Statistics and Probability F-10A

Mathematics

Scope and Sequence

Statistics and Probability Foundation to Level 10

Introduction

What is a Scope and Sequence?

scope

The breadth and depth of content to be covered in a curriculum at any one time (e.g. week, term, year, over a student’s school life.) All that you do in a given period.

sequence

The order in which content is presented to learners over time. The order in which you do it.

Together a scope and sequence of learning bring order to the delivery of content, supporting the maximising of student learning and offering sustained opportunities for learning. Without a considered scope and sequence there is the risk of ad hoc content delivery and the missing of significant learning.

http://activated.act.edu.au/ectl/design/scope_and_sequence.htm

Why does a school need a scope and sequence?

An agreed Scope and Sequence for a Learning Area, provides a sound basis for a school being able to offer a guaranteed and viable curriculum by addressing gaps in students’ leaning and eliminating unnecessary repetition. A shared Scope and Sequence within a school enables teachers to have clarity about the knowledge, skills and dispositions that students will acquire in their learning and what they need to learn next. A Scope and Sequence supports teachers with effective unit and lesson planning and enables teachers to maintain a developmental focus on student learning as students progress through the school.

The Mathematics Scope and Sequence developed by WMR

This document has been developed to support schools with the transition to AusVELS Mathematics for 2013. While it provides examples of yearly overviews and learning sequences based on the content descriptors in the Australian Curriculum, it is not a complete curriculum. Each individual school can use the documents as a basis for developing a guaranteed and viable curriculum that caters for the needs of their school community.

Levels Foundation to 10A each include a set of learning goals/ intentions for each content sub-strand intended to provide a user friendly guide to the essential learnings around which teachers and teams could base their unit and lesson development.

Proficiency strands are listed next to each learning goal / intention as a guide only and teachers / teams are encouraged to consider all proficiencies equally whilst planning units and lessons. Where a particular proficiency is not listed for a content sub-strand teachers and teams should endeavour to contextualise the learning goals to address these proficiencies. Please note the following:

Sequence of teaching

The learning goals/intentions are listed adjacent to the content descriptions to assist teachers when developing a teaching program. They are not necessarily in the order to be taught – teachers /teams will make their own decisions regarding this. The third column has been included to assist teams to develop ideas for unit planning.

A sample Scope and Sequence Overview is also provided for each of the year levels from F to 10A. The number of weeks given to each unit in the overview acts as a guide and the total number of weeks allows for the many interruptions in a typical school year.

Links between the Learning Goals/Intentions and the proficiency strands

(a) The Learning Goals/Intentions have been identified to relate most closely to one of the four proficiency strands (shown in 3 below). This identification is shown in brackets at the end of each Learning Goal/Intention:

· Understanding is identified by (U)

· Fluency is identified by (F)

· Problem Solving is identified by (PS)

· Reasoning is identified by (R)

(b) In this document there are less Problem Solving and Reasoning proficiency strands identified than those for Understanding and Fluency. Should teachers wish to include more of these proficiencies in their curriculum, they are encouraged to emphasise them when teaching, and to develop appropriate learning tasks.

Proficiency strands

The proficiency strands describe the actions in which students can engage when learning and using the content. While not all proficiency strands apply to every content description, they indicate the breadth of mathematical actions that teachers can emphasise. The proficiencies listed next to each learning goal / intention are examples of how students might achieve the goal or what they have demonstrated by achieving the goal but are dependent on the context in which the learning takes place.

Understanding

Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information.

Fluency

Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

Problem Solving

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

Reasoning

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.

Useful references for teams and teachers to use when planning units of work and lessons include the following:

· Ultranet Design Space – DEECD Big Ideas in Number Maps - 128428217

· Ultranet design Space – Mathematics eBookboxes - 66512121

· Teaching Mathematics Foundations to Middle Years

Dianne Siemon, Kim Beswick, Kathy Brady, Julie Clark, Rhonda Faragher and Elizabeth Warren

· Mathematics Domain Page DEECD

· Building Numeracy – George Booker

· Teaching Primary Mathematics George Booker, Denise Bond, Len Sparrow, Paul Swan

· What We Know About Mathematics Teaching and Learning- MCREL

· WMR Numeracy Design Space 106126201

· Acara Scope and Sequence Documents http://www.australiancurriculum.edu.au/Download

· VCAA – resources http://www.vcaa.vic.edu.au/Pages/foundation10/curriculum/index.aspx

Please note: Teachers will be required to join each Ultranet design space before being able to access the resource. The number associated with each space should be entered into the search box in ‘available design spaces’ in order to find the space.

Foundation Level

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

How is the essential learning developed into units of work? How do students make connections between the learning goals and those included in other content strands or sub-strands? How do we ensure students become proficient in fluency, understanding, reasoning and problem solving?

Data representation and interpretation

· Answer yes/no questions to collect information (ACMSP011)

Students will:

· pose questions about themselves and familiar objects and events (U)

· represent responses to questions using simple displays (PS)

· use data displays to answer simple questions (R)

· Problem Solve through modelling authentic problems, using familiar counting sequences to solve unfamiliar problems, and discussing the reasonableness of the answer (PS)

Reason by explaining comparisons of quantities (R)

Achievement Standard: Students answer simple questions to collect information.

Level 1

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

· Identify outcomes of familiar events involving chance anddescribe them using everyday language such as ‘will happen’, ‘won’t happen’ or ‘might happen’ (ACMSP024)

Data representation and interpretation

· Choose simple questions and gather responses (ACMSP262)

· Represent data with objects and drawings where one object or drawing represents one data value. Describe the displays (ACMSP263)

Students will:

· pose questions about themselves and familiar objects and events (U)

· identify outcomes of familiar events involving chance, describing them using everyday language (R)

· represent responses to questions using simple displays such as a probability line (F)

· Problem Solve through modelling authentic problems, using familiar counting sequences to solve unfamiliar problems, and discussing the reasonableness of the answer (PS)

· Reason by explaining comparisons of quantities (R)

· use data displays to answer simple questions, developing the language of least, most, same amount (U)

· Use tally marks to record (F)

· determinine which questions will gather appropriate responses for a simple investigation (R)

· Problem Solve through modelling authentic problems, using familiar counting sequences to solve unfamiliar problems, and discussing the reasonableness of the answer (PS)

· Reason by explaining comparisons of quantities (R)

· Understand that one object or drawing represents on data value (U)

· Explain the display and what it tells us about the data (U)

Compares categories using language such as greatest or least (R

Achievement Standard: Students describe data displays. Students classify outcomes of simple familiar events. They collect data by asking questions and draw simple data displays.

Level 2

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

· Identify practical activities and everyday events that involve chance. Describe outcomes as ‘likely’ or ‘unlikely’ and identify some events as ‘certain’ or ‘impossible’ (ACMSP047)

Data representation and interpretation

· Identify a question of interest based on one categorical variable. (ACMSP048)

· Collect, check and classify data (ACMSP049)

· Create displays of data using lists, table and picture graphs and interpret them (ACMSP050)

Students will:

· Create a probability meter to place everyday events to match the language of chance (R)

· Identify practical activities and everyday events that involve chance. (U)

· Describe outcomes as ‘likely’ or ‘unlikely’ and identify some events as ‘certain’ or ‘impossible’ (U)

· classify a list of everyday events according to how likely they are to happen, using the language of chance, and explaining reasoning

· Make conjectures about chance events and test these (PS)

· Show fluency through listing possible outcomes of chance events (F)

· Formulate problems from authentic situations, (PS)

· determine questions for gathering data (R)

· determine methods for gathering and recording data – including a table (F)

· Gather data relevant to the question (F)

· recognise the usefulness of tally marks (F)

· identifying categories of data and using them to sort data (R)

· check and classify data (F)

· create picture graphs to represent data using one to one correspondence (PS)

· create displays of data using lists, table and picture graphs (PS)

· compare the usefulness of different data displays (R)

· show reasoning by creating and interpreting simple representations of data (R)

Achievement Standard: Students make sense of collected information. They describe outcomes for everyday events. Students collect data from relevant questions

to create lists, tables and picture graphs.

Level 3

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

· Conduct chance experiments, identify and describe possible outcomes and recognise variation in results (ACMSP067)

Data representation and interpretation

· Identify questions or issues for categorical variables. Identify data sources and plan methods of data collection and recording (ACMSP068)

· Collect data, organise into categories and create displays using lists, tables, picture graphs and simple column graphs, with and without the use of digital technologies (ACMSP069)

· Interpret and compare data displays (ACMSP070)

Students will:

· Conduct repeated trials of chance experiments such as tossing a coin or drawing a ball from a bag and identifying the variations between trials. (F,R)

· Collect and record categorical data to answer an identified question. (R)

· Identify efficient ways to record data, and representing and reporting the results of investigations including using digital technologies. (P)

· Compare and contrast between displays of data and make appropriate conclusions. (U)

Achievement Standard: By the end of Level 3, students interpret and compare data displays.Students conduct chance experiments and list possible outcomes. They carry out simple data investigations for categorical variables.

Level 4

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

· Describe possible everyday events and order their chances of occurring (ACMSP092)

· Identify everyday events where one cannot happen if the other happens (ACMSP093)

· Identify events where the chance of one will not be affected by the occurrence of the other (ACMSP094)

Data representation and interpretation

Select and trial methods for data collection, including survey questions and recording sheets (ACMSP095)

Construct suitable data displays, with and without the use of digital technologies, from given or collected data. Include tables, column graphs and picture graphs where one picture can represent many data values (ACMSP096)

Evaluate the effectiveness of different displays in illustrating data features including variability(ACMSP097)

Students will:

Discuss and apply probabilities to everyday situations ranging from Impossible (0) to Certain(1). (R)

· Apply an understanding of mutually exclusive events such as tossing a coin once, which can result in either heads or tails, but not both. (R)

· Apply an understanding of independent events such as the outcome in rolling a die cannot affect the outcome in tossing a coin. (R)

· Choose an effective way to collect and record data for a given investigation. (P)

· Choose appropriate representations for different types of data for interpretation, especially using digital technologies. (P)

· Compare and contrast different displays of data and make appropriate conclusions. (U)

Achievement Standard: By the end of Level 4, students identify dependent and independent events. They describe different methods for data collection and representation, and evaluate their effectiveness. Students list the probabilities of everyday events. They construct data displays from given or collected data.

Level 5

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

· List outcomes of chance experiments involving equally likely outcomes and represent probabilities of those outcomes using fractions (ACMSP116)

· Recognise that probabilities range from 0 to 1(ACMSP117)

Data representation and interpretation

· Pose questions and collect categorical or numerical data by observation or survey (ACMSP118)

· Construct displays, including column graphs, dot plots and tables, appropriate for data type, with and without the use of digital technologies(ACMSP119)

· Describe and interpret different data sets in context (ACMSP120)

Students will:

· Understand the likelihood of winning simple games of chance by considering the number of possible outcomes. (U)

· Represent the likelihood of winning simple games of chance in fractional form. (F)

· Discuss and apply probabilities to everyday situations ranging from 0 to 1. (R)

· Pose questions regarding and collect and interpret both numerical and categorical data. (R,P)

· Choose appropriate representations for different types of data for interpretation, especially using digital technologies. (R)

· Compare and contrast different sets of data and make appropriate conclusions. (U)

Achievement Standard: By the end of Level 5, students compare and interpret different data sets. Students list outcomes of chance experiments with equally likely outcomes and assign probabilities between 0 and 1. Students pose questions to gather data, and construct data displays appropriate for the data.

Level 6

Statistics and Probability

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

· Describe probabilities using fractions, decimals and percentages (ACMSP144)

· Conduct chance experiments with both small and large numbers of trials using appropriate digital technologies (ACMSP145)

· Compare observed frequencies across experiments with expected frequencies (ACMSP146)

Data representation and interpretation

· Interpret and compare a range of data displays, including side-by-side column graphs for two categorical variables (ACMSP147)

· Interpret secondary data presented in digital media and elsewhere (ACMSP148)

Students will:

· Understand use and convert between probabilities using fractions, decimals and percentages. (U,F)

· Conduct trials of chance and identify the variation between trials. (U,F)

· Understand why larger numbers of trials result in more accurate probabilities. (U)

· Compare observed and expected probabilities. (R)

· Select and use a range of appropriate data representations for comparison between sets of similar data. (U,F)

· Discuss and interpret data found in everyday situations. (R)

Achievement Standard: By the end of Level 6, students compare observed and expected frequencies. They interpret and compare a variety of data displays including those displays for two categorical variables. They evaluate secondary data displayed in the media. Students list and communicate probabilities using simple fractions, decimals and percentages.

Level 7 Statistics and Probability

Data Representation and Interpretation and Chance

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Data Representation and Interpretation

· Identify and investigate issues involving numerical data collected from primary and secondary sources (ACMSP169)

· Construct and compare a range of data displays including stem-and-leaf plots and dot plots (ACMSP170)

· Calculate mean, median, mode and range for sets of data. Interpret these statistics in the context of data (ACMSP171)

· Describe and interpret data displays using median, mean and range (ACMSP172)

Chance

· Construct sample spaces for single-step experiments with equally likely outcomes (ACMSP167)

· Assign probabilities to the outcomes of events and determine probabilities for events (ACMSP168)

Students will:

· Describe the difference between primary and secondary data. (U)

· Collect, organise and describe numerical data collected from primary sources. (PS)

· Analyse numerical data collected from secondary sources. (PS)

· Explain why some data representations are more appropriate than others for particular data sets. (U)

· Construct and compare data displays including ordered stem and leaf plots, and dot plots. (F)

· Calculate the mean, median, mode and range for sets of data. (F)

· Explain and interpret data, including referring to the mean, median, mode and range of the data. (R)

· Compare data sets from real life, including using the location of the mean and median on graphs. (R)

· Describe how outliers may affect the comparison of data sets when the mean, median and range are used.(U)

· Define key terms, including probability, sample space, favourable outcomes, trial, events and experiments. (U)

· Identify equally likely outcomes and outcomes that are not equally likely.(F)

· Construct sample spaces for single-step experiments with equally likely outcomes. (U)

· Calculate the probability of an event as a decimal, fraction or percentage. (F)

Achievement Standard:

By the end of Level 7, students determine the sample space for simple experiments with equally likely outcomes and assign probabilities to those outcomes. They calculate mean, mode, median and range for data sets. They construct stem and leaf plots and dot plots.

Level 8 Statistics and Probability

Chance and Data Representations

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Chance

Identify complementary events and use the sum of probabilities to solve problems. (ACMSP204)

Describe events using language of 'at least', exclusive 'or' (A or B but not both), inclusive 'or' (A or B or both) and 'and'. (ACMSP205)

Represent events in two-way tables and Venn diagrams and solve related problems. (ACMSP292)

Data Representations

Investigate techniques for collecting data, including census, sampling and observation(ACMSP284)

Explore the practicalities and implications of obtaining data through sampling using a variety of investigative processes. (ACMSP206)

Explore the variation of means and proportions of random samples drawn from the same population. (ACMSP293)

Investigate the effect of individual data values, including outliers, on the mean and median. (ACMSP207)

Students will:

· Demonstrate that probabilities range between 0 to 1 by convention and that calculating the probability of an event allows the probability of its complement to be identified. (R)

· Identify the complement of familiar events (eg the complement of getting a head on a coin is getting a tail, the complement of winning a game is not winning the game). (R)

· Calculate probabilities for sample spaces for single-step experiments (eg drawing a marble from a bag with 2 black and 3 white marbles with replacement. (F)

· Pose ‘and’, ‘or’, ‘not’ probability questions about objects or people. (R)

· Show that representing data in Venn diagrams or two-way tables facilitates the calculation of probabilities. (U)

· Use Venn diagrams and two-way tables to calculate probabilities for events satisfying ‘and’, ‘or’, ‘given’ and ‘not’ conditions. (R)

· Collect data to answer the questions using Venn diagrams or two-way tables. (P)

· Know the difference between a sample and a census and when each might be appropriate. (U)

· Be able to create, implement and interpret survey data through sampling techniques. (P,R)

· Use sample properties (for example mean, median, range) to predict characteristics of the population acknowledging uncertainty. (U)

· Use displays of data to explore and investigate effects. (R)

Achievement Standard:

By the end of Year 8, students choose appropriate language to describe events and experiments. Students model authentic situations with two-way tables and Venn diagrams. They explain issues related to the collection of data and the effect of outliers on means and medians in that data.

Level 9 Statistics and Probability

Chance and Data Representations

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

Data Representations

· Investigate reports of surveys in digital media and elsewhere for information on how data were obtained to estimate population means and medians (ACMSP227)

· Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources (ACMSP228)

· Construct back-to-back stem-and-leaf plots and histograms and describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bi modal’ (ACMSP282)

· Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread (ACMSP283)

Chance

· List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (ACMSP225)

· Calculate relative frequencies from given or collected data to estimate probabilities of events involving 'and' or 'or' (ACMSP226)

Students will:

· develop a question that will aid the comparison of two or more sets of data. (U)

· use appropriate investigative techniques to collect data. (PS)

· choose appropriate secondary data (F)

· identify everyday questions and issues involving at least one numerical and at least one categorical variable (R)

· display comparative data, such as back-to-back stem-and-leaf plots and histograms. (F)

· explain comparative data, such as back-to-back stem-and-leaf plots and histograms. (R)

· describe data , using spread, mean, medium, outliers, skewed, symmetric . (U)

· explain the data and draw conclusions. (R)

· evaluate media reports and use statistical knowledge to draw conclusions (U)

· list all the outcomes for a two step chance experiment using a tree diagram. (F)

· assign probabilities, using the tree diagram. (F)

· use the tree diagram to solve problems, including and, or and not. (F)

· solve probability questions, using a tree diagram for events ( without replacement). (F)

· use Venn Diagrams to solve problems with ‘and’, ‘or’, and ‘not’. (PS)

Achievement Standard:

By the end of Level 9, students compare techniques for collecting data in primary and secondary sources. They make sense of the position of the mean and median in skewed, symmetric and bi-modal displays to describe and interpret data. Students calculate relative frequencies to estimate probabilities, list outcomes for two-step experiments and assign probabilities for those outcomes. They construct histograms and back-to-back stem-and-leaf plots.

Level 10 Statistics and Probability

Data representation and Interpretation

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Determine quartiles and interquartile range (ACMSP248)

· Construct and interpret box plots and use them to compare data sets (ACMSP249)

· Compare shapes of box plots to corresponding histograms and dot plots (ACMSP250)

· Use scatter plots to investigate and comment on relationships between two continuous variables (ACMSP251)

· Investigate and describe bivariate numerical data where the independent variable is time (ACMSP252)

·
Evaluate statistical reports in the media and other places by linking claims to displays, statistics and representative data (ACMSP253)

Students will:

· determine the minimum and maximum values in a set of data (F)

· determine the range in a set of data (F)

· determine the median in a set of data (F)

· determine the upper and lower quartiles in a set of data (F)

· determine the interquartile range in a set of data (F)

· compare data sets numerically (U) (R)

· represent data by constructing a box plot (F)

· compare data sets visually using box plots (U) (R) e.g. the distribution of Aboriginal and Torres Strait Islander people by age with that of the Australian population as a whole

· understanding that box plots are an efficient and common way of representing and summarising data and can facilitate comparisons between data sets

· represent the same set of data visually in various ways such as box plots, histograms and dot plots(U)

· compare the various visual representations of a set of data and explain their features (U)

· represent data sets as scatter plots (F)

· use authentic data to construct scatter plots, make comparisons and draw conclusions (R)

· comment on relationships between continuous variables using their scatter plots

· investigate and describe relationships between variables (F) (U)

· construct and interpret data displays representing bivariate data over time (F) (U) (R)

· investigate and describe bivariate data where the independent variable is time (F) (U)

· Use real life data to explain and evaluate statistical reports presented in the media (F) (U) (R)

· Investigate data in different ways to make comparisons and draw conclusions (R) (PS) (U) (F)

Achievement Standard:

Students compare data sets by referring to the shapes of the various data displays. They describe bivariate data where the independent variable is time. Students describe statistical relationships between two continuous variables. They evaluate statistical reports. They calculate quartiles and interquartile ranges.

Level 10A Statistics and Probability

Chance

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Investigate reports of studies in digital media and elsewhere for information on the planning and implementation of such studies, and the reporting of variability (ACMSP277)

Students will:

· Evaluate media reports that refer to data from a range of contexts (R)

· Evaluate the visual representation of data in media reports (R)

· Evaluate the size and type of samples used for data collection in a variety of contexts (R)

· Discuss appropriate methods of sampling for data collection (U)

· Generate data by posing appropriate questions and making decisions about sampling a population (U)(R)

Level 10A Statistics and Probability

Data Representation and Interpretation

AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies

Essential Learning

Unit Development Ideas

· Calculate and interpret the mean and standard deviation of data and use these to compare data sets (ACMSP278)

· Use information technologies to investigate bivariate numerical data sets. Where appropriate use a straight line to describe the relationship allowing for variation (ACMSP279)

Students will:

· Calculate the mean for a data set (F)

· Calculate the standard deviation for a data set (F)

· Compare data sets using their standard deviations and mean values (R) (U)

· Discuss data sets by interpreting their mean and standard deviations (R) (U)

· Represent bivariate numerical data sets graphically using digital technologies (F)

· Use straight lines (line of best fit) to represent scatter plots and to describe the relationships between variables (F) (U)

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