______ How can the area of a rectangle be found? Multiply the sides Given the following rectangles and their areas suggest what the sides might be. Hint: What do you know about the sides in relation to the area? they must multiply to equal the area

______ A=42x 3 A=24x A=36x 2 A=18 A=12 A=48x 4 How can the area of a rectangle be found? Multiply the sides Given the following rectangles and their areas suggest what the sides might be. Hint: What do you know about the sides in relation to the area?) they must multiply to equal the area A=42x 3 A=24x A=36x 2 A=18 A=12 A=48x 4

______ Since the sides multiply to equal the area, the sides are the ________ of the area.

Factoring Since the sides multiply to equal the area, the sides are the factors of the area. Factor the following. 6 = 28 = 12x 2 =

Factoring Since the sides multiply to equal the area, the sides are the factors of the area. Factor the following. 6 = 6 x 1 or 2 x 3 28 = 1 x 28 or 2 x 14 or 4 x 7 12x 2 = 1 x 12x 2 or 2x x 6x or 3x2 x 4 … to factor -

Factoring Since the sides multiply to equal the area, the sides are the factors of the area. Factor the following. 6 = 28 = 12x 2 = to factor – to find the ‘things’ that multiply to equal what is being factored - to find the sides of a rectangle ex. 30 =__ x __ ? ? A=20

Factoring Suggest a common factor for each. 9, 24 30, 48, 60 12x3, 18x4 10xy4, 25xy5

Factoring Suggest a common factor for each. 9, 24 - 3 30, 48, 60 – 2, 3, 6 12x3, 18x4 - 2, 2x, 3x3, 4 etc. 10xy4, 25xy5 – x, y, 5xy, 5y4 etc. Give the greatest common factor for each of the above. 3 6 6x3 5xy4

Lets try factoring something more interesting Lets try factoring something more interesting. Lets try factoring some polynomials. For now we are going to factor the polynomials the same way you factored at the beginning of this activity, by determining the sides of rectangles. We will model the polynomials we are trying to factor with our algebra tiles by constructing rectangles made up of the areas of the polynomials being factored and determine the factors by simply taking the sides of our rectangles. As we do, hopefully you will make some connections and conclusions that will allow you to factor without the tiles. We will eventually give this type of factoring a specific name.

_______Factoring Model the following by drawing rectangles with the given areas. Label the sides. Factor the area. (Hint: since the area of a rectangle is found by multiplying the sides, the sides are the factors) 2x + 4

_______Factoring Model the following by drawing rectangles with the given areas. Label the sides. Factor the area. (Hint: since the area of a rectangle is found by multiplying the sides, the sides are the factors) 2x + 4 x 2 2 2x + 4 =2(x + 2)

_______Factoring Model the following by drawing rectangles with the given areas. Label the sides. Factor the area. (Hint: since the area of a rectangle is found by multiplying the sides, the sides are the factors) 2x + 4 x 2 2 2x + 4 =2(x + 2) 4x + 6 c) 6x + 3 d) 3x – 6 e) x2 + 3x f) 2 x2 + 6x 3 x2 + 3x h) 8x + 6 i) 4 x2 + 2x j) 6x + 9 k) 6 x2 + 4x l) 4x + 2 m) 6 x2 + 3x n) 8x + 4

______Factoring Conclusions Have you noticed anything that may help you factor without tiles? Here’s a hint. Look at the sides of your rectangles/your factors. What has been true about one side of the rectangle/one of the factors every time?

Ex. 2 x + 4 x 2 2 2x + 4 =2(x +2) What do you notice about that side/that factor ?

______Factoring Conclusions Have you noticed anything that may help you factor without tiles? Here’s a hint. Look at the sides of your rectangles/your factors. What has been true about one side of the rectangle/one of the factors every time? one side/factor is a monomial and a common factor of the terms in the polynomial being factored

Ex. 2 x + 4 x 2 2x + 4 2 = 2(x + 2) What do you notice about that side/factor ? It is a monomial and a common factor of the terms in the area/polynomial being factored (ex. 2 is a common factor of 2x and 4)

Use the above conclusion to try factoring the following without tiles. 6x + 4 (one factor/one side of the rectangle would be a monomial and a common factor, therefore think of a common factor of 6x and 4) =2( ) (the other factor/side of the rectangle must multiply by the first to equal 6x + 4) =2(3x +2)

b) 6x2 + 3x (think of a common factor) =3x( ) (what would multiply by 3x to equal 6x2 + 3x) =3x(2x + 1)

This type of factoring is called ________ .

Common Factoring This type of factoring is called common factoring.(add that title where common factoring began and at the beginning of the conclusions) To common factor: 1)Find the greatest common factor of the terms in the polynomial/area being factored. That would be one side of the rectangle/one factor. Place it outside the brackets. 2)Find the other factor, knowing it must multiply by the common factor (gcf) to equal the polynomial/area being factored. That would be the other side of the rectangle/the other factor. Place it inside the brackets. ex. 4x2 + 2x = 2x ( 2x + 1) (gcf) (multiplies by the common factor to = the polynomial being factored)