Problem 15.1 Patterns
Designs such as drawings on wallpaper are called patterns. Patterns can be models from which other shapes can be generated.
People
often enjoy exploring patterns. In mathematics, patterns in numbers are
often used to discover a rule for finding the next number.
Finding a Rule
When we look for a pattern in numbers, we try to look at each number and
find out how it is related to other numbers. This enables us to state a rule
in words; and then write it algebraically.
For this problem, you are required to find patterns.
1. The rule for each pattern is given. Use the rule to complete each of
the following patterns:
2. Complete the following multiplication pattern:
3. We can show that:
Similarly, we can show that:
Find the pattern and use it to complete the following:
Problem 15.2 Fermat's Numbers
Fermat's numbers are defined by
Find:
Problem 15.3 Modelling the Number of Pieces of Timber
A fence is to be built around a farm in order to contain sheep within
a certain area of the farm. It needs to be made up of pieces of timber as
shown below.
1. One section of fence requires 6 pieces of timber and two sections of
fence requires 11 pieces of timber.
a. How many pieces of timber are required for a fence made up
of 3 sections?
b. How many pieces of timber are required for a fence made up of 4
sections?
c. How many pieces of timber are required for a fence made up
of 5 sections?
2. Let the number of sections of the fence be denoted by s and the
number of pieces of timber be denoted by p.
a. Complete the following table:
b. In mathematics, patterns in numbers are often used to discover a rule for
finding the next number. When we look for a pattern in numbers, we try to
look at each number and find out how it relates to other numbers. This
enables us to state a rule in words and then write it algebraically. By observing the pattern in the above table, find the
relationship that exists between s and p.
3. Draw a graph of p against s.
4. How many pieces of timber are needed for a fence made up of 100
sections |
