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Chapter 3 Whole Numbers: Operations and Properties Addition of whole numbers Set Model Dora caught 2 stars, and Boots caught 3 stars. How many stars did they catch all together? Addition of whole numbers Set Model Let a and b be any two whole numbers. If A and B are disjoint sets with a = n(A) and b = n(B), then a + b = n(A B) Tween Dora the Explorer How many ducks did they find together? How many newspaper did they collect together? Examples that use Set Model to perform Addition 1. Odelia got 43 candies during “Trick or Treat”, her sister got 38. How many candies did they totally get at that night? 2. On one Wednesday, Kevin borrowed 3 books from the school library, and later that day he borrowed another 4 books from the public library, how many books did he totally borrow on that day? Measurement Model for Addition 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. 0 1 2 3 4 5 6 7 8 9 Examples that use Measurement Model to perform Addition 1. On one Saturday morning, Jonathon walked 2 kilometers from home to his friend Derek’s house. They then walked 3 kilometers together to the beach. How many kilometers did Jonathon walk from home to the beach? 2. Mr. Faber’s house is 23 feet tall, and he put an 8-foot tall flag pole on top of the roof. How high is the top of the flag pole from the ground? Measurement 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. 0 1 2 3 4 5 Click whenever you are ready. 6 7 Example : 2 + 4 1. Our rabbit starts at 0 facing right. 2. It then hops forward 2 units because it sees 2 in the beginning. (Click to see animation.) 0 1 2 3 4 5 6 7 Example : 2 + 4 1. Our rabbit starts at 0 facing right. 2. It then hops forward 2 units because it sees 2 in the beginning. 0 1 2 3 4 5 6 7 Example : 2 + 4 1. Our rabbit starts at 0 facing right. 2. It then hops forward 2 units because it sees 2 in the beginning. 0 1 2 3 4 5 6 7 Example : 2 + 4 1. Our rabbit starts at 0 facing right. 2. It then hops forward 2 units because it sees 2 in the beginning. 3. After this the bunny has to jump 4 more steps forward because it sees + 4. (click when you are ready) 0 1 2 3 4 5 6 7 Example : 2 + 4 1. Our rabbit starts at 0 facing right. 2. It then hops forward 2 units because it sees 2 in the beginning. 3. After this the bunny has to jump 4 more steps forward because it sees + 4. 0 1 2 3 4 5 6 7 Example : 2 + 4 1. Our rabbit starts at 0 facing right. 2. It then hops forward 2 units because it sees 2 in the beginning. 3. 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. 0 1 2 3 4 5 6 7 Properties of Addition 1. Closure property The sum of any two whole numbers is still a whole number. 2. Commutative property For any two whole numbers a and b, a + b = b + a (This can easily be demonstrated by the set model, see next slide.) A closed Ecosystem is a self-replenishing system in which life can be maintained without external factors or outside aid. An ecosphere. Properties of Addition 3. Associative property For any whole numbers a, b, and c, (a + b) + c = a + (b + c) 4. Identity property For any whole number a, a + 0 = a = 0 + a Exercise: What properties of addition are used below? 1. (27 + 89) + 13 = 27 + (89 + 13) associative property _______________ 2. 27 + (89 + 13) = 27 + (13 + 89) commutative property _______________ 3. 27 + (13 + 89) = (27 + 13) + 89 associative property _______________ 4. 2000 + 382 = 2382 identity property _______________ Subtraction of whole numbers Take-away approach 5 baby bears are playing near the river. Later on 2 ran away. How many baby are left behind? 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) The number “a – b” is called the difference and is read “a minus b”, where a is called the minuend and b the subtrahend. Measurement Model for Subtraction Measurement model for Subtraction 1. There is a difference between addition and subtraction. 2. To perform subtraction, the rabbit has to turn around (180 deg) first. 0 1 2 3 4 5 Click whenever you are ready. 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. (Click to see animation.) 0 1 2 3 4 5 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. 0 1 2 3 4 5 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. 3. Now the rabbit has to turn around because it sees the subtraction sign. (click to see animation) 0 1 2 3 4 5 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. 3. Now the rabbit has to turn around because it sees the subtraction sign. 0 1 2 3 4 5 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. 3. Now the rabbit has to turn around because it sees the subtraction sign. 4. Finally the rabbit has to jump forward 3 steps because it sees the number 3. 0 1 2 3 4 5 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. 3. Now the rabbit has to turn around because it sees the subtraction sign. 4. Finally the rabbit has to jump forward 3 steps because it sees the number 3. 0 1 2 3 4 5 6 7 Example 4: 7 – 3 1. Our rabbit still starts at 0 facing right. 2. It then hops forward 7 units because it sees 7 first. 3. Now the rabbit has to turn around because it sees the subtraction sign. 4. Finally the rabbit has to jump forward 3 steps because it sees the number 3. 0 1 2 3 4 5 We now know that 7 – 3 is 4. 6 7 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 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 A set B 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. a. Robby has accumulated a collection of 362 sports cards. Chris has a collection of 200 cards. How many more cards does Robby have than Chris? b. Jack is driving from St. Louis to Kansas City for a meeting, a total of 250 miles. After 2 hours he notices that he has traveled only 114miles. How many more miles does he still have to drive? c. 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? d. 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 far was he from home at that point? Remarks 1. 2. 3. 4. 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. 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 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 a rows and b columns. Example: 5×3 is equal to the number of stars in the following array. 5 rows 3 columns 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. 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 with only one type of meat can we make? Answer: (ham, white) (turkey, white) (roast beef, white) (ham, wheat) (turkey, wheat) (roast beef, wheat) Determine which of the following approach is most appropriate for each situation: repeated addition, rectangular array, or Cartesian product. 1. 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? 2. A can typically contains 12 ounces of soda. How many ounces are in 9 cans? 3. A certain package contains 12 sticks of gum. How many sticks of gum are in 10 packages? 4. In a parking lot, there are 7 rows of cars with 15 cars in each row. How many cars are there? 5. 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? Properties of Multiplication 1. Closure property The product of any two whole numbers is still a whole number. 2. Commutative property For any two whole numbers a and b, a × b = b × a 3. Associative property For any three whole numbers a, b, and c, a × (b × c) = (a × b) × c 4. Identity property For any whole number a, we have a×1=a=1×a 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 hence 3×2 + 3×5 = 6 + 15 which is also equal to 21, 3 × (2 + 5) = 3 × 2 + 3 × 5 2+5 3 Equals to + 3 3 2 5 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 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. 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 in each bag, how many bags will you need? (go to next slide for an animation) Example: If you have baked 54 cookies and you want to put exactly 6 cookies in each bag, then you need 9 bags. Hence 54 ÷ 6 = 9. (click to see animation) 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) 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) Team 1 Team 2 Team 3 Team 4 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. 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. Examples Identify each of the following problems as an example of either sharing or repeated-subtraction division. a.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? b.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? c.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? d. Mr. Roth wants to put 280 chairs in the school auditorium, but he can only put 35 chairs in a row. How many rows of chairs does he need to make? e. There are 2.54cm in an inch, how many inches are in 120cm? f. If a pack of lined paper is $1.80 (including tax), and you have $12.80, how many packs can you buy? g. If you can read 9 pages in 20 minutes, what is your reading rate in pages/minute? Division properties of Zero 1. If a ≠ 0, then 0 ÷ a = 0 2. If a ≠ 0, then a ÷ 0 is undefined 3. 0 ÷ 0 is also undefined 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) Remark: The easiest way to find the remainder is by the repeated-subtraction approach Ordering and Exponents Definition A whole number a is said to be less that another whole number b (written a < b) if there is a positive whole number n such that a + n = b. Example: We say that 3 < 7 because we know that 3 + 4 = 7 and 4 is a whole number. Transitive property of “less than” For all whole numbers a, b, and c, if a < b and b < c, then a < c Properties of “less than” (1) If a < b then a + c < b + c for any whole number c. (2) If a < b and c ≠ 0, then ac < bc. Exponents Repeated multiplication approach For any whole numbers a and n with n ≠ 0, a n = a ×a × ∙∙∙ × a n copies. For all whole numbers a ≠ 0, we define a 0 = 1, and 00 is left as undefined. The number n above is called the exponent or power of a. Examples (1) 25 = 2×2×2×2×2 = 32 (2) 53 = 5×5×5 = 125 (3) 91 = 9 (4) 370 = 1 ( by definition) (5) 04 = 0×0×0×0 = 0 Properties of Exponents (1) For any whole numbers a, m(>0), and n(>0), a m×a n = a m+n Example 34×35 = (3×3×3×3)×(3×3×3×3×3) = 34+5 = 39 Properties of Exponents (2) For any whole numbers a, b, and m(>0), a m×b m = (ab) m Example 34×74 = (3×3×3×3)×(7×7×7×7) = (3×7)×(3×7)×(3×7)×(3×7) = (3×7)4 Properties of Exponents (3) For any whole numbers a, n(>0), and m(>0), (a m) n = a m×n Example (78)4 = (78)×(78)×(78)×(78) = 78+8+8+8 = 732 Caution (2 + 3)5 ≠ 25 + 35 and there is no property involving raising powers of sums or differences. Properties of Exponents (4) For any whole numbers a(>0), n, and m, with m>n>0 a m ÷ a n = a m−n Example 67 ÷ 64 = ? We need to use missing factor approach. If 67 ÷ 64 = x, then x × 64 = 67 , and we can use guess and check to deduce that x = 63. Exercises Rewrite the following with only one exponent a) 23 × 16 b) 37 × 95 × 272 c) 92 × 153 × 54 d) 123 ÷ 26 “Less than” properties for exponents (1) For any whole numbers a, b, and n>0 if a > b then an > bn (2) For any whole numbers a(>1), m and n if m > n then am > an (3) For any whole numbers a, b, and m, n if a > b and m > n, then am > bn Exercise Order the following exponents from smallest to largest without calculators. 322, 414, 910, 810