Graphing and analysing the distribution of superannuation data
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Introduction
This set of resources provides opportunities for students to construct and analyse appropriate data displays to compare data distributions for continuous numerical variables as well as identify the association between 2 variables. They will use technology (spreadsheets) to assist in their calculations and investigate the different ways of representing distributions in the real-world context of superannuation.
Achievement standard
By the end of Year 10, students recognise the effect of approximations of real numbers in repeated calculations. They use mathematical modelling to solve problems involving growth and decay in financial and other applied situations, applying linear, quadratic and exponential functions as appropriate, and solve related equations, numerically and graphically. Students make and test conjectures involving functions and relations using digital tools. They solve problems involving simultaneous linear equations and linear inequalities in 2 variables graphically and justify solutions.
Students interpret and use logarithmic scales representing small or large quantities or change in applied contexts. They solve measurement problems involving surface area and volume of composite objects. Students apply Pythagoras’ theorem and trigonometry to solve practical problems involving right-angled triangles. They identify the impact of measurement errors on the accuracy of results. Students use mathematical modelling to solve practical problems involving proportion and scaling, evaluating and modifying models, and reporting assumptions, methods and findings. They use deductive reasoning, theorems and algorithms to solve spatial problems. Students interpret networks used to represent practical situations and describe connectedness.
They plan and conduct statistical investigations involving bivariate data. Students represent the distribution of data involving 2 variables, using tables and scatter plots, and comment on possible association. They analyse inferences and conclusions in the media, noting potential sources of bias. Students compare the distribution of continuous numerical data, using various displays, and discuss distributions in terms of centre, spread, shape and outliers. They apply conditional probability to solve problems involving compound events. Students design and conduct simulations involving conditional probability, using digital tools.
Content descriptions
Compare data distributions for continuous numerical variables using appropriate data displays including boxplots; discuss the shapes of these distributions in terms of centre, spread, shape and outliers in the context of the data (AC9M10ST02).
Construct scatterplots and comment on the association between the 2 numerical variables in terms of strength, direction and linearity (AC9M10ST03).
Teacher resources
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Graphing distribution and correlation
Student learning resources
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Graphing data distributions and associations
Distribution of income and super in Australia
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Suggested activity sequence
This sequence is intended as a framework to be modified and adapted by teachers to suit the needs of a class group. If you assign this activity to a class, your students will be assigned all student resources on their 'My learning' page. You can also hand-pick the resources students are assigned by selecting individual resources when you add a work item to a class in 'My classes'.
Part A: Calculating super
- If students have not encountered the concept of superannuation in previous maths lessons, , or need to revise the concept play: What is super?.
- Explain to students that the superannuation guarantee is currently 11% of ordinary time earnings and remind them that for every week they work for over 30 hours, they are entitled to the superannuation guarantee (providing they earn more than $450 for the month).
- Explicitly teach students how to calculate the superannuation guarantee using a gradual release of responsibility model.
To calculate the money your employer puts into your super account for you:
$Super=11\%\times salary$
For example: You’re a third year apprentice plumber and your weekly wage is \$669.53 per week. How much super should your employer pay into your super account?
$Super=11\%\times salary$
$=0.11\times\$669.53$
$=\$73.65$
Work backwards: Your employer pays $73.65 into your super account for the week. What was your salary for the week?
$Super=Percentage \times salary$
Dividing both sides of the equation by the percentage:
$\require{cancel}\frac{Super}{Percentage} = \frac{\cancel{Percentage} \times{Salary}}{\cancel{Percentage}}$ $\frac{Super}{Percentage} = {Salary}$
Substitute in known values:
$Salary=\frac{Super}{Percentage}$
$=\ \frac{73.65}{0.11}$
$Salary=\$669.54$
Complete the following table of superannuation calculations.
| Salary | Super |
| $1,250 | |
| $920 | |
| $1,323.50 |
Part B: Modelling use of Excel to find mean and median
- Model the following task. If you’re all working together on this, everyone will need access to a computer with spreadsheet software.
This is the weekly wage of 5 people in a netball team. Two people in the team didn’t want to share their information.
\$1,845 \$1,529 \$2,135 \$1,150 \$986
- Set up a spreadsheet and put the weekly pay data in excel (Include weekly pay, employer superannuation contributions and total cost to employer).
A B C 1 Weekly pay Employers' super contributions (11% of salary) 2 3 4 5 6 7 Average *mean) 8 Median - Sort from lowest to highest.
- Put in a formula to calculate super (=0.11*B2).
- Put in formulas to calculate average and median (for example: =AVERAGE(C2:C6) and =MEDIAN(C2:C6))
- Discuss with students about the average and the median being the same.
- Set up a spreadsheet and put the weekly pay data in excel (Include weekly pay, employer superannuation contributions and total cost to employer).
- The rest of the team have decided they will share what they get paid. One gets \$950 and the other gets \$3,428.
- Add these rows, sort the data, check the formulas and then examine the mean and median. Again, discuss what’s happened to these 2 measures of central tendency.
Set up this table on your spreadsheet.
Category Number of people in this category 0-999 1000-1999 2000-2999 3000-3999 1 Total Put in a sum formula here - Tally the wages and put them in each category ($3000 - $3999 has been done for you).
- Add the formula to calculate the total number of people. Discuss with students what they expect the figure to be (7) and what they should do if it’s not.
- Invite students to complete Part 1 of the worksheet.
Part C: Modelling and practising how to graph and analyse data
- The visualiser includes information that will support you to model how to graph and analyse distribution data.
- Use the visualiser to teach about each type of graphical representation in turn. After you have modelled each, ask students to practice by answering the questions on the worksheet and discuss their answers before moving to the next section. The order is shown below:
Visualiser element Worksheet section Introduction (slide 2)
N/A
Histograms (slides 3-12)
Part 2, questions 1-3
Box plots (slides 13-27)
Part 2, questions 4-6 (questions 7-8 are optional)
Cumulative frequency (slides 28-34)
Part 2, questions 9-15
Scatterplots (slides 35-44) Part 3, questions 1-8 Part D: Investigating super
The investigation is optional but may be used to elicit evidence of student levels of understanding and proficiencies in relation to the achievement standard. It could be set as an assessment task to be completed in the students’ own time.
- Add these rows, sort the data, check the formulas and then examine the mean and median. Again, discuss what’s happened to these 2 measures of central tendency.