An Artist's Delight

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Purpose

This is a level 4 algebra strand activity from the Figure It Out series.

A PDF of the student activity is included.

Achievement Objectives
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Student Activity

Click on the image to enlarge it. Click again to close. Download PDF (709 KB)

Specific Learning Outcomes

use a diagram or table to find a pattern

use a rule to describe a pattern

Required Resource Materials

FIO, Level 4, Algebra, Book Two, An Artist's Delight, page 12

Activity

In this activity, students consider the number of ways that a set of colours can be placed in order.
It might be helpful to use coloured multilink cubes to explore the possible orders that Turi could colour her design. For example, there are 2 orders for blue and green: either blue first and then green or green first and then blue.

cubes.
When Turi adds another colour, red, she could either use a trial-and-error strategy or a systematic strategy to figure out all the 6 possible orders to colour her design. While a trial-and-error strategy is fine when the number of colours is small, the number of ways in which the colours can be ordered increases very quickly.
Encourage the students to develop a systematic strategy so that they can convince themselves and others that they have found all possible orders. Four such systematic strategies are shown below.
Strategy 1: Use one colour in the first, then the second, and finally the third position, and record all the possibilities for the other colours. For example, put red in the first position:

cubes.
Then put red in the second position:

cubes.
Finally, put red in the third position:

cubes.
Strategy 2: Use each colour in the first position.

patterns.
Strategy 3: Use each colour in the second position.

patterns.
Strategy 4: Use each colour in the third position.

patterns.
All the strategies produce the same 6 possible orders of the colours.
In question 2, the students investigate adding a fourth colour, orange. It turns out that there are 24 orders altogether, so it will not be practicable to make them all with multilink cubes. Some students may want to try drawing the different orders on square grid paper, but it will be more profitable for the students to see if they can find a way to record their results systematically. Most students, however, are unlikely to be able to get started on their own initiative. A sensible way to record their results for 4 or more colours involves using the diagram for strategy 2, with each colour in the first position.
Instead of putting the possible arrangements for red first, blue first, and green first side by side, they can be arranged below each other to form the basis of a tree diagram.

cubes.
The tree diagram shows that there are 2 orders for using 3 colours with red first. There are also 2 orders for 3 colours with blue first and 2 orders for 3 colours with green first. So altogether, there are 2 + 2 + 2 = 3 x 2 or 6 orders for 3 colours. A tree diagram for 4 colours might look like this:

tree diagram.
A table can help the students to see the pattern at work here:

table.
There is a shorter way to write numbers in the form 3 x 2 x 1, and so on. For 3 x 2 x 1, this is 3!, and we say “factorial” for the exclamation mark. Factorials are used a lot in probability, and they are also interesting in that they grow very quickly. The following table shows the factorial pattern:

table.

Note that all mathematical and scientific calculators have a factorial function key.

Answers to Activity

1. BRG, BGR, GRB, GBR, RBG, RGB
2. a. 24 different orders. You can generate four lists of equal length:

table.

b.

table.
c. 120. (5 x 24 = 120)
3. The number of different orders is the number of colours multiplied by all the earlier numbers of colours. (For 5 colours, this is 5 x 4 x 3 x 2 x 1 = 120. For 10 colours, it would be 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3 628 800.)

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Level Four