Alg1 Lesson 4.1: Counting Binary Strings

Lesson 4.1: Counting Binary

Strings

Obj: 12C: I can determine whether a relationship is
exponential based on a numerical sequence whose
indices increase by a constant ratio.

EQ: How do I identify and list all possible outcomes?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Draw

Draw an emoji here.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: Exploring Binary

Strings

Binary Strings

Computers transmit and store information using a permutation of
digits with only two choices per spot. A popular way to do this is using a permutation of 0s and 1s, but information can be coded in other ways, too. For example, some hard disk drives use magnetic positives and negatives. DVDs use laser light that either is or is not reflected back. All of these systems are binary, since there are only two options per space. A sequence of symbols that uses only 0s and 1s is called a binary string.

1, 2, 4, 8, 16, 32, 64, 128, 256 …

Fill in the Blanks

How many binary strings have a length of one?

Fill in the Blanks

How many binary strings have a length of two?

Fill in the Blanks

How many binary strings have a length of three?

Word Cloud

What do you notice and wonder about binary strings?

All Permutations of Four 0’s and 1’s

In the common
decimal system,
these binary
numbers represent
numbers 0 through
15.

Drag and Drop

There are ​

Placeholder

strings that start with “0” and ​

Placeholder

strings that start with “1.” The strings that start with “0” can be organized into ​

Placeholder

sets of four, where the second symbol is either a “0” or a “1.” The same thing is true for the strings that start with “1.”

Drag these tiles and drop them in the correct blank above

eight

two

four

six

one

zero

Drag and Drop

There are ​

Placeholder

strings with a length of five. I know because there are ​

Placeholder

length-four strings, and you can put a “0” or a “1” in the front of each string. So, there will be ​

Placeholder

strings that start with a “0” and ​

Placeholder

strings that start with a “1.”

Drag these tiles and drop them in the correct blank above

32

16

8

4

2

Fill in the Blanks

How many binary strings are there of length eight?

Multiple Choice

In general, if we know how many binary strings there are of a certain length, how
could we figure out the number of binary strings when the length is increased by 1?

• Square the known number of binary strings.
• Add 1 to the known number of binary strings.
• Divide the known number of binary strings by 2.
• Multiply the known number of binary strings by 2.

Bytes and Bits

A byte is a unit of storage in a computer
which contains 8-bits (8 binary digits or
length-8 strings) and can store 256
different values: 0 to 255. A bit is the
smallest unit of data in a computer.

Part 2: Creating a Formula

for Binary Strings

A table of
values for
the length

of the

strings, n,

and the

total

number of

binary

strings, b.

Drag and Drop

You ​

Placeholder

by ​

Placeholder

to get to the next row. The number of 2's in the calculation
column ​

Placeholder

by 1.

Drag these tiles and drop them in the correct blank above

multiply

2

increase

divide

decrease

Drag and Drop

You can represent ​

Placeholder

multiplication with an exponent.
To use exponents to determine how many binary strings there are with

Placeholder

to the ​

Placeholder

of ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

repeated

2

12

power

6

Multiple Choice

Are there any advantages to using the exponent formula instead of multiplying each row by 2?

• No, multiplying each row by 2 is faster.
• Yes, using the exponent formula is more efficient than multiplying each row by 2.
• Yes, the multiplication method is always more accurate.
• No, the exponent formula complicates calculations.
• Yes, using the exponent formula is less efficient.

Drag and Drop

If we wrote b_3 = 8, what does that mean in the context of binary strings? There are ​

Placeholder

binary strings with a length of ​

Placeholder

.
How would we use sequence notation to write, “There are 1,024 binary strings with a length of 10”? We would write b_​

Placeholder

=​

Placeholder

.

Drag these tiles and drop them in the correct blank above

8

3

10

1,024

Multiple Choice

What is the general rule (or formula) for determining the number of binary strings
with a length of n written in sequence notation?

• b_n = 2n
• b_n = n^2
• b_n = 2^n

Part 3: Exploring Ternary

Strings

Handout 4.1: Video Game

Cheat Codes

Poll

What is your favorite game genre?

• RPG

• Simulation

• Action

• Sport

• Something else!

Counting Formulas

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.