Number Towers and Pattern Completion Worksheet for Class 4 Math Students

Let's solve the Number Towers and Shape Patterns step by step, based on the worksheet.

---

🔷 Part 1: Number Towers

#### 🔹 Rule:
> Two numbers from the bottom line make the number on top.
> That means: Each number above is the sum of the two numbers directly below it.

We'll use this rule to fill in the missing numbers.

---

✔ Example Given (Leftmost Tower):

```
70
30 40
10 20 30
```

Check:
- 30 + 40 = 70 ✔️
- 10 + 20 = 30 ✔️
- 20 + 30 = 40 ✔️

So, the pattern is: Add adjacent numbers from the bottom row to get the one above.

---

🟦 Tower 1: Top = 90, Middle = ?, Bottom = ?, 20, ?

Let’s denote:

```
90
? ?
? 20 ?
```

Let’s work upward, starting from bottom.

Let the bottom row be: A, 20, B
Then middle row: A+20, 20+B
Top: (A+20) + (20+B) = A + B + 40 = 90

So:
> A + B + 40 = 90 → A + B = 50

Now we need values for A and B such that they are reasonable (likely multiples of 10 or small integers). But since there are multiple solutions unless more constraints exist, let’s assume symmetry or standard values.

But wait — look at the first example: numbers are simple (10, 20, 30), so likely all numbers are multiples of 10.

Try assuming symmetry: A = B → A + A = 50 → A = 25 → not a multiple of 10.

Try A = 10 → B = 40
Then:
- Bottom: 10, 20, 40
- Middle: 10+20=30, 20+40=60
- Top: 30+60=90 ✔️

That works!

So one possible solution:

```
90
30 60
10 20 40
```

✔ This fits the pattern.

Let’s write it clearly:

> Tower 1:
> - Bottom: 10, 20, 40
> - Middle: 30, 60
> - Top: 90

(Other combinations are possible, but this is clean and matches style.)

---

🟨 Tower 2: Top = 80, Middle = 50, ?, Bottom = ?, ?, ?

```
80
50 ?
? ? ?
```

Let’s work upward.

Let the middle row be: 50, X
Then top = 50 + X = 80 → X = 30

So middle row: 50, 30

Now, bottom row: A, B, C
Middle: A+B = 50, B+C = 30

We need to find A, B, C such that:
- A + B = 50
- B + C = 30

Try B = 20 → then A = 30, C = 10

Check:
- Bottom: 30, 20, 10
- Middle: 30+20=50, 20+10=30
- Top: 50+30=80 ✔️

Perfect!

So:

> Tower 2:
> - Bottom: 30, 20, 10
> - Middle: 50, 30
> - Top: 80

Note: The order doesn't matter as long as adjacent sums match.

---

🟩 Tower 3: Circles, Top = 60, Middle has 20, Bottom = ?, ?, ?

```
60
? 20
? ? ?
```

Let’s go step by step.

Let middle row be: A, 20
Then top = A + 20 = 60 → A = 40

So middle row: 40, 20

Now bottom: B, C, D
Then:
- B + C = 40
- C + D = 20

Try C = 10 → then B = 30, D = 10

So bottom: 30, 10, 10

Check:
- 30+10 = 40 ✔️
- 10+10 = 20 ✔️
- 40+20 = 60 ✔️

✔ Works!

> Tower 3:
> - Bottom: 30, 10, 10
> - Middle: 40, 20
> - Top: 60

---

🟪 Tower 4: Top = 100, Middle = ?, ?, Bottom = ?, 30, ?

```
100
? ?
? 30 ?
```

Let middle row: A, B
Then A + B = 100

Bottom: C, 30, D
Then:
- C + 30 = A
- 30 + D = B

So:
- A = C + 30
- B = 30 + D
- A + B = 100 → (C + 30) + (30 + D) = 100 → C + D + 60 = 100 → C + D = 40

Try C = 10 → D = 30
Then:
- A = 10 + 30 = 40
- B = 30 + 30 = 60
- 40 + 60 = 100 ✔️

So:
> Bottom: 10, 30, 30
> Middle: 40, 60
> Top: 100

✔ Valid.

> Tower 4:
> - Bottom: 10, 30, 30
> - Middle: 40, 60
> - Top: 100

---

🟫 Tower 5: Top = 140, Middle = 80, ?, Bottom = ?, ?, ?

```
140
80 ?
? ? ?
```

Let middle: 80, X
Then 80 + X = 140 → X = 60

So middle: 80, 60

Bottom: A, B, C
Then:
- A + B = 80
- B + C = 60

Try B = 30 → A = 50, C = 30

So bottom: 50, 30, 30

Check:
- 50+30 = 80 ✔️
- 30+30 = 60 ✔️
- 80+60 = 140 ✔️

✔ Works!

> Tower 5:
> - Bottom: 50, 30, 30
> - Middle: 80, 60
> - Top: 140

---

## ✔ Summary of Number Towers:

| Tower | Bottom Row | Middle Row | Top |
|-------|------------|------------|-----|
| 1 | 10, 20, 40 | 30, 60 | 90 |
| 2 | 30, 20, 10 | 50, 30 | 80 |
| 3 | 30, 10, 10 | 40, 20 | 60 |
| 4 | 10, 30, 30 | 40, 60 | 100 |
| 5 | 50, 30, 30 | 80, 60 | 140 |

---

## 🔷 Part 2: Complete the next pattern

🔹 Pattern 1:

Sequence:
🔶 (double diamond) → △ (triangle) → 🔷 (hexagon) → ◼ (square) → 🔶 (double diamond)

Looks like shapes with increasing sides, then repeats?

Let’s count sides:

- Double diamond: 4 sides (diamond)
- Triangle: 3 sides
- Hexagon: 6 sides
- Square: 4 sides
- Then back to double diamond (4 sides)

Wait — not consistent.

Wait: perhaps it's outer shape + inner shape?

Actually, observe:

1. Double diamond → 2 diamonds (nested)
2. Triangle → single triangle
3. Hexagon → single hexagon
4. Square → single square
5. Double diamond again → same as first

Wait — maybe it's a repeating cycle?

But the shapes are different.

Wait — look carefully:

- First: Double diamond (a diamond inside another)
- Second: Triangle
- Third: Hexagon
- Fourth: Square
- Fifth: Double diamond again

So the pattern might be: Repeat every 4? But fifth is same as first.

Wait — maybe the pattern is repeating every 4 steps, but 5th is same as 1st → so cycle of 4?

But only 5 shapes shown.

Possibility: It's a sequence of polygons with increasing sides?

List:

1. Diamond → 4 sides
2. Triangle → 3 sides
3. Hexagon → 6 sides
4. Square → 4 sides
5. Double diamond → 4 sides

No clear progression.

Alternative idea: Look at the outer shape and inner shape

But all except first are single.

Wait — perhaps the first and last are the same, suggesting cyclic pattern?

But what comes after?

Wait — perhaps the sequence is:

1. Double diamond (4 sides)
2. Triangle (3)
3. Hexagon (6)
4. Square (4)
5. ??? — should repeat?

But 5th is double diamond → so maybe the pattern is:

Double diamond → Triangle → Hexagon → Square → Double diamond → ...

So it's a cycle of 4?

But then why does it end with double diamond?

Perhaps the next shape after the last (which is double diamond) is Triangle again?

But the question says: "Complete the next pattern."

It shows 5 shapes, and asks to complete the next pattern — probably meaning continue the sequence.

But the sequence is:

1. Double diamond
2. Triangle
3. Hexagon
4. Square
5. Double diamond

So if it's cyclic, then next is Triangle.

But let's check if it's based on number of sides:

- Diamond: 4
- Triangle: 3
- Hexagon: 6
- Square: 4
- Double diamond: 4

Not helpful.

Another idea: Shape types in order:

- Rhombus (diamond) → Triangle → Hexagon → Square → Rhombus

So maybe it's no clear mathematical pattern, but perhaps it's just repeating a set?

But 5th is same as 1st → suggests periodic with period 4?

Wait — positions:

1. Double diamond
2. Triangle
3. Hexagon
4. Square
5. Double diamond

So the fifth is same as first → implies the pattern repeats every 4 steps, so:

- 1st: Double diamond
- 2nd: Triangle
- 3rd: Hexagon
- 4th: Square
- 5th: Double diamond (same as 1st)
- 6th: Triangle (same as 2nd)
- etc.

So the next shape after the last (double diamond) should be Triangle.

But the problem says "complete the next pattern" — and shows only one sequence.

But it says "Complete the next pattern" and gives two sequences.

Wait — actually, the image shows:

> 1) [shape1] [shape2] [shape3] [shape4] [shape5]
> 2) [rect] [hex] [diamond] [circle] [rect]

Ah! There are two patterns.

Let’s analyze both.

---

🔸 Pattern 1: Shapes

Sequence:
1. Double diamond (rhombus with inner rhombus)
2. Triangle
3. Hexagon
4. Square (with border)
5. Double diamond

So:
- Shape 1: Double diamond
- Shape 2: Triangle
- Shape 3: Hexagon
- Shape 4: Square
- Shape 5: Double diamond → same as shape 1

So likely: The pattern repeats every 4 shapes.

Thus, the next shape after the 5th should be same as 2nd → Triangle

But the instruction says “complete the next pattern” — maybe it wants us to predict the next shape in the sequence?

But the sequence ends at 5th, which is same as 1st → so the pattern is cyclic.

So next shape is Triangle.

But perhaps the pattern is based on number of sides?

Let’s list:

1. Double diamond → 4 sides (but nested)
2. Triangle → 3 sides
3. Hexagon → 6 sides
4. Square → 4 sides
5. Double diamond → 4 sides

No clear trend.

Alternatively, could it be alternating between compound and simple shapes?

- 1: Compound (double diamond)
- 2: Simple (triangle)
- 3: Simple (hexagon)
- 4: Simple (square)
- 5: Compound (double diamond)

So maybe: compound → simple → simple → simple → compound → ?

Then next should be simple?

But which one?

Unlikely.

Best guess: The pattern is repeating every 4 shapes, so:

- Position 1: Double diamond
- Position 2: Triangle
- Position 3: Hexagon
- Position 4: Square
- Position 5: Double diamond (same as 1)
- So position 6: Triangle (same as 2)

Therefore, next shape is Triangle.

But since the question says “complete the next pattern”, and only shows 5 shapes, perhaps it wants the 6th shape → Triangle

But maybe it's a typo or misalignment.

Wait — perhaps the last shape is the start of a new cycle, and we're to continue?

But no indication.

Alternatively, maybe the pattern is: Shapes with increasing number of sides?

But 3, 4, 6, 4 — not increasing.

Wait — maybe it's based on letters?

No.

Alternatively, maybe it's a color or style pattern — but all black.

I think the most logical conclusion is:

> The pattern repeats every 4 shapes:
> [Double diamond] → [Triangle] → [Hexagon] → [Square] → [Double diamond] → [Triangle] → ...

So the next shape after the last (double diamond) is Triangle.

But the worksheet may expect you to write the next shape.

But since it's a worksheet, maybe it's just showing the pattern and asking to recognize it.

But the second pattern is clearer.

---

🔸 Pattern 2:

Sequence:
1. Rectangle
2. Hexagon
3. Diamond (rhombus)
4. Circle
5. Rectangle

So:
- Rect → Hex → Diamond → Circle → Rect

Now, Rectangle appears at start and end → suggests cycle of 4

So the pattern is:
Rect → Hex → Diamond → Circle → Rect → Hex → ...

So next shape after the last (rectangle) is Hexagon

But the sequence ends at rectangle, so next is Hexagon

Wait — but the pattern is already shown with 5 shapes.

So if it's a cycle of 4, then:

- 1: Rect
- 2: Hex
- 3: Diamond
- 4: Circle
- 5: Rect → same as 1
- 6: Hex → same as 2

So next shape is Hexagon

But the question says “complete the next pattern” — so likely, for each, we need to predict the next shape.

But since both patterns end with Rectangle and Double diamond, and both were first, likely:

- Pattern 1: Next is Triangle
- Pattern 2: Next is Hexagon

But let's see if there's a better explanation.

Wait — perhaps the pattern is about symmetry?

- Double diamond: symmetric
- Triangle: symmetric
- Hexagon: symmetric
- Square: symmetric
- Double diamond: symmetric

All are symmetric — not helpful.

Alternatively, number of sides:

Pattern 2:
- Rectangle: 4 sides
- Hexagon: 6
- Diamond: 4
- Circle: 0 (infinite?)
- Rectangle: 4

No.

But notice: Rectangle → Hexagon → Diamond → Circle → Rectangle

Could it be increasing sides? No.

Wait — maybe it's polygonal shapes and circle is different.

But circle is included.

Alternatively, maybe it's types of quadrilaterals and others?

- Rectangle: quad
- Hexagon: 6-gon
- Diamond: quad
- Circle: none
- Rectangle: quad

Still messy.

But observe: the last shape is same as first → so it's a cycle of 4.

So:

- Pattern 1: Cycle: [DD, T, H, S] → repeats
- Pattern 2: Cycle: [R, H, D, C] → repeats

So next shape after last:

- For Pattern 1: Triangle
- For Pattern 2: Hexagon

But the worksheet shows:

> 1) [DD] [T] [H] [S] [DD] → next is T
> 2) [R] [H] [D] [C] [R] → next is H

So the answer is:

> 1) Triangle
> 2) Hexagon

But since the worksheet says “complete the next pattern”, and shows the full sequence, maybe it's just asking to recognize the repeating pattern.

But since it says “complete”, likely we are to draw or name the next shape.

But in text, we can say:

---

## ✔ Final Answers:

🔢 Number Towers:

| Tower | Bottom | Middle | Top |
|------|--------|--------|-----|
| 1 | 10, 20, 40 | 30, 60 | 90 |
| 2 | 30, 20, 10 | 50, 30 | 80 |
| 3 | 30, 10, 10 | 40, 20 | 60 |
| 4 | 10, 30, 30 | 40, 60 | 100 |
| 5 | 50, 30, 30 | 80, 60 | 140 |

🔷 Shape Patterns:

1. Sequence: Double diamond → Triangle → Hexagon → Square → Double diamond → Next: Triangle
2. Sequence: Rectangle → Hexagon → Diamond → Circle → Rectangle → Next: Hexagon

---

✔ Final Answer:

Number Towers:
- Tower 1: Bottom: 10, 20, 40; Middle: 30, 60
- Tower 2: Bottom: 30, 20, 10; Middle: 50, 30
- Tower 3: Bottom: 30, 10, 10; Middle: 40, 20
- Tower 4: Bottom: 10, 30, 30; Middle: 40, 60
- Tower 5: Bottom: 50, 30, 30; Middle: 80, 60

Shape Patterns:
1. Next shape: Triangle
2. Next shape: Hexagon

Let me know if you'd like these filled in visually or as a printable answer key!