Understanding Geometric Patterns and Reasoning in Mathematics

1. Unit 1 Geometric Patterns & Reasoning

2. Lesson 1

3. Vocabulary (Logs) • Linear • Like a straight line • Nonlinear • Not forming a straight line • Linear Data • every item is related to its previous and next item • Common Difference • The difference between each number in an arithmetic series

4. Vocabulary (Logs) • Inductive Reasoning • constructs or evaluates propositions that are abstractions of observations of individual instances • a general conclusion is arrived at by specific examples • Deductive Reasoning • reasoning from one or more general statements regarding what is known to reach a logically certain conclusion • using given true premises to reach a conclusion that is also true • Conjecture • Educated Guess

5. Activity • Linear Number Patterns • Relationship between terms • Generate formulas for the following: • 1, 3, 5, 7, 9, … Find the 25th term • 2n-1 • 4, 8, 12, 16, 20, … Find the 100th term • 4n • Arithmetic Sequence • an = a1 + (n - 1)d

6. Lesson 2

7. Vocabulary (Logs) • Figurate Numbers • a number that can be represented by a regular geometrical arrangement of equally spaced points

8. Activity 1 4 9 16 25 n2 n n x n How do the dimensions compare to number of the figure? What would the dimensions of the nth figure be? Is the pattern linear? Enter data from columns 1 and 3 into graphing calculators & plot. Why is this called a square number pattern? Generate a regression equation.

9. Activity 2 6 12 20 30 n2 + n n n x (n + 1) How do the dimensions compare to number of the figure? What would the dimensions of the nth figure be? Is the pattern linear? Why is this called a rectangular number pattern? Enter data from columns 1 and 3 into graphing calculators & plot. Generate a regression equation. How does this equation compare to square pattern equation?

10. Activity How do you find the area of a triangle? 1 3 What is the base and height of these triangles? 6 10 15 n2 x n 2 n Why is this called a triangular number pattern? Is the pattern linear? Enter data from columns 1 and 2 into graphing calculators & plot. Generate a regression equation.

11. Lesson 3

12. Vocabulary (Logs) • Diagonals of Polygons • line segment linking two non-adjacent vertices • How many diagonals can be drawn? • http://www.mathopenref.com/polygondiagonal.html • Graph the pattern for number of sides vs. number of diagonals and create a regression equation • Is the pattern linear?

13. Diagonals of Polygons

14. Vocabulary (Logs) • Sum of the Interior Angles of n-gon • S = 180(n – 2) • http://www.mathsisfun.com/geometry/interior-angles-polygons.html • Graph the pattern for number of sides vs. number of diagonals and create a regression equation • Is the pattern linear?

15. Activity • How many phone calls can be made between two people among a group of six friends? 15 calls

16. Modeling & Counting • What strategies could you use to determine how many games are needed for a tournament or schedule? • Diagonals • Lists • Generate a formula from a table

17. Modeling & Counting • How many games are needed for eight teams to play each either once? • 28 • How may games are needed for a single elimination tournament of eight teams? • 7 • How many games are needed for a double elimination tournament of eight teams? • 15 or 16

18. Double Elimination Tourney Round-Robin Schedule • http://www.devenezia.com/downloads/round-robin/rounds.php • http://quickleague.org/Default.aspx?tabid=909

19. Lesson 4

20. Vocabulary (Logs) • Permutation • When order or position matters (456 ≠ 564) • Can have repetition (lock combination) • (nr) • n = # of things • r = # you are choosing • or not (running order) • (n!) • n = # of things to choose from • Factorial (!) - multiply a series of descending natural numbers

21. Vocabulary (Logs) • Combination • When order or position doesn’t matter (123 = 321) • Can have repetition (choosing ice cream flavors) • n = things to choose from • r = number of things you are getting • Or not (lottery)

22. Activity • How many ways can 3 books be arranged on a shelf if you are choosing from 8 books? • 336 • How many committees of 5 five students can be made from 25 classmates? • Permutation or Combination? • Repetition or No Repetition? • 130

23. Circular Permutation • How many ways can n people sit around a table? • (n – 1)!

24. Review

25. Vocabulary • Linear • Non-linear • Linear Data • Common Difference • Inductive Reasoning • Deductive Reasoning • Conjecture • Figurate Numbers • Diagonals of Polygons • Permutations • Combinations

26. Learning Log Checklist • Vocabulary • Arithmetic Sequences • Permutations • Combinations

27. Assessment & Review 2-3 days

28. Test Questions • Create and/or analyze pictorial or number sequences and solvefor the next three terms and the nth term • Create a tournament schedule and/or round robin schedule • Develop formula for figurate number • Find permutations and/or combinations for situations • Vocabulary matching