Operations & integers

ILLUSTRATING INTEGERS The University of Texas at Dallas

INTRODUCTION TO INTEGERS The University of Texas at Dallas

INTRODUCTION TO INTEGERS Integers are positive and negative numbers. … , -6, -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5, +6, … Each negative number is paired with a positive number the same distance from 0 on a number line. The University of Texas at Dallas -3 -2 -1 2 0 1 3

INTRODUCTION TO INTEGERS We can represent integers using red and yellow counters. Red tiles will represent negative integers, and yellow tiles will represent positive integers. The University of Texas at Dallas Negative integer Positive integer

INTRODUCTION TO INTEGERS The diagrams below show 2 ways to represent -3. Represent -3 in 2 more ways. The University of Texas at Dallas -3 -3 NOTE TO TUTOR

INTRODUCTION TO INTEGERS Students may represent -3 by using 3 red tiles and any number of groups of +1 and -1 tiles The University of Texas at Dallas

INTRODUCTION TO INTEGERS Tell which integer each group of tiles represents. The University of Texas at Dallas -2 +1 ANSWER +2

INTRODUCTION TO INTEGERS If there are the same number of red tiles as yellow tiles, what number is represented? It represents 0. The University of Texas at Dallas

ADDITION AND SUBTRACTION The University of Texas at Dallas

ADDING INTEGERS We can model integer addition with tiles. Represent -2 with the fewest number of tiles Represent +5 with the fewest number of tiles. The University of Texas at Dallas

ADDING INTEGERS What number is represented by combining the 2 groups of tiles? Write the number sentence that is illustrated. -2 + + 5 = + 3 The University of Texas at Dallas + 3

ADDING INTEGERS Use your red and yellow tiles to find each sum. -2 + -3 = ? The University of Texas at Dallas ANSWER + = -5 -2 + -3 = - 5

ADDING INTEGERS The University of Texas at Dallas - 6 + + 2 = ? + = - 4 + = + 1 - 3 + + 4 = ? ANSWER ANSWER - 6 + + 2 = - 4 - 3 + + 4 = + 1

SUBTRACTING INTEGERS We often think of subtraction as a “take away” operation. Which diagram could be used to compute + 3 - + 5 = ? The University of Texas at Dallas + 3 + 3

SUBTRACTING INTEGERS The University of Texas at Dallas We can’t take away 5 yellow tiles from this diagram. There is not enough tiles to take away!! This diagram also represents +3, and we can take away +5. When we take 5 yellow tiles away, we have 2 red tiles left.

SUBTRACTING INTEGERS Use your red and yellow tiles to model each subtraction problem. The University of Texas at Dallas - 2 - - 4 = ? ANSWER

SUBTRACTING INTEGERS The University of Texas at Dallas This representation of - 2 doesn’t have enough tiles to take away - 4. Now if you add 2 more reds tiles and 2 more yellow tiles (adding zero) you would have a total of 4 red tiles and the tiles still represent - 2. Now you can take away 4 red tiles. - 2 - - 4 = + 2 2 yellow tiles are left, so the answer is…

SUBTRACTING INTEGERS The University of Texas at Dallas Work this problem. + 3 - - 5 = ? ANSWER

SUBTRACTING INTEGERS The University of Texas at Dallas Add enough red and yellow pairs so you can take away 5 red tiles. Take away 5 red tiles, you have 8 yellow tiles left. + 3 - - 5 = + 8

SUBTRACTING INTEGERS The University of Texas at Dallas Work this problem. - 3 - + 2 = ? ANSWER

SUBTRACTING INTEGERS The University of Texas at Dallas Add two pairs of red and yellow tiles so you can take away 2 yellow tiles. Take away 2 yellow tiles, you have 5 red tiles left. - 3 - + 2 = - 5

SUBTRACTING INTEGERS The University of Texas at Dallas A fact family gives 4 true equations using the same 3 numbers. For example: 7 + 8 = 15 8 + 7 = 15 15 – 7 = 8 15 – 8 = 7 All of these statements are true.

SUBTRACTING INTEGERS The University of Texas at Dallas We can also use fact family with integers. Use your red and yellow tiles to verify this fact family: - 3 + + 8 = + 5 + 8 + - 3 = + 5 + 5 - + 8 = - 3 + 5 - - 3 = + 8

INTEGERS AND MULTIPLICATION The University of Texas at Dallas

MULTIPLICATION Red and yellow tiles can be used to model multiplication. Remember that multiplication can be described as repeated addition. So 2 x 3 = ? The University of Texas at Dallas 2 groups of 3 tiles = 6 tiles

MULTIPLICATION 2 x - 3 means 2 groups of - 3 The University of Texas at Dallas 2 x - 3 = - 6

MULTIPLICATION Since 2 x 3 = 6 and 3 x 2 = 6, does it make sense that - 3 x 2 = - 6 ? - 3 x 2 = - 6 is true. + 2 x - 3 = - 6 and - 3 x + 2 = - 6 belong to a fact family: The University of Texas at Dallas + 2 x - 3 = - 6 - 3 x + 2 = - 6 - 6 ÷ + 2 = - 3 - 6 ÷ - 3 = + 2

MULTIPLICATION Use your tiles to model each multiplication problem. + 2 x + 4 = ? The University of Texas at Dallas ANSWER 2 groups of + 4 + 2 x + 4 = + 8

MULTIPLICATION + 3 x - 3 = ? The University of Texas at Dallas ANSWER 3 groups of - 3 + 3 x - 3 = - 9

MULTIPLICATION - 2 x + 4 = ? The University of Texas at Dallas ANSWER 4 groups of - 2 - 2 x + 4 = - 8 Use the fact family for - 2 x + 4 = ?  We can’t show - 2 groups of + 4 + 4 x - 2 = ?  we can show 4 groups of - 2

MULTIPLICATION + 1, - 1 are opposites the products are opposite Since + 2 and - 2 are opposites of each other, + 2 x - 3 and - 2 x - 3 have opposite products. The University of Texas at Dallas + 1 x + 3 = + 3 - 1 x + 3 = - 3

MULTIPLICATION To model - 2 x - 3 use 2 groups of the opposite of - 3 The University of Texas at Dallas - 2 x - 3 = + 6

MULTIPLICATION Use tiles to illustrate - 3 x - 4 =? The University of Texas at Dallas ANSWER 3 groups of + 4 (the opposite of - 4) - 3 x - 4 = + 12

INTEGERS AND DIVISION The University of Texas at Dallas

DIVISION Use tiles to model + 12 ÷ + 3 = ? The University of Texas at Dallas Divide 12 yellow tiles into 3 equal groups + 12 ÷ + 3 = + 4 4 yellow tiles in each group.

DIVISION Use tiles to model - 15 ÷ + 5 =? The University of Texas at Dallas Divide - 15 into 5 equal groups - 15 ÷ + 5 = - 3

DIVISION Write the fact family - 4 x + 2 = - 8 Compare your answer. + 2 x - 4 = - 8 - 8 ÷ + 2 = - 4 - 8 ÷ - 4 = + 2 The University of Texas at Dallas

DIVISION Now try these! Use red and yellow tiles to find each answer. 1) - 2 + - 8 5) + 3 x - 4 2) + 6 + - 4 6) - 5 x - 2 3) - 5 - + 4 7) - 12 ÷ + 6 4) - 6 - - 7 8) - 8 ÷ + 2 The University of Texas at Dallas ANSWER

DIVISION The University of Texas at Dallas 1) - 2 + - 8 = - 10 2) + 6 + - 4 = + 2

DIVISION The University of Texas at Dallas 3) - 5 - + 4 = - 9 4) - 6 - - 7 = + 1

DIVISION The University of Texas at Dallas 5) + 3 x - 4 = - 12 6) - 5 x - 2 = + 10 5 groups of the opposite of - 2

DIVISION The University of Texas at Dallas 7) - 12 ÷ + 6 = - 2 8) - 8 ÷ + 2 = - 4 -8 divided into 2 equal groups -12 divided into 6 equal groups

Operations & integers

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Operations & integers