The Budget and forecasting
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Introduction
This set of resources provides opportunities for students to solve problems involving absolute, relative and percentage errors in the context of the Australian Government Budget. Students will also use and extend their data display and analysis skills to test the veracity of claims and investigate the relationship between data and the mean and median in the context of Budget forecasting errors.
Achievement standard
By the end of Year 9, students recognise and use rational and irrational numbers to solve problems. They extend and apply the exponent laws with positive integers to variables. Students expand binomial products and factorise monic quadratic expressions. They find the distance between 2 points on the Cartesian plane, and the gradient and midpoint of a line segment. Students use mathematical modelling to solve problems involving change in financial and other applied contexts, choosing to use linear and quadratic functions. They graph quadratic functions and solve monic quadratic equations with integer roots algebraically. Students describe the effects of variation of parameters on functions and relations, using digital tools, and make connections between their graphical and algebraic representations.
They apply formulas to solve problems involving the surface area and volume of right prisms and cylinders. Students solve problems involving ratio, similarity and scale in two-dimensional situations. They determine percentage errors in measurements. Students apply Pythagoras’ theorem and use trigonometric ratios to solve problems involving right-angled triangles. They use mathematical modelling to solve practical problems involving direct proportion, ratio and scale, evaluating the model and communicating their methods and findings. Students express small and large numbers in scientific notation. They apply the enlargement transformation to images of shapes and objects, and interpret results. Students design, use and test algorithms based on geometric constructions or theorems.
They compare and analyse the distributions of multiple numerical data sets, choose representations, describe features of these data sets using summary statistics and the shape of distributions, and consider the effect of outliers. Students explain how sampling techniques and representation can be used to support or question conclusions or to promote a point of view. They determine sets of outcomes for compound events and represent these in various ways. Students assign probabilities to the outcomes of compound events. They design and conduct experiments or simulations for combined events using digital tools.
Content descriptions
Calculate and interpret absolute, relative and percentage errors in measurements, recognising that all measurements are estimates. (AC9M9M04)
Analyse how different sampling methods can affect the results of surveys and how choice of representation can be used to support a particular point of view. (AC9M9ST02)
Represent the distribution of multiple data sets for numerical variables using comparative representations; compare data distributions with consideration of centre, spread and shape, and the effect of outliers on these measures. (AC9M9ST03)
Choose appropriate forms of display or visualisation for a given type of data; justify selections and interpret displays for a given context. (AC9M9ST04)
Teacher resources
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What is the government Budget?
Absolute, relative and percentage errors
Graphing and interpreting errors in Budget Forecasting - Solutions
Student learning resources
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Graphing and interpreting errors in Budget Forecasting
Budget forecasting - Errors and exaggerations
The Budget and forecasting
Building charts and tables
Budget forecasting - Errors and exaggerations - Solutions
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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.
Part A: Absolute, relative and percentage errors
- Explicitly teach students how to calculate absolute, relative and percentage errors. Refer to the visualiser.
- You might like to give students a practice activity such as the one shown in the table below.
| Scenario | Absolute error or difference with negatives if the scenario is a forecast | Relative error | Percentage error |
| The government estimated it would spend $500 billion, but actually spent $550 billion. | 500 – 550 = - 50 (- $50 billion) | - 50 ÷ 550 = 0.09 | - 50 ÷ 550 x 100 = - 9% |
| They forecasted 20 mm of rain, but we really got 25mm. | |||
| The report said the carpark held 280 cars, but we counted only 220 parking spaces. | |||
| A measurement is 24.44 mm and the true or known value is 25.00 mm |
Part B: Budget forecasting – Errors and exaggerations
- Play: What is the government Budget? as an introduction.
- As a class, read the definitions on the data sheet, clarifying as you read if required.
- Students complete the worksheet, either individually, in pairs or groups depending on students’ level of readiness.
- As a class discuss what students found. Ask students to share their conclusions about how statistics can be selected or exaggerated to support a point of view. Refer to The Budget and forecasting – Errors and exaggerations - Solutions.
Part C: Graphing and interpreting Budget forecasting errors
- As a class, read Part 1 of the investigation. If necessary, clarify the meaning of gross domestic product (GDP) and discuss why forecasting errors are better represented as a percentage of GDP rather than based on absolute dollar values. For example, dollar values fail to take account of:
- The size of the economy
- The size of the population (especially in relation to employment and income tax)
- The rate of employment
- Inflation
- Students complete Part 1 of the investigation.
Part D: Graphing the distribution of forecasting errors
- Revise the concepts of central tendency, mean and median and provide a definition (write them on the board).
Central tendency: The tendency for the values of a random variable to cluster round its mean, mode, or median.
Mean: The average, found by adding the numbers and dividing the sum by the number of numbers in the list.
Median: Is the middle value in a list ordered from smallest to largest.
- Do a simple example as a whole class to model the task. If you’re all working together on this, everyone will need access to a computer with spreadsheet software.
These are the Budget forecasting errors as a percentage of GDP of a hypothetical country between 2012 and 2016. All numbers are positive.
| 0.4 | 0.5 | 0.5 | 0.8 | 0.3 |
Set up a spreadsheet in Excel and arrange the error percentage values numerically from lowest to highest
A B 1 Error percentage 2 0.3 3 0.4 4 0.5 5 0.5 6 0.8 7 Average (mean) 8 Median - Use formulas to calculate average [=AVERAGE(B2:B6)] and median [=MEDIAN(B2B6)]. With students, identify the mode.
- Discuss with students about the average and the median being the same and how a distribution graph of these figures might look (symmetrical).
- The country suffered some setbacks in the following 5 years, including the COVID-19 event. The forecasting errors as a percentage of GDP between 2017 and 2022 were:
- Add rows to your spreadsheet to include these values, correct the formulas and examine the new mean and median.
- Discuss what’s happened to these measures of central tendency.
Set up this table on your spreadsheet.
Weekly Pay ($) Number of years -4 to -3.1 -3 to -2.1 -2 to -1.1 -1 to 0 0 to 1 Total =sum formula - Add the formula to calculate total number of years – discuss with students what they expect the total to be and what should they do if it’s not.
- Add a chart. Try both a line graph and a column graph, change the colours and add titles.
Part E: Investigating
- Walk the students through the instructions of Part 2 of the investigation. reminding them of the similarities to the problem you modelled.
- Students complete the investigation. Students may benefit from working individually and/or in small discussion groups.
Part F: Discussing results
When students have completed their investigations, discuss frequency and distribution and what students have learned, including reasons why the average percentage error is so much higher or lower than the median and how this is represented by the skew of their charts.
Refer to The Budget and forecasting – Errors and exaggerations - Solutions.