Units of Time and Elapsed Time: Fifth Grade Mathematics

Time measurement calculations require managing mixed-base conversion ratios (60 seconds, 60 minutes, 24 hours, 7 days) and finding elapsed time across clock intervals.

Time Units and Non-Base-Ten Conversion Factors

Time does not follow the base-ten metric system. Conversion ratios are based on astronomical and historical cycles:

+--------------------------------------------------------------------------+
|                  TIME EQUIVALENCIES REFERENCE                            |
+---------------------+----------------------------+-----------------------+
| Smaller Unit        | Equivalence                | Larger Unit           |
+---------------------+----------------------------+-----------------------+
| Seconds (sec)       | 60 seconds                 | = 1 minute (min)      |
| Minutes (min)       | 60 minutes                 | = 1 hour (hr)         |
| Hours (hr)          | 24 hours                   | = 1 day (d)           |
| Days (d)            | 7 days                     | = 1 week (wk)         |
| Months (mo)         | 12 months                  | = 1 year (yr)         |
| Days (d)            | 365 days (366 leap year)   | = 1 year (yr)         |
| Weeks (wk)          | 52 weeks                   | = 1 year (yr)         |
+---------------------+----------------------------+-----------------------+

Regrouping with 60s Instead of 10s

A critical error when adding or subtracting time is regrouping by 10 or 100. When you carry or borrow across hours and minutes, 1 hour equals 60 minutes, NOT 100 minutes!
- Subtracting: 4 hr 15 min - 1 hr 45 min.

- Borrow 1 hour from 4 hr (becomes 3 hr).

- Add 60 minutes to 15 min: 15 + 60 = 75 minutes.

- Subtract: 3 hr 75 min - 1 hr 45 min = 2 hr 30 min.

Finding Elapsed Time Using the Mountains, Hills, and Rocks Strategy

Elapsed time is the duration between a start time and an end time. A popular visual method uses:
- Mountains = 1-hour jumps

- Hills = 5-minute or 10-minute jumps

- Rocks = 1-minute jumps

Finding Elapsed Time from 8:45 AM to 11:20 AM:
  Jump 1 (Mountain): 8:45 AM to 9:45 AM (1 hour)
  Jump 2 (Mountain): 9:45 AM to 10:45 AM (1 hour)
  Jump 3 (Hill):     10:45 AM to 11:00 AM (15 minutes)
  Jump 4 (Hill):     11:00 AM to 11:20 AM (20 minutes)

  Total Elapsed Time: 1 hr + 1 hr + 15 min + 20 min = 2 hours 35 minutes.

Exercises and Solutions

Exercise 1: Time Regrouping Calculation. Add 3 hours 48 minutes and 2 hours 35 minutes. Show the regrouping of excess minutes into hours.

Solution 1: Add the hours and minutes separately: Hours: 3 + 2 = 5 hours. Minutes: 48 + 35 = 83 minutes. Notice that 83 minutes exceeds 60 minutes. Divide 83 by 60: 83 / 60 = 1 hour with a remainder of 23 minutes. Add the 1 hour to the hours total: 5 + 1 = 6 hours. Combine: 6 hours 23 minutes.

Exercise 2: Flight Elapsed Time Word Problem. A cross-country airplane flight departs Denver at 9:35 AM and lands in Atlanta at 2:15 PM (all times in the same time zone). What was the total flight duration?

Solution 2: Calculate the elapsed time from 9:35 AM to 2:15 PM. Jump 4 hours from 9:35 AM to 1:35 PM (4 hours). Jump 25 minutes from 1:35 PM to 2:00 PM (25 minutes). Jump 15 minutes from 2:00 PM to 2:15 PM (15 minutes). Sum the increments: 4 hours + 25 minutes + 15 minutes = 4 hours 40 minutes. The flight duration was 4 hours 40 minutes.