Building Patterns Incrementally
This unit aims to give students the opportunity to explore recurrence rules for patterns where the increase from one term to the next is not constant. Encourage them to find the recurrence rule. Actually some students will find it easier to find the general rule here as in some ways it is less complicated than the recurrence rule.
continue a patterns
find the recurrence rule of a pattern
look at relations between two patterns
have some idea of what a general rule is
This unit is the fourth in a sequence of units on patterns. The other units in this sequence are Pede Patterns, Letter Patterns, Level 2, and Building Patterns Constantly, Level 3. Either of these Level 1 and 2 units can be taught at Level 3 by making suitable adjustments and by asking appropriate questions. If you have students who you think might benefit from more practice in this area or need to look at slightly simpler patterns, then these two units are an option.In developing this unit we were conscious of the progression that is shown in the exemplar task in Algebra. We give the six stages of this progression below.
- copy a pattern and create the next element
- continue the pattern with systematic counting
- predict values using relations between successive elements
- predict values using rules
- find an algebraic expression for a relation
- solve linear equations related to patterns
The numbers here represent the approximate Levels of that stage of the progression. Hence a student who can copy a pattern and create the next element is operating at Level 1 unless she can also perform the Level 2 stage. With most tasks in this unit, we would expect students at Level 3 to be able to continue the pattern and know what the relation is between consecutive terms of the sequence.
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 the much more concise 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 about the value of any number on the pattern. So for the pattern above, the general rule is that the number connected to any term of the sequence is 2 plus 3(the number of the term). For instance, the third number in the sequence above is 2 plus 3(3), which equals 11. And the sixth number is 2 plus 3(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.
It should be noted here that there are many rules operating in these more complicated patterns. Encourage students to look for any relation between the numbers involved. Don’t be surprised if they come up with a relation that you hadn’t thought of.
In this unit, we concentrate on patterns with a varying difference between consecutive terms. For a unit with patterns where the difference is constant see the unit Building Patterns Constantly, Level 3. For a more difficult unit on patterns where the recurrence rule is not constant see Building Patterns as a General Rule, Level 4. Some students will find it easier to ‘see’ the general rules than the recurrence rules in this unit.
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.
Links to Numeracy
This unit provides an opportunity to focus on the strategies students are using to solve number problems, particularly in the domains of addition and subtraction and multiplication and division.
As students explore the patterns in the numbers for each sequence encourage them to think about and describe the strategies they are using as they manipulate the numbers involved.
Patterns that involve larger numbers call for more sophisticated strategies. As students discuss the strategies they are using, both to each other and the teacher, there will be an opportunity for students to learn about the strategies of others. If students are not grouped according to strategy stage, they will be exposed to a greater range of strategies for the same problem.
For example in the problem from Copymaster Two could be discussed.
Some students may use an additive strategy. For each consecutive number in the pattern the number that is added to the previous term is increased by 2.
2, add 4, add 6, add 8, add 10, add 12….
Different ways to add the number string could be discussed. Can the students combine the numbers in ways that make it easy to find the total?
eg 2 + 8 + 6 + 4 + 10 + 12 = 10 + 10 + 10 + 12
Some students may use a multiplicative strategy, noting that each building involves a multiplication of two successive numbers.
1 X 2, 2 X 3, 3 X 4, 4 X 5 …
Students that can use this strategy have the advantage of being able to calculate the total number of squares for any building in the sequence. Students using an additive strategy need to calculate every number in the sequence progressively.
Questions to develop strategy use
What is the pattern in these numbers?
How did you work that out? What numbers did you combine?
Could you combine the numbers in another way to make it simpler?
We start off with an activity that may well introduce a new kind of pattern to some students. The patterns in the two examples have recurrence rules that are not constant. In other words the difference in number between consecutive terms of the sequence increases as the number of the term increases.
- Build the sequence of ‘apartments’ in the diagram below using square pieces of paper or magnetic tiles on a whiteboard (see also Copymaster 1). Ask questions like the following:
How many green squares do we need for the first apartment? The second? The third?
How many green squares will we need for the tenth apartment? The twentieth?
How would you find out how the pattern is changing?
- It’s hard to do this in our heads.
What is a useful way to get started here? (Encourage the class to suggest that they should make a table. )
- With the help of the class, construct the table below.
Number of green squares
- Maybe that isn’t enough information.
How many squares will the fourth apartment have? The fifth?
Add these numbers to the table.
How many squares would you need to make the 7th apartment?
Which numbered apartment would have 55 squares?
Can you see from the table by how many squares each apartment is increasing? (In words, the number of squares in an apartment is the number of squares in the last apartment plus one more than the number of the apartment. )
- There are other patterns here. For instance, what happens if you add consecutive numbers? (You get the square of the second term used. ) (This can be seen geometrically be rearranging the squares. )
- In Copymaster 2 there is a sequence of factories (see the picture below). Get the class to work in groups to complete the table for those factories and decide what the number pattern is. When they have done that they should report back. Both here and on the Copymaster we have suggested questions that you might like to consider when the students are altogether. The key idea here is the rate at which the difference in the number of squares between two consecutive factories increases.
- Ask questions such as those below.
How many pink squares do we need for the first factory? The second? The third?
What is the best thing to do at this point? (Make a table)
How many pink squares will we need for the fifth factory? The seventh?
What is the recurrence rule of the number pattern for the pink squares? (How does it change as we go from factory to factory?)
What is the number of the factory if we have to use exactly 42 pink squares to make it?
Which of these numbers are not a number of squares for one of these factories: 71, 72, 73, 74?
Which factory has 12 times as many squares as another?
Can anyone see the general rule for this pattern? That is, given the number of the factory, can you make up a rule that gives the number of the squares?
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. See if some of the class can produce the general rule. Also see what other patterns they have found.
The material for these stations is on Copymaster 3 and 4. The students answer the questions. In Copymaster 3 they are provided a table to complete. They are expected to know that they need to make a table for the work on Copymaster 4.
In this piece of work, pay especial attention to the ability to see the relation between one member of the pattern and the next. However, be alert for those students who can find the general rule or some other relation.
The material for today’s station is on Copymaster 5. The students should answer the various questions here. Make sure that they can see the recurrence rule for the patterns. (They may need to be reminded that a table is useful.)
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 today’s station is on Copymaster 6. Again the students complete the table and look for patterns. There are two tasks here and we would not expect most students to be able to finish the whole task in the time available, so get them to concentrate on Task 1.
We are again interested in the rules that govern the numbers of squares in the pattern. Again check that students can see and use the recurrence rule. Encourage those students that are able, to find the general rule in words.
- On the final day let the students make up their own building patterns. They should concentrate on two tasks. The first is to make a building in one colour so that the difference in the number of squares between consecutive buildings increases each time. Let them make up (and answer) questions about their pattern similar to the ones that have been asked above.
- The second task should involve a building with two colours where the increments in the number of squares change by different (non-constant) amounts. Again ask them to make up a couple of questions of the type above.
- Bring the class together and discuss some of the patterns that they have made. Concentrate on the recurrence rule for those patterns. Some students will be able to put the general rule into words. Ask them to do that. Have they seen any other patterns?