Using Algebra Tiles to Develop Conceptual Understanding

Using Algebra Tiles To Develop Conceptual Understanding

Using Algebra Tiles to

Multiply, Divide, Factor and

Complete the Square

Use algebra tiles to build each array. Sketch the array. Write each solution as a product that is equal to a sum.

1. (x + 2)(x + 5)2. (x + 3)(x + 3)

3. (x + 1)(x + 2)4. (2x + 1)(x + 4)

5. (x + 1)(2x + 3)6. (2x + 1)(2x + 3)

For each problem, build as many rectangular arrays as you can using the given number of “x2"s and “1"s. You may use as many “x” pieces as you need to complete the array. Sketch and list the dimensions of each array. Write your answer as a product equal to a sum.

1. One “x2" and three “1"s

2. One “x2" and eight “1"s

3. Two “x2"s and three “1"s

4. Two “x2"s and four “1"s

Factoring trinomials

Build a rectangle for each problem. Sketch your array. Write the answer as a sum equal to a product.

1. x2 + 5x + 6

2. 2x2 + 3x + 1

3. x2 + 2x + 1

4. 3x2 + 5x + 2

5. 4x2 + 4x + 1

6. x2 + 5x + 4

7. x2 + 6x + 9

8. 2x2 + 7x + 6

9. 3x2 + 7x + 4

Multiplying Polynomials when terms are negative

1.How would you represent (x – 2)? Record your model.

2.How would you represent (x - 2)(x + 3)? Build an algebra tile array to represent the project. Record the area model and the product as a sum.

3. Build algebra tile arrays to represent the following products. Write each answer as a product equal to a sum. Sketch each array.

a. (x - 2)(x - 3) b. (x + 2)(x - 3) c. (x - 1)(x + 3)

d. (x + 1)(x – 3) e. (x + 5)(x – 2) f. (x – 5)(x + 2)

g. (2x + 3)(x – 4) h. (2x – 3)(x + 4) i. (2x – 3)(x – 4)

Describe the patterns you see when these problems are modeled with the algebra tiles. What conclusions can you make from these patterns that would describe the final products?

Factoring when terms are negative

Build rectangles for each of the following trinomials and find their factors. Write each problem as a sum equal to a product. Sketch each array.

1. x2 - 7x + 122. x2 - 4x - 12

3. x2 + 3x - 104. x2+ x - 12

5. 2x2 - 3x – 2 6. 2x2 + x – 3

7. 3x2 - x – 28. 2x2 - x – 3

9. 3x2 - 5x + 210. 2x2 + 7x – 15

Dividing Polynomials

Use algebra tiles to build an array to find each of the following quotients. Sketch each array. Write each answer as a polynomial and express remainders as fractions.

1. x2 + 5x + 42. 3x2 + 8x + 4

x + 2x + 2

3. x2 - 2x – 84. x2 + 7x + 5

x + 2x + 2

5. x2 + 7x + 5 6. x2 - 4x + 3

x + 1x + 1

Completing the Square

If you have 4 “x2” tiles and 8 “x” tiles, how many unit tiles would you need to build a perfect square? Use your tiles to build the model and the sketch it below.

Suppose you have one x2 tile and 9 unit tiles. Is it possible to use some rectangular x tiles to build a composite square? Draw a diagram to show how it is done. Write the area of the composite square as a product and as a sum.

Suppose you have one x2 tile and 12 x tiles. How many small unit tiles will you need to make a composite rectangle that is a square? Sketch the composite square and write its area as a product and a sum.

Complete the following expressions to complete the squares. Sketch the composite square of each one.

1. x2 + 10x + _____2. x2 – 2x + _____

3. x2 + 4x + _____4. x2 – 8x + _____

Use Algebra Tiles to model “completing the square”. Use them to solve the following quadratic equations for x:

1. x2 + 6x = 16

2. x2 + 4x + 1 = 6

Now solve these on your own!

3. x2 - 6x = -5

4. x2 - 10x + 14 = 0

5. x2 - 4x + 3 = 0

6. 2x2 + 3x – 9 = 0