Lesson 6 Home on the Range Solidify Understanding

Jump Start

1.

The Super Discount store is having a sale and all the clearance items are $40 %$ off. Which of the following are correct ways to find the price of an item that is normally $$ 25.00$ ?

A.

Multiply $$ 25.00$ by $40$

B.

Multiply $$ 25.00$ by $0.4$

C.

Multiply $$ 25.00$ by $0.4$ and subtract that amount from $$ 25.00$

D.

Multiply $$ 25.00$ by $0.6$

2.

What is the sale price of the item?

Learning Focus

Understand the characteristics that make sequences decrease.

Compare decreasing arithmetic and geometric sequences.

What makes an arithmetic sequence decrease?

How can a geometric sequence be decreasing?

How are equations of decreasing sequences different than increasing sequences?

Open Up the Math: Launch, Explore, Discuss

Prairie dogs are small mammals that are members of the squirrel family. They average about $14 - 17 inches$ in size and live in underground colonies called prairie dog towns. They are native to the Great Plains region of the United States. The prairie dog population is estimated to have declined by $95 %$ over the past $200 years$ , so prairie dog reserves have been established in some areas. Prairie dog reserves are places where prairie dogs can live in their natural habitat without being disturbed by hunting or other human activities.

One such prairie dog reserve was established on $120 acres$ of land in 2015. As the surrounding land has become more valuable for humans, the size of the reserve has been reduced over time. The following data has been collected about the size of the reserve:

Year	Size (Acres)
$1$	$120$
$2$	$112.5$
$3$	$105$
$4$	$97.5$
$5$	$90$

1.

Write an explicit equation and a recursive equation to model the size of the reserve over time.

2.

What type of sequence does this data represent? Justify your answer.

The sale of land at the reserve was stopped when the reserve was $76.5$ acres, because biologists who study the prairie dogs found that the population was declining at the alarming rate of $20 %$ per year. They determined that the decline was due to several factors, including the reduction of habitat from selling off the reserve land. When the reserve began, the prairie dog population on the reserve was about $600$ individuals.

3.

Model the prairie dog population over time, in years, starting with the beginning population. Create both an explicit equation and a recursive equation.

4.

What type of sequence is represented by the prairie dog population? Justify your answer.

5.

How are decreasing arithmetic sequences different from decreasing geometric sequences?

Ready for More?

What type of relationship exists between the number of acres in the reserve and the population of prairie dogs? Model the relationship with a table and a graph.

Takeaways

Decreasing arithmetic and geometric sequences:

Graph of a decreasing arithmetic sequence:

Graph of a decreasing geometric sequence:

Lesson Summary

In today’s lesson, we modeled a real context with arithmetic and geometric sequences. We found that some arithmetic and geometric sequences decrease. Whether a sequence increases or decreases depends on the common difference or common ratio between terms. Additionally, we used tables, equations, and graphs to compare the behavior of decreasing arithmetic and geometric sequences.

Retrieval

For each sequence:

a. find the missing terms of the sequence,

b. determine if it has a common difference or common ratio,

c. find the common difference or common ratio,

d. determine if it is an arithmetic or geometric sequence.

1.

a.

$3$ , $9$ , $27$ , $\underset{―}{}$ , $243$ , $\underset{―}{}$ , $\dots$

b.

Common difference or ratio?

c.

Common difference or ratio =

d.

Arithmetic or geometric?

2.

a.

$3$ , $9$ , $\underset{―}{}$ , $\underset{―}{}$ , $27$ , …

b.

Common difference or ratio?

c.

Common difference or ratio = $\underset{―}{}$

d.

Arithmetic or geometric?

3.

Graph the two points, find the slope between them, and create an equation of the line through the points in slope-intercept form.

$M (- 2, 0)$ and $R (4, 2)$

Slope:

$m =$

Equation: