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Winter 2013 WORK WITH TILES

Using +/- tiles, build the “4” in at least three different ways. Sketch each model after you have built it.

1) 2) 3)

Again using the +/- tiles, build the number “0” in at least three different ways. Sketch each model after you have built it. Each of these models represents a “NEUTRAL ” or “ZERO PAIR.”

1) 2) 3)

Now build at least three different representations of the number “-3.” Sketch each model after you have built it.

1) 2) 3)

Using +/- tiles, build a model of the following with the smallest number of tiles possible. Sketch each model below.

1) -6 2) 5 3) -1

Now add a neutral field to each model immediately above until you have AT LEAST 10 tiles. Has the value of each model changed? Explain below.

Integer Work with Tiles Page 1 of 6

Winter 2013 INTEGER AND Adapted from CPM Educational Program

Addition Example: -8 + 6 0

Start with a neutral field which is an + + + + equal number of positive and negative - - - - tiles and has a value of zero.

-8 +6

------+ + + + + + Display the two numbers using tiles.

+ + + +

- - - -

Combine the two numbers with the + + + + + + + + + + neutral field. ------

-8 + (+6)

Circle the “zeros.” Record the sum. + + + + + + + + + + ------

-8 + 6 = -2 Subtraction Example: -2 – (-4)

Start with the first number displayed + + + + with the neutral field. ------

-2

Circle the second number in your sketch and show with an arrow that + + + + it will be removed. Remove the second ------number. -2 – (-4)

Circle the “zeros.” Record the difference. + + + +

- -

-2 – (-4) = 2 Integer Work with Tiles Page 2 of 6

Winter 2013 INTEGER Adapted from CPM Educational Program

Multiplication is repeated addition or subtraction in a problem with two factors. The first factor tells us how many groups we are adding (+) or subtracting (-). The second factor tells us how many are in each and whether they are positive (+) or negative (-).

(2)(3) means add 2 groups of 3 positive tiles. (2)(-3) means add 2 groups of 3 negative tiles. (-2)(3) means remove 2 groups of 3 positive tiles. (-2)(-3) means remove 2 groups of 3 negative tiles.

Multiplication Example: (2)(-5) 0 -5 Start with a neutral field. Since the first - - - - - factor is positive, groups will be added to + + + + the neutral field. Build 2 groups of 5 - - - - -5 negative tiles. - - - - -

Combine with the neutral field, circle the + + + + “zeros” and record the product. ------

(2)(-5) = -10

Multiplication Example: (-2)(-3) 0

Start with a neutral field. + + + + + + + + ------

Since the first factor is negative, circle the two groups of negatives that will be removed. + + + + + + + + Use arrow to indicate removal. ------

-3 -3

Remove the groups, circle the “zeros” and + + + + + + + + record the product. - -

(-2)(-3) = 6

Integer Work with Tiles Page 3 of 6

Winter 2013 INTEGER

Division is the inverse of multiplication and can also be seen as repeated addition or subtraction in a problem with two factors. The first number is the dividend and tells us what number is our goal. The second number is the and tells us the size of the groups we are adding or removing to make the dividend or goal number. You will find how many groups of the divisor must be “added” to create the dividend. If this is impossible, you will find how many groups of the divisor must be “removed” to create the dividend.

(6) ÷ (3) means how many groups of positive 3 would need to be added to get 6? (-6) ÷ (-3) means how many groups of negative 3 would need to be added to get “-6?” (6) ÷ (-3) means how many groups of negative 3 would need to be removed to get 6? (-6) ÷ (3) means how many groups of positive 3 would need to be removed to get -6?

Division Example: (-6) ÷ (-3) -3 0 - - - Start with a neutral field and add groups + + + + of negative 3 until you get negative 6. -3 ------

Combine with the neutral field, circle the “zeros” and record the number of groups + + + + you had to add, this is the or answer. ------

(-6) ÷ (-3) = 2

Division Example: (-6) ÷ (3) 0 Start with a neutral field. No matter how you try to add groups of positive 3, + + + + + + + + you will NEVER get negative 6. ------

+3 +3 Therefore, you must take away groups of positive 3 from the neutral field until you + + + + + + + + get negative 6. ------

Circle the neutral fields. Since you had to + + remove 2 groups of positive 3 to get negative ------6 the quotient or answer is negative 2. (-6) ÷ (3) = (-2) Integer Work with Tiles Page 4 of 6

Winter 2013 OPERATIONS WITH

Using the +/- tiles, represent each of the statements below. Sketch each model, show the addition or subtraction, and write your simplified solution below each drawing.

1) -5 + (-2) 2) -5 + 2 3) 5 + (-2)

4) -5 – (-2) 5) -5 – 2 6) 5 – (-2)

7) -3 – (-5) 8) -3 – 5 9) 3 – (-5)

Using +/- tiles represent each of the products and sketch the model below the problem. If the first factor in the product is positive, you add that many groups of the second integer. If the first factor in the product is negative, you remove that many groups of the second integer. Always begin with a neutral pair. Write your solution below each drawing.

1) 3 × 4 2) 3 × (−4) 3) -3 × (−4)

€ 4) 2 × (−6) € 5) - 2 × 6 € 6) -2 × (−6)

€ 7) -3 x 4 € 8) 2 x (-3) € 9) -4 x (-2)

Integer Work with Tiles Page 5 of 6

Winter 2013

Using +/- tiles represent each of the and sketch the model below the problem. You will find how many groups of the divisor must be “added” to create the dividend. If this is impossible, you will find how many groups of the divisor must be “removed” to create the dividend. Begin with a neutral pair.

1) 12 ÷ 3 2) -12 ÷ (-3) 3) -12 ÷ 3

4) 12 ÷ (-3) 5) -9 ÷ (-3) 6) -8 ÷ 4

Integer Work with Tiles Page 6 of 6