Transcript Lesson 1-1 PowerPoint - peacock

Algebra 2
Patterns and Expressions
Lesson 1-1
Goals
Goal
• To identify and describe
patterns.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to solve
simple problems.
Level 4 – Use the goals to solve
more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Essential Question
Big Idea: Variable
• How do variables help you model real-world
situations?
– Students will use an expression to model the nth
term of a pattern.
– Students will use variables to represent unknown
quantities in real-world situations.
Vocabulary
•
•
•
•
•
Constant
Variable quantity
Variable
Numerical expression
Algebraic expression
Patterns
• What is the pattern of geometric figures below?
• How can you identify a pattern?
– Look for the same type of change between consecutive
figures.
• Each figure in the pattern is created by rotating the
previous figure 90˚ clockwise.
Patterns
• What does the eighth figure in the pattern look like?
• Because each figure is rotated 90˚, there are four
figures in the pattern. Therefore, the eighth figure
will be the same as the fourth figure.
Patterns
Patterns can be represented using;
1)
2)
3)
4)
Diagrams (geometric patterns, graphs, etc.)
Words
Numbers
Algebraic expressions
Geometric Patterns
• Look at the figures from left to right.
Hexagon
Triangle hasSquare
3 sides.
has
Pentagon
4 sides.has
5 sides.has 6 sides.
• What is the pattern?
– Look for the same type of change between consecutive
figures.
• The pattern shows regular polygons with the number
of sides increasing by one.
Geometric Patterns
• What would the next figure in the pattern look
like?
• The last figure in the pattern has six sides, so
the next figure would have seven sides, a
heptagon.
Your Turn:
• Look at the figures from left to right.
• What is the pattern? Draw the next figure in the
pattern?
Answer: The pattern shows a red center square and a yellow
square added to each side with the number of squares per
side increasing by one.
Definitions
• Constant – a quantity whose value does not
change.
– Examples: 8, -23, π
• Variable Quantity – a quantity that can have
values that vary.
– Examples: distance, time, cost
Definitions
• Variable – a symbol, usually a letter, that
represents one or more numbers.
– Examples: n, x
• Numerical Expression – a mathematical phrase
that contains numbers and operation symbols.
– Examples: 3 + 5, (8 – 2) + 5
• Algebraic Expression – a mathematical phrase
that contains one or more variables.
– Examples: 3n + 5, (8x – 2) + 5n
Question?
• What is the only difference between algebraic
expressions and numerical expressions?
Answer: An algebraic expression contains one or
more variables, while a numerical expression
contains no variables.
The nth Term
• By "the nth term" of a pattern we mean an
algebraic expression that will allow us to
calculate the term that is in the nth position
(any position) of the pattern.
• For example consider the pattern:
1st term 2nd term 3rd term 4th term …nth term
3
9
27
81
… 3n
Algebraic Patterns
• Use the pattern to determine how many
toothpicks are in the 20th figure?
Algebraic Patterns
• You can use a table to find the pattern that
relates the figure number (term) to the
1
2
3
number of toothpicks? term
term
term
st
4
nd
rd
4
4
4
Term (input)
1
Process
⨯
2
⨯
3
…
⨯
20
…
20 ⨯ 4
4
4
# Toothpicks (output)
4
8
12
…
80
Pattern: To get the number of toothpicks (output), multiply the term (input) by 4.
Algebraic Patterns
Term (input)
Process
# Toothpicks (output)
1
4⨯1
4
2
4⨯2
8
3
4⨯3
12
…
…
…
n
4⨯n
4n
• What is an algebraic expression for the number of
toothpicks in the nth figure (the nth term)?
– Use the pattern from the table. The expression is formed
by multiplying the term number, n, by 4.
– There are 4n toothpicks in the nth figure.
Algebraic Patterns
Two Stage Algebraic Expressions
• Some patterns are not as straight forward and
you may have to use two stages.
• This means that after multiplying your term
number, you may need to add or subtract a
number to reach your answer.
Example
The pattern
1st term
2 nd term
3rd term
4th term
5th term
Pattern table
Term
1
2
3
4
5
# of
Squares
1
3
5
7
9
2
2
2
2
1st term
2 nd term
3rd term
4 th term
5th term
The Process or Rule
1st term
2 nd term
3rd term
4 th term
5th term
Term
1
2
3
4
5
# of
Squares
1
3
5
7
9
• Notice that the pattern is adding 2 to each figure.
• And almost matches the multiples of 2 so lets see
how that works.
The First Stage
Term
1
2
3
4
5
# of
Squares
1
3
5
7
9
Term× 2
2
4
6
8
10
We are not quite there, what do we need to do
to reach the number of squares?
The Second Stage
Term
1
2
3
4
5
# of
Squares
1
3
5
7
9
Term×2
2
4
6
8
10
Subtract 1
1
3
5
7
9
We need to subtract one after we have multiplied.
The Algebraic Expression
• The algebraic expression is:
2 × the term number - 1
Remember to change to algebra we use
variables:
the term number is always "n“
• The algebraic expression is:
2n - 1
Example:
Geometric patterns can be represented
numerically and generalized algebraically.
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Term #
Description
Process
# of blocks
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Term #
1
Description
Process
1 row of 2 plus 1
1(2)+1
Term #1
# of blocks
3
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Term #
Description
Process
1
1 row of 2 plus 1
1(2)+1
3
2
2 rows of 2 plus 1
2(2)+1
5
Term #2
Term #1
# of blocks
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Term #
Description
Process
1
1 row of 2 plus 1
1(2)+1
3
2
2 rows of 2 plus 1
2(2)+1
5
3
3 rows of 2 plus 1
3(2)+1
7
Term #2
Term #1
# of blocks
Term #3
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Term #
Description
Process
1
1 row of 2 plus 1
1(2)+1
3
2
2 rows of 2 plus 1
2(2)+1
5
3
3 rows of 2 plus 1
3(2)+1
7
Term #2
Term #1
# of blocks
Term #3
The number changing in
each process is the
number of rows.
Example Continued
Let’s create a table to see the relationship
between each build and the number of blocks…
Term #
Description
Process
1
1 row of 2 plus 1
1(2)+1
3
2
2 rows of 2 plus 1
2(2)+1
5
3
3 rows of 2 plus 1
3(2)+1
7
Term #2
Term #1
# of blocks
Term #3
The number changing in
each process is the
number of rows.
So the rows are our
variable (n)…
And the nth term is:
2n + 1
Your Turn:
• How many squares are in the 25th figure in this
pattern? Use a table of values. What is the
algebraic expression for the nth term?
Solution:
Term (input)
Process
# Squares (output)
1
2(1) + 2
4
2
2(2) + 2
6
3
2(3) + 2
8
…
…
…
25
2(25) + 2
52
• The pattern is to multiply the term number by 2
and then add 2. So, there are 2(25) + 2 = 52
squares in the 25th figure.
• The algebraic expression for the nth term is:
2n + 2
Graphed Patterns
• You want to set up an aquarium
and need to determine what size
tank to buy. The graph shows
tank sizes using a rule that
relates the capacity of the tank
to the combined lengths of the
fish it can hold.
• If you want five 2-in. red fish,
four 1-in. blue fish, and a 3-in.
green fish, which is the smallest
capacity tank you can buy: 15gal, 20-gal, or 25-gal? Use a table
to find the answer.
Graphed Patterns
Wanted five 2-in. red fish, four 1-in. blue fish, and one 3-in. green fish.
• First, calculate the combined fish
length.
• (5)(2) + (4)(1) + 3 = 17 in.
• Now, how can you use the graph
to find the size tank needed?
• You can use the graph to make a
table and find a pattern relating
combined fish length (input) to
tank capacity (output).
Graphed Patterns
• How do you make a table from
the graph?
• First, chose points that have
whole number coordinates, like
(0, 2), (5, 7), and (10, 12).
• Then make the table using the
input and output values shown
in the ordered pairs.
Length (input)
Process
Cap. (output)
0
2
5
7
10
12
Graphed Patterns
• Next, use the process column of the table to find
the pattern relating length (input, x-coordinate) to
capacity (output, y-coordinate).
Length (input)
Process
Cap. (output)
0
0+2
2
5
5+2
7
10
10 + 2
12
• Therefore, the pattern is; output = input + 2
• Using variables, let L = combined length of fish and
C = capacity of the tank, then
C=L+2
Graphed Patterns
• Finally, we need to answer the question
“which is the smallest capacity tank you can
buy: 15-gal, 20-gal, or 25-gal”.
• Using C = L + 2 and L = 17, C = 19 or the
tank capacity we need is 19-gal. Therefore, the
smallest capacity tank you can buy is 20-gal.
Your Turn:
• The graph shows the
total cost of platys at the
aquarium shop. How
much do six platys cost?
Use a table and find an
equation relating number
platys to total cost.
Solution
• Output = Input • 2
• Using variables, let P = number
platys and C = total cost, then
C = 2P.
• For six platys, P = 6 and
C=2•6
C = 12
• Six platys cost $12.
Number Patterns
• What is the pattern?
– Look for the same type of change between consecutive
numbers or terms.
• Find the next term in each pattern.
21
1) 1, 1, 2, 3, 5, 8, 13, ____
49
2) 1, 4, 9, 16, 25, 36, ____
T
3) O, T, T, F, F, S, S, E, N, ____
Your Turn:
• Find the next term in each pattern.
64
1) 1, 4, 3, 16, 5, 36, 7, ____
18
2) 6, 8, 5, 10, 3, 14, 1, ____
2
3) B, 0, C, 2, D, 0, E, 3, F, 3, G, ____
Assignment
• Section 1-1, Pg 7 – 10; #1 – 7 all, 8 – 48 even.