecture
Spreadsheets always produce incorrect results for patterns
20. A student conjectures "for any linear pattern, the difference between consecutive terms is always constant." Why is testing this on several different linear patterns good practice?
It builds confidence the conjecture holds generally, though a formal algebraic proof would confirm it for certain
One single pattern is always sufficient to prove this for every linear pattern
This conjecture cannot be tested using examples at all
Testing multiple patterns can never add any confidence
21. Understanding how to form and test conjectures about patterns mainly helps you to:
Investigate mathematical relationships rigorously, rather than assuming a pattern is true without checking
Accept every observed pattern as automatically proven
Avoid ever looking for patterns in numbers
Assume a single example is always sufficient evidence
Answer key (parent copy)
1. An educated guess about a pattern, not yet proven
2. A single example that disproves a conjecture
3. Disproven
4. Good evidence, though not full proof
5. Linear pattern
6. Generating more values and checking if it still holds
7. 13
8. y = 3x + 2
9. 17
10. No — y did not double (7 to 11, not 14), disproving the conjecture
11. A conjecture must hold true in every single case to be considered a proven rule
12. 14
13. A conjecture could appear true for a single example by coincidence
14. y = 3x + 4
15. Yes, this example supports it, but more testing (or algebraic proof) would be needed to be fully certain
16. An algebraic rule can predict every value at once, rather than checking cases one at a time
17. There could still be an untested case where the conjecture fails
18. Rule: 4n − 1; 10th term = 39
19. It allows quickly checking far more cases than would be practical by hand
20. It builds confidence the conjecture holds generally, though a formal algebraic proof would confirm it for certain
21. Investigate mathematical relationships rigorously, rather than assuming a pattern is true without checking