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# Multiplying and Factoring Polynomials

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by

## Rani Shaikh

on 16 November 2013

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#### Transcript of Multiplying and Factoring Polynomials

Do you see any relationship between:

the length of the lot
the width of the lot
and the total area of the lot?

Common Core Standards
A-APR 1.

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Please find a buddy to pair up with while we pass out some materials for you to work on your puzzle!

By Rani Shaikh, Caleb Robledo, and Courteney Uraine

Cal State University, Fullerton
Fall 2013

EDSC 542 M: Advanced Methods for Teaching Foundation Level Mathematics

Visual Presentation on Polynomials
You should have:
copy of problem,
graph paper,
a square,
4 rectangles,
and 4 smaller 1x1 squares.

A School District has an empty lot on which it can fit:

1 square building:

each side measures "x" yards long.

4 rectangular bus parking spots:

"x" yards long and 1 yard wide.

4 square bicycle parking spots:

each side measures 1 yard long.

The building will have two entrance doors:

each door will be on an adjacent wall.

The entrance cannot be blocked by any parking spots.

With the materials provided, please glue the building and the parking spots onto the graph paper in the best possible configuration.

HINT:
Parking spots will be adjacent to other two remaining walls.

Now calculate the area of the building.

Area of Building= (fill in the answer)

GOALS AT THIS POINT:

Everyone should have come up with
an expression for the total area of the lot.

Everyone should have noticed that this expression is a polynomial; or more specifically
a trinomial.

Everyone should come to the conclusion that
the length/width of the sides of the lot,
when multiplied give us the area of the lot.

Now find the total area of the lot by adding the individual areas.
In other words:

TOTAL AREA OF LOT =

AREA OF BUILDING
+
TOTAL AREA OF BUS SPACES
+
TOTAL AREA OF BICYCLE SPACES

Do you get an algebraic expression?
What is the expression? What is another name for the expression?
Next calculate the area of a bus parking space
Ok, now that you know the area of one bus parking space;

What is the combined area of ALL bus parking spaces?

Total Area of Bus Spaces =

What is the area of a bicycle parking space?
Ok, now that you know the area of one bicycle parking space;

what is the combined area of ALL bicycle parking spaces?

Total Area of Bicycle Spaces=

WHAT IS THE
L
E
N
G
T
H
OF THE LOT?

WHAT IS THE

W I D T H

OF THE LOT?
What is a factor?

A number may be made by multiplying two or more other numbers together. The numbers that are multiplied together are called factors of the final number

What are factors of an algebraic expression?
In the case of an algebraic expression; Factorization is defined as the process of breaking down an expression into a product of different expressions called factors.
In other words, factorization refers to breaking down large and at times complicated expressions into a product of smaller ones that are then easier to deal with.
You can also think of factorization as the opposite of distribution.
Algebra tiles are hands on tools (manipulatives) used to represent and solve mathematical equations.
You should get:

Representing Polynomials with Algebra Tiles
Try representing the following polynomials using algebra tiles:

x + 5x +6
2
Group#1:
Group#2:

x + 5x + 4
2
Group #3:
x+ 6x + 5
2
Can you find the factors of the polynomial expression?
Here's what your finished puzzle should look like
This is incorrect, because we really don't know what the measurement of "x" is. Therefore, we cannot assume that two bicycle parking spaces would leave enough room for a bus parking space.
Polynomials form a system analogous to integers......
What does that even mean??????

How do we make sense of something that sounds so complicated????

Why don't we solve a puzzle!
And see if we can simplify things a bit!
This activity was designed for ALL students at all learning levels.

The lesson was designed to connect to real world situation in order to connect with an abstract application of the same concepts.

IN
..........
Strategic collaboration between slow learners and gifted students will provide scaffolding.

ELL students will not get confused with the terminology or formulas, but will be able to focus on the hands on learning experience.
Terminology used for the puzzle problem is simple and can be reinforced by posters or bulletins in the classroom with translation of terms like "area", "factors" " multiplication" etc.
this lesson provides a strong foundation for introducing further concepts; such as :

How to represent polynomials with negative terms?

Polynomials that cannot be factored
How to factor polynomials with negative terms
We've built connections
between:
Area models and Multiplication
Algebra tiles and area models
Algebra tiles and factors
We've learned why algebra tiles make sense
Trinomials can be multiplied and factored just like whole numbers!
Area models and factoring
How do we solve for factors if we have a polynomial?
DID YOU MENTION THE WORD "FACTORS" ?
Here's what they look like. Remind you of anything?
ALGEBRA TILES
(hint: the puzzle pieces!)
You're on the right track!
Let's talk a little bit about factors:
ALEGEBRA TILES can be used to find the factors of an expression.
Here's what an area model for multiplication would look like
Algebra tiles are used to depict an area model of multiplication for polynomials

WHAT'S AN AREA MODEL OF MULTIPLICATION???

LET'S TAKE A LOOK:

IMPORTANT QUESTION:

Why do Algebra tiles have to form a perfect rectangle in order to solve or factor the expression?
Or maybe you did it this way..........
Full transcript