Module 3: Investigating measurement and data handling

Section 1: Introducing measurement

1. Using mindmaps

Using a mind map will help you find out what your pupils already know about measurement in everyday life. This information will help you plan activities that will extend their understanding further. See Key Resource: Using mind maps and brainstorming to explore ideas  for ways to do this.

Resource 1: A measurement mind map shows a sample mind map of one group’s ideas.

2. Measuring heartbeats

Consider asking a science teacher to help you show your pupils how to measure their heartbeats or have a go yourself using Resource 3: Heart rate to help you. This is an excellent introduction to actual measurement, as it can be done without any instruments other than one watch with a second hand for the teacher or pupil to use. It can also be integrated with a number of enjoyable exercise and recovery activities, and provide a good basis for cross-curricular work, for example, a ‘healthy bodies’ week. Practical activities such as this will capture pupils’ attention and involve them.

Resource 3 tells you how to measure heart rate, and gives some further information about heart rate, age and exercise to help you with Activity 2.

3. Planning to teach for understanding

Understanding the importance of ‘units’ and the ability to read from scales are central to effective work in measurement. Later sections in this module will deal with length, weight and time: for each, it is important for pupils to understand the correct unit, and how to read correctly from the scales on measuring instruments. This part explores how you can plan activities to help pupils develop these skills. By using practical activities related to their everyday lives, pupils will see a purpose to the work and be more interested.

Resource 1: A measurement mind map

 Example of pupils' work

Measurements that are common to the five situations shown in the mind map below are:

• length, surface or solid size;
• weight;
• capacity;
• time.

Resource 2: How mind maps can help mathematics teachers and pupils

 Background information / subject knowledge for teacher

Mind maps can help mathematics teachers to:

• plan topics and lessons in a way that is logical and systematic;
• identify and plan activities;
• introduce new concepts to pupils in interesting ways;
• promote better understanding of concepts;
• focus pupils’ attention on key aspects of a topic;
• help pupils prepare for tests and examinations;
• organise information that pupils are able to identify and relate to;
• help find out pupils’ misconceptions;
• assess pupils’ understanding of concepts and topics;
• plan extra support for some pupils.

Because concept maps can be drawn by pupils themselves, they are a useful tool for pupil-centred education.

Mind maps can help pupils to:

• summarise their knowledge of a topic and give an overview;
• help order topics according to their importance and relevance;
• link new ideas with previously learned material;
• show prior knowledge of a topic;
• reduce doubt so that the pupil knows what to study and what to leave out.

Resource 3: Heart rate

 Background information / subject knowledge for teacher

Heart rate is a term used to describe the frequency of the cardiac cycle. It is considered one of the four vital signs. Usually it is calculated as the number of contractions (heartbeats) of the heart in one minute and expressed as ‘beats per minute’ (bpm).

When resting, the adult human heart beats at about 70 bpm (males) and 75 bpm (females), but this varies. However, the reference range is nominally between 60 bpm (if less, termed bradycardia) and 100 bpm (if greater, termed tachycardia). Resting heart rates can be significantly lower in athletes. The infant/neonatal rate of heartbeat is around 130–150 bpm, the toddler's about 100–130 bpm, the older child's about 90–110 bpm, and the adolescent's about 80–100 bpm.

The body can increase the heart rate in response to a wide variety of conditions in order to increase the cardiac output (the amount of blood ejected by the heart per unit time). Exercise, environmental stressors or psychological stress can cause the heart rate to increase above the resting rate.

Measuring heart rate

Measuring the pulse at the neck and wrist

The pulse rate (which in most people is identical to the heart rate) can be measured at any point on the body where an artery is close to the surface. Such places are wrist (radial artery), neck (carotid artery), elbow (brachial artery), and groin (femoral artery). The pulse can also be felt directly over the heart. (Remember, never use your thumb to measure your pulse rate, because thumbs have a pulse rate of their own.)

It is also possible to measure heart rate acoustically, by listening to the sounds the heart makes while beating. These sounds can be listened to using a stethoscope.

Resource 4: Units of measurement

 Teacher resource for planning or adapting to use with pupils

Units of time

• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 week = 7 days
• 1 fortnight = 2 weeks
• 1 year = 12 months = 52 weeks = 365 days

How do you think these traders at a market in Knysna measure their goods?

Resource 5: Space and measure

 Teacher resource for planning or adapting to use with pupils

Original source: http://www.bbc.co.uk (Accessed 2008)

Acknowledgements

Grateful acknowledgement is made to the following sources:

Other

Resource 3: Heart rate: Original source: Wikipedia (Accessed 2008)

Resource 5: Space and measure: Original source: http://www.bbc.co.uk (Accessed 2008)

Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.

Section 2: Measuring and handling time

1. Using group work to discuss ‘time’

A good introduction to telling the time is to first discuss the many ways people used to tell the time before the invention of clocks. You could ask your pupils how they think they might be able to tell the time today, without using clocks. Exploring these ideas first and listening to their answers will provide you with evidence of their current understanding. This will help you to judge how much they have learned after undertaking some activities about time.

2. Cross-curricula practical work

‘Some people can tell what time it is by looking at the sun. But I have never been able to make out the numbers’ (Attributed to a primary pupil).

You may find it helpful to work together with the history teacher to explore how time was measured in different cultures throughout history. This could become activity-based – your pupils will probably enjoy experimenting with some of these ancient methods of time telling, such as making a candle clock or sundial. It will show your pupils that mathematics is – and has always been – important in many areas of life and study.

Using other experts in your classroom will help you learn more about a subject and will motivate your pupils. The teacher in Case Study 2 takes this approach.

3. Using local resources to teach telling the time

There are several important facts pupils need to know about time (see Resource 4: Units of time), but one of the most challenging aspects for young children is often being able to ‘read’ a clock face. The use of practical ‘clock hands’ activities should help pupils to be able to read a clock and tell the time.

Once you have a clock or clocks, begin with times that are easier, gradually moving on to the more difficult times:

• ‘on the hour’ (o’clock);
• quarter past, half-past, quarter to the hour;
• five minute intervals;
• one minute intervals.

Case Study 3 and the Key Activity give examples of how you could do this.

Resource 1: Ways of measuring time long ago

 Teacher resource for planning or adapting to use with pupils

Resource 2: Water clocks – ways of measuring time throughout history

 Background information / subject knowledge for teacher

Water clocks were among the earliest timekeepers that didn't depend on the observation of celestial bodies. One of the oldest was found in the tomb of the Egyptian pharaoh Amenhotep I, buried around 1500 BCE. Later named clepsydras (‘water thieves’) by the Greeks, who began using them about 325 BCE, these were stone vessels with sloping sides that allowed water to drip at a nearly constant rate from a small hole near the bottom. Other clepsydras were cylindrical or bowl-shaped containers designed to slowly fill with water coming in at a constant rate. Markings on the inside surfaces measured the passage of ‘hours’ as the water level reached them. These clocks were used to determine hours at night, but may have been used in daylight as well. Another version consisted of a metal bowl with a hole in the bottom; when placed in a container of water the bowl would fill and sink in a certain time. These were still in use in North Africa in the 20th century.

More elaborate and impressive mechanised water clocks were developed between 100 BCE and 500 CE by Greek and Roman horologists and astronomers. The added complexity was aimed at making the flow more constant by regulating the pressure, and at providing fancier displays of the passage of time. Some water clocks rang bells and gongs; others opened doors and windows to show little figures of people, or moved pointers, dials and astrological models of the universe. In the Far East, mechanised astronomical/astrological clock making developed from 200 to 1300 CE. Third-century Chinese clepsydras drove various mechanisms that illustrated astronomical phenomena. One of the most elaborate clock towers was built by Su Sung and his associates in 1088 CE. Su Sung’s mechanism incorporated a water-driven escapement invented about 725 CE. The Su Sung clock tower, over 30 feet tall, possessed a bronze power-driven armillary sphere for observations, an automatically rotating celestial globe, and five front panels with doors that permitted the viewing of changing manikins which rang bells or gongs, and held tablets indicating the hour or other special times of the day. Since the rate of flow of water is very difficult to control accurately, a clock based on that flow could never achieve excellent accuracy. People were naturally led to other approaches.

Original source: http://inventors.about.com (Accessed 2008)

In Ctesibius's clepsydra from the 3rd century BC, the human-shaped hour pointer ascends as water flows in. Outflow drives a series of gears that rotate a cylinder bearing hour lengths appropriate for each day's date.

Resource 3: Sundials

 Background information / subject knowledge for teacher

The Egyptians formally divided their day into parts something like ours. Obelisks (slender, tapering, four-sided monuments) were built as early as 3500 BCE. Their moving shadows formed a kind of sundial, enabling people to partition the day into morning and afternoon. Obelisks also showed the year’s longest and shortest days when the shadow at noon was the shortest or longest of the year. Later, additional markers around the base of the monument would indicate further subdivisions of time.

Another Egyptian shadow clock or sundial, possibly the first portable timepiece, came into use around 1500 BCE. This device divided a sunlit day into ten parts plus two ‘twilight hours’ in the morning and evening. When the long stem, with five variably spaced marks, was oriented east and west in the morning, an elevated crossbar on the east end cast a moving shadow over the marks. At noon, the device was turned in the opposite direction to measure the afternoon ‘hours’.

In Europe, during most of the Middle Ages (roughly 500 CE to 1500 CE), technological advancement virtually ceased. Sundial styles evolved, but didn’t move far from ancient Egyptian principles.

Original source: http://inventors.about.com (Accessed 2008)

Resource 4: Units of time

 Teacher resource for planning or adapting to use with pupils

Time 1 minute = 60 seconds 1 hour = 60 minutes 1 day = 24 hours 1 week = 7 days 1 fortnight = 14 days 1 year = 12 months = 52 weeks = 365 days leap year = 366 days

Many timetables and digital watches use 24-hour clock time. Use this scale to change between 12-hour and 24-hour time.

6:15 am  → 06:15 6:15 pm  → 18:15 Remember: a.m. is morning time (it comes from the Latin ante meridium, meaning 'before midday'); p.m. is afternoon and evening time (it comes from the Latin post meridium, meaning 'after midday'); the 24-hour clock always uses four digits.

Here is a way to remember how many days there are in each month. Start by counting January on the first knuckle of your left hand and February as the gap between the knuckles. All the 'knuckle' months have 31 days. February has 28 days (29 in a leap year). All the other months (April, June, September and November) have 30 days.

Original source: http://www.bbc.co.uk/ schools (Accessed 2008)

Acknowledgements

Grateful acknowledgement is made to the following sources:

Other

Resource 1: Ways of measuring time long ago: Original source: http://inventors.about.com (Accessed 2008)

Resource 2: Water clocks: Original source: http://inventors.about.com (Accessed 2008)

Resource 3: Sundials: Original source: http://inventors.about.com (Accessed 2008)

Resource 4: Units of time: Original source: http://www.bbc.co.uk/ schools (Accessed 2008)

Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.

Section 3: Discussing data

1. Organising surveys to collect data

Pupils are often more interested in working with data that they have collected themselves – they know what the numbers are describing, and where the numbers came from. Surveys help pupils to understand the concept of data collection and pupils are encouraged to continue collecting interesting data outside school.

Organising your class into groups so that everyone is able to contribute is important. Whole-class discussion can be used to share the data the different groups have collected.

2. Interpreting and creating charts in groups

Collecting data is only part of some investigations, as it may need to be analysed and displayed to understand it better or to share the information with others. Pupils can show their data by using pictographs, pie charts, bar graphs, histograms or line graphs. It is important that pupils know which chart or graph is appropriate for which set of data, and so your examples should be clear. An overview of different kinds of charts is given in Resource 2: Charts and graphs.

Again, you will use data from the pupils’ own experiences but it is also a good idea to bring examples to class from newspapers, magazines and government publications.

Helping pupils understand the different kinds of chart takes time and you will need to plan several activities on each method to develop their understanding.

3. Assessing understanding

This final part is on analysis and interpretation of data once it has been displayed. It will enable you to assess the success of your teaching (for more information, see Key Resource: Assessing learning).

Using information from the science teacher or another subject teacher shows that data collection and analysis is important across the curriculum; it also allows you to work with other teachers and gain support. Case Study 3 and the Key Activity show how your pupils can use new data and how you can use this to assess their understanding.

You may decide to use a structured question approach (see Resource 4: Structured questions) for your assessment, so that you can find out exactly how much each of your pupils has learned.

Resource 1: Tally chart

 Teacher resource for planning or adapting to use with pupils

Making tally marks is a useful way to quickly count things in different categories. It involves making a mark or placing a store for each item counted in the particular group. The chart below shows a chart started for siblings in a class of 56 Primary 4 pupils. Each bundle of marks IIII represents five children.

Resource 2: Charts and graphs

 Background information / subject knowledge for teacher

Pie Chart

Bar Chart

Line Graph

Key: 1 - 1991, 2 - 1993, 3 - 1996, 4 - 1998

Resource 3: Data sets

 Teacher resource for planning or adapting to use with pupils

Resource 4: Structured questions

 Background information / subject knowledge for teacher

A structured question is one that has three parts.

• Part 1 is easy and can be answered by all pupils.
• Part 2 is a bit more difficult but can be answered by the majority of pupils.
• Part 3 is more difficult and tests the more able pupils.

Here is an example.

Alice and Mary are having an argument. Alice says that it is hotter this week than it was last week. Mary thinks it was hotter last week. Here are the temperatures for the two weeks.

• a) Draw the temperature for both weeks on the same line graph, using different colours for each week. (Part 1)
• b) Which week showed the highest temperature? (Part 2)
• c) Which week showed the lowest temperature? (Part 2)
• d) Find the mean temperature for each week. (Part 2)
• e) Find the range of temperatures for each week. (Part 2)
• f) In some ways Alice is right and in some ways Mary is right. Explain. (Part 3)

Original source: The New Uganda Primary Mathematics Pupil’s Book 7

Resource 5: Kano’s rainfall

 Teacher resource for planning or adapting to use with pupils

Acknowledgements

Grateful acknowledgement is made to the following sources:

Other

Resource 4: Structured questions: Original source: The New Uganda Primary Mathematics Pupil’s Book 7

Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.

Section 4: Working with weight

1. Doing practical work in groups

Estimation is an important skill in both mathematics and science and a useful skill to cultivate for all pupils. Simple balances can be made with very modest resources that allow pupils to approach measuring and estimating weight through practical investigation. You may like to make simple balances and plan and carry out these activities jointly with a science teacher in your school. This can be done by helping pupils to compare and contrast weights in different ways.

Activity 1: Comparing weight

You will need 5 simple balances (see Resource 1) to carry out this activity and 5 sets of common objects e.g. stones, balls, tins, bottle tops etc that could be used with the balances.

Write instructions for your pupils on the board (see Resource 2: Worksheets) and show the class what you want them to do using any two objects.

Ask them to estimate which is heavier by giving the objects to 2 pupils to answer.

Now ask a pupil to test their idea out by putting them on the pan and let them decide.

Ask them which is the heaviest object and why they think so.

Organise your pupils into 5 groups, giving each group a set of objects and a balance. Ask pupils to find which object is heavier by estimating its weight and then by using the balance. (See Key Resource: Using group work in the classroom)

Ask them to fill in a table of their results to share with the class to see if everyone agrees.

You could challenge older or more able pupils to see if they can order their objects from heaviest to lightest before testing. How could they test their answers using the simple balance?

2. Introducing units to compare weights

When developing understanding of the idea of weighing it is better if non-standard units are used to measure first. If pupils compare and contrast weights against non-standard unit bottle tops or beans they will quickly understand this is not sensible as the weight of different bottle tops and seeds vary. This is made easier by making sure they have sufficient experience weighing objects against different non-standard units.

Only when pupils understand the need for a common unit should the introduction of standard units like grams or kilograms begin.

3. Being resourceful

The previous activities should have shown your pupils that ‘standard units‘ of weight are needed because without these it is not possible to accurately compare or know how much something weighs. This next part explores how you can introduce terms and develop their understanding of kilogram (kg), and gram (g) (1000 gram = 1 kilogram). You may want to bring bags of sugar and rice to class, to show their weight is recorded in grams or kilograms and for them to feel the actual weights or make some mock ones i.e. plastic bags filled with sand, stones etc to the correct weight. If you can, borrow a pair of scales to do this.

If you do not have access to calibrated weighing scales or weights at school, it may still be possible to make approximate measurements of weight using your simple balances, and using some every day objects that have their weight on them to test them against. The second focus in this part is on understanding the terms gram and kilogram and being able to convert one to the other. This kind of activity should only be undertaken when pupils are confident at weighing in grams and kilograms.

Resource 1: Simple balance

 Teacher resource for planning or adapting to use with pupils

Resource 2: Worksheets

 Teacher resource for planning or adapting to use with pupils

1. One member of the group picks up and feels the weight of two objects, e.g. a stone and a bean, one after the other. Which feels heavier? Complete the table below.
2. Another member now uses the balance to compare the weights and completes the table
3. Repeat this for all the objects, comparing two objects at a time, allowing each member of the group to participate.
4. Pin your table on the wall and look at each other's results.

Resource 3: Pupil instructions for weighing activity

 Teacher resource for planning or adapting to use with pupils

1. Put the object to be weighed, say 1 milk tin, in the left scale pan of the balance. Place one bean in the other scale pan. Which is heavier?
2. Add more beans one at a time, in the scale pan on the right until the two sides balance. How many beans do you require for this?
3. Repeat using bottle tops instead of beans.
4. Repeat (1) and (2) for each of the objects to be weighed.
5. Record your observations in the form of the table below
6. Find out how many bottle tops weigh as much as 10 beans.
7. Find out how many beans weigh as much as 10 bottle tops.

Resource 4: Traditional weights used in Ghana

 Background information / subject knowledge for teacher

Traditional Akan weights – Elephant, Bird with body in Knot, and Claw

http://www.metmuseum.org/ search-results?ft=Traditional+Akan+weights&rpp=10&pg=2 (Accessed 2008)

Acknowledgements

Grateful acknowledgement is made to the following sources:

Photographs and images

Traditional Akan weights: http://www.metmuseum.org/ search-results?ft=Traditional+Akan+weights&rpp=10&pg=2 (Accessed 2008)

Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.

Section 5: Investigating distance

1. Organising an investigation

Any investigation needs to be planned and conducted carefully, otherwise the results might be incorrect or unreliable. In Activity 1 you will look at a statement that needs to be shown to be right or wrong. Make sure your pupils have all they need before they start an investigation, and that they understand the task before they begin. Your role is to support them as they work by asking questions to stimulate their thinking and encouraging them to develop their ideas.

2. Letting pupils plan their investigations

When exploring a topic such as measuring, it is important not to rush on to new concepts but to give pupils time to consolidate their learning and practise newly learned skills. This section provides more ways to explore pupils’ understanding and abilities to measure length in different contexts.

Here, you will ask pupils to make comparisons between measures and think about any links. By using the same groups for a series of activities, you can discover whether they see the similarity between the investigations and are able to use the data and the strategies they used before.

3. Assessing understanding of length

It is important that pupils have a real-life ‘feel’ of what different lengths mean and are able to estimate and measure the length or height of an object. This is a very useful skill in real life. For example, will a tree fall on our house if it is chopped or falls down? To assess whether they have this ‘feel’, you can use a question to solve on paper that requires them to use their understanding or give them a final investigation about a real object, such as that in the Key Activity.

When the task is complete, encourage your pupils to find out more about the indigenous trees of your country and have a go at measuring a large tree near your school if there is one. Working in this practical way will build their confidence in dealing with length.

Resource 1: Sample questions

 Background information / subject knowledge for teacher

• What do you understand by the term ‘height of a person’?
• How are you going to measure yourself?
• Should you have shoes on or off, while measuring?
• Should you lie down or stand up against the wall?
• What about those with long hair, should they squash it down?
• Where should you measure from?
• What should you measure with?
• How accurate should you be?
• How do you use a ruler or tape measure?
• How should you collect the measurements?

Resource 2: Two ways to check

 Background information / subject knowledge for teacher

• One way of checking that the statement is true is to make a table with two columns, one for the heights of the pupils going in order from the tallest down and the other for the lengths of the corresponding jumps. Only if the order is the same is the statement true.
• Another way is to make a mark with a cross on a grid using square paper with the pupil’s height on the horizontal axis and the length of jump on the vertical axis. Only if the result of the crosses is a straight line is the statement true.

Resource 3: Question on length

 Teacher resource for planning or adapting to use with pupils

Some pupils in Primary 3 measured different things in school. These are the measurements they wrote down.

• a. 4 metres
• b. ½ metre
• c. 19 metres
• d. 1½ metres

Below is a list of the things they measured. Match each length or height against the most likely object.

1. Florence, who is in Primary 3
2. The length of the school building
3. The length of the teacher’s table
4. The height of the classroom

Resource 4: Big tree in Budongo Forest

 Teacher resource for planning or adapting to use with pupils

Acknowledgements

Every effort has been made to contact copyright holders. If any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangements at the first opportunity.