Stage 5: Mathematics STEM Decision-Makers Unit 4 sample ...

NSW Education Standards Authority – Sample unit Mathematics Stage 5 - STEM Pathway: Unit 4 Decision-Makers of 12

Stage 5: Mathematics STEM Decision-Makers Unit 4 sample program

Overview Duration

In Decision-Makers students will recall, re-learn and develop the following essential skills:

calculate with the fractions associated with probability

calculate measures of centre and spread

create graphical displays

In Decision-Makers students will develop the following essential STEM understandings:

mathematical conventions for use in statistical graphs.

interpreting scale and axis labels.

compare and contrast different presentations of the same data in order to evaluate the effectiveness of each.

recognise statistical analysis as a valuable, and relatively simple, set of tools to gain clarity from data.

10 weeks

Outcomes

A student:

selects and uses appropriate strategies to solve problems (MA5.1-2WM)

provides reasoning to support conclusions that are appropriate to the context (MA5.1-3WM)

uses statistical displays to compare sets of data, and evaluates statistical claims made in the media (MA5.1-12SP)

calculates relative frequencies to estimate probabilities of simple and compound events (MA5.1-13SP)

uses quartiles and box-plots to compare sets of data, and evaluates sources of data (MA5.2-15SP)


Language/Literacy STEM

Negative and positive skew is only referring to direction - not bad and good results.

‘And’, ‘or’, ‘not’ require particular clarity in probability questions and Venn diagrams can assist student understanding.

The vocabulary of similarities and differences may require development.

The vocabulary describing degrees of certainty may need development.

The unit provides opportunities for students to consider probability calculations based on measured data and how they have become an essential component for all STEM pursuits. It is difficult to identify any VET course or associated career that does not rely on data and make decisions based on calculated probabilities.

Forecasting is increasingly important in all aspects of research and business.


Content Teaching and Learning STEM Resources and Stimulus

determine quartiles and interquartile range (ACMSP248)

determine the upper and lower extremes, median, and upper and lower quartiles for sets of numerical data, ie a 'five-number summary'

recognise that the interquartile range is a measure of spread of the middle 50% of the data (Reasoning)

construct a box-plot using the median, the upper and lower quartiles, and the upper and lower extremes of a set of data (ACMSP249)

Activity 1: ‘Five-number summary’

Equipment - paper hats (or equivalent) marked Max (maximum), Min

(minimum), 𝑄1(Lower quartile), 𝑄3 (Upper quartile) and two with Median.

The teacher asks students to write down their specific (ie not rounded) answers for:

1. Global population today. 2. Global number of mobile phones bought today. 3. Money spent on video games today - globally. 4. Years until oil runs out. 5. Babies born across the globe today.

Students discuss whether their answers were guesses or based on facts before comparing answers.

Students focus on the number of mobile phones bought, by:

arranging themselves in the ascending order of their answers, hence establishing the range (Note to teachers: the activity depends on students remaining in order throughout.)

the ‘Max’ and ‘Min’ students putting on their hats

the teacher using the range to establish a number line on the board - marking maximum and minimum values

the students then identifying the median value/answer

‘Median/s’ putting on their hat, and the teacher noting the value on number line. (Note to teachers: averaging two ‘middle’ values may be required - recall Stage 4 data skills.)

Link to learning:

The teacher prompts students to discuss:

STEM: Global statistics, STEM and you Activity.

Go to the data website: ‘Worldometers’: http://www.worldometers.info/.

Scroll down the list deciding whether each is connected to STEM and/or a VET course or career that interests you.

Discussion: Allow students time to digest the figures and their rate of change. Then, discuss in pairs whether any of the values or rates is a surprise and which of those numbers have they personally contributed to today.

http://www.worldometers.info/




‘Are all students represented well by the three values selected (min, max and median)?’

‘If you could have two more students’ answers included to show how all of your guesses were spread out, who would you chose?’ (Note to teachers: students are likely to nominate ‘the middle of the two halves’. If not, teachers guide them towards this.)

𝑄1(Lower quartile) and 𝑄3(Upper quartile) put on their hats and teacher notes values on number line, and instructs students to:

Put their hands up if they are in the middle 50% of data

Put their hands up if they are in the lowest 25% of data

Put their hands up if they are in the upper 50% of data

The teacher explains to students that often it is the ‘middle 50%’ of data that is used for decision-making. The lower and upper quarters are considered more ‘extreme’ while the middle two quarters are a more reliable representation of the whole. Name this as the ‘interquartile range’. (Note to teachers: Interquartile range (IQR) is not calculated until later in this program to avoid entering another ‘value’ into the set of numbers students are considering at this time.)

The teacher then delivers the answer to students as given by data website: ‘Worldometers’: http://www.worldometers.info/ on the day.

The teacher evaluates the accuracy of the class’s answers and offers feedback. Then, students complete the ‘Global statistics, STEM and you’ activity (Under STEM Resources & Stimulus - right column).

Consolidation for skill development:

Students develop the number line created during the activity into a box-plot.

The teacher explicitly teaches:





the names and symbols for each quartile

how to locate the five-number summary for odd and even data sets and for data sets with odd and even halves

how to draw a box-plot

Guided practice:

As a class, students select one more of the original questions as a focus and repeat the process of identifying the five-number summary by lining up in ascending order. Each student then draws their own box-plots and compares class answers to the actual answer.

School-based and online worksheets could be used as resources.

calculate probabilities of events, including events involving 'and', 'or' and 'not', from data contained in two-way tables

describe similarities and differences between two sets of data displayed in parallel box-plots, eg describe differences in spread using interquartile range, and suggest reasons for such differences (Communicating, Reasoning)

describe bias that may exist due to the way in which the data was obtained, eg who instigated and/or funded the research, the types of survey questions asked, the sampling method used (Reasoning)

Activity 2: An activity about building a solar house can be found here: ‘Solar Structure’: http://tryengineering.org/lesson-plans/solar-structures

Note to teachers: The full lesson plan is provided for the STEM activity of designing, building and testing a passive solar house. This activity connects to Activity 1 as those statistics demonstrated the rate of human growth and energy use, and hence the need for ‘environmentally smarter’ building designs.

A suggested sequence for incorporating the mathematics of statistics and probability into this activity is:

The teacher:

shares information about solar building design

discusses or directs students to research: o What is a comfortable temperature range for which neither

heating nor cooling is required in a house? o How does the pathway of the sun change in NSW between

midwinter and midsummer?

Students:

STEM/VET: Build a solar house (Try Engineering: http://tryengineering.org/lesson-plans/solar-structures)

The building could be done in cooperation with Science or Tech faculties or be done as an assignment.

Applies to VET courses related to construction, agriculture, retail, hospitality, human services, etc, as all of these must consider the built environment in which they operate. An understanding of passive solar design will enhance student understanding of their natural and built environment.

http://tryengineering.org/lesson-plans/solar-structures







identify skewed and symmetrical sets of data displayed in histograms and dot plots, and describe the shape/features of the corresponding box-plot for such sets of data

research for the school’s region: o Average number of ‘hot’ summer (December – February)

days o Average number of ‘comfortable’ summer days o Average number of ‘cold’ summer days o Repeat for winter (June - August)

display their findings in a two-way table such as the one below.

More rows and columns could be included or the row/column headings changed.

calculate the probability of: o a hot day o a cold day o a day that is hot and is in summer o a day that is cold and is not in winter o a day that is not hot, nor cold o other combinations of 'and', 'or' and 'not

The teacher:

leads a discussion related to the students calculated probabilities regarding if passive solar design a good idea if it negates need for air conditioners and heaters?

Students:

design and construct their houses

test their houses according (as outlined in the Build a solar house lesson plan) and record results in a table of values

As a group, students collate the class results for each time period

Summer Winter

Hot

Comfortable

Cold





(i.e. hot weather, cold weather) in parallel dot plots, and then: (Note to teachers: a set of magnetic counters adds value here. The number lines can be drawn on a magnetic whiteboard and students add counters for their results, giving a very efficient and accurate whole-class dot plot.)

locate the five-number summary for each dot plot

create parallel box-plots o compare and contrast the box-plots, relating to the house

designs

Special note: many students will naturally assume that the ‘top end’ of the box-plot is the best, as would be the case for exam scores. However, in this case, students need reminding of the objective that good design will keep the house from heating or becoming cold, so they need to carefully consider which quartile is the ‘best’ in this test.

Students:

indicate where their own house is on each box-plot and use this to evaluate the effectiveness of their design

analyse the features of the ‘best-performing’ design

display houses in the classroom along with a copy of the parallel box-plots

plots indicating the position of the house’s data (these will be referred to in later learning)

Link to learning:

Students have:

grouped temperatures into ranges

gathered and collated data

utilised a two-way table

used their own data to create box-plots

statistically analysed their design’s effectiveness

revised area calculations and units of measurement of temperature, length and area





vocabulary for comparing and contrasting parallel dot plots and box-plots

skewed and symmetrical data displayed in dot plots and their corresponding box-plots

how to calculate probability from a two-way table including 'and', 'or' and 'not’ events.

Guided practice:

School-based and online worksheets could be used as resources.

calculate probabilities of events, including events involving 'and', 'or' and 'not', from data contained in Venn diagrams representing two or three attributes (ACMSP226)

design a device to produce a specified relative frequency, eg a four-coloured circular spinner (Problem Solving)

repeat a chance experiment a number of times to determine the relative frequencies of outcomes, eg using random number generators such as dice, coins, spinners or digital simulators

use a tally to organise data into a frequency distribution table

Activity: Predict the weather

Students create a device to use in a chance experiment to predict summer weather, as follows:

Students collate summer weather temperatures from Activity 2. They need to also gather data on wet vs dry days.

The teacher leads a discussion on: How much rain is needed on a day to call it a ‘wet’ day? Is it more meaningful to measure mm or hours of rainfall in this case?

Students: o research average number of ‘wet’ and ‘dry’ summer days in

the school’s region o collate results into a Venn diagram o calculate the probabilities of each region of the Venn diagram o design and create a device to produce these relative

frequencies, eg a six-coloured spinner. (Note to teachers: this will require revision of MA4-13MG – area of circles and sectors.)

Each student uses their spinner once to predict the weather for Christmas Day – the teacher gathers class results.

STEM/VET: Short and long-term weather forecasting is intrinsic to many STEM and VET-related careers. From the field-workers who measure the weather to the mathematicians who create forecasting models and all of the jobs in between, the Science, Technology, Engineering and Mathematics of weather is complex and constantly evolving.

Research/discuss the effect of weather on:

Construction trades, agricultural industries, hospitality industries, human services, retail, aviation industries, automotive trades, entertainment industries, tourism industries, mining, financial

http://syllabus.bostes.nsw.edu.au/glossary/mat/frequency-distribution/?ajax




http://syllabus.bostes.nsw.edu.au/glossary/mat/frequency-table/?ajax



recognise that probability estimates become more stable as the number of trials increases (Reasoning)

explain the relationship between the relative frequency of an event and its theoretical probability (Communicating, Reasoning)

construct frequency histograms and polygons from a frequency distribution table

use the terms 'positively skewed', 'negatively skewed', 'symmetric' or 'bi-modal' to describe the shape of distributions of data

Students repeat the experiment a sufficient number of times for the class to obtain a total of 90 results and use a tally to organise whole-class data into a frequency distribution table. o Compare the frequency distribution table to the original Venn

diagram.

The teacher leads a discussion on: What other models can be constructed for this event that could have more predictive power than the activity the students have completed?

Link to learning:

Students have:

gathered data and collated it in Venn diagrams and frequency distribution tables.

calculated probabilities of combined events.

revised area, percentage and possibly angles. considered experimental probabilities from their device and observed relative frequencies from recorded data for the one situation.

The teacher leads a discussion on:

repeating an experiment increases the likelihood of obtaining results very close to theoretical expectations based on data. Consider the students’ initial individual result for Christmas Day compared to the collation of repeated trials.

‘While not likely, is it possible that every Christmas Day will be cold and wet?’

‘While not likely, is it possible that the ‘spinner’ could predict hot and sunny 1000 times in a row?’



the construction of Venn diagrams and frequency distribution tables

the construction of a frequency histogram from a table

services, electrical and IT industries.

Consider: How might perfect weather predictions change the way we live. Would it change the way you make any of your decisions?




http://syllabus.bostes.nsw.edu.au/glossary/mat/shape-statistics/?ajax









how to recognise and name 'positively skewed', 'negatively skewed', 'symmetric' or 'bi-modal' data in the shape of a histogram

Guided practice: School-based and online worksheets could be used as resources.

recognise randomness in chance situations (Reasoning)

consider the size of the sample when making predictions about the population (Reasoning)

Introduction: ‘Can the toss of a coin or roll of a die decide one’s fate?’

View the video clip from The Big Bang Theory where Sheldon decides to rely on dice to make all his decisions (https://youtu.be/5yk8ixiKUEw).

View this controversial NRL coin toss (https://youtu.be/1Nd9BM8ssYM)

Discuss – ‘Have you ever made a decision by tossing a coin? If so, have you ever seen that first result and decided to go for ‘best of three’?’

The teacher tells class – ‘Let’s use a random chance experiment to determine whether I set homework for tonight.’

Experiment:

The teacher explains that in this experiment two die are rolled and the results are added. The students will get set homework if the teacher can determine the ‘mode’ value of the experiment (plus or minus 1 unit of accuracy). If the teacher ‘guesses’ incorrectly students will not get set homework.

the teacher silently nominates a number between 2 and 12 and writes it on a piece of paper – this number is the ‘mode’ obtained in the experiment when two dies are rolled and their results added

the experiment begins and students roll two dice and add their result

https://youtu.be/5yk8ixiKUEw


https://youtu.be/1Nd9BM8ssYM




as a class: o gather a large sample by collating the results of every

student repeating the experiment at least three times (more for small classes)

o collate results into a frequency table o at its conclusion a tally is used to record all results on the

board o identify the mode

the teacher now reveals their nominated number. If the teacher has ‘guessed’ the mode within one number above or below the class gets homework, if not, the class ‘wins’.

as a class, students consider and discuss - was this a fair game?

Conclusion:

Students discuss:

‘Can understanding data and probability give you an advantage in the world, or at least protect you from being taken advantage of?’

‘What are some everyday examples of when it is important for a young person to understand risk - a word closely linked to probability?’

Assessment strategies

Topic Test - Short answer and multiple choice test.

Student self-evaluation - Students rate their own development through this unit - their understanding and skills, their application to learning and

working mathematically. Students discuss these with one another and then with teacher 'For Learning' in order to identify their readiness to move

on to the next topic and personal learning objectives they might set themselves for the next topic (eg: participation in class, completion of

homework, developing skills).


Resources overview

Additional Teacher Resources URLs:

‘Paper, Scissors, Rock conga line’ (description and clip at: http://www.ultimatecampresource.com/site/camp-activity/rock-paper-scissors-posse.html)

Teaching & Learning URLs of linked resources:

‘Worldometers’: http://www.worldometers.info/

Try Engineering: http://tryengineering.org/lesson-plans/solar-structures)

Sheldon from The Big Bang Theory: https://youtu.be/5yk8ixiKUEw

controversial NRL coin toss: https://youtu.be/1Nd9BM8ssYM

STEM Resources & Stimulus URLs of linked resources:

Build a solar house (Try Engineering): http://tryengineering.org/lesson-plans/solar-structures

Sites showing careers that use maths:

Plus Magazine - career interviews: https://plus.maths.org/content/Career

Get the Math: http://www.thirteen.org/get-the-math/

Teacher Evaluation of Unit

http://www.ultimatecampresource.com/site/camp-activity/rock-paper-scissors-posse.html

http://www.ultimatecampresource.com/site/camp-activity/rock-paper-scissors-posse.html








https://plus.maths.org/content/Career

http://www.thirteen.org/get-the-math/

http://www.thirteen.org/get-the-math/

Stage 5: Mathematics STEM Decision-Makers Unit 4 sample ...

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