This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.
use understanding of place value to estimate an answer
Number Framework Links
Use this activity to help your students to consolidate and apply advanced multiplicative part–whole strategies (stage 7) and knowledge in the domain of multiplication and division.
In this activity, students use different units of time to measure age. They explore the relationships between units such as seconds, minutes, hours, years, centuries, and millennia. The calculations required in the activity involve large numbers, so the use of calculators is appropriate. Recognising and working with the relationships between the units involves multiplicative thinking that may be quite difficult for some students, so they may need extra help from you.
In their calculations, the students will need to use their knowledge of the number of seconds in a minute, the number of minutes in an hour, the number of hours in a day, and the number of days in a year.
What other units can I use? (Centuries, years, months, weeks, minutes, seconds, milliseconds) You may need to define these terms:
• Unit: any standard amount that we use for measuring how much or how little we have of something. (Illustrate this with some examples.) For more on units, see the Measurement section of Book 9 in the Numeracy Project series. • Leap year: a year in which 29 February is an extra day, so there are 366 days instead of 365. (Note that, for this activity, the students are instructed to ignore leap years in their calculations, although the investigation challenges them to find out why leap years are necessary.)
• Millennium: a thousand years.
Divide a guided teaching group into smaller groups of 3–4 students and say: Brainstorm as many units of time as you can with your group. Write each one onto a small piece of paper. Now put your pieces of paper into order from the biggest unit of time to the smallest. (Possible units are: nanosecond [1 thousand millionth of a second], millisecond [1 thousandth of a second], second, minute, hour, day, week,
month, year, decade, century, and millennium.)
While the students are in their small groups, ask the questions in the activity orally, referring to the pictures and speech bubbles in the students’ books.
The calculator has an important role in this activity because it allows the students to focus on the relationships between the units of time rather than on the actual calculations. Estimation and number sense is still vital when using a calculator because it is very easy to make errors entering the numbers. Encourage the students to “estimate in reverse” by looking at the result they have on the calculator and
then judging whether or not it is a sensible answer to the problem they are solving.
In order to develop estimation skills, students need to develop some benchmarks to judge their estimates against. Useful benchmarks for question 1 are the numbers of days, hours, minutes, and seconds there are in 1 year. For example, the students can work out that there are 365 x 24 = 8 760 hours in a year, which is about 9 000. If they use 9 000 as a benchmark for the number of hours in a year, they are better able to estimate how many hours old a 10-year-old would be, as well as to identify which of the students in the activity might be measuring their age using hours.
The students will also need to understand the proportional nature of units. The smaller the units used to measure a given amount, the greater the number required:
After the students have completed the activity, you can promote algebraic thinking by posing questions that focus on how the units are related. The group can then talk about these in a think-pair-share discussion. Possible questions and student ideas (using calculators) are:
If you know how many hours old someone is, what would you do to work out how many seconds old they are?
“I’d multiply the number of hours by 60 (to get minutes) and then by 60 again (to get seconds).”
“I’d multiply by 3 600 because that’s how many seconds there are in an hour. I know there’s 3 600 seconds in an hour because I worked out 60 lots of 60 seconds.”
If you know how many hours old a person is, how would you work out how many centuries old they are?
“I’d divide the number of hours by the number of hours in a year (8 760) to find out how old they are in years and then divide that by 100 (the number of years in a century). I know there’s 8 760 hours in a year because I worked out 24 hours in a day x 365 days.”
If the students are having difficulty, help them build an understanding of the relationships between the units by solving simpler problems first. Diagrams such as those above should help.
Possible questions and student ideas (using calculators) are:
How would you work out how many seconds there are in an hour?
“I know there are 60 seconds in 1 minute and 60 minutes in 1 hour, so I would work out 60 lots of 60 seconds. 60 x 60 = 3 600. There are 3 600 seconds in an hour.”
How could you use that number (3 600) to help you work out how many seconds there are in a week?
“I know there are 24 hours in a day, so there are 24 lots of 3 600 seconds in a day.
24 x 3 600 = 86 400. There are 7 days in a week, so that’s 7 lots of 86 400 seconds. 7 x 86 400 = 604 800 seconds.”
Can you work out how many hours there are in a year?
“I know there are 365 days in 1 year. So I multiplied the number of days in the whole year (365) by the number of hours in 1 day (24). 365 lots of 24 hours = 365 x 24
= 8 760 hours.”
How could you use that number (8 760) to help you work out how many minutes and seconds there are in a year?
“I’d multiply the number of hours in a whole year (8 760) by the number of minutes in 1 hour (60). 8 760 lots of 60 minutes is 8 760 x 60 = 525 600 minutes.”
“I’d multiply the number of minutes in a year (525 600) by the number of seconds in 1 minute (60). 525 600 lots of 60 seconds is 525 600 x 60 = 31 536 000 seconds.”
If you know how many hours there are in 1 year, how could you work out how many hours old a 5-year-old would be?
“I’d work out 5 lots of the number of hours in a year (8 760). 5 x 8760 = 43 800 hours.”
If you know how many hours old a 5-year-old is, how could you use that number to work out how many seconds old he or she is?
“I’d work out 3 600 lots of 43 800, which is the number of seconds in 1 hour multiplied by the 5-year-old’s age in hours. (I know the number of seconds in 1 hour because I went 60 minutes x 60 seconds.) 3 600 x 43 800 = 157 680 000 seconds”
“I’d multiply the number of hours (43 800) by 60 (minutes in each hour) and multiply it again by 60 (seconds in each minute), which would make the 5-year-old 157 680 000 seconds old.”
Possible investigations include:
• How many seconds do we spend at school each year?
• Find out the average life expectancy of a person like yourself living in New Zealand (with the same birth year, gender, and ethnicity). Based on your daily routine now, how long will you spend over your entire life doing activities such as sleeping, watching television, reading, travelling, talking on the phone, playing sport, doing chores, and doing other activities?
• How do scientists measure very short or very long periods of time?
Answers to Activity
1. a. i. Days
b. i. 11 yrs old. 4 200 ÷ 365 = 11.5
ii. 11 yrs old. 96 360 ÷ 365 ÷ 24 = 11
iii. 10 yrs old.
5 523 840 ÷ 365 ÷ 24 ÷ 60 = 10.5
iv. 10 yrs old. 315 532 800 ÷ 365 ÷ 24 ÷ 60 ÷ 60 = 10
2. Answers will vary depending on your age. For example, if you are 10 , your answers will be:
a. 10 yrs
b. 3 833 days
c. 91 980 hrs
d. 5 518 800 min
e. 331 128 000 s
3. a. Centuries. A century is 100 yrs. 0.1 is . of 100 is 10.
b. 0.01. (10 is of 1 000)
4. Answers will vary. To find out your age in centuries, you need to divide your age in yrs by 100. To find out your age in millennia, divide your age in yrs by 1 000. For example:
• 9 yrs old is 0.09 centuries or 0.009 millennia
• 9 yrs old is 0.095 centuries or 0.0095 millennia
• 10 yrs old is the same as James (0.1 centuries or 0.01 millennia)
• 10 yrs old is 0.105 centuries or 0.0105 millennia
• 11 yrs old is 0.11 centuries or 0.011 millennia.
The unit of a year comes from how long the earth takes to circle the sun (approximately 365.25 days). We have leap years because each year is close to 365.25 days long, and we need to catch up on the 0.25 days every 4 years