Free math worksheets

- Think of the full turn as being split into equal one-degree parts. - How many one-degree parts make the whole turn? - The fraction for one equal part has \(1\) in the numerator.

1. A full turn is divided into \(360\) equal one-degree turns. 2. One of those equal parts is \(\frac{1}{360}\) of the full turn.

\(\frac{1}{360}\) of a full turn

- Compare the turn with familiar quarter-, half-, and full-turn benchmarks. - A full turn can be split into four equal quarter-turns. - The requested turn uses three of those four equal parts.

1. A full turn is \(360^\circ\). 2. Three quarters of \(360^\circ\) is \(\frac{3}{4} \times 360^\circ = 270^\circ\).

\(270^\circ\)

- A full circle measures \(360^\circ\). - Equal slices have equal central angles. - For a larger number of slices, think of the slice angle as a missing factor that must make \(360^\circ\).

1. A full circle measures \(360^\circ\). 2. For \(4\) slices, \(360^\circ \div 4=90^\circ\). 3. For \(6\) slices, \(360^\circ \div 6=60^\circ\). 4. For \(9\) slices, \(360^\circ \div 9=40^\circ\). 5. For \(15\) slices, use \(15 \times 24^\circ=360^\circ\), so one slice has a \(24^\circ\) central angle.

a) \(90^\circ\) b) \(60^\circ\) c) \(40^\circ\) d) \(24^\circ\)

- The equal hour spaces divide one full turn into \(12\) equal parts. - Count the number of equal spaces from \(12\) to each target number. - For part c, compare the turn with half of a full turn.

1. The \(12\) equal clock sections make a full turn, so each section is \(360^\circ \div 12 = 30^\circ\). 2. From \(12\) to \(1\) is one section: \(30^\circ\). 3. From \(12\) to \(4\) is four sections: \(4 \times 30^\circ = 120^\circ\). 4. From \(12\) to \(6\) is six sections: \(6 \times 30^\circ = 180^\circ\), a straight angle.

a) \(30^\circ\) b) \(120^\circ\) c) \(180^\circ\), a straight angle

- Determine the angle represented by one hour-to-hour space. - Count the smaller number of spaces between the hands in each panel. - Compare each resulting angle with \(90^\circ\).

1. The \(12\) equal hour sections make a full turn, so each section measures \(30^\circ\). 2. In a), the hands are two sections apart: \(60^\circ\), an acute angle. 3. In b), the hands are five sections apart: \(150^\circ\), an obtuse angle. 4. In c), the smaller separation is three sections: \(90^\circ\), a right angle.

a) \(60^\circ\), acute b) \(150^\circ\), obtuse c) \(90^\circ\), right

- Count the equal hour spaces from the starting hand position to the ending hand position. - One full turn is divided into \(12\) equal hour spaces. - Use the number of spaces and the size of one space to find the turn.

1. The hour hand moves through \(5\) equal hour spaces. 2. Each hour space is \(360^\circ \div 12 = 30^\circ\). 3. The turn is \(5 \times 30^\circ = 150^\circ\).

\(150^\circ\)

- Count how many equal sectors make the complete circle in each panel. - Count how many of those sectors lie on the counterclockwise turn from \(s\) to \(e\). - Use sectors traveled over total sectors, then simplify when possible.

1. In a), the circle has \(4\) equal parts and the turn covers \(1\), so the turn is \(\frac{1}{4}\). 2. In b), the circle has \(6\) equal parts and the turn covers \(2\), so \(\frac{2}{6}=\frac{1}{3}\). 3. In c), the circle has \(8\) equal parts and the turn covers \(3\), so the turn is \(\frac{3}{8}\). 4. In d), the circle has \(3\) equal parts and the turn covers \(2\), so the turn is \(\frac{2}{3}\).

a) \(\frac{1}{4}\) b) \(\frac{1}{3}\) c) \(\frac{3}{8}\) d) \(\frac{2}{3}\)

- Look at the two segments that meet at each named vertex. - Compare a square corner and a straight angle with a complete turn. - Give a fraction for every named angle before deciding which one differs.

1. At \(B\), the horizontal and vertical segments form \(\frac{1}{4}\) of a full turn. 2. At \(C\), the two path segments form a straight angle, which is \(\frac{1}{2}\) of a full turn. 3. At \(D\), the vertical and horizontal segments form \(\frac{1}{4}\) of a full turn. 4. Therefore, \(\angle BCD\) is the one that is not a quarter-turn.

\(\angle ABC=\frac{1}{4}\) turn \(\angle BCD=\frac{1}{2}\) turn \(\angle CDE=\frac{1}{4}\) turn \(\angle BCD\) is not a quarter-turn.

- Compare each pair separately; do not assume all three angles are equal. - A quarter-turn and a half-turn are benchmark fractions of one complete turn. - For the last pair, notice whether the two segments point in opposite directions.

1. The directions of \(\overline{BA}\) and \(\overline{BC}\) differ by a quarter-turn, so a) is \(\frac{1}{4}\). 2. The directions of \(\overline{BC}\) and \(\overline{BD}\) also differ by a quarter-turn, so b) is \(\frac{1}{4}\). 3. The directions of \(\overline{BA}\) and \(\overline{BD}\) are opposite, so c) is a half-turn, \(\frac{1}{2}\).

a) \(\frac{1}{4}\) of a full turn b) \(\frac{1}{4}\) of a full turn c) \(\frac{1}{2}\) of a full turn

- Examine the change in direction at each labeled corner, not the lengths of the segments. - Express one square-corner turn as a fraction of a complete turn. - Combine the three equal turn fractions for the total.

1. Each change in direction is a quarter-turn, so the robot turns \(\frac{1}{4}\) of a full turn at \(B\), \(C\), and \(D\). 2. The total turning is \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\) of a full turn.

a) \(\frac{1}{4}\) turn at each of \(B\), \(C\), and \(D\) b) \(\frac{3}{4}\) of a full turn in total

- Count the equal sectors swept out from spoke \(a\) to spoke \(d\) in the stated direction. - Write the number of swept sectors over the total number of sectors to form the fraction of a full turn. - Divide a full turn equally among the sectors, then scale that angle by the number swept out.

1. The counterclockwise turn from spoke \(a\) to spoke \(d\) covers \(3\) of the \(8\) equal sectors, so it is \(\frac{3}{8}\) of a full turn. 2. Each sector measures \(360^\circ\div8=45^\circ\). 3. Three sectors measure \(3\times45^\circ=135^\circ\).

\(\frac{3}{8}\) of a full turn; \(135^\circ\)

- Think of the full turn as \(12\) equal parts. - Determine the degree measure of one of those equal parts. - Then account for the number of parts named by the numerator.

1. A full turn is \(360^\circ\), so one twelfth is \(360^\circ \div 12 = 30^\circ\). 2. Five twelfths is \(5 \times 30^\circ = 150^\circ\).

\(150^\circ\)

- Put the angle measure over the degree measure of one full turn. - Then simplify the fraction without changing its value. - Check that the result is less than one full turn.

1. Compare the angle with a full turn: \(\frac{120}{360}\). 2. Simplify the fraction: \(\frac{120}{360} = \frac{1}{3}\).

\(\frac{1}{3}\) of a full turn

- Use the circle and the labeled rays to judge each clockwise change in direction. - Compare each turn with benchmark parts of a complete turn. - Combine the two turn fractions only after identifying them from the diagram.

1. From \(s\) to \(a\), the robot turns one quarter of the circle, so the first turn is \(\frac{1}{4}\). 2. From \(a\) to \(b\), the robot turns halfway around the circle, so the second turn is \(\frac{1}{2}\). 3. The total is \(\frac{1}{4}+\frac{1}{2}=\frac{3}{4}\) of a full turn. 4. Three quarters of \(360^\circ\) is \(270^\circ\).

a) \(\frac{1}{4}\) of a full turn b) \(\frac{1}{2}\) of a full turn c) \(\frac{3}{4}\) of a full turn, or \(270^\circ\)

- A straight angle measures \(180^\circ\), and a full turn measures \(360^\circ\). - For part a), first find one ninth of the straight angle. - For part b), look for an angle size that makes equal groups in both \(135^\circ\) and \(360^\circ\).

1. A straight angle measures \(180^\circ\). One ninth of \(180^\circ\) is \(20^\circ\), so \(\frac{4}{9}\) of it is \(4 \times 20^\circ=80^\circ\). 2. A full turn measures \(360^\circ\). 3. Think in \(45^\circ\) parts: \(135^\circ=3 \times 45^\circ\) and \(360^\circ=8 \times 45^\circ\). 4. Therefore, \(135^\circ\) is \(\frac{3}{8}\) of a full turn.

a) \(80^\circ\) b) \(\frac{3}{8}\) of a full turn

- Think of the wheel as a full \(360^\circ\) turn divided into equal sections. - In part a), find the missing angle in \(12 \times ?=360^\circ\). - In part b), find how many \(45^\circ\) sections make \(360^\circ\).

1. A full circle measures \(360^\circ\). 2. Since \(12 \times 30^\circ=360^\circ\), the angle between neighboring cars in part a) is \(30^\circ\). 3. Since \(8 \times 45^\circ=360^\circ\), the smaller wheel has \(8\) cars.

a) \(30^\circ\) b) \(8\) cars

- Connect \(60\) equal one-minute turns with one full turn. - Use the one-minute turn in both forward and reverse directions. - For part c, count equal spaces on the numbered clock face.

1. Sixty equal one-minute turns make \(360^\circ\), so the minute hand turns \(6^\circ\) each minute. 2. In \(10\) minutes, it turns \(10 \times 6^\circ = 60^\circ\). 3. A turn of \(150^\circ\) takes \(150 \div 6 = 25\) minutes. 4. Moving from \(2\) to \(6\) covers four hour spaces; each is \(30^\circ\), so the turn is \(120^\circ\).

a) \(60^\circ\) b) \(25\) minutes c) \(120^\circ\)

- Compare \(45^\circ\) with the full \(360^\circ\) turn. - Recall the degree measure of a quarter-turn as a benchmark. - Form a fraction using the angle measure and the full-turn measure, then simplify.

1. A full turn is \(360^\circ\), so the fraction is \(\frac{45}{360}\). 2. Simplify: \(\frac{45}{360} = \frac{1}{8}\). 3. A quarter-turn is \(90^\circ\), not \(45^\circ\).

Mia is incorrect. A \(45^\circ\) angle is \(\frac{1}{8}\) of a full turn.