Multiply one-digit numbers by two-digit numbers: word problems

Learning Objectives Translate complex word problems involving angular motion into mathematical models that require an initial multiplication step. Calculate the total angle of rotation in both degrees and radians by applying multiplication of one-digit by two-digit numbers as a foundational step. Determine the arc length of a circular path where the total angle is derived from a simple multiplication scenario. Compute the area of a circular sector in problems where the number of rotations or sweeps is a product of a one-digit and a two-digit number. Apply the concept of coterminal angles to find the resulting position of an object after multiple rotations. Analyze and solve multi-step problems in rotational kinematics by first identifying and executing a core multiplication of...

TermDefinitionExample Angle of Rotation (θ)The measure of the amount, in degrees or radians, that a figure is rotated about a fixed point. In complex problems, the total angle is often found by multiplying the number of turns by the angle per turn.If a wheel completes 8 rotations, its total angle of rotation is 8 * 360° = 2880°. RadianA unit of angle measure, where one radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. 2π radians = 360°.To convert 90° to radians, you calculate 90 * (π/180) = π/2 radians. Arc Length (s)The distance along the curved line making up the arc of a circle. It is calculated using the radius and the central angle in radians.For a circle with a 10 cm radius, an angle of 3 radians subtends an arc of length s = 10...

Total Angle of Rotation \theta_{total} = (\text{Number of Rotations}) \times (\text{Angle per Rotation}) Used to find the total angle swept by a rotating object. The angle per rotation is typically 360° or 2π radians. The number of rotations is often found by multiplying a rate (e.g., 15 rev/min) by a time (e.g., 4 minutes). Arc Length Formula s = r\theta Calculates the arc length 's' given the radius 'r' and the central angle 'θ'. CRITICAL: The angle 'θ' must be in radians for this formula to be valid. Sector Area Formula A = \frac{1}{2}r^2\theta Calculates the area 'A' of a circular sector given the radius 'r' and the central angle 'θ'. CRITICAL: The angle 'θ' must also be in radians.