In this unit we introduce some number patterns using buildings constructed with squares. The aim of the unit is for the students to be fluent in constructing tables of patterns and in finding the recurrence rule for a simple pattern where the increase from term to term is a constant.
This unit is the third in a sequence of units on patterns. The earlier units in this sequence are Pede Patterns and Letter Patterns, Level 2. In developing this unit we were conscious of the progression that is shown in the New Zealand Curriculum Exemplar for Algebra. We give the six steps of this progression below.
The numbers on the left above, represent the approximate curriculum level of that step of the progression. Hence a child who can copy a pattern and create the next element is operating at Level 1 unless she can also perform the Level 2 stage.
Patterns are an important part of mathematics. They are one of the over-riding themes of the subject. It is always valuable to be able to tell the relation between two things in order to predict what will happen and understand how they inter-relate.
Patterns also provide an introduction to algebra proper as the rules for simple patterns can be first discovered in words and then be written using algebraic notation. The two useful rules that we concentrate on here are the recurrence rule and the general rule. The first of these tells the way that a pattern is increasing. It tells us the difference between two successive terms. So if we think of the pattern 5, 8, 11, 14, 17, … we can see that this pattern increases by 3 each time. So here the recurrence rule says that the number at any stage in the pattern is 3 more than the previous number.
The general rule tells us the value of any number of the pattern. So for the pattern above the general rule is that the number of any term of the sequence is equal to 2 plus 3 x the number of the term. For instance, the third number in the sequence above is 2 plus 3 x 3, which equals 11. And the sixth number is 2 plus 3 x 6 = 20.
To see why this general rule works it is useful to write the initial term (5) in terms of the increase (3). So 5 = 2 + 3.
In this unit, we concentrate on patterns with a constant difference between consecutive terms.
It should be noted that not all of your students will be able to grasp and use the idea of the general rule at this stage in their development. However, we introduce it here as part of the unit to give students who can grasp the concept some practice in using it. It will also give all students a chance to meet the idea and will be the basis for further work later at Level 4
This unit provides an opportunity to focus on the strategies students are using to solve number problems, in particular their strategies for solving multiplication and division problems.
The numbers involved in the building patterns are all based on multiplicative sequences. Encourage students to think about these patterns by focusing on the different strategies that can be used to calculate successive numbers in the pattern.
For example, the pattern for the third factory on Copymaster Two can be seen as in a variety of ways:
9 X 3 7 X 3 + 2 X 3 7 X 3 + 6
3 X 7 + 2 X 3 3 X 7 + 6 3 X 7 + 3 + 3
Considering these different ways to express the pattern using numbers will help students develop knowledge of the relationships between the number operations.
Questions to develop strategic thinking
What numbers could you use to describe the way the pattern is made?>
How did you work that out?
Can you think of any other numbers you could use?
Encourage students to explain their thinking and to see the patterns in a variety of ways.
squares of coloured paper or grid paper
sequential patterns, multiples, recurrence rule, general rule, successive, prediction
We start off with an activity that should be straightforward for most students. Then we look at another where the key idea is that two numbers in the pattern are a constant number apart.
number of grey squares
number of red squares
number of all squares
Can you see from the table by how many squares each factory is increasing?
How many squares would you need to make the 7th factory?
At the end of the session share findings, asking questions similar to those asked about the first sequence of factories. For example:
How many grey squares do we need for the first factory? The second? The third?
How many grey tiles will we need for the tenth factory? The hundredth? The 3,298th?
What is the number pattern for the grey squares?
How many red squares do we need for the first factory? The second? The third?
If there are 4 grey squares how many red squares are there?
If there are 10 grey squares how many red squares are there?
What is the number pattern for the red squares?
How do we get the number of red squares for one factory from the number of red squares in the factory before that?
If we have to use exactly 33 red squares in making a factory, how many grey squares would we need?
How many squares do we need altogether for the first factory? The second? The third?
Who can tell me how many squares we’ll need for the fourth factory?
Can someone come and show us how to make the fifth factory?
What is the number pattern that we are getting for the total number of squares?
If we had 35 tiles, which numbered factory could we make?
Which of these numbers are not a number of squares for one of these factories: 43, 44, 45, 46?
Over the next 3-4 sessions the students now work at various stations continuing different number patterns and building up the corresponding tables. The station work will take about three days. On each day get the whole class together at an appropriate time to discuss the results of their work. Check that they have been able to answer all of the questions correctly and understand what they have been doing. Place special emphasis on the recurrence rule that exists in each pattern.
The material for these stations is on Copymaster 3. The students complete the table and answer the questions.
In this piece of work, pay especial attention to the ability to see the relation between one member of the pattern and the next (the recurrence rule).
The material for these stations is on Copymaster 4. The students complete the table and use it to answer various questions. Make sure that they can see the recurrence rule for the patterns.
Here you should also consider the overall number patterns – the general rule. If a pattern starts with 5, say and increases by 2 for each building, then the number of squares will build as follows: 5, 5 + 2 = 7, 5 + 2 + 2 = 9, 5 + 2 + 2 + 2 = 11. So a rule for the final pattern is 5 plus 2(one less than the number of building). But we can make this easier if we think of 5 as being 3 + 2. Then the pattern can be thought of as 3 + 2, 3 + 2 + 2, 3 + 2 + 2 + 2, and so on. So a rule for the final pattern is 3 plus 2(number of building). Help the students to see this but realise that not all of them will be able to grasp this idea at their stage of development. These students, however, should be able to see that the number increases by a constant amount each time and be able to use that recurrence rule to predict the number of squares in any future building.
The material for these stations is on Copymaster 5. Again the students complete the table and look for patterns.
We are again interested in the rules that govern the numbers of squares in the pattern. Check that students can see and use the recurrence rule. Encourage those students that are able, to find the general rule in words.
The twist in this station is to look for relations between patterns of the two colours. For instance when is one colour double the other? It is possible to do this by simply increasing the table and checking the numbers out. But it is also possible to do this by using the recurrence relation or the general rule. Encourage them to use these rules.