Question 1:
1.1.1.1 First differences between consecutive terms:
Small triangles: 1, 4, 9, 16, 25, 36
Differences: 3, 5, 7, 9, 11
Black triangles: 1, 3, 5, 7, 9, 11
Differences: 2, 2, 2, 2, 2
Grey triangles: 0, 1, 3, 6, 10
Differences: 1, 2, 3, 4
White triangles: 0, 0, 1, 3, 6, 10
Differences: 0, 1, 2, 3, 4
1.1.1.2 Classification:
Small triangles: Quadratic (second differences are constant)
Black triangles: Linear (first differences are constant)
Grey triangles: Quadratic
White triangles: Quadratic
1.1.1.3 Justification:
Small triangles increase by a changing amount: differences increase by 2, showing a second difference
of 2 (constant). This is a key feature of a quadratic sequence.
Black triangles increase by the same amount each time (2), showing a constant first difference. This is
a linear pattern.
1.1.2.1 Pattern growth description:
Grey triangles: Each new diagram adds one more triangle than the previous (1, 2, 3, etc.).
White triangles: Start from diagram 3, and each new diagram adds one more triangle than the
previous (1, 2, 3, etc.).
1.1.2.2 Flow diagrams:
Grey Triangles: Input (n) → Multiply by (n - 1) → Divide by 2 → Output
White Triangles: Input (n) → Subtract 2 → Multiply by (n - 2) → Divide by 2 → Output
1.1.3.1 Algebraic rule for Black triangles:
Pattern: 1, 3, 5, 7, 9...
Rule: Tn = 2n - 1
Teaching method: Show learners the number line or steps. Let them see each diagram adds 2 more,
starting from 1. Use blocks or counters to show this linear growth.
1.1.3.2 Algebraic rule for Grey triangles:
Grey triangles: 0, 1, 3, 6, 10...
, Rule: Tn = (n-1)n/2
Misconception: Learners may assume the pattern is linear because the numbers increase slowly.
Remedial: Use a triangle dot diagram to show the number of dots in each row is growing (1, 2, 3...),
linking it to triangular numbers.
1.1.3.3 Algebraic rule for White triangles:
White triangles: 0, 0, 1, 3, 6, 10...
Rule: Tn = (n-2)(n-1)/2
Proof: Grey triangles = (n-1)n/2 White triangles = (n-2)(n-1)/2 If we replace n with (n-1) in the grey
formula: (n-2)(n-1)/2 = White triangle formula. So, white triangles at n = grey triangles at n - 1.
1.1.4.1.
Teaching: Show learners both graphs. Use the curve vs. straight line to explain linear vs. quadratic
patterns visually.
1.1.5.1 Diagram 100 Small triangles:
Rule: n^2 = 100^2 = 10 000
1.1.1.1 First differences between consecutive terms:
Small triangles: 1, 4, 9, 16, 25, 36
Differences: 3, 5, 7, 9, 11
Black triangles: 1, 3, 5, 7, 9, 11
Differences: 2, 2, 2, 2, 2
Grey triangles: 0, 1, 3, 6, 10
Differences: 1, 2, 3, 4
White triangles: 0, 0, 1, 3, 6, 10
Differences: 0, 1, 2, 3, 4
1.1.1.2 Classification:
Small triangles: Quadratic (second differences are constant)
Black triangles: Linear (first differences are constant)
Grey triangles: Quadratic
White triangles: Quadratic
1.1.1.3 Justification:
Small triangles increase by a changing amount: differences increase by 2, showing a second difference
of 2 (constant). This is a key feature of a quadratic sequence.
Black triangles increase by the same amount each time (2), showing a constant first difference. This is
a linear pattern.
1.1.2.1 Pattern growth description:
Grey triangles: Each new diagram adds one more triangle than the previous (1, 2, 3, etc.).
White triangles: Start from diagram 3, and each new diagram adds one more triangle than the
previous (1, 2, 3, etc.).
1.1.2.2 Flow diagrams:
Grey Triangles: Input (n) → Multiply by (n - 1) → Divide by 2 → Output
White Triangles: Input (n) → Subtract 2 → Multiply by (n - 2) → Divide by 2 → Output
1.1.3.1 Algebraic rule for Black triangles:
Pattern: 1, 3, 5, 7, 9...
Rule: Tn = 2n - 1
Teaching method: Show learners the number line or steps. Let them see each diagram adds 2 more,
starting from 1. Use blocks or counters to show this linear growth.
1.1.3.2 Algebraic rule for Grey triangles:
Grey triangles: 0, 1, 3, 6, 10...
, Rule: Tn = (n-1)n/2
Misconception: Learners may assume the pattern is linear because the numbers increase slowly.
Remedial: Use a triangle dot diagram to show the number of dots in each row is growing (1, 2, 3...),
linking it to triangular numbers.
1.1.3.3 Algebraic rule for White triangles:
White triangles: 0, 0, 1, 3, 6, 10...
Rule: Tn = (n-2)(n-1)/2
Proof: Grey triangles = (n-1)n/2 White triangles = (n-2)(n-1)/2 If we replace n with (n-1) in the grey
formula: (n-2)(n-1)/2 = White triangle formula. So, white triangles at n = grey triangles at n - 1.
1.1.4.1.
Teaching: Show learners both graphs. Use the curve vs. straight line to explain linear vs. quadratic
patterns visually.
1.1.5.1 Diagram 100 Small triangles:
Rule: n^2 = 100^2 = 10 000
