Joshua Newton

 

Cracking Codes

Objectives:

Students will be able to:

            1. Determine how easy it is to crack a message coded with a shift cipher.

            2. Learn about frequency distributions and be able to use them to break codes.

TEKS:

Algebra I

(b) Foundations for functions: knowledge and skills and performance descriptions.

(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.

(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

(c) Linear functions: knowledge and skills and performance descriptions.

(1) The student understands that linear functions can be represented in different ways and translates among their various representations. Following are performance descriptions.

(A) The student determines whether or not given situations can be represented by linear functions.

 

 

Safety Requirements: 

 

Equipment List:

 

Five-E Organization

 

Teacher Does                                               Student Does

Engage:

Students will be given a message that has been encrypted with a shift cipher and asked to determine the original message.  Students will be told to look for patterns in the encoded message and asked to use those patterns to break the code. After students have been given 5 minutes to break the code, one group will be asked to come up and explain how they arrived at the original message. If other groups used a different strategy, they will be asked to come up and explain it.

 

Students analyze encoded message and try to break the code.

 

Explore:

Hand out Frequency Tally Sheets and ask students to find a paragraph in their book, to count the number of times each letter appears in that paragraph, and to record the numbers on their Tally sheet. Then ask students to get the numbers from 5 other students and to add them up.

 

What do high numbers represent? Low numbers?

 

 

 

 

 

How this table could be used to break a code?

 

Ask students to convert the numbers that they have into frequencies. Explain that a frequency is how often something occurs, and in the case will be the percentage of times that each letter occurs in the English language.

 

Ask students to determine which five letters occur the most according to their results.

 

Pass out a frequency table and have students compare it to their tables.

 

Are the 5 letters that you found to occur the most the same as in the table?

 

Are the values in the table close to the numbers that you got? Are they exactly the same? If not, why do you think they are different?

 

Do you think that shift ciphers are a good choice for a cipher? Why or why not?

 

Tally numbers of letters in paragraphs and share numbers with each other.

 

 

 

 

 

 

High numbers represent letters that occur more often then most letters, and low numbers represent letters that don’t appear as often.

 

 

 

You could do the same thing with the encoded message and compare tables.

 

 

 

 

 

 

 

 

Students should get E, T, N, R, and I.

 

 

 

 

 

 

Yes

 

 

They are not exactly the same because we only used a few paragraphs which may not be representative of normal writing.

 

 

No, because they can be easily cracked by anyone who has a frequency table.

 

Explain:

Why can you use a frequency table to crack a code that has been encoded with a shift cipher?

 

If you do a frequency analysis of an encoded message, will the frequencies match exactly with your frequency table? Why?

 

 

 

Will the order of the letters in the frequency tables always be the same?

 

 

What do you have to do with a frequency table in order to decode a message?

 

 

 

Could you use a frequency table to break a code that only had 5 letters? Why?

 

How many letters do you think are needed in an encoded message to be able to effectively decode it using a shift cipher? Try to find a number that the class can agree upon.

 

What could you do if you knew the message had been encoded by a shift cipher but did not have enough letters to use a frequency table to decode it?

 

Can you write a mathematical equation to describe what a shift cipher does? If so, what would that equation look like?

 

Because the frequencies in the encoded message should be close to the ones in your table.

 

No, the frequencies will not match exactly because there are only a limited number of letters in the encoded message, whereas a frequency table was made from a set of a large number of words.

 

 

No, the order may not be the same because of the limited number of words in the encoded message.

 

You have to do some trial and error by plugging in different letters and seeing which ones give you answers that make sense.

 

Probably not, because you could not generate a good frequency table from it.

 

Students will give various guesses and the reasons behind them.

 

 

 

 

Use trial and error.

 

 

 

 

Yes. Students will offer various formulas.

 

 

Extend:

Discuss arrow diagrams and how they can be used do visually describe a shift cipher.

 

What type of function can a shift cipher be written as? So could the arrow diagram that is used for describing shift ciphers be used to describe other linear functions? If so, how?

 

 

 

 

 

Linear. Yes.

 

 

Source:

Mathematics: Modeling Our World