1. Patterns 7: Reaching Final Conclusions
What have we discovered?

2. Tell you partner everything you know about this table
What are the most important features of this table?
Figure
Number
Number of Toothpicks 4 14 5 17 6 20 7 23

3. Vocabulary The starting value is also called the Initial Value
The recursive pattern is also called the Rate of Change
Figure Number
Number of Toothpicks 4 14 5 17 6 20 7 23
Careful! 14 is NOT the Initial Value

4. Vocabulary The starting value is also called the Initial Value
The recursive pattern is also called the Rate of Change
Figure Number
Number of Toothpicks 4 14 5 17 6 20 7 23
Careful! 14 is NOT the Initial Value

5. Vocabulary The starting value is also called the Initial Value
The recursive pattern is also called the Rate of Change
Figure Number
Number of Toothpicks 4 14 5 17 6 20 7 23
Careful! 14 is NOT the Initial Value

6. 4 14 5 17 6 20 7 23

7. The Initial Value is _________________________2 1 5 8 3 11 4 14 17 6 20 7 23 -3 -3 -3 -3 ? +3 +3 +3
The Initial Value is _________________________
The Rate of Change is ______________________

8. Figure Number
Number of Toothpicks
Recursive Pattern as Repeated Addition
Using Definition of Multiplication
Number of Toothpicks 2 5 8 11 14 17 20 23 2 1 5 8 3 11 4 14 17 6 20 7 23 2
2 + 3∙0
2 + 3
2 + 3∙1
2 + 3∙2
2 + 3∙3
2 + 3∙4
2 + 3∙5
2 + 3∙6
2 + 3∙7 = = = = = = = =

9. What does the 2 represent? What does the 3 represent?
number that’s
changing represent?
What does the 2 represent?
Figure Number
Number of Toothpicks
Recursive Pattern as Repeated Addition
Using Definition of Multiplication
Number of Toothpicks 2 5 8 11 14 17 20 23 2 1 5 8 3 11 4 14 17 6 20 7 23 2
2 + 3∙0
2 + 3
2 + 3∙1
2 + 3∙2
2 + 3∙3
2 + 3∙4
2 + 3∙5
2 + 3∙6
2 + 3∙7 2 2 2 3 =
What does the 3 represent? 1 2 3 2 3 1 = 2 2 3 3 2 3 2 = = 3 2 3 3
What does it
all equal? 3 2 3 3 = 4 2 3 3 3 3 2 3 4 = 5 2 3 3 3 3 3 2 3 5 6 2 3 3 3 3 3 3 2 3 6 = = 7 2 3 3 3 3 3 3 3 2 3 7

10. What does the 2 represent? What does the 3 represent?
number that’s
changing represent?
What does the 2 represent?
Figure Number
Number of Toothpicks
Recursive Pattern as Repeated Addition
Using Definition of Multiplication
Number of Toothpicks 2 5 8 11 14 17 20 23 2 1 5 8 3 11 4 14 17 6 20 7 23 2
2 + 3∙0
2 + 3
2 + 3∙1
2 + 3∙2
2 + 3∙3
2 + 3∙4
2 + 3∙5
2 + 3∙6
2 + 3∙7 2 2 2 3 = 2
What does the 3 represent? 1 5 2 3 2 3 1 = 5 2 8 2 3 3 2 3 2 = 8 = 3 11 2 3 3
What does it
all equal? 3 2 3 3 11 = 4 14 2 3 3 3 3 2 3 4 14 = 5 17 2 3 3 3 3 3 2 3 5 17 6 20 2 3 3 3 3 3 3 2 3 6 = 20 = 7 23 2 3 3 3 3 3 3 3 2 3 7 23

11. ∙ 2 + 3 = Figure Number of number Toothpicks = = = = = = = =
So…What is
changing and what
is staying the same?
Figure Number
Number of Toothpicks
Recursive Pattern as Repeated Addition
Using Definition of Multiplication
Number of Toothpicks 2 5 8 11 14 17 20 23 2 1 5 8 3 11 4 14 17 6 20 7 23 2
2 + 3∙0
2 + 3
2 + 3∙1
2 + 3∙2
2 + 3∙3
2 + 3∙4
2 + 3∙5
2 + 3∙6
2 + 3∙7 2 2 2 3 = 2 1 5 2 3 2 3 1 = 5 2 8 2 3 3 2 3 2 = 8 = 3 11 2 3 3 3 2 3 3 11 = 4 14 2 3 3 3 3 2 3 4 14 = 5 17 2 3 3 3 3 3 2 3 5 17 6 20 2 3 3 3 3 3 3 2 3 6 = 20 = 7 23 2 3 3 3 3 3 3 3 2 3 7 23 ∙
Figure
number
Number of
Toothpicks 2 + 3 =
Starting Value
Initial Value
Recursive Pattern
Rate of Change

12. ∙ 2 + 3 = x y Where x is the Where y is the number of figure number
Toothpicks ∙ 2 + 3 = x y
Starting Value
Initial Value
Recursive Pattern
Rate of Change

13. Use the table of values to write a rule/equation: x y ‒1 ‒5 2 1 9 162 1 9 16
So…What is
changing and what
is staying the same?
𝑺𝒕𝒂𝒓𝒕𝒊𝒏𝒈 𝑽𝒂𝒍𝒖𝒆+ 𝑹𝒂𝒕𝒆 𝒐𝒇 𝑪𝒉𝒂𝒏𝒈𝒆 𝑥=𝑦
𝑦=𝟕𝑥+𝟐