|
Concept map for
patterns
Introduction
The study of patterns is a key
part of algebraic thinking. It is important
that students are able to recognise and analyse
patterns and make generalisations about them.
Patterns can be created from shapes, sounds,
colours or numbers. Patterns can be visual or
spatial. They can be repeating (A B B A B B A B
B A) or they can be growing (1, 4, 7, 10, 13).
Students can continue a given pattern, find the
missing element or elements in a pattern, or
make up their own pattern. Patterns can also be
analysed and general rules created to describe
the patterns. Work with number patterns helps
develop facility, flexibility and familiarity
with numbers as well as building understanding
of general number properties.1 Work with patterns
also allows students to develop logical
reasoning skills, make conjectures, and test
their ideas about them.2
Pattern exploration leads naturally to the
development of rules to describe the pattern and
strategies to work out subsequent members of a
given pattern.
1 :
Stacey, K. and M. Macgregor (1997). "Building
Foundations for Algebra." Mathematics
Teaching in the Middle School 2(4):
252-260
2
:
Ibid, Meeks Roebuck, K. I. (2005). "Coloring
Formulas for Growing Patterns." Mathematics
Teacher 98(7): 472-475.
Recursive (sequential) strategies
Early number pattern work often
focuses on finding the next number in a sequence. Students can use a recursive strategy to work this out. This means that they determine the
next number in the pattern based on the previous number. When asked to find the
next two numbers in the following sequence: 3, 6, 9, 12, __, __ students will
state that the rule is "add 3" and to get the next number they add 3 to 12,
giving 15, then another 3 to that answer, giving 18. This strategy can also be
called the recurrence relation or the iterative strategy.
Research shows that students tend to use the recursive strategy in their
pattern work. This way of working out rules for patterns is a powerful strategy
that is used in computer programming. However its usefulness in the type of
pattern work students will encounter in their school years is limited. For
example, if asked to find the 20th number in the sequence above,
students using a recursive strategy would have to work out all the numbers
between the last given number (12) and the 20th number (60). |
Resources
|
Explicit or functional (direct) strategies
The New Zealand Curriculum
states that, at Level 3, students need to solve problems
that "connect members of sequential patterns with their
ordinal position…" In other words, students need to
develop explicit or functional strategies to work out a rule to ascertain what comes next in a
number pattern. Using the example above, to determine
the next two numbers in the pattern, students might put
the sequence into a table, with the ordinal position of
the number in the pattern shown along the top row.
| Ordinal position: |
1st |
2nd |
3rd |
4th |
5th |
6th |
7th |
nth |
| Number: |
3 |
6 |
9 |
12 |
? |
? |
? |
? |
By exploring the relationship between the ordinal
position of the number in the pattern and the number
itself, a functional rule can be generated. The number
in the pattern is a function of its position.
For example, the 4th number in the sequence
is 12.
12 is 4 (ordinal position) × 3.
In this case the general rule can be
stated as "the position of the number × 3". Pronumerals
such as n can be introduced to develop this general rule for the pattern. In this way n can replace “the position of the number” and stand for
any number, making the rule n × 3 or 3n. The table can then be
completed:
| Ordinal position: |
1st |
2nd |
3rd |
4th |
5th |
6th |
7th |
nth |
| Number: |
3 |
6 |
9 |
12 |
15 |
18 |
21 |
3n |
Functional strategies and rules are very
powerful because they allow the user to work out any number in a pattern without knowing its predecessor.
The information and resources developed
for this concept map explore some ways in which students
can be encouraged to develop functional or explicit strategies in their work with patterns. |
Repeating patterns
Exposure to repeating patterns comes
early - often well before students are at school. It is
important for young students to explore repeating
patterns and to see that patterns continue, sometimes in
more than one direction. However, repeating patterns
can also be used to start thinking in a more functional
way. Simple repeating patterns such as
can initially be used with questions such as:
• |
What comes next? |
| • |
What comes before the first square? |
| • |
What are the next three shapes in this pattern? |
However, they can also be used to ask
questions such as:
• |
What would the 10th shape look
like? |
| • |
What would the 21st shape be? |
or statements such as:
This type of questioning begins to move
students towards developing strategies to look beyond
just working out what the next shape will be. It is
also the beginning of assigning an ordinal number to
each shape, which becomes important later when working
out general rules for number patterns. To assist in
this process, numbers can be placed underneath the
shapes when working out the 10th or 21st shape to start encouraging students to see the
relationship between the ordinal position and the shape.
If students work out the nth shape by drawing all
the shapes in between, get them to find the 40th or 50th or even 100th shape to
encourage them to use a more functional strategy that
does not require counting on.
One common misconception when students are working with
repeating patterns is that they will often repeat what
is given rather than looking at what "chunk" or part of
the pattern is actually being repeated. For example,
given the pattern
and asked to draw the next shape in this pattern, a
student may draw a square because they return to the
beginning of the given pattern and repeat it from
there. While technically not incorrect, the student
should be encouraged to look for the repeating "set" -
in this case the square, circle and triangle. These
could even be circled or shaded to show which part is
the repeating part of the pattern.
Once students are comfortable identifying the repeating
"set" or "chunk" and predicting the nth shape
when there is a single variable (e.g. shape), introduce
other variables such as shading, size or orientation.
Ask students to draw the next three shapes in this
pattern, and then draw the 12th shape, 23rd shape etc.
Size and/or orientation, for example, can also be used
to create increasingly complex patterns. Students can
also create their own repeating patterns. |
Resources
|
Using repeating
patterns to think functionally
Once students are confident and
comfortable recognising the "set" or "chunk" in a
repeating pattern and extending that pattern, a more
functional way of looking at patterns can be introduced.
Using beads or tiles, very simple repeating patterns can
be built up and used as a way of introducing the concept
of n.
Start with a simple repeating pattern such as blue,
blue, red. Give students tiles or beads and threading
string and get them to make several 'sets' of the
pattern.
Either as a class, in pairs, groups, or
individually, construct a table with the 'set' number,
the number of blue beads and red beads (or tiles) needed
to make that set and, if desired, the total number of
beads or tiles for each set. Cover up the ”sets” of
beads or tiles and reveal each set as the table gets
filled in.
| Set |
Blue |
Red |
Total |
|
|
|
|
| 1 |
2 |
1 |
3 |
| 2 |
4 |
2 |
6 |
| 3 |
6 |
3 |
9 |
| 4 |
8 |
4 |
12 |
| ... |
... |
... |
... |
| ... |
... |
... |
... |
Ask students to describe the relationship
between the set number, the number of blues and the
number of reds and the total number of beads (tiles).
Encourage them to look across the table.
The following statements are examples from Year 6 students who
carried out this activity:
Red is half the number of the blue.
The number of red is 1/3 of the total.
The number of blue is 2/3 of the total.
The number of blue is twice the number of red.
One student
said "The blue is the 2 times table." This indicates a student who is looking down the table rather than across it. They may be using a recursive strategy to work out what comes next in the pattern.
Next, ask students a series of questions to encourage them to explore the
relationship across the table between the set number (ordinal position) and the
number of blue and red beads or tiles. Have students explain how they worked
out their answers to see if they are using a functional strategy.
Examples of questions could be:
• |
If I have 10 blue beads (tiles), how many red beads
(tiles) will I need to make the pattern? |
| • |
If I had 48 red beads (tiles), how many blue beads
(tiles) would I have? |
| • |
If I had 212 blue beads (tiles), how many red beads
(tiles) would I have? |
| • |
If I have 12 'sets' of beads (tiles), how many blue
beads (tiles) would I have? How many red beads (tiles)
would I have? |
| • |
If I have 30 beads (tiles) in total, how many are
blue and how many are red? |
If students continue to use a recursive strategy to work out their responses, use
larger and larger numbers to encourage them to look more
functionally at the pattern.
Discuss how a rule could be developed to describe the
pattern. For example, students might say that the
number of red beads (tiles) is the same as the 'set'
number or that the number of blue beads (tiles) is two
times the 'set' number or twice as much as the red beads
(tiles).
Having explored the pattern in this way, an opportunity
now exists to introduce the concept of n.
Explain that n represents 'any number'. Ask
students to explain their rule in terms of ‘any number’
– for example they could state: "For any number of sets,
there will be three lots of red beads." Now have them
substitute n for "any number". Record student
responses as they are stated e.g. "Triple n" or "n × 2" or "n + n". Clarify that mathematicians usually say 2n rather than n × 2 but accept what
students give, modelling the mathematical format where
possible.
This is also an opportune time to show
students that n × 2 is the same as n + n or Triple n and 3n are equal. | Resources
|
Functional
(direct) relationships
There are many ways in which functional
relationships can be explored. In our research we
started by exploring patterns that could be made from
ice block sticks or matchsticks as a way to look at how
the construction of a pattern can inform the rule for
that pattern.
Initially we used simple patterns with rules such as 2n,
3n or 4n because these were the types of
rules we had co-constructed while working with bead
patterns. Students used ice block sticks to physically
build the patterns, then put their findings into a table
and used this information to work out the rule for the
pattern.
For example, the following "Tent" pattern
was used:
Students were asked to construct the pattern using ice block
sticks, then make a table and record how many sticks were needed to make each
tent pattern. They were then asked to continue the pattern in order to work out
what the rule for the nth pattern would be.
Warren and
Cooper (2008) conducted research into patterns with 8 to 9 year old students.
They made the ordinal position of their patterns visually explicit by placing
position cards underneath the pattern, helping to emphasise the link between the
pattern and the pattern number. This is demonstrated below with the matchstick
pattern “Flats”.
As well as working
out how many matchsticks are needed to make a certain pattern number, students
can also be asked questions such as “How many tents can I make with 28
matchsticks?” or “How high would a flat be that had 36 matchsticks in it?” In
this way students are encouraged to look across their tables from number of
matchsticks to pattern number and use their rule in reverse. In our research we
did a little of this but our main focus was on getting students to see the
pattern and make a general rule for it.
When
students have got the idea of simple rules such as 2n, 3n, 5n,
etc., introduce them to patterns that require both multiplying and adding. Some
examples are:
When constructing
the xmas trees, for example, each tree has two sticks for the trunk then a
certain number of triangles, each made with three matchsticks. Encourage
students to see that the number of triangles matches the ordinal number.
Placing ordinal cards underneath their constructions could help with this.
Students could be asked “What would the 6th pattern look like?” and
asked to construct it without making all the patterns in between. Ask the
students to explain how they could work out what the 6th or 10th or 25th patterns would look like. At this point a rule explaining
that the pattern is “the ordinal number times three plus 2” is a good starting
point. From here, students can use n in place of the ordinal number and
then this can be used to describe the rule more succinctly as 3n + 2.
Similarly
with the beds pattern, the bed “head” and “legs” remain the same throughout and
require three sticks in total to construct, while the “base” of the bed
increases – the number of sticks needed to make it is the same as the ordinal
number of the pattern. Therefore, the pattern is n + 3.
The
classic matchstick pattern:
can be used
to explore the different ways that the same pattern can be built up and the rule
described in apparently different ways.
For example, this pattern can be constructed by putting down 1
matchstick
then
adding three more each time to construct a set of 4 squares.
This can be described as:
1 + 3 + 3 + 3 + 3
Another
way to construct the pattern is to make the first square:
Then add three matchsticks each time to make the set of 4
squares:
This
can be described as:
4 + 3 + 3 + 3
A third
way to construct the pattern is to place the 5 vertical matchsticks first:
Then add two rows of 4 matchsticks to construct the set of 4
squares:
This
pattern can be described as:
5 + 4 + 4
Although these pattern descriptions may at first appear different, when they are
presented as a general rule, it can be seen that they are equivalent.
1 + 3 + 3 + 3 + 3 becomes 1 + (4 × 3) which can be generalised as 1 + (n × 3) = 3n + 1
4 + 3 + 3 + 3 becomes 4 + 3 × 3 which generalises to:
4 + 3(n – 1) = 4 + 3n – 3 = 3n + 1
5 + 4 + 4 generalises to (n + 1) + n + n = n + n + n + 1 = 3n + 1
The most
difficult aspect of constructing these rules from the initial description of the
pattern is firstly recognising that n is the number of squares (in this
case, 4) and then using n to describe other numbers. In other words,
understanding that if n is 4, then 5 can be written as n + 1 or 3
is the same as n – 1 can be a difficult concept for students to
understand. However, there are opportunities here to explore some algebraic
generalisations. | Resources
|
Meeks Roebuck,
K. I. (2005). "Coloring Formulas for Growing Patterns."
Mathematics Teacher 98(7): 472-475.
Stacey, K. and M. Macgregor (1997). "Building
Foundations for Algebra." Mathematics Teaching in the
Middle School 2(4): 252-260.
ARB
Home
|
New Zealand Council for Educational Research. All rights reserved 
