Heart Beat - Teacher Notes
Teachers' Notes for Heart Beat
Kate, Aminda and Amadej all had their 14th birthdays last week.
Kate guesses she has had about 20 million heart beats, so far.
Aminda thinks that estimate is too low, but Amadej thinks it is too high. What do you think?
A: GUESS who is right! My guess is........................
B: Now do some measuring - how many times does your heart beat in a minute?
You need a timer and a pulse...
My heart beat at the rate of ............... beats per minute
The lowest pulse rate in class was.............
The highest pulse rate in class was...............
C: Now do an exact calculation - how many minutes are there in 14 years?
D: Now do two plausible estimations
what is the low estimate of the number of heartbeats in 14 years?
what is the high estimate of the number of heartbeats in 14 years?
E: which of Kate, Aminda and Amadej is right?
THROUGHOUT - AVOID SPURIOUS ACCURACY!
Heart Beat [about 40 mins]
Class Organisation
We recommend that pupils work in small groups so they can share ideas about the task, and can check their calculations.
Lesson Activities
A: Guessing: As a whole class activity, ask pupils to guess, and take a vote. [5 minutes]
B: Measuring: Manage the timings for taking pulses, and get pupils to record their results. [5 minutes]
Some teachers get everyone to take their pulse for 2 sessions of 30 seconds, and add them together.
Finding a pulse might not be easy for everyone - but you only need a sample of pulses.
You can tell pupils when to start and stop counting.
Ask around the class for the highest and lowest values. (ask about any values below 50 and above 110. This is an opportunity to talk about 'cleaning the data').
An alternative to measuring is to think of starting points for average heart rates which your class all agree are silly - maybe 20 beats per minute and 200 beats per minute. Everyone raises their hand to agree it is silly. Then move up 25, 30, 35, 40, ..... until you get to a lower value where the class is not so sure it is silly (say about half of the pupils have their hands down). Then do the same for the upper limit, starting at 200 and working down.
C: Exact calculation: encourage pupils to document the time periods they calculate - working out the number of minutes for an hour, then a day, then a year and then 14 years is a sensible approach, but they need to be clear where they are at each stage.
[10 minutes]
D: Plausible estimations:
This activity produces a bounded estimate.
[10 minutes]
E: Your concluding discussion
Key Ideas about Plausible Estimation
- The value of learning to make plausible estimations
- The difference between a guess, an exact calculation, and a plausible estimate
- Key ideas about making plausible estimations
- A guess is just that - and is pretty useless for anything
- The number of minutes in 14 years is an exact figure (perhaps with a [negligible] error associated with leap years - some pupils might multiply beats per hour by 365.25 not 365 to calculate beats per year)
- Counting beats per minute is part of the process of estimation. Making realistic assumptions leads to a plausible estimate
- Calculating the answer with a low estimate of pulse rate and a high estimate of pulse rate produces a bounded estimate.
[10 minutes]
The Heart Beat problem
The average heart rate for a person over 14 years cannot be known exactly because the heart reacts to the needs of the body.
Different people have different heart rates.
It varies with age - see below for some guidelines.
BUT it is still possible to answer the question set with certainty, because none of these effects is big enough to change the answer to the question.
measurement accuracy depends on the purpose of the measurement and on how accurately you know other information. Pupils should be encouraged to think about what accuracy is reasonable when they have had to estimate the heart beats [2 significant figures is probably the most appropriate]
if you can't know the average heart rate exactly then you should not worry about things like the extra days for leap years, how many days ago were their birthdays etc. because these are small effects compared with the estimation problem.
Calculations:
| Rate per minute | per day | per year | in 14 years | sensible accuracy (not more than 2 sf) |
| 80 | 115,200 | 42,048,000 | 588,672,000 | about 600 million |
| 50 | 72,000 | 26,280,000 | 367,920,000 | 370 million |
| 55 | 79,200 | 28,908,000 | 404,712,000 | 400 million |
| 60 | 86,400 | 31,536,000 | 441,504,000 | 440 million |
| 65 | 93,600 | 34,164,000 | 478,296,000 | 480 million |
| 70 | 100,800 | 36,792,000 | 515,088,000 | 520 million |
| 75 | 108,000 | 39,420,000 | 551,880,000 | 550 million |
| 80 | 115,200 | 42,048,000 | 588,672,000 | 590 million |
| 85 | 122,400 | 44,676,000 | 625,464,000 | 630 million |
| 90 | 129,600 | 47,304,000 | 662,256,000 | 660 million |
| 95 | 136,800 | 49,932,000 | 699,048,000 | 700 million |
| 100 | 144,000 | 52,560,000 | 735,840,000 | 740 million |
| 105 | 151,200 | 55,188,000 | 772,632,000 | 770 million |
| 110 | 158,400 | 57,816,000 | 809,424,000 | 810 million |
The pupils will often want to give the exact answer if they use a calculator to work out what the total is by x 60, x 24, x 365, x14 and this is a key opportunity for discussing what accuracy is reasonable: if you say 588 million and 672 thousand then you get much less sense of the size of the number than if you say 600 million [or 590 million] - the detail gets in the way.
Information on heart rates:
http://www.steadyhealth.com/averege_human_heart_rate_t58106.html on May 30th 2007 had the following:
Pupil pulse rate at rest will vary between 60 - 110 beats per minute
Adult average heart beat rates are lower. Pulse depends on age; the average heart beat rate for children of 1-3 years is 130, average for 4-8 years old is 100, and 9-11 is 88. Average heart beat rate for persons of 12-16 is 80, and above 16 is about 70 heartbeats per minute.
