Download packet 2.1: inductive reasoning

Name______________________
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2.1: INDUCTIVE REASONING ]
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Inductive Reasoning is reasoning based on patterns you observe. Let’s look at some examples. Look for a pattern. What are the next two terms in the sequence? 1. 1, 4, 16, 64 … 2. , , ….
You try! 3. , , … 4. 16, 8, 4, 2, 1 … Once we are pretty sure we understand a pattern, we can start to make conjectures. Conjectures are conclusions made using inductive reasoning. Let try and figure out this next pattern. Can you guess how many would be in the 20th picture? . . . You try! Try and figure out the 43rd term of the following sequence: A, E, I, O, U, A, E, …
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2 PACKET 2.1: INDUCTIVE REASONING Write you r q uestions here!
What conjecture can you make about the sum of the first 30 even numbers? Number of Terms 1 2 3 4 Sum (Expanded) 2 2 + 4 2 + 4 + 6 2 + 4 + 6 + 8 Sum (Simplified) 2 6 12 20 As Factors? 1 ∙ 2 2 ∙ 3 3 ∙ 4 4 ∙ 5 ⋮ 30 n 2 + 4 + 6 + 8+ … + 60 So the first 30 even numbers have a sum of _______________. (And the sum of the first n numbers has a sum of ________________. ) J Counterexamples Sadly, not all of your conjectures will be true! All you need to do is find ONE example of when it is not true, and you will know your conjecture is false. A counterexample is an example that shows that a conjecture is false Give a counterexample for each of the false conjectures given. 1.
If the name of a month starts with the letter J, it
is a summer month.
Counterexample:___________________ 2.
Any given three points can form a triangle.
Counterexample:___________________ 3.
Multiplying a number by 2 makes it bigger.
Counterexample:___________________ [ PACKET
2.1: INDUCTIVE REASONING ]
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You try! Give a counter example for each example: 1.
When you multiply a number by 3, the product is
divisible by 6.
Counterexample:___________________ 2.
Counterexample:___________________ 3.
If you teach Flipped-Mastery Geometry, you are
super good-looking.
Counterexample:___________________ Now, summarize
your notes here!
If you are a bird, you can fly!
Find a pattern for each sequence. Use the pattern to find the next two terms. 1. 4, 4.5, 4.56, 4.567… 2. J, F, M, A, M, … 3. O, T, T, F, F, S, S, E, … ! !
4. 1, -­‐1, 2, -­‐2, 3… 5. 1, 2, 3, 5, 8, 13, 21, … 6. 4, 1, ! , !" , … 7. 2, 6, 7, 21, 22, 66, 67… 8. 1, 3, 7, 15, 31… 9. 3
4 PACKET 2.1: INDUCTIVE REASONING 10. , , …. 11. Use the sequence and inductive reasoning to make a conjecture: th
12. What pattern is in the 15 figure? 13. What is the shape of the 12th figure? 14. What pattern is in the 33rd figure? 15. What is the shape of the 41st figure? Give one counterexample to show that each conjecture is false. 16. The sum of any two numbers is always greater than either number. 17. The midpoint formula never gives points with fractions in them. 18. Dividing by a number always gives you a smaller number. Solve each equation for x!
1. -x
– 3 = 12
2. 2x
3. 3x(2x
– 3) Factor!
4. 16x
Algebra Review
+ 1 = 3x - 4
Multiply!
5. Graph the equation: y = 3x -­‐ 4 6. Graph the equation: x = -­‐3 2
– 8x [ PACKET
2.1: INDUCTIVE REASONING ]
1. Use the pattern to find the next two terms: 2. Give one counterexample to show that the conjecture is false: 1,
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0.1, 0.01, 0.001 …
Odd numbers less than 10 are prime.
3. Nicomachus of Gerasa first described pentagonal numbers in about 100 A.D. Draw a diagram to represent the next pentagonal number. What is the next pentagonal number? . . . ?
1 5 4. Graph the following points: A(1, 5) B(2, 2) C(2, 8) D(3, 1) E(3, 9) a.
b.
5. 12 H(7, -­‐1) I(7, 11) J(9, 1) K(9, 9)
L(10, 2) N(11,5) G(6, 10) M(10, 8) F(6, 0) Which points do not fit the same pattern as the others? Describe the figure you would get if you continued graphing those points that fit the pattern. Find the perimeter when 100 triangles are put together in the pattern shown. Assume all of the
triangles are 1 cm long.