1. What comes next?

3. What comes next?
-4, 0, 4, 8…
…12, 16

4. How did we find those answers
How did we find those answers? What went through your mind to figure out the pattern? What was told to you in the beginning of the problem?

5. Inductive Reasoning
The process of reasoning that a rule or statement is true because several specific cases are true. Inductive reasoning can be used to formulate a conjecture about something.

6. Conjecture
A statement that is believed to be true. Also known as your hypothesis. What is a hypothesis? Every example must be true for a conjecture to be true. It is very hard to prove a conjecture true.

7. Counterexample
Example that does not support the conjecture. There only has to be ONE counterexample to prove the conjecture false. Usually, it is easier to prove a conjecture false than to prove it true.

8. Example 1
Formulate a Conjecture Look at the progression of the pattern to formulate a conjecture regarding the number of blocks there will be at the fifth arrangement.

9. Example 2
a. Testing a Conjecture Michelle made the conjecture, “The expressions 6n +1 and 6n -1 will always result in two prime numbers.” Show that this conjecture is true for n=1, 2, and 3, but not true for n=4.

10. Example 2
b. Testing a Conjecture Maria looks at the diagram below and conjectures that the number of triangles in the figure is given by the expression 2n + 1. Is this conjecture true for the four steps of the pattern shown below?
n = 1
n = 2
n = 3
n = 4

11. Example 3
A researcher studying crows for several years made the observation that every crow she studied was black. Her research assistant made this conjecture: “All crows are black.” How can this conjecture be tested? Can it be proved?

12. Your turn to practice!
Working in PAIRS, work on the worksheet. What isn’t finished in class is homework!

13. Perimeter
What is the perimeter of a shape? The perimeter is the sum of the side lengths of a closed geometric figure.

14. Area
What is the area of a figure? The area is the size of the region bounded by the figure.

15. Geoboards!
This diagram shows one unit and one square unit. What do we measure with one unit? What do we measure with square units?

16. Use your geoboard to create a polygon whose vertices are (1,1), (1,4), (5,4), and (5,1) What is the name of this polygon? What is the perimeter of this figure? What is the area of this figure?

17. Use your geoboard and create a triangle whose vertices are (1,8), (4,8), and (4,4). What is the perimeter? How long is each side? What is the area?

18. Make your own shape. It can be any shape you want
Make your own shape. It can be any shape you want. Switch with a partner and find the area and perimeter of your partner’s shape.

19. What do you think of when you hear
FORMULA?

20. Mathematical relationship expressed with symbols.
Formula
Mathematical relationship expressed with symbols.

21. Area and Perimeter Formulas

22. Example 1 Find the perimeter of a star with equal side lengths of 5m.
Find the perimeter of this figure:
20ft
5ft
5ft
4ft
9ft
17ft
9ft
5ft
17ft
7ft
30ft

23. Example 2
a. Find the area of a rectangle whose side lengths are 11cm and 4cm. b. Find the area of a triangle with a base of 9in and height of 14in.

24. Pythagoras

25. Pythagorean Theorem
The sum of the square of the lengths of the legs, a and b, of a right triangle is equal to the square of the length of the hypotenuse c and is written a2 + b2 = c2. c a b

26. Example 3 Using the Pythagorean Theorem
Find the length of the hypotenuse.
Find the area of the triangle.
12cm
5cm
5ft
3ft b

27. Example 4 Measuring Temperature
Different countries use different units to measure the temperature. Much of the world uses degrees Celsius, but a few countries use degrees Fahrenheit. For scientists and travelers, converting between Celsius and Fahrenheit is an important skill.
To convert Celsius from Fahrenheit, use this formula: C = 5/9(F – 32)
If it is 77oF, what is it in Celcius?
If it is 10oC, what is it in Fahrenheit?

28. HOMEWORK
Finish inductive reasoning worksheet and do these problems from the book: pages #2, 6, 13-15, 21, 26, 29

29. (2009). Saxon Geometry: Teacher’s Edition
(2009). Saxon Geometry: Teacher’s Edition. United States of America: HMH Supplemental Publishers.
Geoboard Activities. (2007). Areas and Perimeters. Retrieved from _perimeter.htm