Fill in the missing numbers for counting forwards from 1 to 4 in thirds and backwards from 3 to 0 in halves.

Counting in thirds means adding $\frac{1}{3}$ to the previous number.
The sequence is $1; 1\frac{1}{3}; \square; 2; \square; 2\frac{2}{3}; \square; \square; \square; 4$.

• $1 + \frac{1}{3} = 1\frac{1}{3}$
• $1\frac{1}{3} + \frac{1}{3} = 1\frac{2}{3}$
• $1\frac{2}{3} + \frac{1}{3} = 2$
• $2 + \frac{1}{3} = 2\frac{1}{3}$
• $2\frac{1}{3} + \frac{1}{3} = 2\frac{2}{3}$
• $2\frac{2}{3} + \frac{1}{3} = 3$
• $3 + \frac{1}{3} = 3\frac{1}{3}$
• $3\frac{1}{3} + \frac{1}{3} = 3\frac{2}{3}$
• $3\frac{2}{3} + \frac{1}{3} = 4$

The completed sequence is:

$$1; 1\frac{1}{3}; \boxed{1\frac{2}{3}}; 2; \boxed{2\frac{1}{3}}; 2\frac{2}{3}; \boxed{3}; \boxed{3\frac{1}{3}}; \boxed{3\frac{2}{3}}; 4$$

Counting backwards in halves means subtracting $\frac{1}{2}$ from the previous number.
The sequence is $3; \square; 2; \square; \square; \square; 0$.

• $3 - \frac{1}{2} = 2\frac{1}{2}$
• $2\frac{1}{2} - \frac{1}{2} = 2$
• $2 - \frac{1}{2} = 1\frac{1}{2}$
• $1\frac{1}{2} - \frac{1}{2} = 1$
• $1 - \frac{1}{2} = \frac{1}{2}$
• $\frac{1}{2} - \frac{1}{2} = 0$

The completed sequence is:

$$3; \boxed{2\frac{1}{2}}; 2; \boxed{1\frac{1}{2}}; \boxed{1}; \boxed{\frac{1}{2}}; 0$$

a) 0 to 4 in fifths:
Draw a straight line. Mark the whole numbers $0, 1, 2, 3, 4$ at equal intervals. Between each whole number, divide the segment into five equal parts. Label these divisions as $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}$ (or $1\frac{1}{5}, 1\frac{2}{5}$, etc.).
The points on the number line would be:
$0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1, 1\frac{1}{5}, 1\frac{2}{5}, 1\frac{3}{5}, 1\frac{4}{5}, 2, 2\frac{1}{5}, 2\frac{2}{5}, 2\frac{3}{5}, 2\frac{4}{5}, 3, 3\frac{1}{5}, 3\frac{2}{5}, 3\frac{3}{5}, 3\frac{4}{5}, 4$.

b) 4 to 7 in quarters:
Draw a straight line. Mark the whole numbers $4, 5, 6, 7$ at equal intervals. Between each whole number, divide the segment into four equal parts. Label these divisions as $\frac{1}{4}, \frac{2}{4}, \frac{3}{4}$ (or $4\frac{1}{4}, 4\frac{2}{4}$, etc.).
The points on the number line would be:
$4, 4\frac{1}{4}, 4\frac{2}{4}, 4\frac{3}{4}, 5, 5\frac{1}{4}, 5\frac{2}{4}, 5\frac{3}{4}, 6, 6\frac{1}{4}, 6\frac{2}{4}, 6\frac{3}{4}, 7$.