conjectures. a conjecture is a statement that must be proved or disproved
TRANSCRIPT
![Page 1: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/1.jpg)
CONJECTURES
![Page 2: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/2.jpg)
A conjecture is a statement that must be proved or disproved.
![Page 3: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/3.jpg)
To DISPROVE a statement, provide an example where it is NOT true.
![Page 4: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/4.jpg)
Mathematics students are male.
![Page 5: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/5.jpg)
Men achieve higher marks in Mathematics than women.
![Page 6: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/6.jpg)
Justin Bieber is a responsible adult.
![Page 7: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/7.jpg)
Prime numbers are odd.
![Page 8: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/8.jpg)
Real numbers are counting numbers.
![Page 9: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/9.jpg)
Numbers that are divisible by 4 are also divisible by 8.
![Page 10: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/10.jpg)
What about proving a conjecture true?
Conjecture: Chocolate bars contain peanuts.
Proof: Snickers is a chocolate bar. Snickers bars contain peanuts. The conjecture is true.
![Page 11: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/11.jpg)
What is the problem with that argument?
![Page 12: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/12.jpg)
Chocolate bars contain peanuts.
ALL chocolate bars contain peanuts.
SOME chocolate bars contain peanuts.
![Page 13: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/13.jpg)
VERY IMPORTANT
1 example can DISPROVE.
To PROVE we need to show it is true for ALL cases.
ALGEBRA (your favourite!!)
![Page 14: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/14.jpg)
Let’s try it!
• Investigate the type of number you get if you find the sum of an odd number and an even number.
• Express your answer in words as a conjecture.• Use algebra to prove this conjecture.
![Page 15: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/15.jpg)
THINGS YOU SHOULD KNOW…• Review of Number
Real, integer, whole, counting, rational, irrational, prime, composite…
![Page 16: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/16.jpg)
TWIN PRIME?
SQUARE NUMBER?
TRIANGULAR NUMBER?
![Page 17: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/17.jpg)
How would you write these using algebra?
Even numbers:
Odd numbers:
3 consecutive even numbers:
3 consecutive odd numbers:
![Page 18: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/18.jpg)
How could you represent a two digit number using
algebra?
![Page 19: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/19.jpg)
LOCATING ERRORS
1) Squaring Solve for x: x + 9 - x + 3 = 0
![Page 20: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/20.jpg)
2) Square rooting
- 2 = - 2 1 – 3 = 4 – 6 1 – 3 + 9/4 = 4 – 6 + 9/4 (1 – 3/2)2 = (4 – 3/2)2
1 – 3/2 = 4 – 3/2 1 = 4
![Page 21: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/21.jpg)
3) Dividing by 0
m = p m2 = mp m2 – p2 = mp – p2
(m – p)(m + p) = p (m – p) m + p = p m + m = m 2m = m 2 = 1
![Page 22: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/22.jpg)
4) Inequalities and multiplying or dividing by a negative
10 < 20 - 10 < - 20
![Page 23: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/23.jpg)
SNAP TO IT!!!
![Page 24: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/24.jpg)
Pick a number.Square it.Add ten times the original number.Add 25.Take the square root (rounding to the nearest whole number).Subtract your original answer.What do you get?
![Page 25: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/25.jpg)
Consider the following examples of multiplying any two-digit number by 11. What pattern can you see?
11 x 42 = 46211 x 71 = 78111 x 45 = 495
1. Express your answer as a conjecture.2. Can you find an example that disproves your
initial conjecture? If so, reconsider your conjecture.
3. Use algebra to prove your conjecture.
![Page 26: CONJECTURES. A conjecture is a statement that must be proved or disproved](https://reader035.vdocuments.us/reader035/viewer/2022062806/5697bf7a1a28abf838c82f18/html5/thumbnails/26.jpg)
THIS IS YOUR CHANCE TO BE FAMOUS!!
• Twin Prime conjecture There are an infinite number of twin primes.
• Goldbach’s conjecture Every even number greater than two can be written as a sum of two prime numbers.