Lesson 2 Shh! Please Be Discreet (Discrete)! Solidify Understanding

Jump Start

Which One Doesn’t Belong? Determine which of the numbers below is not like the others and be prepared with a reason for your choice.

$2$

$- 2$

$- \frac{1}{2}$

$\sqrt{2}$

$2. \overset{―}{22}$

Reason:

Learning Focus

Use representations to model situations with linear and exponential functions.

Determine when a discrete model or continuous model is most appropriate.

What is the difference between discrete and continuous functions?

How can I tell if a discrete or continuous model is best for a given situation?

Are all linear functions continuous? Are all arithmetic sequences discrete?

Are all exponential functions continuous? Are all geometric sequences discrete?

How is the domain related to whether the function is continuous or discrete?

Open Up the Math: Launch, Explore, Discuss

1.

The Library of Congress in Washington D.C. is considered the largest library in the world. They often receive boxes of books to be added to their collection. Since books can be quite heavy, they aren’t shipped in big boxes. On average, each box contains $8 books$ .

If, on average, each box contains $8 books$ , how many books are received by the library in $6 boxes$ , $10 boxes$ , or $n boxes$ ?

Use a table, a graph, and an equation to model this situation.

Equation:

Domain:

Table:

2.

Many of the books at the Library of Congress are electronic. About $13 e-books$ can be downloaded onto the computer each hour.

If about $13 e-books$ can be downloaded onto the computer each hour, how many e-books can be added to the library in $3 hours$ , $5 hours$ , or $n hours$ (assuming the computer memory is not limited)?

Use a table, a graph, and an equation to model this situation.

Equation:

Domain:

Table:

3.

The librarians work to keep the library orderly and put books back into their proper places after they have been used. A librarian can sort and shelve $3 books$ in a minute.

If a librarian can sort and shelve $3 books$ in a minute, how many books does that librarian take care of in $3 hours$ , $5 hours$ , or $n hours$ ?

Use a table, a graph, and an equation to model this situation.

Equation:

Domain:

Table:

4.

Would it make sense in any of these situations for there to be a time when $32.5$ books had been shipped, downloaded into the computer, or placed on the shelf? Explain.

5.

Which of these situations (in problems 1–3) represent a discrete function and which represent a continuous function? Justify your answer.

6.

A giant piece of paper is cut into $3 equal pieces$ , and then each of those is cut into $3 equal pieces$ and so forth.

a.

Use a table, a graph, and an equation to model this situation.

How many papers will there be after a round of $10 cuts$ ? $20 cuts$ ? $n cuts$ ?

Equation:

Table:

b.

Identify the domain of the function.

c.

Would it make sense to look for the number of pieces of paper at $5.2 cuts$ ? Why?

d.

Would it make sense to look for the number of cuts it takes to make $53.6 papers$ ? Why?

7.

Medicine taken by a patient breaks down in the patient’s blood stream and dissipates out of the patient’s system. Suppose a dose of $60 milligrams$ of anti-parasite medicine is given to a dog and the medicine breaks down so that $20 %$ of the medicine becomes ineffective every hour.

a.

Use a table, a graph, and an equation to model this situation.

How much of the $60 milligram$ dose is still active in the dog’s bloodstream after $3 hours$ , after $4.25 hours$ , and after $n hours$ ?

Equation:

Table:

b.

Identify the domain of the function.

c.

Would it make sense to look for an amount of active medicine at $3.8 hours$ ? Why?

d.

Would it make sense to look for when there is $35 milligrams$ of medicine? Why?

8.

Which of the functions modeled in problems 6 and 7 are discrete and which are continuous? Why?

9.

What needs to be considered when looking at a situation or context and deciding if it fits best with a discrete or continuous model?

10.

Describe the differences in each representation (table, graph, and equation) for discrete and continuous functions.

11.

Which of the functions modeled in this task are linear? Which are exponential? Why?

Ready for More?

On a separate piece of paper or a pair of index cards, work with your partner to create one continuous linear context and one discrete linear context. Include enough information that the context could be modeled with all the representations. When you finish, trade your cards with another pair of students for feedback. Discuss the following:

Which context is discrete and which context is continuous?
Is the context actually linear?

Takeaways

	Discrete	Continuous
Context Features
Tables
Graphs
Equations
Domain

Adding Notation, Vocabulary, and Conventions

Set Builder Notation:

We write:

We say:

We mean:

Vocabulary

set builder notation
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we modeled linear and exponential functions and learned to identify features that allow us to determine whether a discrete or a continuous model is more appropriate. We discussed number sets and used them to write function domains using set builder notation.

Retrieval

Calculate the slope in problems 1 and 2.

1.

2.

$n$	$f (n)$
$4$	$- 31$
$7$	$- 40$
$11$	$- 52$
$16$	$- 67$

3.

Find the recursive and explicit equation for the table in problem 2.

In problems 4 and 5, solve for $x$ .

4.

$x + 5 = 13$

5.

$\frac{2}{3} x = 12$