Everything about Integers

Interesting Integers! We can represent integers using red and yellow counters. Red tiles will represent negative integers, and yellow tiles will represent positive integers. Negative integer Positive integer

Definition Positive number – a number greater than zero. 1 2 3 4 5 6

Definition Negative number – a number less than zero. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

Definition Opposite Numbers – numbers that are the same distance from zero in the opposite direction -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

Definition Integers – Integers are all the whole numbers their opposites and zero. 7 opposite -7

Example: The opposite of 6 is -6 What are Opposites? Two integers the same distance from the origin, but on different sides of zero Every positive integer has a negative integer an equal distance from the origin Example: The opposite of 6 is -6 The opposite of -2 is 2

Interesting Integers! If there are the same number of red tiles as yellow tiles, what number is represented? zero pair It represents 0.

Definition The absolute value of 9 or of –9 is 9. Absolute Value – The size of a number with or without the negative sign. The absolute value of 9 or of –9 is 9.

What is Absolute Value? Distance a number is from zero on a number line (always a positive number) Indicated by two vertical lines | | Every number has an absolute value Opposites have the same absolute values since they are the same distance from zero Example: |-8| = 8 and |8| = 8 |50| = 50 and |-50| = 50

Examples 7 = 7 10 = 10 -100 = 100 5 - 8 = -3= 3

|7| – |-2| = ? -9 -5 5 9

|-4 – (-3)| = ? -1 1 7 Purple

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level 30 20 10 -10 -20 -30 -40 -50

Negative Numbers Are Used to Show Debt Let’s say your parents bought a car but had to get a loan from the bank for $5,000. When counting all their money they add in -$5.000 to show they still owe the bank.

Hint If you don’t see a negative or positive sign in front of a number it is positive. 9 +

ADDITION AND SUBTRACTION Interesting Integers! ADDITION AND SUBTRACTION

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.

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

= -5 + ADDING INTEGERS Use your red and yellow tiles to find each sum. -2 + -3 = ? ANSWER = -5 + -2 + -3 = -5

+ - 4 = +1 = + ADDING INTEGERS -6 + +2 = ? -3 + +4 = ? -6 + +2 = -4 ANSWER -6 + +2 = ? + = - 4 -6 + +2 = -4 -3 + +4 = ? ANSWER = +1 + -3 + +4 = +1

Solve the Problems -3 + -5 = 4 + 7 = (+3) + (+4) = -6 + -7 = 5 + 9 = -9 + -9 = -8 11 7 -13 14 -18

Solve These Problems 3 + -5 = -4 + 7 = (+3) + (-4) = -6 + 7 = 5 + -9 = -9 + 9 = -2 3 -1 1 -4

Adding Integers Number line T-Method

- 3 + 2 = -1 Adding Integers using the number line Negative three means move 3 places to the left Positive two means move 2 places to the right 1 2 -3 -2 -1 The answer is: -1

8 - 3 = 5 Adding Integers using the number line Positive eight means 8 - 3 = 5 Positive eight means move 8 places to the right Negative three means move 3 places to the left 1 2 3 4 5 6 7 8 5 3 6 2 7 1 8 The answer is: 5

Adding Integers T- Method T-Method

Use your red and yellow tiles to find each sum. 2 + 5 = ? - + 2 + = 5 7 T-Method

Use your red and yellow tiles to find each sum. -2 + (-5) = ? - + 2 + = 5 7 T-Method

Use your red and yellow tiles to find each sum. 2 + (-5) = ? + - 5 2 2 + = 3 T-Method

Use your red and yellow tiles to find each sum. -2 + 5 = ? - + 2 2 5 + = 3 T-Method

Subtracting Integers and T- Method T-Method

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

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

SUBTRACTING INTEGERS -2 - -4 = ? Use your red and yellow tiles to model each subtraction problem. -2 - -4 = ? ANSWER

-2 - -4 = +2 SUBTRACTING INTEGERS Now you can take away 4 red tiles. 2 yellow tiles are left, so the answer is… 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. -2 - -4 = +2

SUBTRACTING INTEGERS Work this problem. +3 - -5 = ? ANSWER

+3 - -5 = +8 SUBTRACTING INTEGERS 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 Work this problem. -3 - +2 = ? ANSWER

-3 - +2 = -5 SUBTRACTING INTEGERS 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

-3 – ( -5) = 4 - 7 = (+3) - (+4) = -6 – (-7) = 5 - 9 = -9 – (-9) = SUBTRACTING INTEGERS -3 – ( -5) = 4 - 7 = (+3) - (+4) = -6 – (-7) = 5 - 9 = -9 – (-9) = 2 -3 -1 1 -4

3 – (-5) = -4 - 7 = (+3) - (-4) = -6 - 7 = 5 – (-9)= -9 - 9 = SUBTRACTING INTEGERS 3 – (-5) = -4 - 7 = (+3) - (-4) = -6 - 7 = 5 – (-9)= -9 - 9 = 8 -11 7 13 14 -18

Keep in mind that subtract means add the opposite. In order to subtract two integers, you would need to re-write the problem as an addition problem. Keep in mind that subtract means add the opposite. - - 2 2 (-5) -5 = = + 5 T-Method

-8 – ( -2) = 5 - 4 = (+7) - (+2) = -8 – (-7) = 8 - 5 = -10 – (-2) = SUBTRACTING INTEGERS -8 – ( -2) = 5 - 4 = (+7) - (+2) = -8 – (-7) = 8 - 5 = -10 – (-2) = 6 1 5 -1 3 -8