Patterns and rules matching | KS3 maths sequences | Teachit - Free Printable

This worksheet is about describing sequences from patterns using a "term-to-term rule". The goal is to identify the pattern in each sequence or visual arrangement and determine how each term is generated from the previous one (i.e., the rule). Let's go through each row and explain the solution step by step.

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🔹 Row 1

- Left: *First term 3, Add 2*
→ Sequence: 3, 5, 7, 9, ...
✔️ This matches the middle box: 3,5,7,...

- Middle: 3,5,7,...
→ First term = 3, add 2 each time → Rule: Add 2

- Right: 8,10,12,...
→ First term = 8, add 2 → So this corresponds to: *First term 8, Add 2*

✔ Match:
- Left → Middle
- Right → "First term 8, Add 2" (which is in the second row left)

Wait — actually, let’s reorganize: Each row has three boxes, and we need to match the description, number sequence, and visual pattern correctly.

Let’s go row by row.

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✔ Row 1:

- Left: *First term 3, Add 2* → Sequence: 3, 5, 7, 9, ...
- Middle: 3,5,7,... → Matches the above
- Right: 8,10,12,... → First term 8, add 2 → Does not match left

So, clearly, the right box should be matched with a different description.

But wait — perhaps the table is meant to be filled in so that each row contains three related items: a description, a number sequence, and a visual pattern — all describing the same sequence.

So our task is to match each cell so that within each row, all three represent the same arithmetic sequence.

Let’s analyze each row:

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## 🟦 Row 1:
- Left: "First term 3, Add 2" → Sequence: 3, 5, 7, 9, ...
- Middle: "3,5,7,..." → Same as above
- Right: "8,10,12,..." → First term 8, add 2 → Different

✘ So this row is inconsistent unless the right box is not part of it.

Wait — perhaps the table is incomplete, and we are supposed to fill in the missing parts? But looking at the image, all cells are filled.

Alternatively, maybe it's a matching exercise where we have to pair up descriptions, sequences, and patterns across rows?

Actually, upon closer inspection, the layout seems like a grid where each row represents a different sequence, and each column represents a different representation:
- Column 1: Text description ("First term X, Add Y")
- Column 2: Number sequence
- Column 3: Visual pattern

So for each row, the three entries should describe the same sequence.

Let’s go row by row and check if they match.

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🔹 Row 1:

- Col 1: First term 3, Add 2 → Sequence: 3, 5, 7, 9, ...
- Col 2: 3,5,7,... → ✔ Matches
- Col 3: 8,10,12,... → First term 8, add 2 → ✘ Doesn’t match

✘ Mismatch!

So either the right box is wrong, or this isn't the correct pairing.

Wait — maybe the visual pattern in the right box is 8,10,12,...? But there's no visual in the right box of row 1.

Wait — look again:

The visuals are only in some cells.

Let’s list the actual contents:

| Row | Col 1 (Text) | Col 2 (Numbers/Pattern) | Col 3 (Visual/Text) |
|-----|-------------------------------|-------------------------------|-------------------------------|
| 1 | First term 3, Add 2 | 3,5,7,... | 8,10,12,... |
| 2 | First term 8, Add 2 | 1,3,5,... | First term 5, Add 3 |
| 3 | First term 1, Add 2 | [Green dots pattern] | [Green rectangles pattern] |
| 4 | 5,8,11,... | First term 4, Add 4 | [Green triangles pattern] |
| 5 | [Dots pattern] | 1,5,9,... | [Cross pattern] |
| 6 | [Square frame pattern] | 4,8,12,... | First term 1, Add 4 |

Ah! So some cells have visuals, others have text or numbers.

So the idea is: Each row contains three representations of the same sequence — but currently, they’re mixed up.

Our job is to correctly match them so that in each row, the text description, number sequence, and visual pattern all represent the same arithmetic sequence.

So we need to identify which items belong together.

Let’s extract all the elements and categorize them.

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## 🔍 Step 1: List All Items

📌 Text Descriptions (Col 1):

1. First term 3, Add 2 → Sequence: 3,5,7,9,...
2. First term 8, Add 2 → 8,10,12,14,...
3. First term 1, Add 2 → 1,3,5,7,...
4. 5,8,11,... → Common difference = 3 → First term 5, Add 3
5. [No text in col 1 of row 5]
6. [No text in col 1 of row 6]

But row 4 has: 5,8,11,... → So this is a sequence, not a description.

Let’s separate:

✔ Type A: Text Descriptions (Rule-based)

- A1: First term 3, Add 2 → 3,5,7,...
- A2: First term 8, Add 2 → 8,10,12,...
- A3: First term 1, Add 2 → 1,3,5,...
- A4: First term 5, Add 3 → 5,8,11,...
- A5: First term 4, Add 4 → 4,8,12,...
- A6: First term 1, Add 4 → 1,5,9,...

✔ Type B: Number Sequences

- B1: 3,5,7,... → Add 2, start 3
- B2: 1,3,5,... → Add 2, start 1
- B3: 5,8,11,... → Add 3, start 5
- B4: 1,5,9,... → Add 4, start 1
- B5: 4,8,12,... → Add 4, start 4

✔ Type C: Visual Patterns

Now examine the visuals:

#### C1: Row 3, Col 2 → Green dots forming L-shapes?
- Term 1: 3 dots (maybe?)
- Term 2: 5 dots?
- Term 3: 7 dots?

Wait, let's count:

- First shape: 3 dots (like a small L)
- Second: 5 dots (bigger L)
- Third: 7 dots (even bigger L)

→ So: 3,5,7,... → Add 2 → First term 3 → So this matches A1 and B1

So this visual = 3,5,7,...

#### C2: Row 3, Col 3 → Rectangles made of dots
- First: 4 dots (2x2 square)
- Second: 6 dots (2x3 rectangle)
- Third: 8 dots (2x4 rectangle)

→ 4,6,8,... → Add 2, start 4 → But no such sequence listed? Wait — but we have 4,8,12,... → That’s +4

Wait: 4,6,8,... → Add 2, first term 4 → Not in list.

But look at Row 6, Col 1: Square frames

Let’s check that.

#### C3: Row 6, Col 1 → Square frames
- First: 4 squares (a cross-like shape, or a square with center missing?)
- Actually: looks like a square outline:
- Term 1: 4 squares (a 2x2 square, but just the border?) → No, it's a plus sign? Wait.

Wait, let’s re-express:

#### Visual 1: Row 3, Col 2 → Dots in L-shape
- Shape 1: 3 dots (L-shaped)
- Shape 2: 5 dots (larger L)
- Shape 3: 7 dots (even larger L)

→ Counts: 3,5,7 → Add 2, start 3 → So this is 3,5,7,...

Matches: A1 and B1

#### Visual 2: Row 3, Col 3 → Rectangular blocks
- First: 4 dots in a line? No — looks like 4 dots in a 2x2 block?
- Wait: It shows three rectangles:
- First: 4 dots (2x2 square)
- Second: 6 dots (2x3 rectangle)
- Third: 8 dots (2x4 rectangle)

→ So: 4,6,8,... → Add 2, start 4 → But this sequence is not in the number list.

But Row 4, Col 1 says: 5,8,11,... → Add 3

Hmm.

Wait — Row 4, Col 3: Triangles
- Three green triangles:
- First: 1 triangle
- Second: 3 triangles
- Third: 5 triangles

→ 1,3,5,... → Add 2, start 1 → So this is 1,3,5,...

So this visual = 1,3,5,...

Matches A3 and B2

So now:

- Visual: 1,3,5,... → Row 4, Col 3

Then Row 4, Col 1: 5,8,11,... → Add 3 → First term 5 → So this is A4

But what visual matches 5,8,11,...?

Look at Row 5, Col 1: Dots in clusters
- First: 4 dots (2x2)
- Second: 8 dots (2x4)
- Third: 12 dots (2x6)

→ 4,8,12,... → Add 4, start 4 → So this is 4,8,12,...

But Row 6, Col 2 says: 4,8,12,... → So matches.

And Row 6, Col 3: First term 1, Add 4 → 1,5,9,...

So let’s collect all:

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🔍 Final Matching

Let’s assign each sequence to its three representations.

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#### 🔸 Sequence 1: 3,5,7,... (Add 2, start 3)
- Text: "First term 3, Add 2" → Row 1, Col 1
- Numbers: "3,5,7,..." → Row 1, Col 2
- Visual: Dots in L-shapes → Row 3, Col 2

So this sequence belongs in Row 3? But Row 3 has:
- Col 1: First term 1, Add 2 → 1,3,5,...
- Col 2: L-shape dots → 3,5,7,...
- Col 3: Rectangles → 4,6,8,...

So mismatch.

Wait — perhaps the rows are scrambled, and we need to rearrange them so that each row has matching items.

So the task is likely to correctly pair each description, sequence, and visual into consistent rows.

Let’s build the correct rows.

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✔ Correct Groupings:

#### ✔ Group 1: 3,5,7,... (Add 2, start 3)
- Text: "First term 3, Add 2" → Row 1, Col 1
- Numbers: "3,5,7,..." → Row 1, Col 2
- Visual: L-shape dots → Row 3, Col 2

So these three should be in the same row.

But currently, they are split.

Similarly:

#### ✔ Group 2: 1,3,5,... (Add 2, start 1)
- Text: "First term 1, Add 2" → Row 3, Col 1
- Numbers: "1,3,5,..." → Row 2, Col 2
- Visual: Triangle pattern → Row 4, Col 3

So this group has:
- Text: Row 3, Col 1
- Numbers: Row 2, Col 2
- Visual: Row 4, Col 3

#### ✔ Group 3: 8,10,12,... (Add 2, start 8)
- Text: "First term 8, Add 2" → Row 2, Col 1
- Numbers: "8,10,12,..." → Row 1, Col 3
- Visual: ? → None shown yet

But no visual for this.

Wait — Row 5, Col 1: Dots in clusters
- First: 4 dots
- Second: 8 dots
- Third: 12 dots → 4,8,12,... → Add 4, start 4

So this is 4,8,12,...

So:

#### ✔ Group 4: 4,8,12,... (Add 4, start 4)
- Text: "First term 4, Add 4" → Row 4, Col 2
- Numbers: "4,8,12,..." → Row 6, Col 2
- Visual: Dots in clusters → Row 5, Col 1

Yes!

#### ✔ Group 5: 1,5,9,... (Add 4, start 1)
- Text: "First term 1, Add 4" → Row 6, Col 3
- Numbers: "1,5,9,..." → Row 5, Col 2
- Visual: Cross shapes → Row 5, Col 3

Yes! Because:
- Term 1: 1 square
- Term 2: 5 squares (cross)
- Term 3: 9 squares (larger cross)

→ 1,5,9,... → Add 4

Perfect.

#### ✔ Group 6: 5,8,11,... (Add 3, start 5)
- Text: "First term 5, Add 3" → Row 2, Col 3
- Numbers: "5,8,11,..." → Row 4, Col 1
- Visual: ? → Missing

But we have Row 3, Col 3: Rectangular blocks
- First: 4 dots
- Second: 6 dots
- Third: 8 dots → 4,6,8,... → Add 2, start 4

Not matching.

Wait — Row 3, Col 3 is 4,6,8,... → Add 2, start 4

But no such sequence listed.

But we have Row 4, Col 1: 5,8,11,... → Add 3, start 5

And Row 2, Col 3: "First term 5, Add 3" → matches

So we need a visual for 5,8,11,...

But none is given.

Wait — perhaps the square frame pattern?

Row 6, Col 1: Square frames
- First: 4 squares (a 2x2 square with hole? Or just a frame?)
- Second: 8 squares
- Third: 12 squares

→ 4,8,12,... → Add 4, start 4 → already used

Wait — no.

Wait — Row 5, Col 1: Dots in clusters → 4,8,12,...

Already used.

But Row 3, Col 3: Rectangular blocks → 4,6,8,... → Add 2

But no text or number for that.

Wait — Row 1, Col 3: 8,10,12,... → Add 2, start 8

So that’s a valid sequence.

But no visual.

So perhaps the only missing visual is for 5,8,11,...

But it's not present.

Wait — let’s list all existing visuals:

1. Row 3, Col 2: L-shape dots → 3,5,7,... → Add 2, start 3
2. Row 3, Col 3: Rectangular blocks → 4,6,8,... → Add 2, start 4
3. Row 4, Col 3: Triangles → 1,3,5,... → Add 2, start 1
4. Row 5, Col 1: Clusters of dots → 4,8,12,... → Add 4, start 4
5. Row 5, Col 3: Crosses → 1,5,9,... → Add 4, start 1
6. Row 6, Col 1: Square frames → 4,8,12,... → Add 4, start 4 → same as above?

Wait — Row 6, Col 1 shows:
- First: a cross-like shape with 4 squares
- Second: a larger square frame with 8 squares
- Third: even larger with 12 squares

→ So 4,8,12,... → Add 4, start 4 → same as Row 5, Col 1

But Row 5, Col 1 also shows 4,8,12,...

So both Row 5, Col 1 and Row 6, Col 1 show the same sequence?

But they are different visuals.

Wait — Row 5, Col 1: Dots in clusters (possibly 4,8,12) → 4,8,12,...

Row 6, Col 1: Square frames → 4,8,12,...

So both represent 4,8,12,... → Add 4, start 4

So two visuals for the same sequence.

But then Row 4, Col 2: "First term 4, Add 4" → matches

Row 6, Col 2: "4,8,12,..." → matches

So sequence: 4,8,12,... → represented by:
- Text: "First term 4, Add 4" → Row 4, Col 2
- Numbers: "4,8,12,..." → Row 6, Col 2
- Visuals: Row 5, Col 1 AND Row 6, Col 1

So two visuals for one sequence.

Now back to 5,8,11,...

- Text: "First term 5, Add 3" → Row 2, Col 3
- Numbers: "5,8,11,..." → Row 4, Col 1
- Visual: ??? → None provided

But Row 3, Col 3: 4,6,8,... → Add 2, start 4 → not matching

Row 1, Col 3: 8,10,12,... → Add 2, start 8 → no visual

So only one visual is missing: for 5,8,11,...

But it's not in the image.

Wait — perhaps Row 2, Col 2: "1,3,5,..." → Add 2, start 1

And Row 3, Col 1: "First term 1, Add 2" → matches

And Row 4, Col 3: Triangles → 1,3,5,... → matches

So that’s complete.

Now let’s compile the correct matches.

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✔ Final Answer: Correct Pairings

We need to form six rows, each with:
- Description
- Number sequence
- Visual

Here are the correct groupings:

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#### 🟩 Row 1: 3,5,7,... (Add 2, start 3)
- Text: "First term 3, Add 2" → Original Row 1, Col 1
- Numbers: "3,5,7,..." → Original Row 1, Col 2
- Visual: L-shape dots → Original Row 3, Col 2

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#### 🟨 Row 2: 1,3,5,... (Add 2, start 1)
- Text: "First term 1, Add 2" → Original Row 3, Col 1
- Numbers: "1,3,5,..." → Original Row 2, Col 2
- Visual: Triangle pattern → Original Row 4, Col 3

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#### 🟪 Row 3: 8,10,12,... (Add 2, start 8)
- Text: "First term 8, Add 2" → Original Row 2, Col 1
- Numbers: "8,10,12,..." → Original Row 1, Col 3
- Visual: ??? → Not provided

But no visual given — so perhaps it's omitted.

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#### 🟫 Row 4: 5,8,11,... (Add 3, start 5)
- Text: "First term 5, Add 3" → Original Row 2, Col 3
- Numbers: "5,8,11,..." → Original Row 4, Col 1
- Visual: ??? → Not provided

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#### 🟬 Row 5: 4,8,12,... (Add 4, start 4)
- Text: "First term 4, Add 4" → Original Row 4, Col 2
- Numbers: "4,8,12,..." → Original Row 6, Col 2
- Visuals:
- Dots in clusters → Original Row 5, Col 1
- Square frames → Original Row 6, Col 1

(Both visuals represent the same sequence)

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#### 🟭 Row 6: 1,5,9,... (Add 4, start 1)
- Text: "First term 1, Add 4" → Original Row 6, Col 3
- Numbers: "1,5,9,..." → Original Row 5, Col 2
- Visual: Cross shapes → Original Row 5, Col 3

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✔ Summary of Corrections

To fix the worksheet, the rows should be rearranged so that each row contains the correct triple.

For example:

| Correct Row | Description | Sequence | Visual |
|-------------|-------------|----------|--------|
| 1 | First term 3, Add 2 | 3,5,7,... | L-shape dots |
| 2 | First term 1, Add 2 | 1,3,5,... | Triangles |
| 3 | First term 8, Add 2 | 8,10,12,... | (no visual) |
| 4 | First term 5, Add 3 | 5,8,11,... | (no visual) |
| 5 | First term 4, Add 4 | 4,8,12,... | Dots or frames |
| 6 | First term 1, Add 4 | 1,5,9,... | Crosses |

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✔ Conclusion

The worksheet contains mixed-up representations. The task is to recognize that:

- Each sequence follows a term-to-term rule (add a constant).
- The first term and common difference define the sequence.
- Visual patterns can be counted to find the sequence.

By matching the number of elements in each visual, you can determine the sequence.

For example:
- L-shape dots: 3,5,7,... → Add 2, start 3
- Triangles: 1,3,5,... → Add 2, start 1
- Crosses: 1,5,9,... → Add 4, start 1
- Dots/clusters: 4,8,12,... → Add 4, start 4
- Square frames: 4,8,12,... → Add 4, start 4

And so on.

The solution is to reorganize the table so that each row has a consistent sequence across all three columns.

If this is a student activity, they should:
1. Identify the sequence from the numbers or visuals.
2. Write the rule (e.g., "Add 2").
3. Find the matching description and visual.

This helps reinforce understanding of arithmetic sequences and pattern recognition.