## Pareto's rule

This is a level 3 to 4 mathematics in science contexts activity from the Figure It Out series. It is focused on working with very large numbers, express large number fractions as percentages, use ratios in different contexts. It also focuses on identifying the appropriate numerical information and operations needed to solve a problem or answer a question.

## About this resource

*Figure It Out* is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.

This resource provides the teachers' notes and answers for one activity from the *Figure It Out* series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.

Specific learning outcomes:

- Work with very large numbers.
- Express large number fractions as percentages.
- Use ratios in different contexts.
- Identify the appropriate numerical information and operations needed to solve a problem or answer a question.
- Students will discover that:
- it is possible to have a percentage of a percentage
- it is possible to convert from ratio to percentage, and vice versa.

# Pareto's rule

## Achievement objectives

NA4-1: Use a range of multiplicative strategies when operating on whole numbers.

## Required materials

*Figure It Out, Using Resources,*Levels 3+–4+, "Pareto's rule", pages 6–7- a globe
- a computer spreadsheet or calculator

See **Materials that come with this resource** to download:

*Paretos rule activity*(.pdf)

## Activity

### Preparation and point to note

These activities work best if students have completed "Ecological Footprints" on pages** **4–5 and understand the concept of resource use and Adena’s eco-footprint.

As well as fostering "thinking", the challenges in "Pareto’s rule" will build new knowledge and require the use of precise language and appropriate tools – all part of the key competency, using language, symbols, and texts.

Pareto’s rule is an example of a rule of thumb. The questions in the activities compare his rule with actual data and discuss the usefulness of such rules. These activities can also help students deal with the numbers and operations they need to understand and find meaning in actual situations.

Introduce and discuss rules of thumb that the students may be familiar with, for example, the New Zealand Police slogan “The faster you go, the bigger the mess”. A rule of thumb is simply a shorthand way of predicting an outcome.

In these activities, students need to use percentages and ratios in various combinations (percentages are actually ratios too: so many out of 100) to make meaningful comparisons possible. Percentages and ratios require students to think proportionally.

### Points of entry: Mathematics

Given that proportional measures depend on the context for their meaning, you need to emphasise reading and interpretation to your students. Often the most important question to be asked is “What is the whole?”

Review your students’ understanding of percentages and give them practice at converting percentages to fractions (and vice versa) and applying percentages in different situations.

The quantities in these activities are large: in the millions and up to hundreds and thousands of millions. Have the students practise reading them. Discuss the strategies that people use to break up and manage big numbers; for example, thinking of 6 778 070 000 as 6 778 millions.

The answer to Activity 1, question 1a, is found by dividing the number of people living in the developed world by the number of people in the whole world and expressing this as a percentage. Basic calculators may not accept this number of digits. If this is a problem, the students can divide both numbers by 1 000 or 10 000 before they enter them. Encourage them to think about why this works. Link back to previous discussions about really big numbers, for example, global hectares in "Ecological Footprints".

Question 1b** **requires the students to use their answer for the previous question. If the entire human population were to live like Adena, 3.6 Earths would be needed to support them. But “just” 17% live, like Adena, in the developed countries.

- 17% x 3.6 = 0.61.

So this minority segment of Earth’s population consumes the bulk of its resources.

Questions** **1c-d involve simple subtractions, but question 1e involves a level of abstraction that will challenge most students. They may need help to interpret and answer it. One way of working this out is shown in the answers. Another way of describing the mathematics is:

- 83% of Earth’s population live in developing countries.
- These people consume resources at the rate of 0.39 Earths.
- 1% of the population would consume at just 1/83 this rate (0.39 x 1/83).
- 100% of the population (“everyone”) would consume at 100 times this rate (0.39 x 1/83 x 100 = 0.47 Earths).

Questions 2–3 compare the rule of thumb (Pareto’s rule) with the data from question 1.

The diagram at the top of page 7 of the students’ book illustrates the ratios used in this activity. People in developed countries have a large eco-footprint; those in developing countries have a much smaller footprint. Compared with developing countries, there are relatively few people in the countries that are developed. The diagram invites questions such as:

- What happens when a small number of people have a large footprint?
- What happens when a large number have a small footprint?
- Which is large: a lot of small amounts or a few big amounts?

The answer to the last question is “It depends on the actual figures.” This can be easily demonstrated with pairs of examples such as 1/5 of 40 and 3/4 of 8. In this case, the small fraction of the larger number is greater (8 versus 6), but with 1/5 of 40 and 3/4 of 12, the large fraction of the smaller number is greater (8 versus 9).

### Points of entry: Science

Many of the resources that sustain our lifestyle are fi nite, and even renewable resources (for example, fresh water, food, and wood) are often only available in strictly limited supply. These resources are typically unevenly distributed and unequally shared.

A graphic illustration of the inequities in resource use is “the world seen in terms of use of resources”. (See the map URL in Related information above.)

Compare this map with a globe and focus on countries that the students are familiar with, for example, New Zealand, Australia, the Pacific, and other countries of students’ births. Discuss the shape and size of these countries as they appear in this map. Ask:

- Why does New Zealand appear to be its correct size and shape but Australia looks smaller, even though both are developed countries that use a large per-person share of Earth’s resources? (Encourage them to think about population density.)

The students could research the area of land used by people in different countries, developed and developing. The pictures at the bottom right of page 7 of the students’ book depict graphically the difference between the amount of land required to support 1 New Zealander and that for 1 person from a developing country. You could use them as a starting point for discussion on diet and lifestyle.

For example:

- In New Zealand, we have become used to lots of space. We like plenty of personal space at home [including our own bedroom, if possible] and expect to be able to “get away from it all” on holiday. We have so much space outside cities and towns that we can allow animals such as sheep and cows to occupy huge parts of it!)

The students could select a densely populated developing country, do a point-to-point comparison, and present their findings to the class. Have them discuss whether their investigation suggests actions that they could take to make their lifestyle more sustainable.

### Activity 1

**1.**

17%. (1 168 530 000 ÷ 6 778 070 000 x 100)

b. 0.61 Earths. (If 100% of the world lived like Adena, in a developed country, 3.6 Earths are needed. But only 17% of people live in developed countries, so they are using “just” 17% of 3.6 Earths. 17% = 17/100 = 17 ÷ 100. So 17% x 3.6 = 17 ÷ 100 x 3.6 = 0.61 Earths.)

c. 0.39 Earths. (1 Earth – 0.61 = 0.39 Earths)

d. 83%. (100% – 17% = 83%)

e. At a rate of 0.47 Earths. (83% have the use of 0.39 Earths; 0.39 x [100 ÷ 83] = 0.47 Earths)

**2.**

a. 80% or 0.8 Earths

b. According to the information in question 1, 17% of people live in developed countries and consume 0.61 Earths, leaving 0.39 Earths for the 83% who live in developing countries. This is a 60:40 split rather than Pareto’s 80:20. So Pareto’s rule is not exact, but it does give some indication of the imbalance.

**3. **

Pareto’s rule provides a quick, reasonable approximation in many circumstances. Nobody pretends that it is a substitute for careful data gathering and calculation.

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