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1-3 Properties of Numbers Warm Up Problem of the Day Lesson Presentation Lesson Quizzes 1-3 Properties of Numbers Warm Up Simplify. 1. 2. 3. 4. 10 · 7 + 7 · 10 140 15 · 9 + 61 196 (41 + 13) + (13 + 41) 108 4(32) – 16(8) 0 1-3 Properties of Numbers Problem of the Day Ms. Smith wants to buy each of her 113 students a colored marker. If the markers come in packs of 8, what is the least number of packs she could buy? 15 1-3 Properties of Numbers Learn to apply properties of numbers and to find counterexamples. 1-3 Properties of Numbers 1-3 Properties of Numbers Vocabulary conjecture counterexample 1-3 Properties of Numbers Reading Math Equivalent expressions have the same value, no matter which numbers are substituted for the variables. 1-3 Properties of Numbers Additional Example 1A: Identifying Equivalent Expressions Use properties to determine whether the expressions are equivalent. 7 · x · 6 and 13x 7·x·6=7·6·x Use the Commutative Property. = (7 · 6) · x Use the Associative Property. = 42x Follow the order of operations. The expressions 7 · x · 6 and 13x are not equivalent. 1-3 Properties of Numbers Additional Example 1B: Identifying Equivalent Expressions Use properties to determine whether the expressions are equivalent. 5(y – 11) and 5y – 55 5(y – 11) = 5(y) – 5(11) = 5y – 55 Use the Distributive Property. Follow the order of operations. The expressions 5(y – 11) and 5y – 55 are equivalent. 1-3 Properties of Numbers Check It Out: Additional Example 1A Use properties to determine whether the expressions are equivalent. 2(z + 33) and 2z + 66 2(z + 33) = 2(z) + 2(33) = 2z + 66 Use the Distributive Property. Follow the order of operations. The expressions 2(z + 33) and 2z + 66 are equivalent. 1-3 Properties of Numbers Check It Out: Additional Example 1B Use properties to determine whether the expressions are equivalent. 4 · x · 3 and 7x 4·x·3=4·3·x Use the Commutative Property. = (4 · 3) · x Use the Associative Property. = 12x Follow the order of operations. The expressions 4 · x · 3 and 7x are not equivalent. 1-3 Properties of Numbers Additional Example 2A: Consumer Math Applications During the last three weeks, Jay worked 26 hours, 17 hours, and 24 hours. Use properties and mental math to answer the question. How many hours did Jay work in all? 26 + 17 + 24 26 + 24 + 17 (26 + 24) + 17 Add to find the total. Use the Commutative and Associative Properties to group numbers that are easy to add mentally. 50 + 17 = 67 Jay worked 67 hours in all. 1-3 Properties of Numbers Additional Example 2B: Consumer Math Applications Jay earns $7.00 per hour. How much money did he earn for the last three weeks? 7(67) Multiply to find the total. 7(70 – 3) Rewrite 67 as 70 – 3 so you can use the Distributive Property to multiply mentally. 7(70) – 7(3) Multiply from left to right. 490 – 21 = 469 Subtract. Jay made $469 for the last three weeks. 1-3 Properties of Numbers Check It Out: Additional Example 2A During the last three weeks, Dosh studied 13 hours, 22 hours, and 17 hours. Use properties and mental math to answer the question. How many hours did Dosh study in all? 13 + 22 + 17 13 + 17 + 22 (13 + 17) + 22 Add to find the total. Use the Commutative and Associative Properties to group umbers that are easy to add mentally. 30 + 22 = 52 Dosh studied 52 hours in all. 1-3 Properties of Numbers Check It Out: Additional Example 2B Dosh tutors students and earns $9.00 per hour. How much money does he earn if he tutors students for 21 hours a week? 9(21) Multiply to find the total. 9(20 + 1) Rewrite 21 as 20 + 1 so you can use the Distributive Property to multiply mentally. 9(20) + 9(1) Multiply from left to right. 180 + 9 = 189 Add. Dosh makes $189 if he tutors for 21 hours a week. 1-3 Properties of Numbers A conjecture is a statement that is believed to be true. A conjecture is based on reasoning and may be true or false. A counterexample is an example that disproves a conjecture, or shows that it is false. One counterexample is enough to disprove a conjecture. 1-3 Properties of Numbers Additional Example 3: Using Counterexamples Find a counterexample to disprove the conjecture, “The product of two whole numbers is always greater than either number.” 2·1 2·1=2 Multiply. The product 2 is not greater than either of the whole numbers being multiplied. 1-3 Properties of Numbers Check It Out: Additional Example 3 Find a counterexample to disprove the conjecture, “The product of two whole numbers is never equal to either number.” 9·1 9·1=9 Multiply. The product 9 is equal to one of the whole numbers being multiplied. 1-3 Properties of Numbers Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 1-3 Properties of Numbers Lesson Quiz Use properties to determine whether the expressions are equivalent. not equivalent equivalent 1. 3x – 12 and 3(x – 9) 2. 11 + y + 0 and y + 11 3. Alan and Su Ling collected canned goods for 4 days to donate to a food bank. The number of cans collected each day was: 35, 4, 21, and 19. Use properties and mental math to answer each question. a. How many cans did they collect in all? 79 b. If each can contains 2 servings, how many servings of food did Alan and Su Ling collect? 158 4. Find a counterexample to disprove the conjecture, “The quotient of2two numbers is always 1whole = 2; the quotient 2 is notless lessthan thaneither eithernumber.” of the whole numbers. 1-3 Properties of Numbers Lesson Quiz for Student Response Systems 1. Which of the following expresssions are equivalent? A. 2x – 4 = 2(x – 4) B. 2x – 4 = 2x – 2 + 2 C. 2x – 4 = 2x – 2 – 2 D. 2x – 4 = 2(x + 4) 1-3 Properties of Numbers Lesson Quiz for Student Response Systems 2. Which of the following expresssions are equivalent? A. 3x + 4 = 2 + 2 + 3x B. 3x + 4 = 2 + 2 + 3 + x C. 3x + 4 = 3(x + 4) D. 3x + 4 = 3(x + 2) 1-3 Properties of Numbers Lesson Quiz for Student Response Systems 3. Find a counterexample to disprove the conjecture, “Any number that is divisible by 2 is also divisible by 4.” A. 20 2 = 10 and 20 4 = 5 B. 18 2 = 9 and 18 4 = 4.5 C. 20 2 = 20 and 20 4 = 80 D. 18 2 = 36 and 18 4 = 72