Worksheet - Creating algebraic rules for geometric patterns

Table of Contents
1. Instructions to the Learner
2. Section A: Multiple Choice
3. Section B: Short Answer and Structured Questions
4. Section C: Problem Solving and Word Problems
5. Bonus / Challenge Question
6. Answer Key
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Instructions to the Learner

1. Read all instructions carefully before beginning the worksheet.
2. All questions require you to analyze geometric patterns and create algebraic rules.
3. Show all working in the space provided. Partial marks may be awarded for correct methods.
4. Simplify all algebraic expressions where possible.
5. Write clearly and check your answers before submitting.
6. The bonus question is optional and is intended for learners who finish early.

Section A: Multiple Choice

Q1. A pattern is made from matchsticks. Pattern 1 has 4 matchsticks, Pattern 2 has 7 matchsticks, and Pattern 3 has 10 matchsticks. Which algebraic rule describes the number of matchsticks in Pattern \(n\)? [2 marks]

1. \(n + 3\)
2. \(3n + 1\)
3. \(4n\)
4. \(3n + 4\)

Q2. A sequence of squares is arranged in rows. Pattern 1 has 1 square, Pattern 2 has 4 squares, Pattern 3 has 9 squares. The general rule for Pattern \(n\) is: [2 marks]

1. \(n + 3\)
2. \(n^2\)
3. \(2n\)
4. \(n^2 - 1\)

Q3. Thabo draws a pattern using dots. Pattern 1 has 5 dots, Pattern 2 has 8 dots, Pattern 3 has 11 dots. What is the constant difference between consecutive patterns? [2 marks]

1. 2
2. 3
3. 5
4. 8

Q4. A geometric pattern follows the rule \(T = 2n + 3\), where \(T\) is the total number of shapes and \(n\) is the pattern number. How many shapes are in Pattern 6? [2 marks]

1. 9
2. 12
3. 15
4. 18

Q5. Lindiwe creates a pattern using hexagons. The first pattern has 6 sides showing, the second has 11 sides showing, and the third has 16 sides showing. Which algebraic expression represents the number of sides showing in Pattern \(n\)? [2 marks]

1. \(5n + 1\)
2. \(6n\)
3. \(n + 5\)
4. \(5n + 6\)

Section B: Short Answer and Structured Questions

Q1. A pattern is made from toothpicks forming triangles.

• Pattern 1: 3 toothpicks
• Pattern 2: 5 toothpicks
• Pattern 3: 7 toothpicks
• Pattern 4: 9 toothpicks

(a) Describe the pattern in words. [2 marks]

(b) How many toothpicks would be needed for Pattern 5? [2 marks]

(c) Determine the algebraic rule for Pattern \(n\). [3 marks]

(d) Use your rule to calculate the number of toothpicks in Pattern 20. [2 marks]

Q2. A sequence of rectangular tables is arranged in a line. Each table seats 2 people on each long side and 1 person on each short end.

• Pattern 1 (1 table): 6 people
• Pattern 2 (2 tables): 10 people
• Pattern 3 (3 tables): 14 people

(a) What is the constant difference in this pattern? [2 marks]

(b) Create an algebraic rule for the number of people that can be seated when \(n\) tables are placed in a line. [4 marks]

(c) How many people can be seated when 15 tables are arranged in a line? [2 marks]

Q3. Consider the following pattern made from square tiles:

• Pattern 1: 1 tile in the center, 8 tiles around it = 9 tiles total
• Pattern 2: 4 tiles in the center, 12 tiles around it = 16 tiles total
• Pattern 3: 9 tiles in the center, 16 tiles around it = 25 tiles total

(a) How many tiles would Pattern 4 have in total? [2 marks]

(b) Write down the algebraic rule for the total number of tiles in Pattern \(n\). [3 marks]

(c) Verify your rule using Pattern 2. [2 marks]

Q4. A growing pattern of circles is shown below:

• Pattern 1: 2 circles
• Pattern 2: 5 circles
• Pattern 3: 8 circles
• Pattern 4: 11 circles

(a) Identify the first term and the constant difference. [2 marks]

(b) Write the general rule in the form \(T = an + b\), where \(T\) is the number of circles and \(n\) is the pattern number. [4 marks]

(c) Which pattern will have exactly 32 circles? [3 marks]

Section C: Problem Solving and Word Problems

Q1. Sipho is building a fence using wooden posts and horizontal rails. Each section of fence requires 1 post at the start, and then each additional section adds 1 more post. The horizontal rails connect the posts. He observes the following pattern:

• 1 section: 2 posts and 3 rails
• 2 sections: 3 posts and 6 rails
• 3 sections: 4 posts and 9 rails
• 4 sections: 5 posts and 12 rails

(a) Write an algebraic rule for the number of posts needed for \(n\) sections of fence. [3 marks]

(b) Write an algebraic rule for the number of rails needed for \(n\) sections of fence. [3 marks]

(c) Sipho needs to build a fence with 25 sections. How many posts and how many rails will he need in total? [4 marks]

Q2. Nomsa designs beaded necklaces using a geometric pattern. She uses small beads and large beads in each necklace. She records her pattern:

• Design 1: 7 small beads and 3 large beads = 10 beads total
• Design 2: 12 small beads and 5 large beads = 17 beads total
• Design 3: 17 small beads and 7 large beads = 24 beads total

(a) What is the pattern for the total number of beads? Identify the constant difference. [2 marks]

(b) Determine the algebraic rule for the total number of beads in Design \(n\). [4 marks]

(c) Nomsa wants to create Design 12. How many beads in total will she need? [2 marks]

(d) If Nomsa has 150 beads available, which is the highest design number she can complete? [4 marks]

Bonus / Challenge Question

This question is optional and intended for learners who finish early.

Q1. Zanele creates a complex pattern using black and white tiles. The pattern follows these rules:

• Pattern 1: 1 black tile in the center, surrounded by 4 white tiles = 5 tiles total
• Pattern 2: 4 black tiles in the center, surrounded by 8 white tiles = 12 tiles total
• Pattern 3: 9 black tiles in the center, surrounded by 12 white tiles = 21 tiles total

(a) Determine separate algebraic rules for the number of black tiles and the number of white tiles in Pattern \(n\). [5 marks]

(b) Write an algebraic rule for the total number of tiles in Pattern \(n\), and simplify your expression completely. [4 marks]

(c) In which pattern will there be exactly 100 black tiles? How many white tiles will there be in that pattern? [4 marks]

Answer Key

Section A: Multiple Choice

Q1. B: \(3n + 1\)

Explanation: The pattern increases by 3 each time (constant difference = 3). Pattern 1: 4 matchsticks. Using the rule \(3n + 1\): when \(n = 1\), \(3(1) + 1 = 4\) ✓; when \(n = 2\), \(3(2) + 1 = 7\) ✓; when \(n = 3\), \(3(3) + 1 = 10\) ✓. Option A gives 4, 5, 6 (incorrect). Option C gives 4, 8, 12 (incorrect). Option D gives 7, 10, 13 (incorrect starting value).

Q2. B: \(n^2\)

Explanation: Pattern 1 has 1 = \(1^2\) square, Pattern 2 has 4 = \(2^2\) squares, Pattern 3 has 9 = \(3^2\) squares. The rule is \(n^2\). Option A gives 4, 5, 6 (incorrect). Option C gives 2, 4, 6 (incorrect). Option D gives 0, 3, 8 (incorrect).

Q3. B: 3

Explanation: The difference between consecutive patterns is: 8 - 5 = 3, and 11 - 8 = 3. The constant difference is 3. Options A, C, and D do not represent the constant difference between consecutive terms.

Q4. C: 15

Explanation: Using the rule \(T = 2n + 3\) with \(n = 6\):
\(T = 2(6) + 3\)
\(T = 12 + 3\)
\(T = 15\)
Options A, B, and D result from calculation errors or using wrong values of \(n\).

Q5. A: \(5n + 1\)

Explanation: The pattern increases by 5 each time (11 - 6 = 5, 16 - 11 = 5). Using \(5n + 1\): when \(n = 1\), \(5(1) + 1 = 6\) ✓; when \(n = 2\), \(5(2) + 1 = 11\) ✓; when \(n = 3\), \(5(3) + 1 = 16\) ✓. Option B gives 6, 12, 18 (incorrect). Option C gives 6, 7, 8 (incorrect). Option D gives 11, 16, 21 (incorrect starting value).

Section B: Short Answer and Structured Questions

Q1.

(a) The pattern starts at 3 toothpicks and increases by 2 toothpicks each time. [2 marks: 1 mark for starting value, 1 mark for constant difference]

(b) Pattern 5 would have 11 toothpicks.
Pattern 4 has 9 toothpicks
Add 2 more: 9 + 2 = 11 toothpicks
Answer: 11 toothpicks [2 marks: 1 mark for method, 1 mark for correct answer]

(c) First term = 3, constant difference = 2
General form: \(T = dn + c\) where \(d\) is the constant difference
\(T = 2n + c\)
Using Pattern 1: \(3 = 2(1) + c\)
\(3 = 2 + c\)
\(c = 1\)
Algebraic rule: \(T = 2n + 1\) [3 marks: 1 mark for identifying constant difference, 1 mark for correct method, 1 mark for correct rule]

(d) Using \(T = 2n + 1\) with \(n = 20\):
\(T = 2(20) + 1\)
\(T = 40 + 1\)
\(T = 41\)
Answer: 41 toothpicks [2 marks: 1 mark for substitution, 1 mark for correct answer]

Q2.

(a) Constant difference = 10 - 6 = 4
Check: 14 - 10 = 4 ✓
Constant difference = 4 [2 marks: 1 mark for calculation, 1 mark for correct answer]

(b) First term = 6, constant difference = 4
General form: \(T = 4n + c\)
Using Pattern 1 (\(n = 1\)): \(6 = 4(1) + c\)
\(6 = 4 + c\)
\(c = 2\)
Algebraic rule: \(T = 4n + 2\)
Verification with Pattern 2: \(T = 4(2) + 2 = 10\) ✓ [4 marks: 1 mark for identifying pattern, 1 mark for setting up rule, 1 mark for finding constant, 1 mark for correct final rule]

(c) Using \(T = 4n + 2\) with \(n = 15\):
\(T = 4(15) + 2\)
\(T = 60 + 2\)
\(T = 62\)
Answer: 62 people [2 marks: 1 mark for substitution, 1 mark for correct answer]

Q3.

(a) Pattern sequence: 9, 16, 25, ...
This represents: \(3^2\), \(4^2\), \(5^2\)
Pattern 4 would be \(6^2 = 36\)
Answer: 36 tiles [2 marks: 1 mark for identifying pattern, 1 mark for correct answer]

(b) Pattern 1 = \((1 + 2)^2 = 3^2 = 9\)
Pattern 2 = \((2 + 2)^2 = 4^2 = 16\)
Pattern 3 = \((3 + 2)^2 = 5^2 = 25\)
Pattern \(n\) = \((n + 2)^2\)
Algebraic rule: \(T = (n + 2)^2\) or expanded: \(T = n^2 + 4n + 4\) [3 marks: 1 mark for recognizing square pattern, 1 mark for correct rule, 1 mark for proper notation]

(c) Using \(T = (n + 2)^2\) with \(n = 2\):
\(T = (2 + 2)^2\)
\(T = 4^2\)
\(T = 16\) ✓
This matches Pattern 2 from the given information.
Rule is verified [2 marks: 1 mark for substitution, 1 mark for verification]

Q4.

(a) First term = 2
Constant difference: 5 - 2 = 3, 8 - 5 = 3, 11 - 8 = 3
First term = 2, Constant difference = 3 [2 marks: 1 mark for each correct value]

(b) Using general form \(T = an + b\)
Constant difference = 3, so \(a = 3\)
\(T = 3n + b\)
Using Pattern 1 (\(n = 1\)): \(2 = 3(1) + b\)
\(2 = 3 + b\)
\(b = -1\)
General rule: \(T = 3n - 1\)
Verify with Pattern 2: \(T = 3(2) - 1 = 5\) ✓ [4 marks: 1 mark for identifying \(a\), 1 mark for setting up equation, 1 mark for finding \(b\), 1 mark for correct final rule]

(c) Using \(T = 3n - 1\) with \(T = 32\):
\(32 = 3n - 1\)
\(32 + 1 = 3n\)
\(33 = 3n\)
\(n = 33 ÷ 3\)
\(n = 11\)
Answer: Pattern 11 [3 marks: 1 mark for setting up equation, 1 mark for correct solving method, 1 mark for correct answer]

Section C: Problem Solving and Word Problems

Q1.

(a) Number of posts: 2, 3, 4, 5, ...
Pattern increases by 1 each time
First term = 2, constant difference = 1
\(P = 1n + c\)
Using \(n = 1\): \(2 = 1(1) + c\), so \(c = 1\)
Rule for posts: \(P = n + 1\) [3 marks: 1 mark for identifying pattern, 1 mark for method, 1 mark for correct rule]

(b) Number of rails: 3, 6, 9, 12, ...
Pattern increases by 3 each time
First term = 3, constant difference = 3
\(R = 3n + c\)
Using \(n = 1\): \(3 = 3(1) + c\), so \(c = 0\)
Rule for rails: \(R = 3n\) [3 marks: 1 mark for identifying pattern, 1 mark for method, 1 mark for correct rule]

(c) For 25 sections:
Posts: \(P = n + 1 = 25 + 1 = 26\) posts
Rails: \(R = 3n = 3(25) = 75\) rails
Total items: 26 + 75 = 101
Sipho will need 26 posts and 75 rails [4 marks: 1 mark for posts calculation, 1 mark for rails calculation, 1 mark for both correct answers, 1 mark for clear working]

Q2.

(a) Total beads: 10, 17, 24, ...
Constant difference: 17 - 10 = 7, 24 - 17 = 7
The total increases by 7 beads each time. Constant difference = 7 [2 marks: 1 mark for identifying pattern, 1 mark for constant difference]

(b) First term = 10, constant difference = 7
\(T = 7n + c\)
Using Design 1 (\(n = 1\)): \(10 = 7(1) + c\)
\(10 = 7 + c\)
\(c = 3\)
Algebraic rule: \(T = 7n + 3\)
Verify with Design 2: \(T = 7(2) + 3 = 17\) ✓ [4 marks: 1 mark for identifying values, 1 mark for setting up rule, 1 mark for finding constant, 1 mark for correct rule]

(c) Using \(T = 7n + 3\) with \(n = 12\):
\(T = 7(12) + 3\)
\(T = 84 + 3\)
\(T = 87\)
Nomsa will need 87 beads for Design 12 [2 marks: 1 mark for substitution, 1 mark for correct answer]

(d) Using \(T = 7n + 3\) with \(T = 150\):
\(150 = 7n + 3\)
\(150 - 3 = 7n\)
\(147 = 7n\)
\(n = 147 ÷ 7\)
\(n = 21\)
Check: \(T = 7(21) + 3 = 147 + 3 = 150\) ✓
Nomsa can complete Design 21 (the highest design number) [4 marks: 1 mark for setting up equation, 1 mark for solving, 1 mark for correct answer, 1 mark for verification or clear interpretation]

Bonus / Challenge Question

Q1.

(a) Black tiles:
Pattern 1: 1 = \(1^2\)
Pattern 2: 4 = \(2^2\)
Pattern 3: 9 = \(3^2\)
Rule for black tiles: \(B = n^2\)

White tiles:
Pattern 1: 4 white tiles
Pattern 2: 8 white tiles
Pattern 3: 12 white tiles
Constant difference = 4
\(W = 4n + c\)
Using \(n = 1\): \(4 = 4(1) + c\), so \(c = 0\)
Rule for white tiles: \(W = 4n\) [5 marks: 2 marks for black tiles rule, 2 marks for white tiles rule, 1 mark for showing working clearly]

(b) Total tiles: \(T = B + W\)
\(T = n^2 + 4n\)
This can be factored (optional): \(T = n(n + 4)\)
Rule for total tiles: \(T = n^2 + 4n\) or \(T = n(n + 4)\)
Verify with Pattern 2: \(T = 2^2 + 4(2) = 4 + 8 = 12\) ✓ [4 marks: 1 mark for adding the two rules, 1 mark for correct simplified form, 1 mark for alternative factored form if shown, 1 mark for verification]

(c) For 100 black tiles:
\(B = n^2 = 100\)
\(n^2 = 100\)
\(n = \sqrt{100}\)
\(n = 10\)
So Pattern 10 will have 100 black tiles.
Number of white tiles: \(W = 4n = 4(10) = 40\) white tiles
Pattern 10 will have 100 black tiles and 40 white tiles
This question tests higher-order thinking by requiring learners to work backwards from a given value to find the pattern number, then use that to find a related quantity. [4 marks: 2 marks for finding pattern number, 2 marks for calculating white tiles]

Total Marks Summary