Dear Family, your Operation Maths guide to Data

Dear Family, your Operation Maths guide to Data

Category : Uncategorized

Dear Family, given below is a brief guide to understanding the topic of data as well as some practical suggestions as to how you might support your children’s understanding at home. Also below, are a series of links to digital resources that will help both the children, and you, learn more about data. The digital resources are organised according to approximate class level:

Understanding Data

Data, as the name suggests, is all about information, and in maths it is about organising information in such a way that it is easy to read and interpret. Most of us are quite familiar with information from surveys, voting etc., presented in graphs, charts and tables in various print and digital media. But graphing is only one part of the data presentation and analysis process, and this process is essentially the same, whether at the junior or senior end of primary school, or even at a more advanced level of statistics:

• It starts when someone ask a question, that doesn’t have an obvious and/or immediate answer. This could be a question like who do most people intend to vote for in the next election or what is the favourite colour of a group of people or which sweet occurs most often in a box.
• Information is then collected relevant to the question. This may be collected via a digital or face-to-face survey. It may be collected from a large or small representative sample of people.
• This collected information or data is represented in a structured way that makes it easier to read. This might be a type of graph, pie chart or table.
• This represented data is then examined and compared (analysed and interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question; if the question remains unanswered, it may be necessary to re-start the process again, perhaps using different methods.

In the senior end of primary school the children will encounter more complex data and charts/graphs, while also analysing data in more complex ways, such as calculating the average (also know as mean), in 5th & 6th class, and identifying the most frequently occurring value in a data set (also known as mode) in 6th class.

Practical Suggestions for Supporting Children

• Let’s get organised! As mentioned earlier, data is all about organising information in an easy-to-interpret way. So any activities which involve sorting or organising can become a data analysis activity, for example:
• What lollipop flavour/colour occurs most often in the bag (see image)? Ask your child to organise the lollipops in such a way that we can see the answer, without the need to count. This could be done with a box of wrapped sweets also, for example, Quality Street, Heroes, Celebrations etc. For more challenging questions, ask your child to tell you how many more/fewer of one type than another type.
• What toy type do you have most of? When tidying up the toys, lay them out in rows alongside each other (parallel rows, similar to above), with the same type in each row. Of what toy type is there the most? The least?
• Hat sort: Organise your hats into rows of winter hats and summer hats or hats with rims and hats without rims or even just according to colour. You can do something similar with other clothes types also.
• You can also organise buttons or Lego pieces or building blocks in a similar way …. or any suitable material you may have at home.
• Real-world examples: Anytime you come across any examples of the data process, share these experiences with your children. It could be completing a review (survey) for an online purchase or a holiday stay. It could be survey or election results you come across on the internet, radio or TV. If a graph is used, ask your child to tell you the type of graph it is and to tell you what they notice, or can tell, from the information shown.
• League tables (soccer, GAA, rugby), are an ideal example of data presented in a table. Look at a table of results together, ask your child to interpret the information given, what it tells us, and what the various headings mean. Discuss an upcoming game: if your preferred team wins, how will that affect the table?
• Planning a party and not sure what to do or where to go? Why not ask your child to survey his/her playmates with 3 or 4 possible options and then use the collated results to determine the destination?
• Do a survey: You could do a traffic survey outside your house or a bird watch survey in your back garden. Or just encourage your child to come up with their own questions that they would like to answer. Survey your friends and family and then graph/present the collected information. Digital technologies (for example Microsoft Excel and Google Docs/Sheets) make it very easy to create a variety of very effective graph types.

Digital Resources for Infants

Fruit Fall: A simple game where the fruit that is caught is laid out in rows on a grid.

Curious George – Hat Grab: Help George grab hats to make a graph

Fishing Game: Catch the fish which are then arranged as a pictogram

Matholia – Data: A number of video lessons that include Picture Graphs and Making a Picture Graph.

I Know It – Reading Picture Graphs:  A review game/quiz. You can also try out a similar quiz here on block graphs.

Graphs: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum.)

Digital Resources for First and Second Classes

Khan Academy – Picture Graphs: Watch the videos and then answer the practice questions. You can also register for a free Khan Academy account to record your progress and explore other topics.

Matholia – Data: A number of video lessons that cover Picture Graphs, Making a Picture Graph and Making a Bar Chart.

White Rose Maths – Statistics: a series of lessons for Year 2 Week 1 and Year 2 Week 2.

Pictograph Game

I Know It – Reading Picture Graphs:  A review game/quiz. You can also try out a quiz here on basic bar graphs and more advanced bar graphs.

Handling Data – Quiz: Test yourself on what you know about data. Another similar quiz is also available here.

That Quiz – Graphs: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. From the options on the left hand side select pictogram, how many, difference, minimum, maximum, easier content. Do the set 10 questions, if you get 10 or 9 correct go up a level, and/or choose normal content.

IXL.com – Graphs: a selection of interactive quizzes. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum.)

Math Games – Graphing: games to practice graphing skills; go to the activities for your class level.

Digital Resources for Third to Sixth Classes

Maths is Fun – Data: Background information on using and handling data.

Matholia – Data: A number of video lessons that cover Making a Bar Chart and Interpreting Line Graphs.

White Rose Maths – Statistics: a series of lessons for Year 3 Week 1, Year 3 Week 2, Year 5 Week 1 and Year 5 Week 2.

Khan Academy – Data: A unit of work including video tutorials and practice questions. You can also register for a free Khan Academy account to record your progress and explore other areas and/or try more difficult material.

That Quiz – Graphs: This quiz has lots of options, on the left hand side, that can be changed to suit the ability of the child. Ensure that the level is set to 1. Each time do the set 10 questions, if you get 10 or 9 correct go up a level, if not stay at that level. There are lots of different types of activities: it automatically starts on bar charts, and you can choose pictogram, line (trend graph), circle (pie chart), multi-bar also. There are many question options also: plot, how many, difference, minimum, maximum, mean (average, 5th up) and mode (6th class).

Softschools.com – Tally Chart Game:  on this site you can also answer questions on a Favourite Colours Bar Chart, and Favourite Vegetables Bar Chart

I Know It – Graphing: A bar graph interactive quiz

Bar Charts: From Maths Frame, answer the questions on both vertical and horizontal bar charts; it also has both one-step and two-step questions.

How to make a graph using MS Excel: a tutorial

How to make a graph using Google Docs/sheets: a video tutorial.

Interactive programme to create line (trend) graphs

Create a Graph: Online graph creation facility that also allows you to print finished product.

Splash Learn – Data Games: These games cover Data on a Bar Graph, Read Line Plots and Represent Data on Line Plots.

Mashup Math – Video Lessons: includes Tally Charts and Picture Graphs, and for fifth and sixth classes, Mean, Median, Mode and Range and Line Plots.

Averages and Bar Models: Video tutorial on how bar models can be used to solve problems involving averages.

I Know It – Averages: A quiz on calculating averages

Handling Data – Quiz: Test yourself on what you know about data

IXL.com – Graphs: a selection of interactive quizzes. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum.)

Math Games – Graphing: games to practice graphing skills; go to the activities for your class level.

Digging Deeper into … Chance (3rd – 6th)

For practical suggestions for families, and helpful links to digital resources, to support children learning about the topic of chance, please check out the following post: Dear Family, your Operation Maths Guide to Chance

Chance is one of the most fascinating areas of primary mathematics, since it is concerned with the outcomes of random processes. Thus, the conceptual foundations for areas of mathematics such as probability and combinatorics, can be found in this strand unit.

• When considering random events and/or processes, we can use what we know (eg past experience and/or knowledge of the variables involved) to estimate/predict the likely outcome(s).
• If we identify all the possible outcomes in advance,  we can refine and/or express our prediction using mathematical language.
• However, no matter how accurate the mathematical prediction, the actual outcome(s) is not certain (except in the unlikely case where there is only one possible outcome); that is the element of chance!
• If we collate the results from repeated identical investigations of a specific random process, the actual outcomes (experimental probability) are more likely to reflect the original mathematical predictions (theoretical probability).

Predicting Outcomes: Terminology

When beginning to discuss and predict the likelihood of various outcomes,  the initial focus should be on the language of chance, and the terminology that accompanies it.

It can be very useful for the children to identify the various terminology, to discuss their interpretation of it and to explore the contexts in which the terminology is used in everyday parlance.

And while some of the phrases are more objective (e.g. impossible, never, certain, sure, definite), much of the language can be more ambiguous and is open to personal interpretation (possible, might, there’s a chance, (highly) likely, (highly) unlikely, not sure, uncertain).

FACT: To avoid ambiguity, some organisations have agreed on a consensus that equates this terminology with a fractional expression or percentage; you can view one such consensus here.

It can be helpful to try to organise this language across a continuum for the children to interpret and establish their meanings in relation to the other phrases. Ask the children to identify terminology that is used when describing the likelihood of something occurring. Use questions/statements to elicit from the children the vocabulary for chance that they already have; this can be the language that they would use to answer the questions from the text above or could be from their responses to questions such as the following:

• What is the chance that it will rain today?
• What is the chance that it will be hot today?
• What is the chance that it will be dark tonight?
• What chance does my team have of winning the league?
• What chance does my county have of winning the All-Ireland Championship?

Ask the children to write this terminology on pieces/slips of paper. Sort the pieces of paper into groups and/or order them along a line (continuum), as shown in the images below, with words that have similar or identical meaning together.

This task is a perfect example of a low threshold, high ceiling task, in that all children can participate and there is no limit to the complexity of terminology that can be incorporated. If mathematical values such as percentages and/or fractions (eg 1 in 2 chance) are suggested, the children should be encouraged to incorporate these, as they see fit.

Indeed, in fifth and sixth class the children should be encouraged to use a continuum which is graded from 0-100% and/or 0-1, and to associate and align the vocabulary with mathematical values (eg impossible/never =0%, might or might not/even chance = 50%, definite/certain = 100% etc).

Predicting Outcomes Mathematically

Irrespective of whether it is tossing a coin, rolling a dice, spinning a spinner, picking from a bag, choosing a card, etc., the children should always be encouraged to identify all the possible outcomes, to predict outcomes that are more or less likely, and to justify their predictions.

The children can also be encouraged to make more mathematical predictions based on their understanding of the variables involved e.g. if we repeated this investigation 30 times, how many times would you expect each colour would be picked? What about 60 times? 120 times? Express the fraction of the total number number of “picks”, that you would expect for each colour. Can you express any of these as a percentage?

When predicting the outcomes of random processes that involve a combination of variables, it can be very useful to use a type of pictorial structure, such as branching (NB these can also be referred to as tree diagrams), to illustrate the possible outcomes. For example, when predicting the outcomes of a double coin toss, children will often think that each of the three outcomes have an equal chance, when in fact there is double the chance (ie 2 in 4 or 1 in 2 chance) of getting a heads and tails combination, than either both heads or both tails (see diagram below).

However, it is worth noting that, unless the children come up with a similar structure to predict outcomes of combinations, it is preferable to hold back on showing such a structure until they have conducted an investigation, similar to above, where their predicted outcomes did not align to the actual outcomes.

Conducting the investigations

Once all appropriate predictions have been recorded, we can move on to the most exciting part, the investigating! When conducting chance investigations, it is important that the children recognise that that they need to be conducted fairly and recorded clearly, similar to scientific investigations.

Encourage the children to consider what factors need to be kept the same each time, and how practices could affect the reliability of the results eg:

• When picking items (eg cubes from a bag, cards from a deck) does the chosen item need to be returned each time? Why/why not?
• How many times does an investigation needs to be repeated in order to get a reliable result?

To generate sufficient data, while not spending too much time on each investigation, ten can be a suitable number of turns per child. It can also be a good idea to organise the children into groups of three with rotating roles eg the first child has their turn, the second child records the outcome of each turn and the third child keeps count of how many turns the first child has had, and roles are rotated after ten turns.

Recording and reflecting on results

As mentioned previously, the children should be encouraged to consider how best to record results. Tally charts and frequency tables can be very useful and link in well with the strand unit of Representing and Interpreting Data. Results of investigations can be displayed in various types of graphs and charts. Children in fifth and sixth classes could also be asked to calculate the average value for each outcome, when all the results of a class group are considered; for example, in the double coin toss, what was the average number of heads, tails and heads-tails combination per group.

Once the results have been collated, it is very important that the children be given time to reflect on the results and to compare them to their predictions. While we would expect an equal number of heads and tails in a single coin toss (ie theoretical probability), the actual results may not resemble these predictions (experimental probability). Such is the element of chance! And this can be a difficult concept for the children to accept, particularly the notion that, even though the mathematics behind their predictions was accurate, the actual outcomes are different.

To explore this further, using a spreadsheet, such as Google Sheets or Microsoft Excel, to collate the results of the entire class can be a great way demonstrate, that when we combine all the investigations, experimental probability (ie the results) is more likely to mirror theoretical probability (the predictions). This can often help reassure the children that the “maths” behind this does indeed work!

TIP: To make life easier for you, we have created a sample spreadsheet for the Double Coin Toss, please click on the link to view (and save/copy). For further information on the values of using spreadsheets to record results please check out this informative article on Probability Experiments with Shared Spreadsheets from NCTM.

• Dear Family, your Operation Maths Guide to Chance includes practical suggestions for supporting children, and links to a huge suite of digital resources, organised according to class level.
• The Operation Maths Digital Resources have specific resources designed to support this strand unit. Full details of these can be found in your TRB. Click here for the Quick Start Guide to the Digital Resources.
• Virtual Maths Manipulatives for Chance: Lots of tools that can be used in many different ways to explore this concept.
• PDST Data and Chance Handbook for Teachers
• Data and chance in the world around us: materials from the PDST workshop
• NRICH: selection of problems, articles and games for chance
• Playing dice, card, spinner games, or indeed any type of chance-based games, can be a great way to get students thinking about probability, while also providing practice with mental computation, estimation, subitising and experience of problem-solving via strategic thinking.
• Don’t forget to check out the games bank in your Operation Maths TRB and/or the last page of the Number Facts books for examples and ideas.
• Check out this Mathwire page for more games that focus on probability.
• For a fifth and sixth class who are exploring combinations, Mashup Math has two excellent videos (view both below) which demonstrate how tree diagrams and area models can be used to identify all possible combinations; both video use contexts to which the children could readily relate.

Digging Deeper into … Representing and Interpreting Data (infants to second class)

Category : Uncategorized

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of data, please check out the following post: Dear Family, your Operation Maths Guide to Data

Data Analysis Process

Data analysis, whether at lower primary, upper primary, or even at a more specialised level of statistics, is essentially the same process:

• It starts with a question, that doesn’t have an obvious and/or immediate answer. Information is then collected relevant to the question.
• This collected information or data is represented in a structured way that makes it easier to read.
• This represented data is then examined and compared (interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question (if the question remains unanswered it may be necessary to re-start the process again, perhaps using different methods).

When choosing a question it is worth appreciating that some questions might not lend themselves to rich answers. Take, for example, the first question above; once the data is collected, and represented, there is not that much scope for interpretation of results other than identifying the most common eye/hair colour and comparing the number of children with one colour as being more/less than another colour. However, other questions might lead to richer answers, with more possibilities to collect further information, to make predictions and to create connections with learning in other areas. Take, for example, the question above about travel; the children could be asked to suggest reasons for the results e.g. can they suggest why they think most children walked/came by car on the day in question, whether weather/season/distance from school was a factor and to suggest how the results might be different on another day/time of year. Even in a very simple way, the children are beginning to appreciate that data analysis has a purpose i.e. to collect, represent and interpret information, so as to answer a question.

From Operation Maths Jr Infs TRB p. 147

Sets and Data

Data is very closely related to sorting and classifying sets:

• The initial question may focus on a particular set in the classroom e.g. identifying the most common/frequent occurring item in the set of farm animals, the set of buttons in our button box, the shoes that the children are wearing, the nature items collected on the walk etc
• Information is then collected by sorting and classifying the items in the original set according to the target attribute.
• This collections of items are represented in a structured way that makes it easier to compare e.g. items put in lines of same type, use cubes or drawings to represent the actual items.
• This represented data is interpreted to answer the question and to make other statements about  relationships e.g. which group has more, less etc

Thus sorting and classify activities should be viewed as potential springboards into data activities and it is important that the children realise that they can represent and compare the size of the sets within each sort by graphing them.

CPA Approach

Even as the children move into first and second classes, it is important that their data activities continue to follow a CPA approach:

Concrete: Continuing to use real objects initially to sort and classify ) e.g. the number of different colour crayons in a box, the different type of PE equipment in the hall , the different fruit we brought for lunch etc), progressing towards using unifix cubes, blocks, cuisinere rods etc to represent the same data. Indeed, the children themselves could be used at this stage; sort the children into groups according to eye colour, hair colour,  age etc and get them to organise themselves into lines that represent the same criterion. This is turn can be very useful for the children to realise that how they are lined up is crucial to being able to interpret the data easily and correctly. If you have visible tiles/markings as flooring on the classroom/hall/corridor, these can be used to organise the “data” accurately!

The children can build block graphs using cubes or blocks, laid flat on a piece of paper or their Operation Maths MWBs.

Pictorial: using multiple copies of identical images to make pictograms and/or using identical cut out squares/rectangles of paper on which the children draw an image that represents the data as it relates to them (e.g. how I traveled to school today). These can then be collected and organised into lines, so that it is easier to read the data. As a development, identical cut out squares/rectangles of paper of different colours can be used with the children taking the correct colour as it relates to them (e.g. choosing the colour for their eyes/hair colour etc.) while also progressing towards using a specific colour for a specific criterion (“Take a blue square if you walked to school today”). Thus, the children should begin to appreciate the need to label the graph, axes etc so that the meaning of the represented data can be correctly interpreted.

HINT: A common confusion among children when making vertical graphs of any type is that the pictures/blocks start at the top and go down; an understandable misconception when you consider that in most other activities we work from the top down! A simple way to show how vertical graphs are formed, is to demonstrate, using a concrete Connect 4 type game, how the first counter in each column falls to the bottom and subsequent counters in that column build up from there. If you don’t have an actual Connect 4 game in your classroom you could use an interactive type such as this one here

Abstract: the final stage, where the focus is primarily on numbers and/or digits e.g. identifying how many, how many more prefer this than that etc.

Digging Deeper into … Representing and Interpreting Data (3rd-6th)

Category : Uncategorized

For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of data, please check out the following post: Dear Family, your Operation Maths Guide to Data

Data Analysis Process

Data analysis, whether at lower primary, upper primary, or even at a more specialised level of statistics, is essentially the same process:

• It starts with a question, that doesn’t have an obvious and/or immediate answer. Information is then collected relevant to the question.
• This collected information or data is represented in a structured way that makes it easier to read.
• This represented data is then examined and compared (interpreted) in such a way as to be able to make statements about what it reveals and, in turn, to possibly answer the initial question (if the question remains unanswered, it may be necessary to re-start the process again, perhaps using different methods).

• What is the most common eye/hair colour in our class?
• Which fruit/pet do we prefer?
• How did we come to school today?
• What candidate would we vote for?
• What is the temperature each day?
• How many children are absent from school every day?

When choosing a question it is worth appreciating that some questions might not lend themselves to rich answers. Take, for example, the first question above; once the data is collected, and represented, there is not that much scope for interpretation of results other than identifying the most common eye/hair colour and comparing the number of children with one colour as being more/less than another colour. However, other questions might lead to richer answers, with more possibilities to collect further information, to make predictions and to create connections with learning in other areas. Take, for example, the question above about travel; the children could be asked to suggest reasons for the results e.g. can they suggest why they think most children walked/came by car on the day in question, whether weather/season/distance from school was a factor and to suggest how the results might be different on another day/time of year. Thus, the children are beginning to appreciate that data analysis has a purpose i.e. to collect, represent and interpret information, so as to answer a question. And, in the case of the questions about temperature and number of absences, the children may begin to appreciate that it is too much to give the specific details for each individual day and that a figure to represent a larger set of numbers (eg the average) is preferable in some situations.

Content overview

A quick glance at the curriculum content for representing and interpreting data for these classes, reveals the following:
3rd class: pictograms, block graphs, bar charts
4th class: pictograms, block graphs, bar charts and bar-line graphs incorporating the scales 1:2, 1:5, 1:10, and 1:100
5th class: pictograms, single and multiple bar charts and simple pie charts; calculating averages
6th class: pie charts and trend graphs; calculating averages

In Operation Maths for 3rd and 4th classes, representing and interpreting data is specifically taught in September, at the beginning of the school year, so that the children are enabled to incorporate these skills into other subject areas where possible e.g. reading and interpreting tables and graphs, collecting and displaying data in science, geography etc.

In 5th and 6th classes, representing and interpreting data is taught later in the school year, after the children have encountered degrees in lines and angles and the circle in 2D shapes, as this content is necessary prerequisite knowledge for creating pie charts. In these classes, representing and interpreting data is also taught as a double chapter (two week block), to allow for the extra time required to explore averages.

CPA

As with every topic in Operation Maths, a CPA approach is also recommended for representing and interpreting data:

Concrete: Using real objects to sort and classify eg the number of different colour crayons in a box, the different type of PE equipment in the hall etc; using unifix cubes, blocks, cuisinere rods etc to represent data; using cubes to introduce and explore the calculation of averages.
Pictorial: using multiple copies of identical images to make pictograms; using identical cut out squares/rectangles to make block graphs etc, using folded circles to make pie charts, using bar models to calculate averages.
Abstract: the final stage, where the focus is primarily on numbers and/or digits eg reading and interpreting the scale on a graph where all the scale intervals are not given; calculating averages without pictorial or concrete supports.

Interpreting data

For children to become comfortable interpreting tables and graphs it is vital that they have plenty of opportunities to look at and read a variety of tables and graphs. This shouldn’t be limited to just the tables and graphs in their maths books. In particular, data sets that are relevant to them, such as soccer league tables can be a great way to encourage the children to appreciate how relevant this strand units is to them.

Utilize every opportunity to expose them to real-life examples of data from print and digital media and use purposeful questions to highlight the features of the graph:

• What is the title of this graph/chart?
• How is the information displayed? Horizontally or vertically?
• What type of chart/graph was used?
• Why do you think this graph type was chosen? What other types would have been suitable?
• What key information is required to interpret the data (eg scale intervals, labels on the axes, a key for piecharts)?
• Is there information missing that would have been useful to get a better insight into the data?

The children can be asked to create questions based on the graph/chart and swap with a partner to answer. When they become adept at producing charts themselves (see next section) they can also be asked to represent the data using a different chart type.

One of the most common mistakes that children make when interpreting graphs is misreading the scale. Always draw children’s attention to this first, and ask them to identify the scale interval and what it means for the bars/blocks/points etc on the graph. The graph quiz on That Quiz provides lots of extra practice for this skill. The quizzes are also very customisable, with options to show pictograms, bar charts, trend graphs (line) and pie charts (circle), easier or normal content, and a variety of question types. Another similar activity is this one from MathsFrame which offers three different levels of questions on bar charts.

A very interesting  and very different way to explore interpreting data is to show the children graphs where much of the key information is missing initially, but is then slowly revealed as the children share their thoughts and ideas. Following on from Brian Bushart’s work on numberless word problems, many teachers have used graphs to create “slow reveal” activities or “notice and wonder graphs”, and have very generously shared these online for other teachers to use. Some of these include:

Representing data

As mentioned previously, where suitable children should begin to represent data themselves using concrete materials. They can build block graphs using cubes or blocks, laid flat on a piece of paper or their Operation Maths MWBs. These should all start from the same baseline and the children should also write in labels for the axes and a title.

As a development, they can then trace around the stacks of cubes and remove the cubes to have a pictorial representation of the concrete. Using cubes like this to represent 1:1 quantities can in turn lead children to see a need for one cube to represent more than one, ie scales of 1:10, 1:5 etc, especially if there are not enough cubes to represent the data or there is not enough space.

The next step could be to have small squares or rectangles of identical pieces of paper which can then be pasted onto a page to display the information. This can work particularly well for pie charts; cut out a circle of paper and divide it by folding into eighths; the circle can be left whole and the folds outlined in pencil/marker or the eighths can be cut up. A groups of eight children can then use either of these to show data like their favourite ice-cream flavour or TV programme. In this case, because the amount of data gathered is limited, the choices/categories should be limited, also, to three or four.

If the children are also collecting the data to make a graph or chart, they will need to come up with a system to accurately collect and record this data. This will usually involve compiling a type of table with three columns; the first column to list the categories, the second to record tally marks and the third to total the tally marks. When introducing tally systems the children could use lollipop sticks to explore and make tally marks.

For children, drawing their own graphs can present many difficulties. Some common mistakes that can be made include:

• Incorrectly transferring the data from the table to the graph.
• Omitting the graph title and/or category titles on the axes.
• Using an inappropriate scale for a specific graph.
• Not setting the scale at regular, even intervals
• Zero being incorrectly located somewhere other than at the base line/axis.

And in other cases, it can just be a lack of neatness and exactness that reduces the quality, and readability, of a hand-draw graph. To overcome the difficulties associated with hand-draw graphs, the children could use either an online or offline computer application, all of which can produce very impressive results. Listed below are a small sample of those available; click on any of the links to access a tutorial or the application itself.

Calculating averages

Averages are introduced for the first time in 5th class and the children should have ample opportunities to explore this concept concretely and pictorially, before being given the formula to calculate the average of a set of numbers. Initially, the concept should be introduced as sharing amounts out to be fair/balanced:

Through plenty of concrete and pictorial opportunities to balance these separate quantities, it is hoped that the children begin to see a connection between the total number of items and the balanced quantity or average:

Bar models, one of the key problem-solving strategies used in Operation Maths, are very useful here, where comparison models can be used to compare the total of the averaged quantities with the total of the individual quantities. They are also used in Operation Maths 6 to calculate the extra number(s) when the average increases or decreases, a concept which can be very difficult to reason if no pictorial structures are used to help visualise the relationships.

You can also check out this video to see how bar models can be used to solve averages: