Difference+of+Squares


 * A.SSE.2 Use the structure of an expression to identify ways to rewrite it and factor a difference of squares.**

**Note to teachers:** All images appear on the worksheets and teacher's copy which are in the attachments.

 * Student’s Background Knowledge:** Students have already learned to factor quadratic expressions.


 * Mathematical Goal:** Students will understand the structure of the difference of two squares and extend that knowledge to factor differences with coefficients other than one and exponents other than 2.


 * //__Identifying Differences of Squares:__//**


 * __Develop__**


 * Launch:** Bell work on factoring: Factor the following polynomials:


 * 1) [[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]]


 * 1) [[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]]

Introduce the task – Please sort the following polynomials into two or more groups. Be ready to justify the decisions for your groups.


 * Explore:** Sort the following polynomials into 2 or more groups and will provide justification for the groups you chose.

    

What name could you give to each group? Within the discussion, teacher guides students to realize that in math we call this the “difference of squares”.
 * Discuss**: Why did you choose the groups you created?

In groups, come up with a definition for the difference of squares. Discuss results as a class and come up with a class definition.


 * __Solidify__**


 * Launch:** Students will sort the following polynomials into 2 or more groups based on the definition we found as a class.


 * Explore:** Sort the following polynomials into 2 or more groups and provide justification for the groups you chose based on the definition we just discussed.

    

Do all of the equations in the difference of perfect squares group meet the definition?
 * Discuss:** Why did you choose the groups you created?


 * __Practice__**


 * //Part I//**


 * Launch:** Instruct students to answer the questions on the following two problems.


 * Explore:** Are these two polynomials the difference of perfect squares? Explain using the definition we came up with to defend your reasoning.


 * 1) [[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]]


 * 1) [[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]]


 * Discuss:** How do these fit in our difference of squares definition?


 * //Part II//**


 * Launch:** Instruct students to answer the following questions:


 * Explore:**


 * 1) Create your own difference of squares polynomial that has one variable and a coefficient other than one.

coefficient other than one.
 * 1) a. Create your own difference of squares polynomial that has two variables and a

b. Create your own difference of squares polynomial that has two variables whose exponents are both greater than 2.


 * Discuss:** Choose some of the examples to discuss with the class. Emphasize with the class that each term of the polynomial is a perfect square and the difference is subtraction.


 * //__Factoring:__//**


 * __Develop__**


 * Launch:** Instruct students to complete part E of the worksheet.


 * Explore:**

Find the area of the rectangle below.


 * Discuss:** What is the total length of each side of the rectangle? Explain how you found your area.


 * __Solidify__**


 * Launch:** Instruct students to complete part F & G of worksheet. Discuss as you go along.


 * Explore:** Given the area of a rectangle is [[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]], can you fill in the dimensions on the following shape.




 * Discuss:** What are the lengths of each side?


 * Explore:** How can you represent [[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]] ?



Discuss the various representations.
 * Discuss:** What does the final area look like? Draw the picture.


 * Explore:** What do you get when you multiply <span style="font-family: 'Calibri','sans-serif'; font-size: 14.6667px;">[[image:http://secondaryiiinutah.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]] ?

How many terms to you get?

How is this similar to a trinomial?

How is this different than a trinomial?

How are the original product and your result related?

Can you represent this area in another way? Show your work.

Can you think of another example of factors that produce similar results?


 * __Solidify__**


 * 1) Factor all of the problems in Part A that are a difference of squares.


 * __Practice__**


 * 1) Factor all of the problems in Part B that are difference of squares.


 * Final Discussion:** Wrap up final results and clarify any questions that are remaining.


 * __Extension__**

This comes from NCTM:


 * Step 1:** Pick any two consecutive numbers.
 * Step 2:** Square each and find the difference.
 * Step 3:** Add the two original numbers.
 * Step 4:** Explain why steps 2 and 3 give the same result.