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Visualizing Multiplication of Binomials and Factoring with Algebra Tiles. Jim Rahn www.jamesrahn.com [email protected] Multiplication Using the Area Model. Write the dimensions of both sections and find the area of each. .

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Visualizing Multiplication of Binomials and Factoring with Algebra Tiles

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Visualizing multiplication of binomials and factoring with algebra tiles l.jpg

Visualizing Multiplication of Binomials and Factoring with Algebra Tiles

Jim Rahn

www.jamesrahn.com

[email protected]


Multiplication using the area model l.jpg

Multiplication Using the Area Model


Slide3 l.jpg

Write the dimensions of both sections and find the area of each.

Draw a rectangle that measures 3 by 8 using the grid at the top of the page.

9

3

3

15

3

5

Complete the following statement:

What is the area of the rectangle?

Separate the rectangle into two parts by drawing a vertical line so that one rectangle has an area of 9.

Left Rectangle

Right Rectangle

Whole

Rectangle


Slide4 l.jpg

Write the dimensions of both sections and find the area of each.

Draw a rectangle that measures 5 by 9 using the grid at the top of the page.

5

15

30

5

Complete the following statement:

3

6

What is the area of the rectangle?

Separate the rectangle into two parts by drawing a vertical line so that one part is 5 by 3

Left Rectangle

Right Rectangle

Whole

Rectangle


Slide5 l.jpg

Draw a rectangle that measures 6 by 10 using the grid at the top of the page.

6

24

36

6

Complete the following statement:

What is the area of the rectangle?

4

6

Separate the rectangle into two parts to illustrate 6 x 4 and 6 x 6.

Left Rectangle

Right Rectangle

Whole

Rectangle


Slide6 l.jpg

Use the partial rectangle at the bottom of the page to draw a rectangle that is 12 x 15.

Left Rectangle

Right Rectangle

Whole

Rectangle

Complete the following statement:

What is the area of the rectangle?

15

Separate the rectangle into two parts to illustrate 12 x 4 and 12 x 11

12

11

4


Slide7 l.jpg

Use the partial rectangle at the bottom of the page to draw a rectangle that is 13 x 18.

10

8

10

3

10

3

3

8

10

10

10

8

100 + 80+30+24

234

Separate the 18 side into two parts 10 and 8 using a vertical line.

18

Study your picture and complete the statement at the top of this page.

10

8

Separate the 13 side into two parts 10 and 3 by using a horizontal line.

10

13

3


Multiplying with variables l.jpg

Multiplying with Variables


Slide9 l.jpg

  • To complete 2(x+1) show two groups of (x + 1). Form a rectangle with the pieces on the table.

  • How long is your rectangle? How wide is your rectangle? Notice how the length and width of the rectangle are part of 2(x+1).

  • What is the algebraic name for the inside of your rectangle?


Slide10 l.jpg

  • To completeshow three groups of .

  • Form a rectangle with the pieces on the table.

  • How long is your rectangle?

  • How wide is your rectangle?

  • What is the algebraic name for the inside of your rectangle?


Slide11 l.jpg

To completewhat should you show on the table?

What is this equal to?


Slide12 l.jpg

To completewhat should you show on the table?

What is this equal to?


Multiplying binomials l.jpg

Multiplying Binomials


Slide14 l.jpg

  • Use the multiplication rectangle to make a rectangle whose dimensions are x by 2x+ 1.

  • Place an x tile on the left side and tiles that represent 2x + 1 across the top as illustrated at the right.

  • Fill in the rectangle to show it’s area.

  • Draw a picture of your rectangle at the right. What is the algebraic name for the inside of your rectangle?


Slide15 l.jpg

  • Fill in the area of the rectangle.

  • Draw a picture of your rectangle.

  • What is the algebraic name for the inside of your rectangle?

  • Use the multiplication rectangle to multiply .


Slide16 l.jpg

  • Use the multiplication rectangle to multiply .

  • Fill in the rectangle.

  • Draw a picture of your rectangle.

  • What is the algebraic name for the inside of your rectangle?


Slide17 l.jpg

  • Use the multiplication rectangle to multiply .

  • Fill in the rectangle.

  • Draw a picture of your rectangle.

  • What is the algebraic name for the inside of your rectangle?


Slide18 l.jpg

Multiply


Slide19 l.jpg

  • Study the picture of the last few problems. Is there a way you can predict how many x rectangles will be in your final rectangle?

  • Can you predict their location?

  • Can you predict the number of blue unit squares that will be in your final rectangle?

  • Can you predict their location?

  • Try to predict what the rectangle will look like for .

  • Draw the picture without using the tiles.

  • Write the algebraic expression for the rectangle.


Slide20 l.jpg

  • Draw the picture for the multiplication of .

  • Write the algebraic expression for the rectangle.


Slide21 l.jpg

  • Recall that each of the following is a representation for zero

  • Show zero using 4 rectangles.

  • Show zero using 6 small squares


Slide22 l.jpg

  • Use the multiplication rectangle to multiply (x + −1)(x + 2) .

  • Once you have set up the dimensions fill in the rectangle. Watch the colors of the tiles. This problem involves a negative sign.

  • Draw a picture of your rectangle. Can you simplify the rectangle by using zero pairs? What is the algebraic name for the inside of your rectangle?


Slide23 l.jpg

  • Use the multiplication rectangle to multiply (x + 1)(2x + -3) .

  • Once you have set up the dimensions fill in the rectangle. Watch the colors of the tiles. This problem involves a negative sign.

  • Draw a picture of your rectangle. Can you simplify the rectangle by using zero pairs? What is the algebraic name for the inside of your rectangle?


Slide24 l.jpg

  • Use the multiplication rectangle to multiply (x + −1)(2x + −1) .

  • Use the multiplication rectangle to multiply (2x + −1)(x + 3) .

  • Use the multiplication rectangle to multiply (x + −2)(2x + 3).


Slide25 l.jpg

  • Study the picture from the last few problems.

  • Is there a way you can predict how many x rectangles will be in your final rectangle?

  • Can you predict their location?

  • Can you predict the number of blue unit squares that will be in your final rectangle?

  • Can you predict their location?

  • Try to predict what the rectangle will look like for (2x + −3)(x + 4) .

  • Draw the picture without using the tiles. Write the algebraic expression for the rectangle.


Slide26 l.jpg

  • Draw the picture for the multiplication of (3x + −1)(x + 2) .

  • Write the algebraic expression for the rectangle.


Extending the algebra tile model to an rectangle diagram l.jpg

Extending the Algebra Tile Model to an Rectangle Diagram


Using a rectangle diagram l.jpg

Using a Rectangle Diagram

Multiply (2x + 1)(x + 1)

Subdivide the rectangle

2x

1

Label the dimensions

x

x

2x2

Calculate the areas

Write the expression

1

1

2x


Using a rectangle diagram29 l.jpg

Using a Rectangle Diagram

Multiply (3x + 1)(x + -2)

Subdivide the rectangle

3x

1

Label the dimensions

x

x

3x2

Calculate the areas

Write the expression

-2

-2

-6x


Using a rectangle diagram30 l.jpg

Using a Rectangle Diagram

Multiply these:

(4x +-1)(x + -3)

(1+-2x)(-1+2x)

(1+-x)(2+3x)


Factoring polynomials l.jpg

Factoring Polynomials


Slide32 l.jpg

  • Use the following pieces: one x2 piece, four x pieces, and three unit pieces.

  • Form a rectangle from these eight pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these eight pieces.

  • Describe the dimensions of your rectangle.


Slide33 l.jpg

  • Use the following pieces: one x2 piece, four x pieces, and four unit pieces.

  • Form a rectangle from these nine pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these nine pieces.

  • Describe the dimensions of your rectangle.


Slide34 l.jpg

  • Use the following pieces: one x2 piece, five x pieces, and six unit pieces.

  • Form a rectangle from these twelve pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these twelve pieces.

  • Describe the dimensions of your rectangle.


Slide35 l.jpg

  • Use the following pieces: one x2 piece, seven x pieces, and six unit pieces.

  • Form a rectangle from these fourteen pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these fourteen pieces.

  • Describe the dimensions of your rectangle.


Slide36 l.jpg

  • Use the following pieces: one x2 piece, nine x pieces, and eight unit pieces.

  • Form a rectangle from these eighteen pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these eighteen pieces.

  • Describe the dimensions of your rectangle.


Slide37 l.jpg

  • Use the following pieces: one x2 piece, eight unit pieces, a different number of x pieces.

  • How many x pieces will you need to be able to form a rectangle from these pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these more than nine pieces.

  • Describe the dimensions of your rectangle.


Slide38 l.jpg

  • Use the following pieces: two x2 piece, seven x pieces pieces, and three unit pieces.

  • Form a rectangle from these pieces.

  • What polynomial is represented by the rectangle?

  • Describe the polynomial represented by these pieces.

  • Describe the dimensions of your rectangle.


Slide39 l.jpg

  • Suppose you want to determine the two factors whose product is 2x2 +11x + 9 .

  • Describe how you think about the arrangement of the twenty-two tiles so it will make a rectangle.


Slide40 l.jpg

  • Find the factors for

  • Find the factors for

  • Find the factors for

  • Find the factors for


Slide41 l.jpg

  • Find the factors for

  • Find the factors for

  • Find the factors for

  • Find the factors for


Slide42 l.jpg

  • Suppose you want to determine the two factors whose product is .

  • Describe how you think about the arrangement of the six tiles so it will make a rectangle.


Slide43 l.jpg

  • Suppose you want to determine the two factors whose product is .

  • Describe how you think about the arrangement of the six tiles so it will make a rectangle.


Extending the algebra tile model to an rectangle diagram44 l.jpg

Extending the Algebra Tile Model to an Rectangle Diagram


Using a rectangle diagram45 l.jpg

Using a Rectangle Diagram

Factor 2x2 + 3x + 1

Subdivide the rectangle

2x

1

Label the area at the bottom right

x

x

2x2

Label the area at the top left

Think how you can separate the area of 3x

1

1

2x

Label the dimensions


Using a rectangle diagram46 l.jpg

Using a Rectangle Diagram

Factor 4x2 + 3x + -1


Using a rectangle diagram47 l.jpg

Using a Rectangle Diagram

Factor 2x2 + 7x +3


Using a rectangle diagram48 l.jpg

Using a Rectangle Diagram

Factor 2x2 + 7x +3


Using a rectangle diagram49 l.jpg

Using a Rectangle Diagram

Factor 2x2 + 5x +3


Think about it l.jpg

Think about it

  • Did the Distributive Property Template build understanding for the Distributive Property?

  • Did the Algebra Tiles help you give meaning to 2(x+1)?

  • Did the Algebra Tiles help you visualize multiplication of binomials?

  • Did the Algebra Tiles help you visualize factoring trinomials?


Think about it51 l.jpg

Think about it

  • Did the Multiplication Rectangle help you multiply binomials?

  • Did the Multiplication Rectangle help you factor trinomials?


Other activities for algebra tiles and rectangle diagrams l.jpg

Other activities for Algebra Tiles and Rectangle Diagrams

  • Completing the Square

  • Solving Quadratic Equations

  • Developing the Quadratic Formula


Activities on www jamesrahn com l.jpg

Activities on www.jamesrahn.com

  • Algebra Tiles

  • Cutout set of algebra tiles


Visualizing multiplication of binomials and factoring with algebra tiles54 l.jpg

Visualizing Multiplication of Binomials and Factoring with Algebra Tiles

Jim Rahn

www.jamesrahn.com

[email protected]


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