- Level 9
- Number and Algebra
- Patterns and algebra
- Set structures

###### Patterns and algebra • Level 9

# Set structures

VCMNA307

## Teaching Context

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At this level students will use sort algorithms to investigate population samples proportion and population proportions for examples of sets of data in the real world. They will use sorting techniques to find the median of a data sets and find the mean of a random sample and explore proportional variations in the sample data.

Encourage the use of correct mathematical terminology such as set and structure throughout this area of learning. A set is a collection of data that have properties in common. For example, even numbers between 0 and 20 will form a set. A structure gives a set additional meaning. For example, applying order will provide a specific structure to a set.

Ask students to brain storm examples where sorting is required. For example, organising books in a library, sorting names of students at school, sorting by price on an online store. They could also investigate different sorting techniques such as bubble sort, insertion sort and quick sort or they may also create their own.

Students show their learning by applying sorting algorithms to a set of data. Once the data is ordered, students answer questions such as; What is the first item?and What is the median item?

Students will learn about random samples and investigate the impact of different sample sizes on sets of data. For example, if there 100 people are surveyed to collect data about their age, how would the mean age of a sample of 5 compare to the mean age of a sample of 30?

They can also compare sample proportion.

$$sampleproportion=\frac{numberofsuccessfuloutcomes}{samplesize}$$ For example if 25 people are aged 15 years, then the sample proportion is $\frac{25}{100}=0.25$. This means 25% or a quarter of the sample was aged 15 years.

Students will learn that the larger sample size, the more representative it is of the whole population.

### Victorian Curriculum

Apply set structures to solve real-world problems (VCMNA307)

VCAA Sample Program: A set of sample programs covering the Victorian Curriculum Mathematics.

VCAA Mathematics glossary: A glossary compiled from subject-specific terminology found within the content descriptions of the Victorian Curriculum Mathematics.

##### Achievement standards

Students apply the index laws using integer indices to variables and numbers, express numbers in scientific notation, solve problems involving very small and very large numbers, and check the order of magnitude of calculations. They solve problems involving simple interest.

Students use the distributive law to expand algebraic expressions, including binomial expressions, and simplify a range of algebraic expressions. They find the distance between two points on the Cartesian plane and the gradient and midpoint of a line segment using a range of strategies including the use of digital technology.

Students sketch and draw linear and non-linear relations, solve simple related equations and explain the relationship between the graphical and symbolic forms, with and without the use of digital technology.

## Online Resources

## Teaching ideas

### Median age

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Working in small groups, students will find the median age of the sample group they are given. Before beginning the task discuss the meaning of ‘median’. i.e. the middle value of a set of data, when the data is placed from smallest to largest. Provide some examples.

Assign each group a different sorting technique. Provide an explanation of their assigned technique, and allow them time to discuss it within their group so that all members understand. When they are ready, provide the data (collection of ages) and instruct them to begin.

Here are the ages of people in a town:

{2, 5, 9, 19, 24, 54, 5, 87, 9, 10, 44, 32, 21, 13, 24, 18, 26, 16, 19, 25, 39, 47, 56, 71, 91, 61, 44, 28} Explain that this is a set of ages which is represented using curly brackets.

Each group will use their technique to sort the data and find the median. Once they have completed this give each group a new set of numbers to sort. They may either use the same technique or they could select a technique.

Explanations of different techniques:

*Two sorters: *

This sorting technique uses two people (or two groups of people) to sort the data. The first group finds the lowest number and places it at the start. This process is repeated. The second group finds the highest number and places it at the end and then repeats this process. Once in order, they are able to find the median.

*Easy quick sort:*

First choose an age that you think is close to the middle. Then organise the numbers according to whether each is “higher” or “lower” than that number. Now put the numbers in order. Once in order, they are able to find the median.

*Insertion sort: *

Select one data item and write it in the middle of a piece of paper. Considering each data value one at a time decide where it belongs in relation to the list established and place it accordingly. Repeat this until all data is sorted. Once in order, they are able to find the median.

*Individual: *

Give each person in the group the same (or similar) number of items to sort. Each person sorts their data. They then come together to place all the data in order and consequently find the median.

As a class, discuss the pros and cons of each technique.

### Book sorting

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Watch the video ‘What’s the fastest way to alphabetiseyour bookshelf?’

Ask each student to bring 3 to 4 novels from home, or alternative bring a collection of different books into the class from the school library.

In small groups, ask students to use one of the sorting techniques from the video to arrange the books in alphabetical order. After they have completed one sort, ask them to do the same task using a different technique from the video.

Once students are familiar with one or more different sorting techniques, ask the class to work together to sort other things. For example, sort the class into height order, muesli bars into number of kilojoules, or the names of students in the class into alphabetical order.

As a class discuss which sorting technique worked best for each task.

### Sample proportions

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Students will look at sample proportion for a large number of samples of various sizes.

Before you begin this activity review definitions of the following terms:

- Sample – part of a population
- Population – all the items being considered
- Proportion – a number of a particular item in comparison to the whole. For example, I have 100 marbles and 31 of them are red. The proportion of red marbles = 31/100 = 0.31 or 31%.

Gather a large selection (around 100) of a common object (e.g. marbles). Define a success (e.g. red marble) and a failure (e.g. not a red marble).

Ask students to perform many sampling experiments using various sample sizes. For example, students do 10 experiments with a sample size of 5, then 10 experiments with a sample size of 10 and so on. Record the results in a table. See the example below.

Discuss their findings. Which sample size is more accurate? Why? Students will learn that the closer a sample size is to the population size the more closely it represents the population.

Discuss how this activity is connected to real world sampling using examples such as television ratings and political polls. These values are used to represent the population, but only a sample of the population is used. Are they accurate?