1 / 18

Understanding Properties of Numbers and Finding Counterexamples

Learn how to apply properties of numbers and find counterexamples in math. Explore vocabulary like conjecture, counterexample, and equivalent expressions. Practice using properties to determine if expressions are equivalent or not. See examples and learn how to identify counterexamples to disprove conjectures.

lhylton
Download Presentation

Understanding Properties of Numbers and Finding Counterexamples

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Warm Up Evaluate. 1. 10 · 7 + 7 · 10 2. 15 · 9 + 61 3. (41 + 13) + (13 + 41)‏ 4. 4(32) – 16(8) 140 196 108 0

  2. Learn to apply properties of numbers and to find counterexamples.

  3. Vocabulary conjecture counterexample

  4. Reading Math Equivalent expressions have the same value, no matter which numbers are substituted for the variables. Ex. a+6 is equivalent to 2+a+4 because 2+a+4 a+(2+4) a+6

  5. 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.

  6. Additional Example 1B: Identifying Equivalent Expressions Use properties to determine whether the expressions are equivalent. 5(y – 11) and 5y – 55 Use the Distributive Property. 5(y – 11)= 5(y) – 5(11) Follow the order of operations. = 5y – 55 The expressions 5(y – 11) and 5y – 55are equivalent.

  7. Check It Out: Additional Example 1A Use properties to determine whether the expressions are equivalent. 2(z + 33) and 2z + 66 Use the Distributive Property. 2(z + 33)= 2(z) + 2(33) Follow the order of operations. = 2z + 66 The expressions 2(z + 33) and 2z + 66are equivalent.

  8. 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.

  9. 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 Add to find the total. Use the Commutative and Associative Properties to group numbers that are easy to add mentally. 26 + 24 + 17 (26 + 24) + 17 50 + 17 = 67 Jay worked 67 hours in all.

  10. 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. Rewrite 67 as 70 – 3 so you can use the Distributive Property to multiply mentally. 7(70 – 3)‏ 7(70)– 7(3) Multiply from left to right. Subtract. 490 – 21 = 469 Jay made $469 for the last three weeks.

  11. 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 Add to find the total. Use the Commutative and Associative Properties to group umbers that are easy to add mentally. 13 + 17 + 22 (13 + 17) + 22 30 + 22 = 52 Dosh studied 52 hours in all.

  12. 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. Rewrite 21 as 20 + 1 so you can use the Distributive Property to multiply mentally. 9(20 + 1)‏ 9(20) + 9(1) Multiply from left to right. Add. 180 + 9 = 189 Dosh makes $189 if he tutors for 21 hours a week.

  13. 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.

  14. 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 Multiply. 2 · 1 = 2 The product 2 is not greater than either of the whole numbers being multiplied.

  15. 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 Multiply. 9 · 1 = 9 The product 9 is equal to one of the whole numbers being multiplied.

  16. Lesson Quizzes Standard Lesson Quiz

  17. Lesson Quiz Use properties to determine whether the expressions are 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? b. If each can contains 2 servings, how many servings of food did Alan and Su Ling collect? 4. Find a counterexample to disprove the conjecture, “The quotient of two whole numbers is always less than either number.”

More Related