Teaching and Learning

Sample learning activity – graph theory constructions and proofs

Introduction

The module on the Australian Mathematical Sciences page provides an excellent introduction to the beginning student of graph theory.

The following collections of related results can be proved after the initial exploration of ideas using a collection of sample graphs constructed by hand and/or using technology, for example: National Council of Teachers of Mathematics or VISUALGO.net

Part 1

Prove that the sum of the degrees of the vertices of any finite graph is even.

1. Show that every simple graph has two vertices of the same degree.
2. Show that if n people attend a party and some shake hands with others (but not with themselves) then at the end, there are at least two people who have shaken hands with the same number of people.
3. Prove that a complete graph with n vertices contains edges.
4. Prove that a finite graph is bipartite if and only if it contains no cycles of odd length.

Part 2

1. Show that any graph where the degree of every vertex is even has an Euler circuit.
2. Show that if there are exactly two vertices of odd degree, there is an Euler trail from one to the other.
3. Show that if there are more than two vertices of odd degree, it is impossible to construct an Euler trail.

Areas of study

The following content from the areas of study is addressed through this learning activity.

Outcomes

The following outcomes, key knowledge and key skills are addressed through this task.

Sample learning activity – number proofs by induction

Introduction

This learning activity provides the opportunity to develop proofs for several elementary number theory results using mathematical induction. In this activity take the natural numbers to be {1, 2, 3 …}.

Part 1

1. Construct a table of values for the sum of the first n natural numbers from 1 to 100 and write down a formula for this sum in terms of n.
2. Use mathematical induction to prove this formula.
3. Repeat parts a. and b. for the sum of the first n odd natural numbers.
4. Repeat parts a. and b. for the sum of the first n even natural numbers.

Part 2

1. Construct a table of values for the sum of the squares of the first n natural numbers from 1 to 100 and write down a formula for this sum in terms of n.
2. Use mathematical induction to prove this formula.
3. Repeat parts a. and b. for the sum of the cubes of the first n natural numbers.

Part 3

1. Construct a table of values for n! and n² for n from 1 to 100.
2. Use mathematical induction to prove that n! ≤ n² for all natural numbers n ≥ 2.
3. Randomly generate several six-digit natural numbers and express them as products of powers of prime numbers.
4. Use mathematical induction to prove that every natural number greater than or equal to two can be expressed as a product of one or more prime numbers.

Areas of study

The following content from the areas of study is addressed through this learning activity.

Outcomes

The following outcomes, key knowledge and key skills are addressed through this task.

Sample learning activity – distribution of sample means

Introduction

There are many ways of simulating sampling, including the use of technology.

This learning activity uses technology to explore the distribution of sample means from a normal (bell shaped) population with mean 200 and standard deviation 10.

A practical context could be a manufactured component which has a mean length of 2 metres with standard deviation 10 cm.

The samples are of size 20 initially and the student is asked to observe what happens to the distribution as the sample size increases.

The mean and standard deviation of the sample can be found.

Repeat with increased sample size.

For large n and more samples, we have a dot plot appearing as shown.

Areas of study

The following content from the areas of study is addressed through this learning activity.

Outcomes

The following outcomes, key knowledge and key skills are addressed through this task.

Sample learning activity – arc lengths of sections of curves

Introduction

Length is one of the basic forms of measurement. This learning activity investigates arc lengths of sections of curves defined by the graph of a function.

Consider a continuous function that satisfies the following conditions: it is non-negativeon the interval [0,1], it has the value 0 at x = 0 and x = 1; and the area bounded by the graph of the function and the x-axis over the interval [0, 1] is one square unit.

For each of the following functions draw the corresponding graphs.

Part 1

1. Let f(x) = ax for 0 ≤ x ≤ and f(x) = –ax + a for ≤ x ≤ 1, where a is a non-zero real constant. Find the value of a so that f satisfies the conditions and find the length of the graph of y = f(x).
2. Repeat this analysis for f(x) = ax(x –1),  0 ≤ x ≤ 1.

Part 2

1. Show that if f is any continuous function that satisfies the conditions then
   so does the function g(x) = (f(x) + f(1–x)).
2. Determine the rule for g(x) if f is a quadratic polynomial function satisfying the conditions.
3. Determine the rule for g(x) if f is a cubic polynomial function satisfying the conditions.

Part 3

1. Let f(x) = a sin nx. Find the values of a and n so that f satisfies the conditions, and determine the corresponding length of the arc from (0, 0) to (1, 0).
2. Repeat this analysis for f(x) = ax arccos (x) and a quartic polynomial function.

Areas of study

The following content from the areas of study is addressed through this learning activity.

Outcomes

The following outcomes, key knowledge and key skills are addressed through this task.

Sample learning activity – quadratic reciprocal and rational functions

Introduction

This learning activity provides the opportunity to explore key features of graphs of quadratic reciprocal and rational functions, that is functions with rules of the form

and  respectively.

Simple cases can be sketched by hand, with more general investigation assisted by technology.

Part 1

Consider the function with rule  where .

1. Find the derivative of the function and investigate the location and nature of any stationary points for different combinations of values of the coefficients.
2. Plot the corresponding graphs, and state the equations of any asymptotes and the maximal domain and range.
3. Classify the forms of the graphs in terms of the discriminant of the quadratic function.

Part 2

Consider the function with rule  where .

Carry out similar analysis for this type of function, including consideration of horizontal axis intercepts and points of inflection.

Areas of study

The following content from the areas of study is addressed through this task.

Outcomes

The following outcomes, key knowledge and key skills are addressed through this task.