Section 4 (Four Credits)

Sizing Things Up – Measurement


Measurement is the most expensive of all mathematics. Mistakes are often made with measurement in the building industry which require more time and more material to repair or fix. There is a fundamental truth in measurement as well! “Measure twice, cut once.”

The history of measurement is a fascinating study. As civilisations became more agricultural, more technical and more trade based so the need to quantify length, weight, capacity and time  became more and more essential. The History of the Metric System.

Time – Unit is 1 second

It became pretty obvious from very early times that time cycled and involved the Sun, the Moon and the Stars. The four seasons became a year. Human culture and religions embraced the seasons and remain so today. The Harvest Festival for example happens around the end of summer and autumn every year, or harvest time.  

A solar year is the time it takes the Earth to complete its orbit around the Sun — about one year. But the actual time it takes for the Earth to travel around the Sun is in fact a little longer than that—about 365 ¼ days (365 days, 5 hours, 48 minutes, and 46 seconds, to be precise).


Time becomes a study all on its own. How to measure small time intervals was a problem solved first by Galileo in the 1600’s and is now measured with incredible accuracy using atomic vibrations. The Global Positioning System in use today measures time to 14 nanoseconds or 14 billionths of a second.



Length – Unit is 1 metre

Original measures of length include an inch, a fathom, a cubit and a chain. unfamiliar names now but they all made sense at the time of use.


Mass – Unit is 1 kilogram

The term mass is correct rather than the common term called weight. The value of gravity on Earth is the connection between Weight and mass where w = mg. Mass is the amount of matter in something. Like length there were some strange units in early days.


Capacity – 1 litre

This is the term for liquid measure and is distinct from volume.


There are also units for temperature, charge, light, sound, radiation, magnetism. The system we now us is the SI or System International. The US does not use this system.


Task 1

Find twenty unusual measures not in common and everyday use.


Derived Measures

These compare two measures and there are many in common usage.


When traveling on the road speed is measured as a rate and expressed in kph or kilometres per hour. In one hour this is how far the car travels. The speed limit on most NZ roads is 100km/hr. Notice the different way of writing the unit for speed. A rate compares by division two measures that are not the same. A ratio compares in the same way two measures that are the same.


Task 2

Name three other derived units in common use.


Measurements need Axiom 1 and Axiom 2.

Axiom 1 is “One can be anything I choose one to be”.

Axiom 2 is “When combining or comparing in mathematics we do so using the same size units.”


The UNITS of measurement we choose are a direct result of Axiom 1. We can use anything but it is wise we all use the same unit!


Axiom 2 allows combining and comparing of these units so that we get sensible answers at all times. Adding half a small pie to half a big pie does not make 1 pie of either size! We must use the same size pie to get a sensible addition.


Task 3

(a)   Add 1mm, 1cm and 1m

(b)   How much bigger or smaller is 1 pound compared to 1 kg?

(c)   How old are you in seconds?


Converting between Units of Measure

These questions can be typed directly into most search engines such as Google.


It is useful to be able to convert common units.


Task 4

(a)   Convert 680 seconds to minutes

(b)   Convert 12345m to kilometres

(c)   Convert 1 day to seconds

(d)   300 degrees Fahrenheit is how many degrees Celsius?

(e)   1 cup is how many millilitres?


A more detailed explanation for conversions is available on the COVID FILES #5 and #6.

Tools of Measurement

Common tools used for measuring units and derived units.

A protractor used to measure angles.

Task 5

Complete the table










Lux meter








Tape Measure


There is a job called a tool maker who specialises in making tools for measurement.


Accuracy and Precision

Quite quickly when measuring it becomes apparent that one person’s answer might not be what another person expects. This is heavily connected to number, rounding and eyesight.


Task 6

Using a ruler or tape measure have someone measure the height of a door. Now repeat the process yourself. How close are the measures?


Task 7

Measure the height of someone or yourself. What difficulties did you encounter?


Task 8

Throw three darts at the bull’s eye on a dartboard. What happened?


The darts are accurate if they are all near the bull’s eye.


The darts are precise if they are close to each other. They do not have to be near the bull’s eye!


Finally another Axiom that is almost never talked about.


“If you require a more precise and accurate measure then purchase a more expensive tool”


There is no point in trying to measure the size of an atom with a 50m tape measure!

Electronic scales now weigh to the nearest gram which is usually unnecessary precision when cooking.


Error in Measurement

A general guide is the error is ±half of the smallest division. This could also be an axiom but is more of a guide to using a ruler. 

I use the term "ruler" in a generic sense here meaning any measuring device. One of my Year 9 students given the task of making a "ruler" built and calibrated a bendy wire contraption that weighed the fishing flies he used and labelled them fast sinking, medium sinking and slow sinking according to their weight. Very clever and used.


    Task 9

        Here is a measurement in progress. Read the scales to find the mass of the person.


    The smallest division is 1kg so the answer should be either 52 kg or 53kg and both are correct.
[There could be a "who dunit problem here to try and figure more about the person. More fun!"


    A standard 50kg mass carefully made to be as close to 50kg as possible could be used to calibrate a set of scales such as those shown in the picture.


    It is doubtful cheap bathroom scales are very accurate at all! Spend a bit more money if accuracy is important!



Section 4 (Four Credits)

Sizing Things Up – Measurement


1. Measure the length of this line.


       2.     Find the weight of a typical grain of dried rice.

(a)   Describe your method or strategy

(b)  Complete the task.


3.     Use the stopwatch feature on a cell phone.

(a)   Shut your eyes for ten seconds. Repeat 10 times. Did you improve?

(b)  Shut your eyes for 1 minute. Repeat a few times. Did you improve?


4.     Locate a water jog and a cup. Estimate how many cups of water there are in the water jug. Check your answer!


5.     Draw a line 15cm long.


6.     Cut a block of butter that weighs 150gm.


7.     Mix hot and cold water to get a cup of water at 35C. Check with a thermometer.


8.     What is the density of milk?

(a)   Describe you strategy or method.

(b)  Complete the task.


9.     Find the mass of 1 litre of water.

(a)   Describe you strategy or method.

(b)  Complete the task



10.  The cold tap (15C) turned on full fills a bath in 30 minutes. The hot tap (60C) fills the empty bath in 1 hr. If both taps are turned on full…

(a)   How long does it take to get half a bath of water?

(b)  What temperature is the bath water after filling?



To gain 4 Credits for this section screenshot or photograph your answers or otherwise email a copy to for registering(first time), checking and getting feedback. This is intended to be a painless process and all questions are accepted. Once you have been awarded the 3 credits a fee of $5 will be requested for this section.


Well done, four sections over! On to Section 5! Is the course fun? Use the Navigator to find the next section.