Lesson 1 Piggy Banks and Pools Develop 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?

Technology guidance for today’s lesson:

Create a Graph From a Table: Casio ClassPad Casio fx-9750GIII
Create a Graph From an Equation: Casio ClassPad Casio fx-9750GIII

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 $5$ pennies, and each day when I come home from school, she is excited when I give her $3$ 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 $n$ .

Table:

Equations:

2.

Our family has a small pool for relaxing in the summer that holds $1, 500 gallons$ 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 $2 gallons$ every minute. When I had $5 gallons$ 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 $t minutes$ .

Equation:

Table:

3.

I’m more sophisticated than my little sister, so I save my money in a bank account that pays me $3 % interest$ 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 $$ 50$ 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 $m months$ .

Equation:

Table:

4.

At the end of the summer, I decided to drain the $1, 500 -gallon$ 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 $3 %$ 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 $t minutes$ .

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:

Ready for More?

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:

Vocabulary

continuous function/discontinuous function
discrete function
domain
exponential function
linear function
Bold terms are new in this lesson.

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.

a.

$4, 8, 16, \dots$

b.

$2, 5, 10, \dots$

c.

$6, 9, 12, \dots$

Find the unit rate for each of the items.

2.

A dozen ears of corn for $$ 5$ .

3.

Three t-shirts for $$ 10$ .

Solve each of the equations.

4.

$7 x = 42$

5.

$\frac{1}{5} x = 4$