Let me analyze the contents of the image and provide solutions to the questions shown.

Observations:

The worksheet contains multiple sections:

1. A bar model and a question asking for the value of $A$ and other missing numbers.
2. A number line with equal jumps and calculations to determine the value of each jump.
3. Arithmetic operations involving fractions.
4. Repeating patterns and addition of fractions with an investigation into patterns greater than a specific value.

Let’s go through the key questions step by step:

Section Analysis and Solutions

1. Bar Model (Value of $A$)

• The bar is divided into segments labeled $5, 5, 1, 5$.
• Total length = Sum of all segments.
  • Total = $5 + 5 + 1 + 5 = 16$.
  • Therefore, $A = 16$.

2. Number Line (Value of Each Jump)

• Jumps start at $\frac{1}{3}$ and end at $1$.
• Number of jumps = 3.
• Value of each jump:
  • Total range = $1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$.
  • Value of each jump = $\frac{2}{3} \div 3 = \frac{2}{9}$.

3. Fraction Arithmetic

• $\frac{1}{4} + \frac{2}{7}$:

  • LCM of 4 and 7 is 28.
  • Convert fractions: $\frac{1}{4} = \frac{7}{28}$, $\frac{2}{7} = \frac{8}{28}$.
  • Sum = $\frac{7}{28} + \frac{8}{28} = \frac{15}{28}$.

• $7 \div \frac{3}{4}$:

  • Division of fractions: $7 \div \frac{3}{4} = 7 \times \frac{4}{3} = \frac{28}{3}$.

• $\frac{11}{18} - \frac{7}{72}$:

  • LCM of 18 and 72 is 72.
  • Convert fractions: $\frac{11}{18} = \frac{44}{72}$.
  • Difference = $\frac{44}{72} - \frac{7}{72} = \frac{37}{72}$.

4. Fraction Addition Patterns

• $1 + \frac{1}{2} = \frac{3}{2}$.

• $1 + \frac{1}{2} + \frac{1}{3}$:

  • LCM of 2 and 3 is 6.
  • Convert: $\frac{1}{2} = \frac{3}{6}$, $\frac{1}{3} = \frac{2}{6}$.
  • Sum = $1 + \frac{3}{6} + \frac{2}{6} = 1 + \frac{5}{6} = \frac{11}{6}$.

• Observing the pattern, we add decreasing fractions $\frac{1}{n}$ successively.

• Eventually, the sum will exceed 100 because the series $1 + \frac{1}{2} + \frac{1}{3} + \ldots$ diverges.

Would you like further explanations or visualizations for any part?

Here are five related questions to expand on this content:

1. How do you calculate the least common multiple (LCM) for fractions effectively?
2. Can you explain why $1 + \frac{1}{2} + \frac{1}{3} + \ldots$ diverges?
3. How do we interpret bar models for larger data sets?
4. Why does dividing by a fraction involve multiplication?
5. What are alternative methods for comparing fractions in subtraction?

Tip: Always simplify fractions at each step to ensure clarity in operations.