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# Early Stage 1

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Early Stage 1

Mathematics Years 710 Syllabus

9.4 Data

In our contemporary society, there is a constant need for all people to understand, interpret and analyse information displayed in tabular or graphical forms. Students need to recognise how information may be displayed in a misleading manner resulting in false conclusions.

The Data strand extends from Early Stage 1 to Stage 5.2 and includes the collection, organisation, display and analysis of data. Early experiences are based on real-life contexts using concrete materials. This leads to data collection methods and the display of data in a variety of ways. Students are encouraged to ask questions relevant to their experiences and interests and to design ways of investigating their questions. Students should be aware of the extensive use of statistics in society. Print and Internet materials are useful sources of data that can be analysed and evaluated. Tools such as spreadsheets and other software packages may be used where appropriate to organise, display and analyse data.

This strand links to the topic Probability in the interpretation of the relative frequency of an event.

This section presents the outcomes, key ideas, knowledge and skills, and Working Mathematically statements from Stages 2 and 3 in one substrand. The Stage 4 content is presented in the topics: Data Representation and Data Analysis and Evaluation. The content for Stage 5.1 is represented in the topic Data Representation and Analysis while Stage 5.2 is represented in the topic Data Analysis and Evaluation.

Summary of Data Outcomes for Stages 2 to 5 with page references

Data

DS2.1 Gathers and organises data, displays data using tables and graphs, and interprets the results (p 112) DS3.1 Displays and interprets data in graphs with scales of many-to-one correspondence (p 113) Data Representation

DS4.1 Constructs, reads and interprets graphs, tables, charts and statistical information (p 114) Data Analysis and Evaluation

DS4.2 Collects statistical data using either a census or a sample and analyses data using measures of

location and range (p 115) Data Representation and Analysis

DS5.1.1 Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs

(p 116) Data Analysis and Evaluation

DS5.2.1 Uses the interquartile range and standard deviation to analyse data (p 117)

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Mathematics Years 710 Syllabus

Data Stage 2

DS2.1 Key Ideas

Gathers and organises data, displays data using tables and Conduct surveys, classify and organise data using tables

graphs, and interprets the results Construct vertical and horizontal column graphs and

picture graphs

Interpret data presented in tables, column graphs and Working Mathematically picture graphs Students learn to

Knowledge and Skills ? pose a suitable question to be answered using a survey

eg ‘What is the most popular playground game among Red Blue Yellow Green Students learn about students in our class?’ (Questioning) 5 2 7 1 ? conducting surveys to collect data ? pose questions that can be answered using the ? interpreting information presented in simple tables ? creating a simple table to organise data information from a table or graph (Questioning) ? constructing vertical and horizontal column graphs and eg ? create a table to organise collected data, using a picture graphs on grid paper using one-to-one computer program eg spreadsheets correspondence (Applying Strategies) ? marking equal spaces on axes, labelling axes and ? use simple graphing software to enter data and create a naming the display graph (Applying Strategies) ? interpreting information presented in column graphs and ? interpret graphs found on the Internet, in media and in picture graphs factual texts (Applying Strategies, Communicating) ? representing the same data in more than one way ? discuss the advantages and disadvantages of different eg tables, column graphs, picture graphs representations of the same data ? creating a two-way table to organise data (Communicating, Reflecting) eg ? compare tables and graphs constructed from the same

data to determine which is the most appropriate method Drinks Boys Girls of display (Reasoning) Milk 5 6

Water 3 2

Juice 2 1

? interpreting information presented in two-way tables

Background Information

This topic provides many opportunities for students to collect Data could also be collected from the Internet. information about a variety of areas of interest and can be readily

linked with other key learning areas such as Human Society and

Its Environment (HSIE) and Science.

Language

Column graphs consist of vertical columns or horizontal bars.

However, the term ‘bar graph’ is reserved for divided bar graphs

and should not be used for a column graph with horizontal bars.

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Mathematics Years 710 Syllabus

Data Stage 3

DS3.1 Key Ideas

Displays and interprets data in graphs with scales of Determine the mean (average) for a small set of data

many-to-one correspondence Draw picture, column, line and divided bar graphs using

scales of many-to-one correspondence

Read and interpret sector (pie) graphs

Read and interpret graphs with scales of many-to-one

correspondence

Knowledge and Skills Working Mathematically

Students learn about Students learn to

? using the term ‘mean’ for average ? pose questions that can be answered using the

information from a table or graph (Questioning) ? finding the mean for a small set of data

? collect, represent and evaluate a set of data as part of an Picture Graphs and Column Graphs investigation, including data collected using the Internet ? determining a suitable scale for data and recording the (Applying Strategies) scale in a key eg ?= 10 people ? use a computer database to organise information ? drawing picture or column graphs using a key or scale collected from a survey (Applying Strategies) ? interpreting a given picture or column graph using the ? use a spreadsheet program to tabulate and graph key or scale collected data (Applying Strategies) Line Graphs ? determine what type of graph is the best one to display a ? naming and labelling the horizontal and vertical axes set of data (Reflecting)

? drawing a line graph to represent any data that ? explain information presented in the media that uses the demonstrates a continuous change term ‘average’ eg ‘The average temperature for the eg hourly temperature month of December was 24 degrees.’ (Communicating)

? determining a suitable scale for the data and recording ? discuss and interpret graphs found in the media and in the scale on the vertical axis factual texts (Communicating, Reflecting)

? using the scale to determine the placement of each point ? identify misleading representations of data in the media when drawing a line graph (Reflecting)

? interpreting a given line graph using the scales on the ? discuss the advantages and disadvantages of different axes representations of the same data

(Communicating, Reflecting) Divided Bar Graphs and Sector (Pie) Graphs

? naming a divided bar graph or sector (pie) graph

? naming the category represented by each section

? interpreting divided bar graphs

? interpreting sector (pie) graphs

Background Information

In picture graphs involving numbers that have a large range, one Sector (pie) graphs and divided bar graphs are used to show how

symbol cannot represent one real object. a total is divided into parts. A key is used for convenience eg ? = 10 people. Column graphs are useful in recording the results obtained from Line graphs should only be used where meaning can be attached simple probability experiments. to the points on the line between plotted points. Advantages and disadvantages of different representations of the

same data should be explicitly taught.

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Mathematics Years 710 Syllabus

Data Representation Stage 4

DS4.1 Key Ideas

Constructs, reads and interprets graphs, tables, charts and Draw, read and interpret graphs (line, sector, travel, step,

statistical information conversion, divided bar, dot plots and stem-and-leaf

plots), tables and charts

Distinguish between types of variables used in graphs

Identify misrepresentation of data in graphs

Construct frequency tables

Draw frequency histograms and polygons

Knowledge and Skills Working Mathematically

Students learn about Students learn to

? drawing and interpreting graphs of the following types: ? choose appropriate forms to display data

(Communicating) - sector graphs

? write a story which matches a given travel graph - conversion graphs (Communicating) - divided bar graphs ? read and comprehend a variety of data displays used in - line graphs the media and in other school subject areas - step graphs (Communicating)

? choosing appropriate scales on the horizontal and ? interpret back-to-back stem-and-leaf plots when vertical axes when drawing graphs comparing data sets (Communicating)

? drawing and interpreting travel graphs, recognising ? analyse graphical displays to recognise features that concepts such as change of speed and change of may cause a misleading interpretation eg displaced zero, direction irregular scales (Communicating, Reasoning)

? using line graphs for continuous data only ? compare the strengths and weaknesses of different

forms of data display (Reasoning, Communicating) ? reading and interpreting tables, charts and graphs

? interpret data displayed in a spreadsheet ? recognising data as quantitative (either discrete or (Communicating) continuous) or categorical

? identify when a line graph is appropriate ? using a tally to organise data into a frequency (Communicating) distribution table (class intervals to be given for

grouped data) ? interpret the findings displayed in a graph eg the graph

shows that the heights of all children in the class are ? drawing frequency histograms and polygons between 140 cm and 175 cm and that most are in the ? drawing and using dot plots group 151155 cm (Communicating) ? drawing and using stem-and-leaf plots ? generate questions from information displayed in graphs ? using the terms ‘cluster’ and ‘outlier’ when describing (Questioning) data

Background Information

The construction of scales on axes can be linked with the Data may be quantitative (discrete or continuous) or categorical

drawing of similar figures in Space and Geometry. eg gender (male, female) is categorical

height (measured in cm) is quantitative, continuous It is important that students have the opportunity to gain quality (poor, average, good, excellent) is categorical experience with a wide range of tabulated and graphical data. school population (measured in individuals) is quantitative, Advantages and disadvantages of different representations of the discrete. same data should be explicitly taught.

Language

Students need to be provided with opportunities to discuss what Language to be developed would include superlatives,

comparatives and other language such as ‘prefer ….over’ etc. information can be drawn from the data presented. Students need

to think about the meaning of the information and to put it into

their own words.

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Mathematics Years 710 Syllabus

Data Analysis and Evaluation Stage 4 DS4.2 Key Ideas

Collects statistical data using either a census or a sample, Use sampling and census

and analyses data using measures of location and range Make predictions from samples and diagrams

Analyse data using mean, mode, median and range

Working Mathematically Knowledge and Skills Students learn to Students learn about ? work in a group to design and conduct an investigation ? formulating key questions to generate data for a eg - decide on an issue problem of interest - decide whether to use a census or sample ? refining key questions after a trial - choose appropriate methods of presenting

questions (yes/no, tick a box, a scale of ? recognising the differences between a census and a 1 to 5, open-ended, etc) sample

- analyse and present the data ? finding measures of location (mean, mode, median) for - draw conclusions (Questioning, Reasoning, small sets of data Applying Strategies, Communicating) ? using a scientific or graphics calculator to determine the ? use spreadsheets, databases, statistics packages, or other mean of a set of scores technology, to analyse collected data, present graphical ? using measures of location (mean, mode, median) and displays, and discuss ethical issues that may arise from the range to analyse data that is displayed in a frequency the data distribution table, stem-and-leaf plot, or dot plot (Applying Strategies, Communicating, Reflecting) ? collecting data using a random process ? detect bias in the selection of a sample eg numbers from a page in a phone book, or from a (Applying Strategies) random number function on a calculator ? consider the size of the sample when making ? making predictions from a sample that may apply to the predictions about the population (Applying Strategies) whole population ? compare two sets of data by finding the mean, mode

and/or median, and range of both sets ? making predictions from a scatter diagram or graph

(Applying Strategies) ? using spreadsheets to tabulate and graph data ? recognise that summary statistics may vary from sample ? analysing categorical data eg a survey of car colours to sample (Reasoning)

? draw conclusions based on the analysis of data (eg a

survey of the school canteen food) using the mean,

mode and/or median, and range

(Applying Strategies, Reasoning)

? interpret media reports and advertising that quote

various statistics eg media ratings (Communicating)

? question when it is more appropriate to use the mode or

median, rather than the mean, when analysing data Background Information (Questioning) Many school subjects make use of graphs and data eg in PDHPE climatic change, greenhouse gas emission, ozone depletion, acid students might review published statistics on road accidents, rain, waste management and carbon emissions. drownings etc. In Science, students carry out investigations to test or research a In Stage 4 Design and Technology, students are required, in problem or hypothesis; they collect, record and analyse data and relation to marketing, to ‘collect information about the needs of identify trends, patterns and relationships. consumers in relation to each Design Project’. Many opportunities occur in this topic to implement aspects of The group investigation could relate to aspects of the PDHPE the Key Competencies (see Cross-curriculum Content): syllabus eg ‘appraise the values and attitudes of society in - collecting, analysing and organising information relation to lifestyle and health’. - communicating ideas and information In Geography, range is used when discussing aspects such as - planning and organising activities temperature and is given by stating the maximum and minimum - working with others and in teams values. This is different to the use of ‘range’ in mathematics - using mathematical ideas and techniques where the difference is calculated for the range. - solving problems, and In Geography, use is made of a computer database of local - using technology. census data. Also, students collect information about global

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Mathematics Years 710 Syllabus

Data Representation and Analysis Stage 5.1 DS5.1.1 Key Ideas

Groups data to aid analysis and constructs frequency and Construct frequency tables for grouped data cumulative frequency tables and graphs Find mean and modal class for grouped data

Determine cumulative frequency

Find median using a cumulative frequency table or

polygon

Knowledge and Skills Working Mathematically Students learn about Students learn to

? constructing a cumulative frequency table for ? construct frequency tables and graphs from data

ungrouped data obtained from different sources (eg the Internet) and

discuss ethical issues that may arise from the data ? constructing a cumulative frequency histogram and (Applying Strategies, Communicating, Reflecting) polygon (ogive)

? read and interpret information from a cumulative ? using a cumulative frequency polygon to find the frequency table or graph (Communicating) median

? compare the effects of different ways of grouping the ? grouping data into class intervals same data (Reasoning) ? constructing a frequency table for grouped data ? use spreadsheets, databases, statistics packages, or other ? constructing a histogram for grouped data technology, to analyse collected data, present graphical ? finding the mean using the class centre displays, and discuss ethical issues that may arise from

the data ? finding the modal class (Applying Strategies, Communicating, Reflecting)

Background Information

For grouped data, the mode becomes the ‘modal class’ and the

mean is estimated using the class centre.

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Mathematics Years 710 Syllabus

Data Analysis and Evaluation Stage 5.2

DS5.2.1 Key Ideas

Uses the interquartile range and standard deviation to Determine the upper and lower quartiles of a set of scores

analyse data Construct and interpret box-and-whisker plots

Find the standard deviation of a set of scores using a

calculator Working Mathematically Use the terms ‘skew’ and ‘symmetrical’ to describe the Students learn to shape of a distribution

? compare two or more sets of data using box-and-Knowledge and Skills whisker plots drawn on the same scale

(Applying Strategies) Students learn about ? compare data with the same mean and different standard ? determining the upper and lower quartiles for a set of deviations (Applying Strategies) scores ? compare two sets of data and choose an appropriate way ? constructing a box-and-whisker plot using the median, to display these, using back-to-back stem-and-leaf plots, the upper and lower quartiles and the extreme values histograms, double column graphs, or box-and-whisker (the ‘five-point summary’) plots (Communicating, Applying Strategies) ? finding the standard deviation of a set of scores using a ? analyse collected data to identify any obvious errors and calculator justify the inclusion of any scores that differ remarkably - range ? using the mean and standard deviation to compare two from the rest of the data collected - interquartile range sets of data (Applying Strategies, Reasoning) - standard deviation ? comparing the relative merits of measures of spread: ? use spreadsheets, databases, statistics packages, or other ? using the terms ‘skewed’ or ‘symmetrical’ when technology, to analyse collected data, present graphical describing the shape of a distribution displays, and discuss ethical issues that may arise from

the data

(Applying Strategies, Communicating, Reflecting)

? use histograms and stem-and-leaf plots to describe the

shape of a distribution (Communicating)

? recognise when a distribution is symmetrical or skewed,

and discuss possible reasons for its shape

(Communicating, Reasoning) Background Information

It is intended that students develop a feeling for the concept of Graphics calculators will display box-and-whisker plots for standard deviation being a measure of spread of a symmetrical entered data.

distribution without going into detailed analysis. When using a No specific analysis of the relative positions of mean, mode and calculator the button for standard deviation of a population ?median in skewed distributions is required. Recognition of the n

general shape and lack of symmetry (only) needs to be will suffice. considered. Use of technology such as computer software and graphics

calculators enables ‘what if’ questions to be asked and explored

eg what happens to the standard deviation if a score of zero is

added, or if three is added to each score, or if each score is

doubled?

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