What Are Patterns?

Pattern: A change that occurs in a predictable way.

Examples:checkerboard, days of the weeks, even numbers, etc.

Arithmetic Pattern/Sequence: A sequence in which the same number is added or subtracted repeatedly.

Example: 1, 5, 9, 13, 17 . . . . or 0, -5, -10, -15 . . . . .

Geometric Pattern/Sequence: A sequence in which the numbers are multiplied by the same factor.

Example: 2, 4, 8, 16, 32 . . . .

Visual Patterns

1-step

2-step

3-step

Patterns in Graphs

Patterns in Tables

x

y

1

5

2

7

3

9

4

14

n

2n + 3

x

y

1

1

2

4

3

9

4

16

n

n2

Patterns in Equations

Examples: one step n + 3

two-step2n -5

Arithmetic or Geometric Sequences

-4 -4 -4 -4

1. 9, 5, 1, -3, -7, ____, _____, _____ Pattern is Subtract 4

2. 2, 6, 18, 54, 162, ______, _______, _______ Pattern is _____________________

3. -4, 1, 6, 11, 16, ______, _______, _______ Pattern is _____________________

4. 1, 3, 4, 7, 8, 11, ______, _______, _______ Pattern is _____________________

5. 1, 2, 4, 7, 11, 15, ______, _______, _______ Pattern is _____________________

6. 10, 12, 16, 24, ______, _______, _______ Pattern is _____________________

7. 9, 6, 3, 0, -3, -6, ______, _______, _______ Pattern is ____________________

8. Ian filled a jar with 60 gumballs, and he and his family helped themselves to them whenever they liked. Ian counted the second day and found 53 gumballs, and on the third day, there where 46 gumballs. How many gumballs will be in the jar on the fifth day?

a. Whats the question? _________________________________________

b. List the key facts below:

Day 1 ______Day 2 ______Day 3 ______

Pattern is: __________ Arithmetic or Geometric? ________________

c. Strategy: Looking for a Pattern and Making a Table

Day

1

2

3

4

5

#

60

53

46

d.

e. There were _____ gumballs left on the 5th day.

9. Mrs. Sacco in the Counseling Office noticed a pattern in the number of new students that came to our school from month to month. In August 2 new students came, September 4 new students, October 8 new students, November 16 new students, etc. How many new students came by the end of the year in June?

a. Whats the question? _________________________________________

b. List the key facts below:

Aug _____ stud.Sept _____ stud.Oct _____ stud. Nov _____ stud.

Pattern is: __________ Arithmetic or Geometric? ________________

c. Strategy: Looking for a Pattern and Making a Table

Month

Aug

Sept

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

June

d.

e. There were __________ students in June.

Visual Patterns

1.

lst Term2nd Term 3rd Term

Term

# Pieces

Pattern

1

2

2

5

3

4

5

6

n

2.

lst Term 2nd Term 3rd Term

Term

# Pieces

Pattern

1

2

3

4

5

6

n

3.

Term

# Pieces

Pattern

1

2

3

4

5

6

n

Function Machines

In the real world function has several connected meanings. Function can mean the purpose or job of something, such as the function of homework is to practice what you already know. As a verb, function means to work. This computer functions well.

In math, the word takes on a slightly different meaning. The mathematical definition of function is the dependent relationship between two or more variables. For example, if a car is traveling at 80 km/hour, the distance the car travels is dependent on the amount of time.

Function Machines are another way of demonstrating and solving patterns. Function Machines have an input and an output. Something goes in and a different thing comes out.

Machine

Input

Output

Washing Machine

dirty clothes

clean clothes

Oven

cookie dough

cookies

Wind Mill

wind energy

electricity

Directions

(Possible Functionsx + 72x + 23x 3) ( 5)

(MysteryFunction)

(12)

In order to recognize a pattern, you have to have enough evidence. In the example above, the function can not be defined because there are not enough examples.

However, if another number, such as 7, is put into the machine and 16 comes out, then there is enough evidence to say that the pattern is 2x + 2.

Online Computer Practice with function machines (best ones are first):

http://www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines.html

http://images.google.com/imgres?imgurl=http://www.mathwire.com/images2/shipsahoy.jpg&imgrefurl=http://www.mathwire.com/archives/may06.html&usg=__ePJcMeU3TCxaMVuNdf4cKwfNMqU=&h=246&w=330&sz=19&hl=en&start=62&um=1&itbs=1&tbnid=cMt9I8CDI6zvxM:&tbnh=89&tbnw=119&prev=/images%3Fq%3Dfunction%2Bmachines%2Belementary%26start%3D54%26um%3D1%26hl%3Den%26client%3Dfirefox-a%26sa%3DN%26rls%3Dorg.mozilla:en-US:official%26ndsp%3D18%26tbs%3Disch:1

http://teams.lacoe.edu/documentation/classrooms/amy/algebra/3-4/activities/functionmachine/functionmachine3_4.html

http://nlvm.usu.edu/en/nav/frames_asid_191_g_3_t_1.html

http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=1035

Input/Output Machines

Make Up Your Own Rules

Function Machines and Substitutions

fdf

(y = x 53; (3 5)-24; (4 5)-18; (8 - 5)39; (9 - 5)4)

Writing Equations from a Table

Complete the table and write the equation. Make sure to check your pattern works for all values of x!

x

y

5

25

6

36

7

49

8

64

9

81

x

x2

x

y

5

0

6

1

7

2

8

3

9

4

x

x

y

0

7

1

8

2

9

3

10

4

11

x

x

y

1

1

2

8

3

27

4

64

5

125

x

x

y

1

7

2

17

3

27

4

37

5

47

x

x

y

8

20

9

22.5

10

25

11

27.5

12

30

x

x

y

5

-7

6

-6

7

-5

8

-4

9

-3

x

x

y

4

1

5

1.25

6

1.5

7

1.75

8

2

x

x

y

7

-3

8

-2

9

-1

10

0

11

1

x

Complete the Table From the Graph

Fill in the values for y and write an equation, if you can.

(Hint: It helps to look at the x-intercept and y-intercept if you get stuck.)

((1) xy-45-34-23-1201Equation: y = x(-1) +1)

Complete the Table from the Word Problem

Graph the Table

Graph the 4 of the tables from the previous page. You may use a separate sheet of graph paper if you wish.

Hints: Dont forget your labels and plan your intervals ahead of time.

Translate the Word Expressions into Number Expressions

Key Words

Addition

sum

increased by

Subtraction

difference

decreased by

Multiplication

product

twice

Division

Quotient

half

1. a number v increased by 6 v + 6

2. 43 minus a number g________________

3. difference of 47 and a number u________________

4. quotient of a number t and 5________________

5. sum of 22 and a number f________________

6. 11 decreased by a number r________________

7. 4 less a number m________________

8. 38 less than a number q________________

9. take away 16 from a number j________________

10. 35 divided into a number p________________

11. a number w multiplied by 26________________

12. subtract 45 from a number u________________

13. a number h plus 30________________

14. 34 more than a number v________________

15. a number z times 9________________

16. product of a number x and 33________________

17. quotient of a number f and 27________________

Translate the Number Expressions & Equations into Word Expressions and Equations

Challenge yourself to use a variety of different vocabulary words.

1. 6 + b = 25The sum of 6 and b is 25.

2. 5x 2 = 24____________________________________________________

3. + 2 = 14____________________________________________________

4. = 8____________________________________________________

5. t + 8.5 = 12____________________________________________________

6. 45 2y = 35____________________________________________________

7. y