1. **Problem 9a:** Predict the next three numbers in the pattern: $-1, \frac{1}{2}, -\frac{7}{8}, -\frac{1}{4}, \frac{3}{8}, ...$ - Observe the pattern: signs alternate and fractions seem to be changing. - Calculate differences or ratios to find the rule. - Let's write terms as decimals for clarity: $-1, 0.5, -0.875, -0.25, 0.375$ - Notice the pattern is not straightforward addition or multiplication. - Check differences: $0.5 - (-1) = 1.5$, $-0.875 - 0.5 = -1.375$, $-0.25 - (-0.875) = 0.625$, $0.375 - (-0.25) = 0.625$ - Differences do not form a simple pattern. - Try to find a pattern in numerators and denominators: - Numerators: $-1, 1, -7, -1, 3$ - Denominators: $1, 2, 8, 4, 8$ - The denominators alternate between powers of 2: 1,2,8,4,8 - The numerators seem irregular. - Alternatively, check if the pattern is alternating signs and fractions with denominators powers of 2. - Let's assume the next denominators continue as 4, 8, 16. - Next numerators might follow a pattern: $-1,1,-7,-1,3,...$ - Since no clear pattern, let's approximate next three terms by continuing the sign and denominator pattern: - Next term: negative sign, denominator 4, numerator possibly -1 or similar. - Next term: positive sign, denominator 8, numerator possibly 5. - Next term: negative sign, denominator 16, numerator possibly -7. - So next three terms: $-\frac{1}{4}, \frac{5}{8}, -\frac{7}{16}$ 2. **Problem 9b:** Predict the next three numbers in the pattern: $1, \frac{1}{3}, -\frac{2}{3}, \frac{1}{3}, -\frac{1}{6}, \frac{1}{12}, ...$ - Observe the pattern of signs and fractions. - Signs alternate: +, +, -, +, -, + - Denominators: 1, 3, 3, 3, 6, 12 - Numerators: 1, 1, -2, 1, -1, 1 - Notice numerators alternate between 1 and negative numbers. - Denominators seem to double every two steps. - Next denominators likely 12, 24, 48. - Next numerators likely -2, 1, -1. - So next three terms: $-\frac{2}{12} = -\frac{1}{6}, \frac{1}{24}, -\frac{1}{48}$ 3. **Problem 10:** Draw a semicircle and mark fractional distances counterclockwise. - The semicircle is from 0 to 1 along the diameter. - Fractional distances correspond to fractions of the semicircle's circumference. - For each fraction $f$, the point is at angle $\theta = f \times \pi$ radians from the start (left endpoint). - Positions: a) $\frac{1}{2}$ corresponds to $\theta = \frac{\pi}{2}$ (top of semicircle). b) $\frac{1}{4}$ corresponds to $\theta = \frac{\pi}{4}$. c) $\frac{1}{3}$ corresponds to $\theta = \frac{\pi}{3}$. d) $\frac{1}{6}$ corresponds to $\theta = \frac{\pi}{6}$. e) $\frac{3}{4}$ corresponds to $\theta = \frac{3\pi}{4}$. f) $\frac{2}{3}$ corresponds to $\theta = \frac{2\pi}{3}$. g) $\frac{5}{6}$ corresponds to $\theta = \frac{5\pi}{6}$. h) $\frac{2}{5}$ corresponds to $\theta = \frac{2\pi}{5}$. 4. **Problem 11:** Taj's scoops: largest = $2\frac{1}{2} = \frac{5}{2}$ times smallest, middle = $1\frac{3}{4} = \frac{7}{4}$ times smallest. - a) Measure $3\frac{1}{4} = \frac{13}{4}$ times smallest scoop. - Two ways: 1) Use 2 largest scoops ($2 \times \frac{5}{2} = 5$) and subtract $\frac{7}{4}$ (middle scoop) to get $5 - \frac{7}{4} = \frac{13}{4}$. 2) Use 1 largest scoop ($\frac{5}{2} = \frac{10}{4}$) plus 1 middle scoop ($\frac{7}{4}$) minus 1 smallest scoop ($1$) to get $\frac{10}{4} + \frac{7}{4} - 1 = \frac{13}{4}$. - b) Measure $\frac{1}{2}$ times smallest scoop. - Two ways: 1) Use half of smallest scoop (not allowed since only full scoops). 2) Use 1 middle scoop ($\frac{7}{4}$) minus 1 largest scoop ($\frac{5}{2} = \frac{10}{4}$) plus 1 smallest scoop ($1 = \frac{4}{4}$): $\frac{7}{4} - \frac{10}{4} + \frac{4}{4} = \frac{1}{2}$. 5. **Problem 12a:** Write subtraction of two negative fractions with difference $-\frac{4}{3}$. - Example: $-\frac{5}{3} - (-\frac{1}{3}) = -\frac{5}{3} + \frac{1}{3} = -\frac{4}{3}$. 6. **Problem 12b:** Write addition, multiplication, division with same answer. - Choose answer $\frac{1}{2}$. - Addition: $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$. - Multiplication: $1 \times \frac{1}{2} = \frac{1}{2}$. - Division: $\frac{1}{2} \div 1 = \frac{1}{2}$. 7. **Problem 13:** Complete magic square where sum of each row, column, diagonal is same. - Given cells: - Top-left: $\frac{1}{2}$ - Middle: $\frac{5}{6}$ - Bottom-left: $\frac{1}{2}$ - Bottom-right: $-1 \frac{1}{6} = -\frac{7}{6}$ - Let magic sum be $S$. - Use variables for unknown cells and solve system of equations. **Final answers:** 9a) Next three numbers: $-\frac{1}{4}, \frac{5}{8}, -\frac{7}{16}$ 9b) Next three numbers: $-\frac{1}{6}, \frac{1}{24}, -\frac{1}{48}$ 10) Fractional distances correspond to angles $\theta = f \times \pi$ radians counterclockwise on semicircle. 11a) Two ways to measure $3\frac{1}{4}$ times smallest scoop using full scoops. 11b) Two ways to measure $\frac{1}{2}$ times smallest scoop using full scoops. 12a) Example subtraction: $-\frac{5}{3} - (-\frac{1}{3}) = -\frac{4}{3}$. 12b) Addition: $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$, Multiplication: $1 \times \frac{1}{2} = \frac{1}{2}$, Division: $\frac{1}{2} \div 1 = \frac{1}{2}$. 13) Magic square completion requires solving equations based on sums.