Add and subtract mixed time units

Learning Objectives Convert fluently between different units of time (seconds, minutes, hours, days). Add multiple time intervals involving mixed units, correctly regrouping values across units. Subtract time intervals, correctly borrowing from larger units when necessary. Solve multi-step word problems involving elapsed time, project durations, and time differences. By the end of of this lesson, students will be able to normalize non-standard time expressions (e.g., 75 minutes) into standard mixed-unit format (1 hour, 15 minutes). Apply the principles of base-60 (sexagesimal) and other base systems (e.g., base-24) to time calculations. A rocket launch is scheduled for 14:00:00 UTC. If the final countdown sequence takes 3 hours, 47 minutes, and 52 seconds, at what exact time...

TermDefinitionExample Mixed Time UnitsA measurement of time expressed as a combination of different, related units, such as hours, minutes, and seconds.3 hours, 25 minutes, and 10 seconds is a value in mixed time units. Sexagesimal SystemA numeral system with a base of 60. This is the foundation for our system of time, where 60 seconds make a minute and 60 minutes make an hour.When we calculate 30 seconds + 45 seconds = 75 seconds, we convert it to 1 minute and 15 seconds, which is a base-60 operation. Regrouping (Carrying)In addition, this is the process of converting a sum of a smaller unit into one or more of the next larger unit when the sum equals or exceeds the base of that unit (e.g., 60 for minutes).Adding 40 minutes and 35 minutes gives 75 minutes. We regroup this as 1 hour and 1...

Addition with Regrouping Given two times T_1 = H_1:M_1:S_1 and T_2 = H_2:M_2:S_2. Let S_{sum} = S_1+S_2, M_{sum} = M_1+M_2, H_{sum} = H_1+H_2. The final time is: H_{final} = H_{sum} + \lfloor \frac{M_{final}}{60} \rfloor, M_{final} = (M_{sum} + \lfloor \frac{S_{sum}}{60} \rfloor) \pmod{60}, S_{final} = S_{sum} \pmod{60} Add each column of units (seconds, minutes, hours) independently. Then, starting from the smallest unit (seconds), normalize the result by dividing by the base (60) and carrying the quotient to the next larger unit. Subtraction with Borrowing To calculate T_1 - T_2 where T_1 = H_1:M_1 and T_2 = H_2:M_2. If M_1 < M_2, we borrow: H_{final} = (H_1 - 1) - H_2, M_{final} = (M_1 + 60) - M_2. Subtract each column of units independently, starting from the smallest...