5.1C Homework: Linear and Non-linear Patterns - Google Sites

5.1C Homework: Linear and Non-linear Patterns* Name:

Period:

Directions: For each of the following patterns, draw the next stage and determine whether or not the relationship between the stage number and the number of boxes in a stage can be represented by a linear function. Justify your answer by making a table and using the model. Stage 1

# of boxes 1

2

3

3

6

4

10

Stage 1

# of boxes 6

2

8

3

10

4

12

Stage 1

# of boxes 2

2

4

3

8

4

16

5

32

1. Stage 1

Stage 2

Stage 3

Stage 4

No, this relationship is not a linear function. The number of boxes added each time is not constant. The model grows wider and taller at the same time. In the table, there is not a constant rate of change. 2. In this pattern, count the gray boxes only.

Stage 4 Yes, this relationship is a linear function. The number of boxes added each time is 2. The model grows in one direction. In the table, the first difference is 2.

3.

Stage 5

No, this relationship is not a linear function. The number of boxes added each time is not constant. In the table, there is not a constant rate of change.

4.

Stage 4

No, this relationship is not a linear function. The number of boxes added each time is not constant. The model grows taller and wider at the same time. In the table, there is not a constant rate of change.

Stage 1

# of boxes 5

2

10

3

17

4

26

(problem #5 is adapted from Tennessee Department of Education)

5. The pattern below relates the number of hexagons (n) to the perimeter (p) of the total figure.

a. Complete the table.

# of hexagons 1

Perimeter of total figure 6

2

10

3

14

b. Write an equation that relates , the number of hexagons, to , perimeter of the total figure. p = 4n + 2

4 18 c. Does the pattern represent a linear function? Justify your reasoning. The pattern represents a linear function because as one hexagon is being added each time, the total perimeter consistently increases by 4.

d. What would be the total perimeter of a figure with 10 hexagons? Show all of your work. p = 4n + 2 p = 4(10) + 2 p = 42 The total perimeter for a figure with 10 hexagons would be 42. 6. When Michael opened his email this morning, he had 180 unread emails. The table shows the number of unread emails remaining in Michael’s inbox. Assume that Michael does not receive any new emails while he is reading his email. Consider the relationship between time and the number of unread emails.

Time (hours)

a. Is the data linear? Explain your reasoning. Yes, Michael reads his email at a constant rate of 40 emails/hour.

b. If linear, define your variables and write an equation. t = time in hours u = # of unread emails 𝑢 = −40𝑡 + 180

0

# of Unread Emails 180

0.5

160

1

140

2

100

2.5

80

4

20

4.5

0

7. Suppose a certain bank pays 4% interest at the end of each year on the money in an account. When Devon was born, his parents put $100 in the account and will leave it there until he goes to college. Is the relationship between time (in years) and the amount of money in the account (in dollars) linear or not? Explain your reasoning. No, students can show that the rate of change in a table is not constant.

Height

8. Suppose you drop a basketball from a height of 60 inches. The table shows the height of the basketball after b bounces. Graph the data on the coordinate plane below. Is the data linear? Explain your reasoning.

# of bounces 0

Height (inches) 60

2

37

4

25

70

6

15

60

8

10

50

10

7

40 30

The data is non-linear because there is not a constant rate of change in the table of values. The graph is not a line.

20 10 0

2

4

6 8 10 12 14 # of bounces

9. Justine and her family are floating down a river on a sailboat. After 1 hour, they have floated 1.25 miles, after 4 hours they have floated 5 miles, and after 6 hours they have floated 7.5 miles. a. Is the relationship between time (in hours) and distance (in miles) linear? Explain your reasoning. Yes, they float at a constant rate of 1.25 miles/hr. b. If linear, define your variables and write an equation. t = time in hours d = distance in miles 𝑑 = 1.25𝑡 c. What physical features of the river might explain the linearity? The linearity might be caused if the river has the same width, topography, and depth.

d. What features of a river would cause non-linearity? The non-linearity might be caused if the river runs down a mountain, has varying width or depth, or has many rocks or dams.

Spiral Review: 10. The lengths of two sides of a rectangle are consecutive even integers. The perimeter of the rectangle is196 cm. Find the area. 2400 11. Solve:

=− =−

3