The document discusses different ways of representing integers using red and blue chips, with red representing positive integers and blue representing negative integers. It introduces the concept of the "distinguished representation" and has examples of representing various integers in their distinguished representation and other representations. It then discusses how to perform addition and subtraction of integers by taking the union or difference of the chip representations and finding the distinguished representation of the result.

1. Integer Representations History

2. Integer Representations Taken together, these chips add to zero. Think of a red chip as positive one and a blue chip as negative one. Example +1 -1 Red Blue

3. Represent a positive three:

4. Representation There are many ways to represent any integer. But the representation below will be referred to as the Distinguished Representation.

5. Representation This is an example of a Distinguished Representation. This is not an example of a Distinguished Representation.

6. Representation Example Non Example What is the definition of a Distinguished Representation?

7. Integer Representations Represent each of these numerals physically (with collections of red and blue markers). For each numeral make 3 different representations, including the distinguished representation. a) -4 b) +5 c) 0 d) -1

8. Addition of Integers

9. Addition of Integers To add 3 + 4. To add two numbers, model each number and take the union of the models. Then find the distinguished representation.

10. Addition of Integers To add two numbers, model each number and take the union of the models. Then find the distinguished representation. To add 3 + -4. -1

11. Addition of Integers Addition Practice. Use the Set Model to find the sums. a) +3 + (+4) b) -3 + (-4) c) 3 + (-4) d) -4 + (5)

12. Addition of integers 5 + -3 = ?? 2 + 3 + -3 = ?? How can we approach addition of integers by their symbols? For example: What are two numbers that add to 5? 2 + 0 = 2 2 2

13. Addition of Integers 5 + 5= 5 + 4 = 5 + 3 = 5 + 2 = 5 + 1 = 5 + 0 = 5 + -1 = 5 + -2= 5 + - 3= 5 + - 4= 5 + - 5= 5 + - 6= 5 + - 7= 5 + - 8= 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3

14. Subtraction of Integers

15. Subtraction of Integers To subtract two numbers, model the first number and take-away the second number. Then find the distinguished representation. To subtract 5 - 3.

16. Subtraction of Integers 2 To subtract two numbers, model the first number and take-away the second number. Then find the distinguished representation. To subtract 5 - 3.

17. Subtraction of Integers To subtract two numbers, model the first number and take-away the second number. Then find the distinguished representation. Example: 5 - - 3. But there are no blue chips to take-away! What is another representation for 5?

18. Subtraction of Integers 8 To subtract two numbers, model the first number and take-away the second number. Then find the distinguished representation. Example: 5 - -3.

19. Subtraction of Integers Use the Take-away Model. a) +5 - (+4) b) -5 - (-4) c) +4 - (+5) d) +4 - (-5) e) -4 - (+5) f) -4 - (-5) -1 1 9 -9 -1 1

20. Subtraction of Integers 5 - -4 = 5 - -3= 5 - -2 = 5 - -1= 5 - 0 = 5 - 1 = 5 - 2 = 5 - 3= 5 - 4= 5 - 5= 5 - 6= 5 - 7= 5 - 8= 9 8 7 6 5 4 3 2 1 0 -1 -2 -3