V

jput/Output Machines

Quick Review

This Input/Output machine divides each input number by 2, then adds 3

Input^ 2 + 3

Output

The pattern rule forthe output is:

Start at 13. Add 5 each time.

Input Output

20 13

30 18

40 23

50/WVVXA/WW

28VVWWWNAA

Try These . , , _ ; ^ _ ^ ^

Complete the table for each Input/Output machine.Write the pattern rule for each output.

1. Input+ 7

Output 2. Input

Input Output

20 17 .

18 16

16

14 14Iaaaaaaaaaa/VWWWWVN

X 4 - 1Output

Input Output

12 47

13 51

14 55

15 59k/wvwwwvWWXAAAAA

Start at 17. Subtract 1 Start at 47. Add 4

each time. each time.

a

Practice

Complete the table for each Input/Output machine.Write the pattern rule for each output.

Input+ 5 X 4

Output

Input Output

9 56

10 60

11 64

12/\AyVVvVWVv

68\AAA/WVVV\A

b) InputX 7 - 3

Output

Input Output

6 3Qigir

7 46

8 53

9VVvVVVVVVV

60WWWWWN

each timi

The table shows the input and output froma machine with two operations.

a) Write the numbers and the operationsused by the Input/Output machine.

b) Write the next 3 input and output numbers.

Input: 45,50,55: Output: 27,30,33

Input Output

25 15

30 18

35 . 21

40AAAAA/VWVv

24WWVWvA/V\

Input

Stretch Your Thinking

The first 5 input numbers for a machineare 2527,2577,2627,2677, and 2727. 'The first 5 output numbers for the machineare 5061,5161,5261,5361, and 5461.Record the numbers and the operations in the machine.

x 2 J- "7T /Output

m mm&m Patterns

Quick Review

In a recursive pattern, each term can be found by applyingthe pattern rule to the previous term.

To find the pattern rule for the recursive pattern: 4,7,13,25,49,...,find the difference between each pair of consecutive terms.

^497-4 = 3 13 - 7 = 6 25 - 13 = 12 49 - 25 = 24

Each difference is double the previous difference.This suggests that X 2 is part of the pattern rule.

The first term is 4. 4X2 = 8The second term is 7. Subtract 1 from 8 to get 7. 8-1=7

This pattern rule is: ,

Start at 4. Multiply by 2, then subtract 1 each time.

To extend the pattern, continue to multiply by 2, then subtract 1.The next 3 terms are 97,193, and 385.

Try These ^ ^ ^ ; ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ & o ^ ^ a ^ ^ ^ !s ^ ^ # ^ ei

1. Write the first 5 terms of each pattern. Start at 6 each time.

a) Add 2, then multiply by 3 each time. 6,24,78,240,726

b) Subtract 1, then multiply by 2 each time. 6,10,18,34,66

c) Add 4, then multiply by 2 each time. 6,20,48,104,216

d) Subtract 3, then multiply by 7 each time. 6,21,126,861,6006

Use a calculator when it helps.

1. Write the first 4 terms of each pattern.

a) Start at 9. Multiply by 4, then add 2 each time. 9.38,154,618

b) Start at 12. Add 3, then multiply by 4 each time. 12.60.252.1020

c) Start at 200. Add 25, then subtract 7 each time. 200.218,236,254

d) Start at 4. Multiply by 8, then subtract 2 each time. 4,30,238,1902

2. Write the next 4 terms in each pattern. Write each pattern rule.

a) 460,435,410,385,360, 335 , 310 285 260

Pattern Rule: Start at 460. Subtract 25 each time.

b) 7,25,79,241,727, 2185 , 6559 , 19 681, 59 047

Pattern Rule: Start at 7. Multiply bv 3. then add 4 each time.

C) 12,36,84,180,372, 756 , 1524 , 3060 , 6132

Pattern Rule: Start at 12. Add 6. then multiply bv 2 each time.

d) 8,12,24,60,168, 492 , 1464 , 4380 , 13 128

Pattern Rule: Start at 8. Multiply bv 3. then subtract 12 each time.

3. Write the 8th term of each pattern.

a) 1,5,33,229,1601, ... 549 029 b) 12,59,294,1469,7344, ... 917 969

c) 15,24,42,78,150, ... 1158 d) 3,19,99,499,2499, ... 312 499

Stretch Your Thinking . . . . .,.,,.

1. Write the first 6 terms of 2 different recursive patterns beginning with 2,6.Then write each pattern rule. Sample Answers

a) 2.6. 14 , 30 , 62 , 126_

Pattern Rule: Start at 2. Multiply bv 2. then add 2 each time.

b) 2.6. 26 . 126 , 626 , 3126

Pattern Rule: Start at 2. Multiply bv 5. then subtract 4 each time.

5

~ ^presenting Patterns

Quick Review

Here are some ways to represent a growing pattern.> Model the pattern on a grid.

Frame 1 Frame 2

> Make a table.

Frame 3 Frame 4

> Draw a graph.

FrameNumber

Number of Squaresin a Frame

OrderedPair

1 5 (1,5)2 6 (2,6)3 7 (3,7)4 8 (4,8)

| Nwmbor of Squares ¦ra in a Growing Pattern

10-u.

•s 8m

§• 4-M !

- 2

2 0& 8 10

ifel'me Num"'

> Write a pattern rule.

^^The pattern rule for the number of squares in a frame is:Start at 5. Add 1 each time.

Irw Thesea :>:• a <-5 O & Q Q O © 0 ^ <

1 e a) Write the pattern rule for this growing pattern

• • •

Frame 1 Frame 2 Frame 3 Frame 4 Frame 5

The number of dots in a frame is: Start at 1. Add 3 each time.

b) How many dots are needed for Frame 10?

110

28

Prac

1. a) Write a pattern rule for the table.

Start at 1. Add 2 each time.Frame Number 1 2 3 4 5Number of Squares 1 3 5 7 9in a Frame

b) Shade squares on the grid to model the pattern. Sample Answer

: : i ;I rm bjb1

iiirn:e 1 Frame 1 ji:rame)LL

i1 ;1 j 'illbb1 1 "~in n b— :rame 4 i ) l ii i i iiirn: 5i \ i— r-l—1— _

c) Graph the pattern.

d) How many squares are needed forFrame 10?

19

e) Which frame has 29 squares?Frame 15

f) Which frame has 51 squares?

Frame 26

2. Draw a growing pattern to modelthe data in the table.Sample Answer

i ®-I i! (0

WMmb@r@ff Squaresin a Growing Pattern

-10-

{-•«} IB-J ci ii «aI Li S N1 1 i a 14..wj f*

IMm -!i 01

ill! 1 b-¦fef 1-2"

~1 ' I 1 I ' j i—

4 ) f f 4 | ^ 1 i|e-Frarrte Number i I

Frame Number 1 2 3 4Number of Triangles 1 2 4 8in a Frame

'A' • • '/V • ' ' A7V ' ' AAAAJ 'Frame 1 Frame 2 Frame 3 Frame 4

Stretch Your Thinking9 « ® a ® « » # i

Use the table in question 2.How many triangles are in Frame 10? 512

Which frame has 8192 triangles? Frame 14

ill

UNIT lOmSKEmm. lating Graphs and

Input/Output Machines

n>

\

w I WW JTo draw a graph for thisInput/Output machine:Make an Input/Output table.

InputX 2 - 1

Output

Then draw and labela coordinate graph.

Input 1 2 3 4

Output 1 3 5 7

j-lf^• b

i 4si 0J

To draw an Input/Output machine for thegraph to the left:

2 4~ 6 8 To|

nt/Guiput Pattern

f 8| 6-

Q

Make a table.

Input 1 3 5 7

Output 1 2 3 4

2 4 6 8 10Input Number

Then draw and label anInput/Output machine.

Input+ 1 - 2

Output

These

1. a) Make a table of input and output numbers for the graph below,

b) Write the numbers and operations in the Input/Output machine.

Input Output

1 62 83 104 12

Inputx 2 + 4

Outputi Input^Oirfpuf PatterrT142J10-

£ 86

O 20 -!—I—i—(—'¦—! !—I—I—!—r

2 4 6 8 10Input Number

1

Practice

For each graph, make a table for the input and output numbers.Write the numbers and operations in the Input/Output machine.

•{ t-j fnpijrt! | ;OMput Pattern

,18-

4^,I12

W p-Ift—o

Input Output

1 62 %3 124 155 18

Inputx 3 + 3P

Output

Input/Output Pajttem2B-

^ 24-

20-:e3 16-

5 ifeft, i3.—8- -a ,. .p.

1 r—r-* T2 -42 4 1\J

.{ jllfp mmb

Input Output

_

4 106 ::!6

8 2210

Inputx 3

Output

, Stretch Your Thinking ^

The graph shows how much water is lost by a dripping faucet. Make a table todisplay the data. Draw an Input/Output machine that will produce the graph.

| qq!filer List ( > Tim© Amount nf Input

wl 75- -

''fW>

" 2 3

2-4SH

4. r?,.-1 1 1 i ! I I

Time Amount of(min) Water (ml)

1 _

2 30qt 454 60C3 75

x 15Output

113

Name Date

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Sample Answers

Extra Practice 1 – Master 10.18

Lesson 1 1. a) Pattern 1: Start at 2. Add 2 each time.

Pattern 2: Start at 3. Add 4 each time.

b) Pattern 1:

Pattern 2:

c)

2. Pattern 1: Frame 27 has 54 tiles; the

number of squares in each frame is double

the frame number.

Pattern 2: No; each frame has an odd

number of squares.

Lesson 2 1.

2. a)

b)

Master 10.21