# Lesson 1Piggies and PoolsDevelop Understanding

## Learning Focus

Represent situations with different types of growth.

Compare models for situations that occur over time.

What type of situation can be modeled by a continuous graph? When is a graph of only separate points appropriate?

What are the similarities and differences between an arithmetic sequence and a linear function?

Can a geometric sequence be continuous?

## Open Up the Math: Launch, Explore, Discuss

### 1.

My little sister, Savannah, is three years old. She has a piggy bank that she wants to fill. She started with pennies, and each day when I come home from school, she is excited when I give her pennies that are left over from my lunch money. Use a table, a graph, and an equation to create a mathematical model for the number of pennies in the piggy bank on day .

Table:

Equations:

### 2.

Our family has a small pool for relaxing in the summer that holds of water. I decided to fill the pool for the summer. I was getting bored just standing there watching water flow and began to think about a mathematical model for the time it takes to fill the pool while I was waiting. I checked the flow on the hose and found that it was filling the pool at a rate of every minute. When I had of water in the pool, I started the timer. Use a table, a graph, and an equation to create a mathematical model for the number of gallons of water in the pool at .

Equation:

Table:

### 3.

I’m more sophisticated than my little sister, so I save my money in a bank account that pays me on the money in the account at the end of each month. (If I take my money out before the end of the month, I don’t earn any interest for the month.) I started the account with that I got for my birthday. Use a table, a graph, and an equation to create a mathematical model of the amount of money I will have in the account after .

Equation:

Table:

### 4.

At the end of the summer, I decided to drain the swimming pool. I noticed that it drains faster when there is more water in the pool. That was interesting to me, so I decided to measure the rate at which it drains. I found that was draining out of the pool every minute. Use a table, a graph, and an equation to create a mathematical model of the gallons of water in the pool at .

Equation:

Table:

### 5.

Compare problems 1 and 3. What similarities do you see? What differences do you notice?

Similarities:

Differences:

### 6.

Compare problems 1 and 2. What similarities do you see? What differences do you notice?

Similarities:

Differences:

### 7.

Compare problems 3 and 4. What similarities do you see? What differences do you notice?

Similarities:

Differences:

Use your model in problem 4 to find when the pool will be empty. Justify your answer.

## Takeaways

A geometric sequence

Exponential functions

Discrete functions

Continuous functions

Arithmetic and geometric sequences are

## Adding Notation, Vocabulary, and Conventions

Domain of a function:

## Lesson Summary

In this lesson, we learned that the possible inputs for a function are called the domain. We found that some situations are best described using a discrete model and others are represented better with a continuous model. Arithmetic sequences are part of the linear family of functions and geometric sequences are part of the exponential family of functions.

## Retrieval

### 1.

For each sequence, find the next two terms in the sequence and then state whether the sequence is arithmetic, geometric, or neither. Justify your answer.

#### c.

Find the unit rate for each of the items.

### 2.

A dozen ears of corn for .

### 3.

Three t-shirts for .

Solve each of the equations.