How wrong?

Download the teacher’s slides found in the ‘What you need’ section.

Coastline paradox

• (Slide 2) Distribute post-it notes and ask students to write their best estimate in kilometres to answer the question: ‘How long is the coastline of Australia?’
• Draw a rough number line on the whiteboard. Mark around five equal intervals, labelling the left-hand end ‘0’ and marking up to ‘50,000’. Students can use the scale on the number line to place their estimate in intervals of 10,000.
• Ask students what they notice. Identify the most common estimate and the range.
• Refer students to these measurements and references to compare to their estimate:
  • 19,320km, The Australia Handbook
  • 25,760km Wikipedia
  • 34,000km (including islands), Geoscience Australia
  • 30,270km or 47,070km (including islands), CSIRO Division of Land Use Research in Canberra.
• Ask: Why do you think there are so many different answers? Answers may include different definitions (including or excluding islands, distance measured inland for estuaries), changes in coastline owing to erosion or land reclamation, better measuring tools and technologies. All of these are valid explanations.
• Use slide 3 to explain that this issue is known as the ‘coastline paradox’. Explain that one of the biggest factors that contributes to the variance is the scale of the measuring instrument (visualise this as the size of the ruler) you use to measure the coast. A smaller ‘ruler’ is better able to get into the nooks and crannies of the coastline and so will record a longer measurement than a longer ‘ruler’. This is because the coastline has an irregular curved shape, similar to a mathematical concept known as a fractal curve.
• Key point: How we measure things matters. The tools we use, the units we use and the degree of accuracy or rounding we employ can give rise to different answers. Emphasise that any measurement is necessarily rounded, as measured variables such as length, mass and time are examples of continuous data.

• Ask students: Why does measurement accuracy matter? Looking at the images, can you explain how inaccuracies in measurement could be an issue? Use the examples to discuss errors in measurement. (Slide 4)
• Use worked examples to investigate the effect of compounding measurement errors in volume calculations.
• Explain that we are going to explore how rounding in measurements can have a compounding impact once rounded numbers are included into calculations, especially involving multiplication. Volume is important when measuring, for example, the capacity of an aeroplane’s fuel tank. We’ll use volume as our example, as calculating volume typically involves multiplying together three sets of measurements (as it relates to a three-dimensional shape).
• Distribute Worksheet 1 and show slides 5 and 6. Give students a few minutes to read and absorb the calculations for finding the volume of the following rectangular prism using the given measurements, rounding to two decimal places, one decimal place and the nearest whole number. They then determine the percentage errors generated using the rounded figures.
• Walk through the calculations using questions to determine and consolidate student understanding such as: ‘What did I do at this step?’ ‘Where has this number come from?’ ‘Why am I subtracting 1.76?’ ‘Why did I divide by 1.81504378?’
• Students may need a refresher and practice in finding the volume of a rectangular prism, rounding, or expressing an amount as a percentage of another amount if they are struggling to understand aspects of the worked example.
• Display slide 7, which shows a different rectangular prism. Ask students to vote on what they think is the most likely answer to the question: ‘Will we get the biggest error when we round to two decimal places, one decimal place or to the nearest whole number?’
• Instruct students in pairs to repeat the process shown in the worked example to answer the question and to notice how Worksheet 1 breaks the task down into the required steps.

Differentiation (support): There is also a heavily scaffolded version of the question for students who might initially struggle with the task.

• Check students’ answers using slide 8 and explore the answer to question 4. A typical student response might be: ‘Rounding to the nearest whole number gives a much bigger error and % error because the numbers change quite significantly, for example, 4.574 ≈ 5. You would expect this would always be the case as the more you round a number the less accurate your rounded value becomes.’

Differentiation (extension): Display slide 9: An unexpected result. Ask different groups of students to calculate the volume using the given values – values rounded to two decimal places, one decimal place and to the nearest whole number. Have a student from each group present their calculations on the board and calculate the various errors or display the table on the slide that shows the different volumes and errors. The unexpected result is that the values rounded to the nearest whole number give a more accurate estimate of the volume (1.04% error) versus those rounded to one decimal place (4.41% error).

• This reflects the impact of compensating errors: the length and width are both close to 3.95m and 3.05m and when rounded to one decimal place they both are rounded up. However, when rounded to the nearest whole number, the first rounds up to 4m and the second down to 3m so the differences compensate to reduce the net error when calculating the volume.
• This is a relatively rare phenomenon and, in most cases, the greater the degree of accuracy in your measurements, the closer to the actual volume your calculation will be.
• Students could use a spreadsheet such as the Excel calculator to experiment with finding different dimensions that give similar results.

Paired work

• Display slide 10 and explain that there are various different 3D solids with measurements given to two or three decimal places. Students choose at least three different solids and find the volumes. They then repeat this process using rounded values and calculate the error and percentage error.
• Ask students to name the different solids and explain how they would find the volume. In general, the volume of a prism is found using the formula V = Ah where A is the cross-sectional area and is the height of the prism. Refer to slide 10 notes for relevant formulae.
• Distribute Worksheet 2 and monitor students as they proceed with the activity.

Differentiation (support): student choice allows for students to choose more accessible shapes such as the cube and rectangular prisms or more challenging shapes such as the trapezoidal prism or annular cylinder. Students can also choose whether they explore using only one set of rounded dimensions (or more) for their chosen solid.

Review answers using the Excel calculator: Worksheet solutions tab. Show students slide 11 and ask them to write down three things they learned today. Monitor students’ responses and select three students to contribute their ideas to be recorded on the board. Key points you may want to highlight include:

• Measurement errors arise because all measurements are rounded (continuous data).
• More accurate instruments reduce measurement error (the coastline paradox).
• There are many practical situations where accurate measurement is important.
• The more you round a number (i.e. to fewer decimal places) the less accurate it becomes.
• In volume calculations, measurement errors compound as rounded numbers are multiplied together.
• Errors can become quite significant (e.g. 10% or even 20%) when small numbers are rounded to the nearest whole number and then multiplied.
• When calculating a % error you divide by the original (most accurate) volume.