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INTRODUCING INTEGERS

INTRODUCING INTEGERS. Unit Essential Questions. What are integers? When do we work with integers in the real world? How do we add and subtract integers?. Lesson Essential Questions. What are integers? What situations can be represented with integers?

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INTRODUCING INTEGERS

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  1. INTRODUCING INTEGERS

  2. Unit Essential Questions • What are integers? • When do we work with integers in the real world? • How do we add and subtract integers?

  3. Lesson Essential Questions • What are integers? • What situations can be represented with integers? • In what ways can we order and compare integers?

  4. Key Vocabulary • Integers • Positive Numbers • Negative Numbers • Positive Integers • Negative Integers • Addends • Sum • Difference • Opposites

  5. Integers • Whole numbers and their opposites are integers such as 6 and -6. • Zero is also a whole number. 0 -6 6 Opposites

  6. Definition • Positive number – a number greater than zero. 0 1 2 3 4 5 6

  7. Definition • Negative number – a number less than zero. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  8. Definition • Opposite Numbers – numbers that are the same distance from zero in the opposite direction -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

  9. Negative Numbers Are Used to Measure Temperature

  10. Negative Numbers Are Used to Measure Under Sea Level 30 20 10 0 -10 -20 -30 -40 -50

  11. 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.

  12. 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. 0 1 2 -3 -2 3 -1

  13. 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. Positive integer Negative integer

  14. INTRODUCTION TO INTEGERS • The diagrams below show 2 ways to represent -3. • Represent -3 in 2 more ways. -3 -3

  15. INTRODUCTION TO INTEGERS • Students may represent -3 by using 3 red tiles and any number of groups of +1 and -1 tiles

  16. INTRODUCTION TO INTEGERS • Tell which integer each group of tiles represents. +1 -2 +2

  17. INTRODUCTION TO INTEGERS • If there are the same number of red tiles as yellow tiles, what number is represented? • It represents 0.

  18. Adding and Subtracting Integers

  19. Absolute Value -5 0 5 -10 10 -7 +7 A number line has many functions. Previously, we learned that numbers to the right of zero are positive and numbers to the left of zero are negative. By putting points on the number line, we can graph values. If one were to start at zero and move seven places to the right, this would represent a value of positive seven. If one were to start at zero and move seven places to the left, this would represent a value of negative seven.

  20. Absolute Value -5 0 5 -10 10 -7 +7 Both of these numbers, positive seven (+7) and negative seven (-7), represent a point that is seven units away from the origin. The absolute value of a number is the distance between that number and zero on a number line. Absolute value is shown by placing two vertical bars around the number as follows: 5  The absolute value of five is five. -3  The absolute value of negative three is three.

  21. Adding Integers - Same Sign What is 5 + 7? We can show how to do this by using algebra tiles. + = 5 + 7 = 12 What is –5 + -7? = + -12 = + -7 -5

  22. 9 -5 0 5 -10 10 Adding Integers - Same Sign We can show this same idea using a number line. What is 5 + 4? Move five (5) units to the right from zero. Now move four more units to the right. The final point is at 9 on the number line. Therefore, 5 + 4 = 9.

  23. -9 -5 0 5 -10 10 Adding Integers - Same Sign What is -5 + (-4)? Move five (5) units to the left from zero. Now move four more units to the left. The final point is at -9 on the number line. Therefore, -5 + (-4) = -9.

  24. Adding Integers - Same Sign To add integers with the same sign, add the absolute values of the integers. Give the answer the same sign as the integers. Solution Examples

  25. Additive Inverse -5 0 5 -10 10 What is (-7) + 7? To show this, we can begin at zero and move seven units to the left. Now, move seven units to the right. Notice, we are back at zero (0). For every positive integer on the number line, there is a corresponding negative integer. These integer pairs are opposites or additive inverses. Additive Inverse Property – For every number a, a + (-a) = 0

  26. Additive Inverse When using algebra tiles, the additive inverses make what is called a zero pair. For example, the following is a zero pair. 1 + (-1) = 0. This also represents a zero pair. x + (-x) = 0

  27. Adding Integers - Different Signs -5 0 5 -10 10 Add the following integers: (-4) + 7. Start at zero and move four units to the left. Now move seven units to the right. The final position is at 3. Therefore, (-4) + 7 = 3.

  28. Adding Integers - Different Signs -5 0 5 -10 10 Add the following integers: (-4) + 7. Notice that seven minus four also equals three. In our example, the number with the larger absolute value was positive and our solution was positive. Let’s try another one.

  29. Adding Integers - Different Signs -5 0 5 -10 10 Add (-9) + 3 Start at zero and move nine places to the left. Now move three places to the right. The final position is at negative six, (-6). Therefore, (-9) + 3 = -6.

  30. Adding Integers - Different Signs -5 0 5 -10 10 Add (-9) + 3 In this example, the number with the larger absolute value is negative. The number with the smaller absolute value is positive. We know that 9-3 = 6. However, (-9) + 3 = -6. 6 and –6 are opposites. Comparing these two examples shows us that the answer will have the same sign as the number with the larger absolute value.

  31. Adding Integers - Different Signs To add integers with different signs determine the absolute value of the two numbers. Subtract the smaller absolute value from the larger absolute value. The solution will have the same sign as the number with the larger absolute value. Solution Example Subtract

  32. Subtracting Integers We can show the addition of numbers with opposite signs by using algebra tiles. For example, 3 + (-5) would look like this: + Group the zero pairs. Remove the zero pairs

  33. Subtracting Integers We can show the addition of numbers with opposite signs by using algebra tiles. For example, 3 + (-5) would look like this: + = Group the zero pairs. Remove the zero pairs The remainder is the solution. Therefore, 3 + (-5) = -2

  34. Subtracting Integers Let’s see how subtraction works using algebra tiles for 3 - 5. Begin with three algebra tiles. Now, take away or remove five tiles from these three. It can’t be done. However, we can add zero pairs until we have five ones because adding zero doesn’t change the value of the number.

  35. Subtracting Integers Let’s see how subtraction works using algebra tiles for 3 - 5. Begin with three algebra tiles. Now remove the five ones.

  36. Subtracting Integers Let’s see how subtraction works using algebra tiles for 3 - 5. Begin with three algebra tiles. Now remove the five ones. The remainder is negative two (-2). Therefore, 3 – 5 = -2. This is the same as 3 + (-5). To subtract a number, add its additive inverse. For any numbers a and b, a – b = a + (-b).

  37. Find each sum or difference. • -24 – 11 2. 18 + (-40) 3. -9 + 9 • 4. -16 – (-14) 5. 13 – 35 6. -29 + 65 Simplify each expression. 7. 18r – 27r 8. 9c – (-12c) 9. -7x + 45x Evaluate each expression. • 89  11. -14  • 12. - -7  13. c + 5  if c = -19

  38. Solutions

  39. Solutions

  40. Solutions 13. c + 5  if c = -19

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