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Transcript
Day 10: Precious Conjectures Grade 7 Description some conjectures about powers and primes. • Examine factors, multiples and products of prime numbers, square numbers, and odd numbers as they prove or disprove a variety of conjectures. • Explore Materials • BLM 10.1, 10.2 Assessment Opportunities Minds On ... Action! Consolidate Debrief Concept Practice Whole Class Æ Discussion Describe how some mathematicians spend years attempting to prove or disprove a conjecture or theory. Sometimes practical applications of these theories are not evident for years afterward. Students may recognize Einstein’s Theory of Relativity, E = MC2 as an example of a conjecture. Lead a discussion on Goldbach’s Conjecture: “Every even number greater than 2 can be written as the sum of two prime numbers.” Students provide numeric examples to prove or disprove the conjecture as they answer the three questions on BLM 10.1. Examples to support Goldbach’s Conjecture: 4=2+2 6=3+3 8=3+5 10 = 3 + 7 Students work in pairs to prove or disprove further conjectures. To do so, they must understand certain terminology. Bronzebach’s Conjecture requires knowing that perfect squares are 1, 4, 9, 16, 25, 36, 49, … and that these can be written as 12, 22, 32, 42, 52, … Tinbach’s Conjecture refers to the prime numbers 2, 3, 5, 7, 11, 13, 17… Brassbach’s Conjecture refers to factors or divisions of a number. Ensure that students understand that every number has 1 and itself as factors. Pairs Æ Activity Curriculum Expectations/Observation/Mental Note: Observe students’ mental arithmetic skills, appropriate use of calculators, and problem solving skills as they work on the activity. Working in pairs or groups of three, students complete BLM 10.2 to prove or disprove each of the remaining conjectures. Students complete each of the three questions for each conjecture, as modelled for Goldbach’s Conjecture. Whole Class Æ Debrief Students present their findings. Lead a discussion about what would be enough to prove a conjecture true or to disprove or refute a conjecture. Most students will agree that you can prove a conjecture if you show all possible cases to be true. Many students will not think it possible to establish a proof if there are an infinite number of cases. A few students may be interested in researching “proof by mathematical induction.” The following counter-example is sufficient to disprove or refute Brassbach’s Conjecture: 16 has 5 factors 1, 2, 4, 8, 16, not just 3 factors. A conjecture is a statement that may appear to be true, but has not yet been proven. In honour of Goldbach, mathematicians humorously named several other prime number conjectures after metals and minerals. (Tinbach’s, Copperbach’s, Aluminumbach’s Conjecture) A prime number has only 2 different factors: 1 and itself. “There are an infinite number of prime numbers” is a conjecture that many mathematicians have worked on. Prime numbers are of particular interest to those who create security codes. Select a student to add vocabulary to the Word Wall: power, prime, product, and factor. Home Activity or Further Classroom Consolidation Complete worksheet 10.2 for Precious Conjectures about Powers and Primes. TIPS: Section 3 – Grade 7 © Queen’s Printer for Ontario, 2003 Page 46 10.1: Goldbach’s Conjecture Name: Date: “Every even number greater than 2 can be written as the sum of two prime numbers.” 4=2+2 1. Verify that the conjecture is true for the first ten numbers in the pattern. 2. Select five additional numbers (not necessarily consecutive or immediately following the first 10) and verify that the conjecture is also true for them. 3. Suppose that you were able to write valid statements for the first 1 000 000 numbers in the pattern. Would this be enough to prove that the conjecture is true? Explain your reasoning. TIPS: Section 3 – Grade 7 © Queen’s Printer for Ontario, 2003 Page 47 10.2: Precious Conjectures about Powers and Primes Name: Date: For each of the conjectures listed: • verify that the conjecture is true for the first ten numbers in the pattern. • select five additional numbers and verify that the conjecture is true for them. These numbers do not have to be consecutive or immediately follow the first ten numbers. • suppose that you were able to write valid statements for the first 1 000 000 numbers in the pattern. Would this be enough to prove that the conjecture is true? Explain your reasoning. Conjectures Bronzebach’s Conjecture Every natural number is the sum of four or fewer perfect squares. 10 = 32 + 12 Tinbach’s Conjecture Every number can be expressed as a difference of two prime numbers. 14 = 17 – 3 Copperbach’s Conjecture The product of any number of prime numbers is odd. 5 × 3 × 7 = 105 Aluminumbach’s Conjecture Every odd number can be expressed as the sum of three primes. 11 = 3 + 3 + 5 Challenge Brassbach’s Conjecture states that “Every square number has exactly 3 factors.” One example is 22, with factors 1, 2, and 4. Investigate this conjecture using the first 10 perfect squares. Write a brief explanation of your results and explain how they confirm or refute the conjecture. TIPS: Section 3 – Grade 7 © Queen’s Printer for Ontario, 2003 Page 48