b Hence, prove that the product of two odd numbers must be an odd number.
There are many problems in mathematics that can be solved by looking for patterns, then finding rules that describe them. In this section, we will extend our study of patterns in linear relationships to the solution of more general problems. This may involve the use of various problem-solving strategies as well as algebra.
Example
Cubes, similar to the one shown, but of any size, are constructed from small cubes.
The faces of the large cube are then painted blue.
Find an algebraic rule for the number of cubes that are painted, P, and the number that remain unpainted, U.
Solution
The large cube contains 4 × 4 × 4 = 43= 64 small cubes.
The cubes on the inside, which remain unpainted, form a smaller cube of side 2.
Thus, there are 23= 8 cubes which are not painted.
The remainder, 64 − 8 = 56 will all be painted on at least one face.
If this case is generalised to a cube of side length s, then the inner cube of small cubes which are not painted will have side length s− 2.
There will be (s− 2)3 unpainted cubes.
The remaining cubes s3− (s − 2)3 will have been painted.
Painted cubes: P= s3− (s − 2)3 Unpainted cubes: U= (s − 2)3
Generalising solutions to problems using patterns 2.12
EG+S
1 We are interested in finding the minimum number of straight lines of any length that are required to draw each figure.
a Complete the table.
b Describe, in words, the relationship between the number of rows of squares and the minimum number of lines in each figure.
c Write an algebraic statement linking l and n.
d What is the minimum number of lines that are required to draw a figure with 30 rows of squares?
2 A sheet of writing paper is folded in half horizontally, then folded again and again.
The number of creases is recorded at each stage.
a Copy and complete the following table.
b Write down a formula to describe the relationship between the number of creases and the number of folds.
c How many creases would there be if the paper had been folded 7 times?
■ Consolidation
3 Square rooms are tiled with white and black square tiles as shown.
There are x tiles along each side of the room, and the top left tile is always white.
a By considering square rooms of various sizes, find the number of tiles N that are needed to tile a square room of any size.
b Find expressions for the number of white tiles that are needed to tile a square room of any size. (HINT: Consider separately squares with odd and even numbers of tiles on each side.)
c How many black squares would be needed to tile a square room with a side length of 50 tiles?
Number of rows of squares (n) 1 2 3 4
Minimum number of lines (l) 6
Number of folds (f) 1 2 3 4
Number of creases (c) Exercise 2.12
4 This triangular pattern is made up of black and white triangular tiles as shown. There are t black tiles along each side of the triangle. The top tile is always black.
a Find an expression for the total number of tiles in a triangle of any size.
b Find an expression for the number of black tiles in a triangle of any size.
c How many white tiles would there be if there were 10 black tiles along each side of a triangle?
5 A chess board is in the shape of a square with a side length of 8 units. Consider the following problem.
How many squares of any size are there on a standard chess board?
This problem can be made simpler by first drawing smaller diagrams such as those below.
a Copy and complete this table of values from the diagrams above.
b Describe in words the relationship that exists between the side length and the total number of squares in the diagram.
c Write this relationship as a formula linking N and x.
d Use this formula to find the number of squares on a standard chess board.
6 Consider the following three-dimensional models.
a Copy and complete the following table.
Side length (x) 1 2 3 4
Number of squares (N)
Number of rows 1 2 3 4
Number of cubes in bottom layer Number of cubes in second layer Total number of cubes
b Show by substitution that the number of cubes in the bottom layer is , where n is the number or rows.
c Find an expression for the total number of cubes in each figure.
d Hence, find an expression for the number of cubes in the second layer.
e How many cubes will there be in the second layer of the 25th figure?
7 A circle has been divided into a number of regions by drawing several straight lines.
No more than two lines can intersect at any one point and the number of regions is to be a maximum.
a Form a table of values and use it to find a relationship between the number of lines (l) and the number of regions (r).
b How many regions would be formed by the intersection of 10 straight lines?
8 The pyramid shown has 3 storeys. Consider a similar pyramid with n storeys.
a Write down a number pattern that shows the number of cubes in each storey.
b Find an expression for the number of cubes on the bottom layer of a pyramid with n storeys.
c How many cubes would there be in the bottom layer of a pyramid that is 10 storeys high?
d Use the number pattern in a to find the total number of cubes in the pyramid.
■ Further applications
9 Square rooms are to be tiled using two colours as shown.
Taking separate ‘odd’ and ‘even’ cases, establish a rule for the number of white tiles (T) needed, where there are n tiles along each side.
10 For this question refer to the diagrams in Q5.
A person wishes to produce a 10 × 10 square on a computer by repeating a small square.
For example, the 3 × 3 square could be produced using 8 small squares, as the central square is outlined by its surrounding squares.
a Find the minimum number of small squares that can be used to produce the 10 × 10 square.
b Taking ‘odd’ and ‘even’ as separate cases, form equations to find the minimum number of small squares for an n× n square.
Let S represent the total number of small squares required.
n n( +1) ---2
A binomial is an expression that contains two terms. Some examples of binomial expressions are a+ 7, 3m − 4 and x2+ 2x. The product of two binomials is called a binomial product.
Some examples of binomial products are (x+ 6)(x − 4), (2t − 1)(t − 3) and (3a − b)(2a + 7b).
There are several methods that can be used to expand a binomial product.