Chapter 3 Whole Numbers: Operations and Properties

1. Chapter 3Whole Numbers:Operations and Properties

2. Addition of whole numbers Set Model To find “2 + 3”, we find two disjoint sets, one with 2 objects and the other with 3 objects, form their union, and count the total. (click to see animation) It is very important to use disjoint sets.

3. Tween Dora the Explorer

4. 0 1 2 3 4 5 6 7 8 9 Measurement Model In this model, the addition is performed on a number line. For example, to calculate 3 + 4, (1) draw an arrow with length 3 starting at 0. (2) draw an arrow with length 4 starting from the end of the previous arrow. (3) the sum is then the length of the two arrows combined together this way.

5. Alternative number line model for addition In this model, a bunny is sitting above the number 0 (which is its home base) facing right (which is the positive direction), and will jump forward if it has to perform an addition problem. 3 4 5 0 1 2 6 7 Click whenever you are ready.

6. Example : 2 + 4 • Our rabbit starts at 0 facing right. • It then hops forward 2 units because it sees 2 in the beginning.(Click to see animation.) 3 4 5 0 1 2 6 7

7. Example : 2 + 4 • Our rabbit starts at 0 facing right. • It then hops forward 2 units because it sees 2 in the beginning. 3 4 5 0 1 2 6 7

8. Example : 2 + 4 • Our rabbit starts at 0 facing right. • It then hops forward 2 units because it sees 2 in the beginning. 3 4 5 0 1 2 6 7

9. Example : 2 + 4 • Our rabbit starts at 0 facing right. • It then hops forward 2 units because it sees 2 in the beginning. • After this the bunny has to jump 4 more steps forward because it sees + 4. (click when you are ready) 3 4 5 0 1 2 6 7

10. Example : 2 + 4 • Our rabbit starts at 0 facing right. • It then hops forward 2 units because it sees 2 in the beginning. • After this the bunny has to jump 4 more steps forward because it sees + 4. 3 4 5 0 1 2 6 7

11. Example : 2 + 4 • Our rabbit starts at 0 facing right. • It then hops forward 2 units because it sees 2 in the beginning. • After this the bunny has to jump 4 more steps forward because it sees + 4. Since the bunny finally stops at 6, we know that 2 + 4 = 6. 3 4 5 0 1 2 6 7

12. Properties of Addition • Closure propertyThe sum of any two whole numbers is still a whole number. • Commutative propertyFor any two whole numbers a and b, a + b = b + a(This can easily be demonstrated by the set model, see next slide.)

13. A closed Ecosystem is a self-replenishing system in which life can be maintained without external factors or outside aid. An ecosphere.

15. Properties of Addition • Associative propertyFor any whole numbers a, b, and c, (a + b) + c = a + (b + c)

16. 4. Identity property For any whole number a, a + 0 = a = 0 + a

17. Subtraction of whole numbers Take-away approach Let a and b be whole numbers. Let A be a set with a elements and B a subset of A with b elements, then a – b = n(A – B) Example: 5 – 2 can be modelled by The number “a – b” is called the difference and is read “a minus b”, where a is called the minuend and b the subtrahend.

18. Number line model for Subtraction • There is a difference between addition and subtraction. • To perform subtraction, the rabbit has to turn around (180 deg) first. 3 4 5 0 1 2 6 7 Click whenever you are ready.

19. Example 4: 7 – 3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first.(Click to see animation.) 3 4 5 0 1 2 6 7

20. Example 4: 7 – 3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first. 3 4 5 0 1 2 6 7

21. Example 4: 7 – 3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first. • Now the rabbit has to turn around because it sees the subtraction sign. • (click to see animation) 3 4 5 0 1 2 6 7

22. Example 4: 7 – 3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first. • Now the rabbit has to turn around because it sees the subtraction sign. 3 4 5 0 1 2 6 7

23. Example 4: 7 –3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first. • Now the rabbit has to turn around because it sees the subtraction sign. • Finally the rabbit has to jump forward 3 steps because it sees the number 3. 3 4 5 0 1 2 6 7

24. Example 4: 7 –3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first. • Now the rabbit has to turn around because it sees the subtraction sign. • Finally the rabbit has to jump forward 3 steps because it sees the number 3. 3 4 5 0 1 2 6 7

25. Example 4: 7 –3 • Our rabbit still starts at 0 facing right. • It then hops forward 7 units because it sees 7 first. • Now the rabbit has to turn around because it sees the subtraction sign. • Finally the rabbit has to jump forward 3 steps because it sees the number 3. 3 4 5 0 1 2 6 7 We now know that 7 – 3 is 4.

26. Missing addend approach Let a and b be whole numbers. Then a – b is the number that when added to b equals to a. i.e. a – b = c if and only if c + b = a. Example: 7 – 4 = 3 because 3 + 4 = 7. Example: Nicole wants to buy a $13 DVD. She has saved only $5. How much more does she need to save? This is obviously a subtraction problem: 13 – 5 = ?But we can think this way: ? + 5 = 13

27. Comparison approach is used to determine how many more or fewer when two known quantities are compared. Example: Victoria has 4 bunnies, and she bought them 7 carrots. How many more carrots did she buy than the bunnies she had? set B set A

28. Each situation described next involves a subtraction problem. In each case, determine what approach best describes the situation. Typical answers may be take-away, measurement, missing addend, comparison. • Robby has accumulated a collection of 362 sports cards. Chris has a collection of 200 cards. How many more cards than Chris does Robby have? • Jack is driving from St. Louis to Kansan City for a meeting, a total of 250 miles. After 2 hours he notices that he has traveled only114miles. How many more miles does he still have to drive? • An elementary school library consists of 1095 books. As of may 8, 105 books were checked out of the library. How many books were still available for check out on May 8? • Sam drove 20 miles in the morning from home to his office. At noon, he drove back 5 miles along the same road to see a client. How fast was he from home at that point?

29. Remarks • The set of whole numbers is not closed under subtraction. • Subtraction is not commutative. • Subtraction is not associative. • There is no two-sided identity for subtraction.

30. Multiplication of whole numbers Repeated-addition approach Let a and b be whole numbers where a≠ 0, then a×b = b + b + … + b a addends Example: A dragonfly has 4 wings, how many wings do 12 dragonflies have? If a = 0, then a×b = 0

32. Rectangular array approach Let a and b be whole numbers, the product a×b is defined to be the number of elements in a rectangular array having arows and bcolumns. Example: 5×3 is equal to the number of stars in the following array. 5 rows 3 columns

33. Another example of Rectangular Arrays It is much faster to use multiplication to find out the number of seats in a section of a baseball stadium.

34. Cartesian Product approach Let a and b be whole numbers. Pick a set A with a elements and a set B with b elements. Then a×b is the number of elements in the set A×B, i.e.number of ordered pairs whose first component is from A and whose second component is from B. This approach is best for counting combinations. Example: There are 3 kinds of meat: ham, turkey, and roast beef. And there are 2 kinds of bread: white and wheat. How many different types of sandwiches can we make? Answer: (ham, white) (turkey, white) (roast beef, white) (ham, wheat) (turkey, wheat) (roast beef, wheat)

35. Determine which of the following approach is most appropriate for each situation: repeated addition, rectangular array, or Cartesian product. • A store employee taking inventory counted 8 rows of cans of tomato soup. Each row had seven cans of tomato soup. How many cans of soup were there altogether? • A can typically contains 12 ounces of soda. How many ounces are in 9 cans? • A certain package contains 12 sticks of gum. How many sticks of gum are in 10 packages? • In a parking lot, there are 7 rows of cars with 15 cars in each row. How many cars are there? • A small restaurant makes 6 types of salads and 5 types of soup. Each lunch consists of a soup and a salad. How many different lunches can be ordered?

36. Properties of Multiplication • Closure propertyThe product of any two whole numbers is still a whole number. • Commutative propertyFor any two whole numbers a and b, a×b = b×a

37. 3. Associative propertyFor any three whole numbers a, b, and c,a × (b ×c) = (a ×b) ×c4. Identity propertyFor any whole number a, we have a × 1 = a = 1 ×a

38. 5. Distributive Property of Multiplication over addition For any whole numbers a, b, and c,a × (b + c) = a×b + a×c Example: we know that 3 × (2 + 5) = 3 ×7 = 21, but 3×2 + 3×5 = 6 + 15 which is also equal to 21, hence 3 × (2 + 5) = 3 ×2 + 3 ×5

39. 2 + 5 3 3 3 Equals to + 5 2

40. 6. Distributive property of Multiplication over subtraction Let a, b, and c be whole numbers, then a × (b − c) = a×b−a×c Example: In order to calculate 3×14 – 3×9, We can use the property that 3×14 – 3×9= 3×(14 – 9) = 3×5 = 15

41. 7. Multiplication property of Zero.For any whole number a, we have a × 0 = 0. 8. Zero Divisors Property For any whole numbers a and b, if a×b = 0, then either a = 0 or b = 0.

42. Division of whole numbers Repeated-subtraction approach(also called measurement approach) For any whole numbers m and d, m ÷ d is the maximum number of times that d objects can be successively taken away from a set of m objects (possibly with a remainder). Example: If you have baked 54 cookies and you want to put exactly 6 cookies on each plate, how many plates will you need? (go to next slide for an animation)

43. Example: If you have baked 54 cookies and you want to put exactly 6 cookies on each plate, then you need 9 plates. Hence 54 ÷ 6 = 9. (click to see animation)

44. Partition approach ( or Sharing approach) If m and d are whole numbers, then m ÷ d is the number of objects in each group when m objects are separated into d equal groups. Example: If we have 20 children and we want to separate them into 4 teams of equal size, then each group will have 5 children because 20 ÷ 4 = 5 (go to next slide for animation)

45. Team 1 Team 2 Team 3 Team 4 Example: If we have 20 children and we want to separate them into 4 teams of equal size, how many children will be in each group? (click to see animation)

46. Example • Identify each of the following problems as an example of either sharing or repeated-subtraction division. • Gabriel bought 15 pints of paint to redo all the doors in his house. If each door requires 3 pints of paint, how many doors can Gabriel paint? • Hideko cooked 12 tarts for her family of 4. If all of the family members receive the same amount, how many tarts will each person have? • Ms. Ivanovich need to give 3 straws to each student in her class for their art activity. If she uses 51 straws, how many students does she have in her class?

47. Remarks The above two approaches are very similar and are interchangeable when we are working with whole numbers. However, when we need to divide fractions or decimals, only the repeated subtraction approach has a realistic meaning.

48. Missing factor approach If a and b (b>0) are whole numbers, then a ÷ b = c if and only if c × b = a Example: 91 ÷ 13 = ? Think ? × 13 = 91 Remark: This approach is most useful when we divide numbers with exponents.

49. Division properties of Zero 1. If a≠ 0, then 0 ÷ a = 02. If a ≠ 0, then a ÷ 0 is undefined3.0 ÷ 0 is also undefined

50. The Division Algorithm If a and b are whole numbers with b≠ 0, then there exist unique whole numbers q and r such that a = bq + r where 0 ≤ r < b (r is called the remainder) Example: Let a = 57 and b = 9 then clearly b≠ 0, and we can write 57 = 6 × 9 + 3 and this expression is unique if we require that the remainder is between 0 and 9 (not including 9)