Time and Temperature. Grade 7 Mathematics: Time and Temperature Subtopic Navigator Understanding Time and Temperature Reading and Interpreting Time Time Conversions and Calculations Elapsed Time Problems Reading Temperature Scales Temperature Conversions Temperature Changes and Differences Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Read and interpret analog and digital time displays accurately Convert between different units of time (seconds, minutes, hours, days) Calculate elapsed time for complex scenarios Read and interpret Celsius and Fahrenheit temperature scales Convert temperatures between Celsius and Fahrenheit Calculate temperature changes and differences Apply time and temperature concepts to real-world problems Time and Temperature Time and temperature are fundamental measurements in our daily lives. Time helps us organize our activities and understand duration, while temperature measures thermal energy and affects everything from weather to cooking. Mastery of reading, interpreting, and calculating with these measurements is essential for scientific understanding and practical problem-solving. Reading and Interpreting Time Time can be displayed in analog format (clock face with hands) or digital format (numerical display). Understanding both 12-hour and 24-hour formats is important. The 12-hour format uses AM (ante meridiem, before noon) and PM (post meridiem, after noon), while the 24-hour format runs from 00:00 to 23:59. Example 1: Complex Time Reading Convert these times between 12-hour and 24-hour formats: a) 14:45 to 12-hour format b) 8:15 PM to 24-hour format c) 23:30 to 12-hour format d) 11:45 AM to 24-hour format Solution: a) 14:45 = 2:45 PM (subtract 12 from hours > 12) b) 8:15 PM = 20:15 (add 12 to PM times, except 12 PM) c) 23:30 = 11:30 PM d) 11:45 AM = 11:45 (AM times stay the same in 24-hour, except 12 AM becomes 00:00) Example 2: Reading Analog Clocks An analog clock shows the hour hand at 3 and the minute hand at 9. What time is it? If the hour hand is exactly halfway between 3 and 4, and the minute hand is at 6, what time is it? Solution: First clock: Hour hand at 3, minute hand at 9 (45 minutes) Time = 3:45 or 15:45 Second clock: Hour hand halfway between 3 and 4 = 3:30 position Minute hand at 6 = 30 minutes Time = 3:30 or 15:30 Note: When minute hand moves, hour hand also moves proportionally: At 3:30, hour hand is exactly halfway between 3 and 4 Time Reading Problems Convert to 12-hour format: 16:30, 08:45, 00:15, 23:59 Convert to 24-hour format: 3:45 PM, 11:30 AM, 12:00 AM, 12:00 PM If hour hand is at 4 and minute hand is at 8, what time is it? What time is shown when hour hand is at 2 and minute hand is at 11? Express 17:23 in 12-hour format with AM/PM Time Conversions and Calculations Time units convert using specific relationships: 1 minute = 60 seconds, 1 hour = 60 minutes, 1 day = 24 hours. When performing calculations with time, we must be careful with base-60 arithmetic rather than base-10. Example 1: Complex Time Conversions Convert 2 days, 5 hours, 45 minutes, and 30 seconds to: a) Total hours b) Total minutes c) Total seconds Solution: a) Total hours: 2 days = 48 hours 5 hours = 5 hours 45 minutes = 45 ÷ 60 = 0.75 hours 30 seconds = 30 ÷ 3600 = 0.00833 hours Total = 48 + 5 + 0.75 + 0.00833 = 53.75833 hours b) Total minutes: 2 days = 2 × 24 × 60 = 2,880 minutes 5 hours = 5 × 60 = 300 minutes 45 minutes = 45 minutes 30 seconds = 30 ÷ 60 = 0.5 minutes Total = 2,880 + 300 + 45 + 0.5 = 3,225.5 minutes c) Total seconds: 2 days = 2 × 24 × 60 × 60 = 172,800 seconds 5 hours = 5 × 60 × 60 = 18,000 seconds 45 minutes = 45 × 60 = 2,700 seconds 30 seconds = 30 seconds Total = 172,800 + 18,000 + 2,700 + 30 = 193,530 seconds Example 2: Time Arithmetic Calculate: 3 hours 45 minutes 30 seconds + 2 hours 30 minutes 45 seconds Express answer in hours, minutes, and seconds. Solution: Add seconds: 30 + 45 = 75 seconds = 1 minute 15 seconds Add minutes: 45 + 30 + 1 = 76 minutes = 1 hour 16 minutes Add hours: 3 + 2 + 1 = 6 hours Total: 6 hours 16 minutes 15 seconds Alternative method: Convert all to seconds: First: (3×3600) + (45×60) + 30 = 10,800 + 2,700 + 30 = 13,530 seconds Second: (2×3600) + (30×60) + 45 = 7,200 + 1,800 + 45 = 9,045 seconds Sum: 13,530 + 9,045 = 22,575 seconds Convert back: 22,575 ÷ 3600 = 6.27083 hours = 6 hours Remainder: 0.27083 × 60 = 16.25 minutes = 16 minutes Remainder: 0.25 × 60 = 15 seconds Time Conversion Problems Convert 3.75 hours to hours and minutes How many seconds are in 2 days, 3 hours, and 15 minutes? Calculate: 4h 25m 30s + 2h 45m 50s Convert 10,000 seconds to hours, minutes, and seconds What fraction of a day is 6 hours 45 minutes? Elapsed Time Problems Elapsed time is the amount of time that passes between a start time and an end time. Calculating elapsed time requires careful attention to crossing hour boundaries, AM/PM changes, and day boundaries. Example 1: Complex Elapsed Time A train departs at 8:45 AM and arrives at 3:30 PM. How long is the journey? If there's a 45-minute stop during the journey, what is the actual travel time? Solution: From 8:45 AM to 12:00 PM (noon): 8:45 to 9:00 = 15 minutes 9:00 to 12:00 = 3 hours = 180 minutes Total to noon: 15 + 180 = 195 minutes = 3 hours 15 minutes From 12:00 PM to 3:30 PM: 3 hours 30 minutes Total journey: 3h 15m + 3h 30m = 6h 45m With 45-minute stop: Actual travel time = 6h 45m - 45m = 6 hours Example 2: Elapsed Time Across Midnight A movie starts at 10:45 PM and ends at 1:15 AM. How long is the movie? Solution: From 10:45 PM to 12:00 AM (midnight): 10:45 to 11:00 = 15 minutes 11:00 to 12:00 = 1 hour = 60 minutes Total to midnight: 15 + 60 = 75 minutes = 1 hour 15 minutes From 12:00 AM to 1:15 AM: 1 hour 15 minutes Total: 1h 15m + 1h 15m = 2 hours 30 minutes Elapsed Time Problems From 9:15 AM to 2:45 PM, how much time elapses? A flight departs at 11:30 PM and arrives at 6:45 AM. How long is the flight? If a class starts at 1:30 PM and lasts 2 hours 45 minutes, when does it end? From 8:15 AM to 5:30 PM with 1 hour lunch break, what is working time? A process takes 3h 25m to complete. If it starts at 14:45, when does it finish? Reading Temperature Scales Temperature is measured using different scales, primarily Celsius (°C) and Fahrenheit (°F). Celsius is based on water's freezing point (0°C) and boiling point (100°C). Fahrenheit uses 32°F for freezing and 212°F for boiling. Understanding both scales is important for scientific and everyday applications. Example 1: Interpreting Temperature Readings Interpret these temperatures in practical terms: a) 25°C b) 98.6°F c) -10°C d) 212°F Solution: a) 25°C = Room temperature, pleasant day b) 98.6°F = Normal human body temperature (37°C) c) -10°C = Cold winter day, freezing temperature d) 212°F = Water boiling point at sea level (100°C) Example 2: Temperature Comparisons Arrange these temperatures from coldest to warmest: 15°C, 59°F, 10°C, 68°F, 20°C Solution: Convert all to Celsius for comparison: 59°F = (59 - 32) × 5/9 = 15°C 68°F = (68 - 32) × 5/9 = 20°C Now we have: 15°C, 15°C, 10°C, 20°C, 20°C From coldest to warmest: 10°C, 15°C (twice), 20°C (twice) Original order: 10°C, 15°C and 59°F (both 15°C), 20°C and 68°F (both 20°C) Temperature Reading Problems Which is warmer: 25°C or 77°F? At what temperature does water freeze in Fahrenheit? Normal room temperature is about 20-25°C. What is this in Fahrenheit? Arrange from coldest to warmest: 0°C, 32°F, 10°C, 50°F A fever is 38°C. What is this in Fahrenheit? Temperature Conversions To convert Celsius to Fahrenheit: °F = (°C × 9/5) + 32 To convert Fahrenheit to Celsius: °C = (°F - 32) × 5/9 These formulas are exact and must be applied carefully, especially with negative temperatures. Example 1: Complex Temperature Conversions Convert these temperatures: a) 25°C to Fahrenheit b) 95°F to Celsius c) -40°C to Fahrenheit d) 0°F to Celsius Solution: a) °F = (25 × 9/5) + 32 = (45) + 32 = 77°F b) °C = (95 - 32) × 5/9 = 63 × 5/9 = 35°C c) °F = (-40 × 9/5) + 32 = (-72) + 32 = -40°F (Interesting: -40°C = -40°F) d) °C = (0 - 32) × 5/9 = -32 × 5/9 = -17.78°C (approximately) Example 2: Precision in Conversions Convert 98.6°F to Celsius with two decimal places. Convert 37.5°C to Fahrenheit with one decimal place. Solution: 98.6°F to Celsius: °C = (98.6 - 32) × 5/9 = 66.6 × 5/9 = 333/9 = 37°C exactly Actually: 66.6 × 5 = 333, 333 ÷ 9 = 37.0°C 37.5°C to Fahrenheit: °F = (37.5 × 9/5) + 32 = (67.5) + 32 = 99.5°F Temperature Conversion Problems Convert 100°C to Fahrenheit (water boiling point) Convert 32°F to Celsius What is 20°C in Fahrenheit? Convert -20°C to Fahrenheit At what temperature do Celsius and Fahrenheit have the same numerical value? Temperature Changes /Differences Temperature changes can be calculated by finding the difference between initial and final temperatures. When temperatures are in different scales, they must be converted to the same scale before finding differences. Temperature change in Celsius is numerically different from the same change in Fahrenheit. Example 1: Complex Temperature Change The temperature rises from 15°C to 25°C. What is the temperature change in: a) Celsius b) Fahrenheit Solution: a) Change in Celsius = 25°C - 15°C = 10°C b) Convert initial and final to Fahrenheit: 15°C = (15 × 9/5) + 32 = 27 + 32 = 59°F 25°C = (25 × 9/5) + 32 = 45 + 32 = 77°F Change in Fahrenheit = 77°F - 59°F = 18°F Note: A 10°C change equals an 18°F change (ratio 9:5) Example 2: Mixed Scale Temperature Differences Temperature A is 68°F and Temperature B is 20°C. Which is warmer and by how much in Celsius? Solution: Convert 68°F to Celsius: °C = (68 - 32) × 5/9 = 36 × 5/9 = 20°C Both are 20°C, so they are equal. Difference = 0°C In Fahrenheit: Both are 68°F, difference = 0°F Temperature Change Problems Temperature changes from 10°C to 25°C. What is the change in Celsius and Fahrenheit? Morning temperature is 5°C, afternoon is 68°F. Did temperature increase? By how much in °C? A 15°F temperature change equals how many degrees Celsius change? If temperature drops from 86°F to 50°F, what is the change in Celsius? Which represents a larger change: 10°C or 20°F? Real-World Applications Time and temperature calculations are essential in everyday life for scheduling, cooking, weather forecasting, travel planning, and scientific experiments. These applications demonstrate the practical importance of mastering these measurement skills. Example 1: Cooking Time Application A turkey needs to cook for 15 minutes per kg plus 30 minutes extra. If the turkey is 4.5 kg, a) How long should it cook? b) If it goes in the oven at 2:45 PM, when should it be checked? c) The recipe says to cook at 180°C. What is this in Fahrenheit? Solution: a) Cooking time = (15 × 4.5) + 30 = 67.5 + 30 = 97.5 minutes = 1 hour 37.5 minutes = 1 hour 37 minutes 30 seconds b) Start: 2:45 PM Add 1 hour: 3:45 PM Add 37 minutes: 4:22 PM Add 30 seconds: 4:22:30 PM Should be checked around 4:22 PM c) 180°C to Fahrenheit: °F = (180 × 9/5) + 32 = 324 + 32 = 356°F Example 2: Weather and Travel Application A flight departs at 8:45 AM from a city at 25°C and arrives at 2:30 PM at a destination at 15°C. a) What is the flight duration? b) What is the temperature difference in Celsius and Fahrenheit? c) If you need to be at the airport 2 hours before departure, when should you arrive? Solution: a) Flight duration: 8:45 AM to 2:30 PM 8:45 to 12:00 = 3 hours 15 minutes 12:00 to 2:30 = 2 hours 30 minutes Total = 5 hours 45 minutes b) Temperature difference: In Celsius: 25°C - 15°C = 10°C drop In Fahrenheit: Convert both: 25°C = 77°F, 15°C = 59°F Difference = 77°F - 59°F = 18°F drop c) Airport arrival: 2 hours before 8:45 AM = 6:45 AM Real-World Application Problems A cake bakes for 45 minutes at 350°F. If started at 3:30 PM, when is it done? What's 350°F in Celsius? A meeting starts at 1:45 PM and lasts 2h 15m. When does it end in 24-hour format? Morning temp: 12°C, rises by 18°F by afternoon. What is afternoon temp in °C? A train journey takes 3h 45m. If departure is 14:30, what is arrival time? Water boils at 100°C. What is this in Fahrenheit? At high altitude, it boils at 95°C. What's the difference in °F? Cumulative Exercises Convert 16:45 to 12-hour format with AM/PM Calculate elapsed time from 9:15 AM to 3:45 PM Convert 3.25 hours to hours and minutes Convert 25°C to Fahrenheit Convert 95°F to Celsius Temperature changes from 10°C to 25°C. What is the change in Fahrenheit? A process takes 2h 45m 30s. If it starts at 14:30, when does it finish? How many seconds are in 1 day, 5 hours, and 30 minutes? Which is warmer: 20°C or 68°F? Show your work A flight departs at 23:45 and arrives at 06:15. How long is the flight? Show/Hide Answers Problem: Convert 16:45 to 12-hour format with AM/PM Answer: 16:45 = 4:45 PM (16 - 12 = 4) Problem: Calculate elapsed time from 9:15 AM to 3:45 PM Answer: From 9:15 AM to 12:00 PM: 2 hours 45 minutes From 12:00 PM to 3:45 PM: 3 hours 45 minutes Total: 2h 45m + 3h 45m = 6 hours 30 minutes Problem: Convert 3.25 hours to hours and minutes Answer: 3 hours + (0.25 × 60) = 3 hours 15 minutes Problem: Convert 25°C to Fahrenheit Answer: °F = (25 × 9/5) + 32 = 45 + 32 = 77°F Problem: Convert 95°F to Celsius Answer: °C = (95 - 32) × 5/9 = 63 × 5/9 = 35°C Problem: Temperature changes from 10°C to 25°C. What is the change in Fahrenheit? Answer: Change in Celsius = 15°C Change in Fahrenheit = 15 × 9/5 = 27°F Or: 10°C = 50°F, 25°C = 77°F, difference = 27°F Problem: A process takes 2h 45m 30s. If it starts at 14:30, when does it finish? Answer: 14:30 + 2h = 16:30 16:30 + 45m = 17:15 17:15 + 30s = 17:15:30 or 5:15:30 PM Problem: How many seconds are in 1 day, 5 hours, and 30 minutes? Answer: 1 day = 86,400 seconds 5 hours = 5 × 3600 = 18,000 seconds 30 minutes = 30 × 60 = 1,800 seconds Total = 86,400 + 18,000 + 1,800 = 106,200 seconds Problem: Which is warmer: 20°C or 68°F? Show your work Answer: Convert 68°F to Celsius: °C = (68 - 32) × 5/9 = 36 × 5/9 = 20°C They are equal: 20°C = 68°F Problem: A flight departs at 23:45 and arrives at 06:15. How long is the flight? Answer: From 23:45 to 00:00: 15 minutes From 00:00 to 06:15: 6 hours 15 minutes Total: 15m + 6h 15m = 6 hours 30 minutes Conclusion/Recap Time and temperature are essential measurements that permeate every aspect of our lives. Mastering the reading, interpretation, and calculation of time in various formats, along with understanding and converting between temperature scales, provides crucial skills for scientific inquiry, daily planning, and practical problem-solving. These skills enable us to navigate schedules, understand weather patterns, follow recipes accurately, and make informed decisions based on temporal and thermal information. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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