Class 4 Math worksheet on Number Towers and patterns from Grade1to6.com.

Problem Analysis and Solution

The worksheet provided involves two main sections: Number Towers and a Pattern Completion task. Let's solve each section step by step.

---

#### Section 1: Number Towers

The task is to complete the number towers based on the given hint: "Two numbers from the bottom line make the number on the top, and so on."

##### Example Tower
The first tower is already completed:
```
70
/ \
30 40
/ \ / \
10 20 30
```
Here, the rule is clear:
- The top number (70) is the sum of the two numbers directly below it (30 + 40).
- Each subsequent level follows the same rule.

##### Tower 2
```
90
/ \
? ?
/ \ / \
? 20 ?
```
Using the rule:
1. The top number (90) is the sum of the two numbers below it.
2. Let the two numbers below 90 be \( x \) and \( y \). Then:
\[
x + y = 90
\]
3. The number 20 is part of the bottom row. Let the other two numbers in the bottom row be \( a \) and \( b \). Then:
\[
a + 20 = x \quad \text{and} \quad 20 + b = y
\]

To solve:
- Assume \( x = 50 \) and \( y = 40 \) (since \( 50 + 40 = 90 \)).
- Then:
\[
a + 20 = 50 \implies a = 30
\]
\[
20 + b = 40 \implies b = 20
\]

So, the completed tower is:
```
90
/ \
50 40
/ \ / \
30 20 20
```

##### Tower 3
```
80
/ \
50 ?
/ \ / \
? ? ?
```
Using the rule:
1. The top number (80) is the sum of the two numbers below it.
2. Let the two numbers below 80 be \( x \) and \( y \). Then:
\[
x + y = 80
\]
3. The number 50 is part of the bottom row. Let the other two numbers in the bottom row be \( a \) and \( b \). Then:
\[
a + b = 50
\]

To solve:
- Assume \( x = 50 \) and \( y = 30 \) (since \( 50 + 30 = 80 \)).
- Then:
\[
a + b = 50 \implies a = 20 \quad \text{and} \quad b = 30
\]
- For the right side:
\[
c + d = 30
\]
Assume \( c = 10 \) and \( d = 20 \).

So, the completed tower is:
```
80
/ \
50 30
/ \ / \
20 30 10 20
```

##### Circle Tower 1
```
60
/ \
? 20
/ \ / \
? ? ? ?
```
Using the rule:
1. The top number (60) is the sum of the two numbers below it.
2. Let the two numbers below 60 be \( x \) and 20. Then:
\[
x + 20 = 60 \implies x = 40
\]
3. The number 20 is part of the bottom row. Let the other three numbers in the bottom row be \( a \), \( b \), and \( c \). Then:
\[
a + b = 40 \quad \text{and} \quad b + c = 20
\]

To solve:
- Assume \( a = 20 \) and \( b = 20 \) (since \( 20 + 20 = 40 \)).
- Then:
\[
b + c = 20 \implies 20 + c = 20 \implies c = 0
\]

So, the completed tower is:
```
60
/ \
40 20
/ \ / \
20 20 20 0
```

##### Circle Tower 2
```
100
/ \
? ?
/ \ / \
? ? ? ?
```
Using the rule:
1. The top number (100) is the sum of the two numbers below it.
2. Let the two numbers below 100 be \( x \) and \( y \). Then:
\[
x + y = 100
\]
3. The number 80 is part of the bottom row. Let the other three numbers in the bottom row be \( a \), \( b \), and \( c \). Then:
\[
a + b = x \quad \text{and} \quad b + c = y
\]

To solve:
- Assume \( x = 60 \) and \( y = 40 \) (since \( 60 + 40 = 100 \)).
- Then:
\[
a + b = 60 \implies a = 30 \quad \text{and} \quad b = 30
\]
\[
b + c = 40 \implies 30 + c = 40 \implies c = 10
\]

So, the completed tower is:
```
100
/ \
60 40
/ \ / \
30 30 30 10
```

##### Circle Tower 3
```
140
/ \
? ?
/ \ / \
? ? ? ?
```
Using the rule:
1. The top number (140) is the sum of the two numbers below it.
2. Let the two numbers below 140 be \( x \) and \( y \). Then:
\[
x + y = 140
\]
3. The number 80 is part of the bottom row. Let the other three numbers in the bottom row be \( a \), \( b \), and \( c \). Then:
\[
a + b = x \quad \text{and} \quad b + c = y
\]

To solve:
- Assume \( x = 80 \) and \( y = 60 \) (since \( 80 + 60 = 140 \)).
- Then:
\[
a + b = 80 \implies a = 40 \quad \text{and} \quad b = 40
\]
\[
b + c = 60 \implies 40 + c = 60 \implies c = 20
\]

So, the completed tower is:
```
140
/ \
80 60
/ \ / \
40 40 40 20
```

---

#### Section 2: Pattern Completion

The task is to complete the next pattern.

##### Pattern 1
The given sequence is:
```
1. ◇ △ ○ □ ◇
```
The pattern alternates between different shapes. The next shape should be a triangle (△).

##### Pattern 2
The given sequence is:
```
2. ■ ⬡ ◇ ○ ▢
```
The pattern alternates between different shapes. The next shape should be a diamond (◇).

---

Final Answers

#### Number Towers
1. Second tower:
```
90
/ \
50 40
/ \ / \
30 20 20
```
2. Third tower:
```
80
/ \
50 30
/ \ / \
20 30 10 20
```
3. Circle tower 1:
```
60
/ \
40 20
/ \ / \
20 20 20 0
```
4. Circle tower 2:
```
100
/ \
60 40
/ \ / \
30 30 30 10
```
5. Circle tower 3:
```
140
/ \
80 60
/ \ / \
40 40 40 20
```

#### Pattern Completion
1. Next shape for Pattern 1: △
2. Next shape for Pattern 2: ◇

---

Final Boxed Answers

\[
\boxed{
\begin{array}{l}
\text{Second tower:} \\
\begin{array}{c}
90 \\
/ \ \backslash \\
50 \quad 40 \\
/ \ \backslash \quad / \ \backslash \\
30 \quad 20 \quad 20
\end{array} \\
\text{Third tower:} \\
\begin{array}{c}
80 \\
/ \ \backslash \\
50 \quad 30 \\
/ \ \backslash \quad / \ \backslash \\
20 \quad 30 \quad 10 \quad 20
\end{array} \\
\text{Circle tower 1:} \\
\begin{array}{c}
60 \\
/ \ \backslash \\
40 \quad 20 \\
/ \ \backslash \quad / \ \backslash \\
20 \quad 20 \quad 20 \quad 0
\end{array} \\
\text{Circle tower 2:} \\
\begin{array}{c}
100 \\
/ \ \backslash \\
60 \quad 40 \\
/ \ \backslash \quad / \ \backslash \\
30 \quad 30 \quad 30 \quad 10
\end{array} \\
\text{Circle tower 3:} \\
\begin{array}{c}
140 \\
/ \ \backslash \\
80 \quad 60 \\
/ \ \backslash \quad / \ \backslash \\
40 \quad 40 \quad 40 \quad 20
\end{array} \\
\text{Next shape for Pattern 1: } \triangle \\
\text{Next shape for Pattern 2: } \diamondsuit
\end{array}
}
\]