Download Pattern 3

Name:
Patterns and Sequences
When working with a sequence of figures, it’s good to draw the next several shapes in the
sequence to make sure that you are clear about how the sequence is changing.
Instructions For each pattern below,
a) draw shapes 4 and 5; and
b) write out in words how one shape in the pattern is created from the previous one.
Pattern 1
[The shapes from Building with Toothpicks.]
Pattern 2
Shape 1
Shape 2
Shape 3
Pattern 3
Shape 1
Shape 2
Shape 3
Pattern 4
Shape 1
Shape 2
Shape 3
The Point When looking for a pattern in a sequence of shapes, the shapes must change in the
same way from one shape to the next in the sequence, no matter where you are in the sequence.
Name:
Translating a Sequence of Shapes into a Number Sequence
When looking at geometric patterns, we often want to know things like:
 How many toothpicks are in the perimeter of the 100th shape from pattern 1?
 How many dots are there in the 31st shape in pattern 2?
 How many squares are there in shape n in Pattern 3?
Each of these questions identifies some aspect of the shape that could be counted and ask us to
count it. In Building with Toothpicks, the aspect of the shapes we are interested in is the number
of toothpicks in the perimeter of the shape. To organize this information about the perimeter, we
create a T-chart with the shape number in the first column and the number of toothpicks in the
perimeter of that shape in the second column.
Instructions Fill-in the following T-chart for the perimeter of each shape in Building with
Toothpicks.
Shape number
Number of toothpicks in the
perimeter of that shape
1
2
3
4
5
Fill in the following T-chart for the number of squares in each shape of pattern 2.
Shape number
1
2
3
4
5
Area of the shape
Fill in the following T-chart for the number of squares in each shape of pattern 3.
Shape number
1
2
3
4
5
Area of the shape
Name:
Exploring Number Patterns in a T-chart
Now we want to explore number patterns in the T-chart for Building with Toothpicks. In
particular, we are interested in how the shape number is related to the number of toothpicks in
each shape and in how the sequence of numbers in the second column behaves.
Instructions Answer the following questions with WORDS.
1. How is the shape number related to the number of toothpicks in the perimeter for that
shape?
2. Copy the T-chart for Building with Toothpicks below. In words, how does the sequence
of numbers in the second column change?
Shape number
1
2
3
4
5
6
…
n
n 1
Number of toothpicks in the
perimeter of that shape
…
?
??
Instructions Now answer the following questions using SYMBOLS.
3. How is the shape number related to the number of toothpicks in the perimeter for that
shape?
4. How does the sequence of numbers in the second column change?
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Using Notation to Describe Number Patterns in a T-chart
We’d like to begin using symbols to talk about the number patterns in the T-chart. Let n
represent a general shape number, so n  1 represents the next shape number and n 1 represents
the previous shape number. So if we let n  10 we are talking about the 10th shape. Then n  1
represents the 11th shape and n 1 represents the 9th shape.
Subscript Notation
Now we need some notation to describe the number of toothpicks in shape n. We are going to
use subscript notation, that is, we are going to let the symbol tn represent the number of
toothpicks in shape n. Here we use the letter t to remind us that we are talking about toothpicks.
The n in the symbol tn is called a subscript and we say “t sub n” to refer to the symbol tn . So,
t10 refers to the number of toothpicks in shape 10 and we read t10 as “t sub 10.”
Function Notation
We could have used function notation instead, that is, we could refer to the number of toothpicks
in shape n with the symbol t  n  . In both cases, the notation is designed to reflect the fact that
the number of toothpicks in shape n is somehow related to the number of the shape. We say “t of
n” to refer to the symbol t  n  . So, t 10  refers to the number of toothpicks in shape 10 and we
read t 10  as “t of 10.”
›› Common Confusion
When first encountering the symbol t  n  , it is natural to think that it means to
multiply t by n since we often indicate multiplication by parentheses and writing two
terms next to each other. For example, when we see 3  x  5 , we know that this
means to distribute the 3 through the parentheses, that is, multiply 3 by x and 3 by 5 to
get 3x 15 .
Be conscious of what symbols like t  n  and tn mean as you work through the activities and
exercises.
Two ways to talk about tn : the recursive equation or the general equation
Now we have a choice in how we describe tn : we can write tn in terms of n or we can write tn 1
in terms of tn . When we write tn in terms of n, we only need to know the shape number to
determine the number of toothpicks in shape n. Writing tn 1 in terms of tn , tells us how to
determine the number of toothpicks in shape n  1 if we already know the number of toothpicks
in shape n.
The general equation is an equation in n that tells us how to compute the number of toothpicks
in the perimeter of shape n. We name this equation tn in subscript notation and t  n  in function
notation.
Name:
1. Write an equation that tells us how to compute the number of toothpicks in the perimeter
of shape n.
The recursive equation describes how the number of toothpicks in shape n  1 is related to the
number of toothpicks in shape n. Using subscript notation, we want to describe how tn 1 is
related to tn . Using function notation, we want to describe how t  n  1 is related to t  n  .
2. Write a recursive equation that relates tn 1 and tn .
Summary
The recursive formula tells you the number of new objects to add to the previous one to get the
new one. The recursive formula encodes how the number pattern changes from one entry in the
table to the next. This change in the number pattern corresponds to a change in the figures from
one shape to the next. We want a systematic way to create shape n  1 from shape n. The
recursive description describes in words how to systematically represent the change in the
shapes. The recursive formula describes how the number pattern is changing. This corresponds
to the general algebra standard of analyzing change in various contexts.