1. Modeling and Equation Solving
1.1
Modeling and Equation Solving

2. What you’ll learn about
Numeric Models
Algebraic Models
Graphic Models
The Zero Factor Property
Problem Solving
Grapher Failure and Hidden Behavior
A Word About Proof
… and why
Numerical, algebraic, and graphical models provide different methods to visualize, analyze, and understand data.

3. Mathematical Modeling
To mathematically model a situation means to represent it in mathematical terms. The particular representation used is called a Mathematical model of the situation.

4. Party Handshakes
Each of the 30 people at a party shook hands with everyone else. How many handshakes were there altogether?
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

5. Party Handshakes
Get into groups of four students.
Draw the table below in your Notes.
Fill in the table. Under the table, briefly describe how your group figured out the answers.
# of People 1 2 3 4 5
# of Handshakes
01:39
01:45
01:40
01:38
01:34
01:33
01:41
01:36
01:37
01:43
01:48
01:49
01:55
01:47
01:46
01:32
01:44
01:42
01:35
01:25
01:20
01:21
01:19
01:18
01:14
01:16
01:17
01:22
01:23
01:29
01:50
01:30
01:28
01:27
01:24
01:26
01:31
01:53
02:17
02:18
02:19
02:16
02:14
02:11
02:12
02:13
02:25
02:20
02:26
02:27
02:28
02:24
02:23
02:21
02:22
02:10
02:15
01:58
01:59
02:05
01:57
01:56
01:52
01:13
01:54
03:00
02:01
02:07
02:08
02:09
02:06
02:04
02:02
02:03
01:51
01:11
00:23
00:24
00:26
00:22
00:21
00:19
00:25
00:20
00:27
00:28
00:32
00:33
00:34
00:31
00:30
00:29
00:35
00:18
00:17
00:04
00:06
00:07
00:03
00:02
00:05
01:00
00:01
00:08
00:09
00:13
00:14
00:16
00:12
00:11
00:15
00:10
00:36
00:37
02:00
01:01
01:02
01:05
00:59
00:56
00:57
00:58
01:03
01:04
01:15
01:10
02:29
01:09
01:08
01:06
01:07
00:54
00:53
00:41
00:42
00:43
00:40
00:45
00:38
00:39
00:44
00:46
00:50
00:51
00:52
00:55
00:49
00:47
00:48
01:12
02:31
04:15
04:10
04:11
04:09
04:08
04:04
04:06
04:07
04:12
04:13
04:19
04:25
04:20
04:18
04:17
04:14
04:16
04:03
04:02
03:50
03:51
03:52
03:55
03:49
03:46
03:47
03:48
03:53
03:54
04:05
05:00
04:01
03:59
03:58
03:56
03:57
04:21
04:22
04:48
04:49
04:55
04:47
04:46
04:42
04:43
04:44
04:50
04:51
04:57
04:58
04:59
04:56
04:54
04:52
04:53
04:41
04:40
04:29
04:35
04:30
04:28
04:27
04:23
04:24
04:26
04:31
04:32
04:38
04:39
04:45
04:37
04:36
04:33
04:34
03:44
03:43
02:54
02:56
02:57
02:53
02:52
02:55
02:50
02:51
02:58
02:59
03:03
03:04
03:06
03:02
03:01
03:05
04:00
02:49
02:48
02:36
02:37
02:38
02:34
02:33
02:30
02:32
02:39
02:45
02:44
02:46
02:47
02:43
02:42
02:40
02:41
03:07
03:08
03:31
03:32
03:33
03:30
03:35
03:27
03:28
03:29
03:34
03:36
03:40
03:41
03:42
03:45
03:39
03:37
03:38
02:35
03:26
03:12
03:13
03:14
03:11
03:10
03:09
03:15
03:16
03:17
03:21
03:22
03:23
03:24
03:20
03:18
03:19
03:25
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

6. Party Handshakes
# of People 1 2 3 4 5
# of Handshakes 1 3 6 10 +1 +2 +3 +4
Answer the questions on your notes.
What happens to the number of handshakes as the number of people increase?
Is this a linear function? In other words, are the differences between the number of handshakes the same?
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

7. 1 2 3 4 Party Handshakes 1 2 3 4 5 6 10 How are these numbers related?
Points 1 2 3 4 5
Segments 6 10
Segments Per Vertex 1 2 3 4
How are these numbers related?
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

8. Party Handshakes 1 2 3 4 5 6 10 Points Segments Segments Per Vertex6 10
Segments Per Vertex
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

9. Party Handshakes
Each of the 30 people at a party shook hands with everyone else. How many handshakes were there altogether?
Serra - Discovering Geometry
Chapter 2: Reasoning in Geometry

10. Numerical Models uses data, charts and tables
sometimes the trends are easy to see but sometimes we need a computer to do the analysis

11. Numeric Model

12. Algebraic Models (equation that fits the data)
formulas and equations are created from numbers to make further predictions for unknown values
some of the formulas we will use are ones we learned from other math courses or science courses (d = rt, A = lw, F = ma)

13. Example Comparing Pizzas

14. Solution

15. Graphical Models
a visible representation of a numerical or algebraic model
examples of graphical models: scatter plots, function graphing, charts, etc.

16. Example 1
Numerical model and its corresponding graphical model and algebraic model. 1.
a p. x 2 4 6 8 10 12 y 16 22 28 34

17. Example 2
Numerical model and its corresponding graphical model and algebraic model. 2.
a s. x 5 7 9 11 13 15 y 1 2 3 4 6

18. The Zero Factor Property
A product of real numbers is zero if and only if
at least one of the factors in the product is zero.

20. Solutions

21. Example Solving an Equation

22. Solution

23. Fundamental Connection
If a is a real number that solves the equation f (x) = 0,
then these three statements are equivalent:
1. The number a is a root (or solution) of the equation f (x) = 0.
2. The number a is a zero of y = f (x).
3. The number a is an x-intercept of the graph of y = f (x). (Sometimes the point (a, 0) is referred to as an x-intercept.)

24. Problem Solving Polya’s 4-step process: Understand the problem.
Devise a plan.
Carry out the plan.
Look back.

25. Problem Solving Polya’s 4-step process: Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Understand

26. Problem Solving Polya’s 4-step process: Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Understand
Think

27. Problem Solving Polya’s 4-step process: Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Understand
Think Do

28. Problem Solving Polya’s 4-step process: Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Understand
Think Do
Check

29. Problem Solving Polya’s 4-step process: Understand the problem.
Devise a plan.
Carry out the plan.
Look back.
Understand
Think Do
Check

30. A Problem-Solving Process
Step 1 – Understand the problem.
Read the problem as stated, several times if necessary.
Be sure you understand the meaning of each term used.
Restate the problem in your own words. Discuss the problem with others if you can.
Identify clearly the information that you need to solve the problem.
Find the information you need from the given data.

31. HW: Section 1.1 / Problem Set A
a. If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
b. Describe how Polya’s four steps were used to solve part a.

32. Step 1: Understand the problem (understand).
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Draw a picture of a square with the diagonals drawn in.
The question asks, “how many triangles?”
“CLUE”: how many triangles of all sizes ?

33. Step 1: Understand the problem.
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Draw a picture of a square with the diagonals drawn in.
The question asks, “how many triangles?”
“CLUE”: how many triangles of all sizes ?

34. Step 1: Understand the problem.
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Draw a picture of a square with the diagonals drawn in.
The question asks, “how many triangles?”
“CLUE”: how many triangles of all sizes ?

35. Step 1: Understand the problem.
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Draw a picture of a square with the diagonals drawn in.
The question asks, “how many triangles?”
“CLUE”: how many triangles of all sizes ?

36. Step 1: Understand the problem.
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Draw a picture of a square with the diagonals drawn in.
The question asks, “how many triangles?”
“CLUE”: how many triangles of all sizes ?

37. A Problem-Solving Process (think)
Step 2 – Develop a mathematical model of the problem.
Draw a picture to visualize the problem situation. It usually helps.
Introduce a variable to represent the quantity you seek. (There may be more than one.)
Use the statement of the problem to find an equation or inequality that relates the variables you seek to quantities that you know.

38. Step 2: Devise a plan. How to count the # of triangles of all sizes?
Plan: Figure out how many sizes there are and count them from smallest to largest.

39. A Problem-Solving Process
Step 3 – Solve the mathematical model and support or confirm the solution.
Solve algebraically using traditional algebraic models and support graphically or support numerically using a graphing utility.
Solve graphically or numerically using a graphing utility and confirm algebraically using traditional algebraic methods.
Solve graphically or numerically because there is no other way possible.

40. Step 3: Carry out plan.
First, how many sizes of triangles are there?

41. Step 3: Carry out plan (Do).
First, how many sizes of triangles are there?

42. Step 3: Carry out plan. First, how many sizes of triangles are there?
‘little’
‘big’

43. Step 3: Carry out plan. Next, count how many of each size: # little?
# big?

44. A Problem-Solving Process
Step 4 – Interpret the solution in the problem setting.
Translate your mathematical result into the problem setting and decide whether the result makes sense.

45. 4 4 Step 4: Look back. We’ve counted 8 triangles:
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Answer: 8 4 4

46. 4 4 Step 4: Look back. We’ve counted 8 triangles:
If the diagonals of a square are drawn in, how many triangles of all sizes are formed?
Answer: 8 4 4

47. Grapher Failure and Hidden Behavior
sometimes the graphing calculator will have problems correctly showing the graph you want to see for a function
maybe the window is too large or too small
the grapher has trouble with functions with a vertical asymptote
maybe there are “hidden” zeros for a graph

48. Example Seeing Grapher Failure
 of y = 3/(2x – 5) on a graphing calculator. Is there an x-intercept?

49. Solution
 of y = 3/(2x – 5) on a graphing calculator. Is there an x-intercept?
Notice that the graph appears to show an x-intercept between 2 and 3. Confirm this algebraically.
This statement is false for all x, so there is no x-intercept. The grapher shows a vertical asymptote at x = 2.5.