Download Function Tables and Patterns, Linear to Quadratic Equations with

2008 USM Summer Math Institute
Co-Sponsored by
Institutions of Higher Learning (IHL)
U.S. Department of Education
(No Child Left Behind Funding)
The Center for Science and Mathematics
Education
The Department of Mathematics
College of Science and Technology
College of Education and Psychology
Function Tables and Patterns, Linear to
Quadratic Equations with profit,
tessellations
Day
Thirteen
1
2008 USM Summer Math Institute
Your Mathematics Instructors, Staff and Partners
• Ms. Michelle Green, Co-Director and CoInstructor for (SM)2I
– Stringer Attendance Center
– National Board Certified/Early Adolescence
– Email [email protected] , Phone 601428-5508
• Dr. Myron Henry, Director and Co-Instructor for
(SM)2I
– Department of Mathematics and the Center for Science
and Mathematics Education
– Johnson Science Tower 314
– Email [email protected], Phone 601-266-4739 or
266-6516
• All Participants (that’s you)
2
Expanded Form and Decimal
Numbers
1.
2.
3.
4.
Sequences and Function Tables
Linear Regression
Quadratic Regression
Tessellations
3
Sequences, Function Tables, and
Number Patterns
•
•
•
•
Squares of numbers ending in five
Adding three each time
Sum of the first n whole numbers
Fibonacci Sequence, Da Vinci Code, and
the divine proportion or golden ratio
A
B
C
D
E
F
1
Whole
Number
Fn
=
Fn = xxx…
(Fn) = zzz…
An
Algorithm
2
0
F0
=
5
25
25
3
1
F1
=
15
225
225
4
2
F2
=
25
625
625
5
3
F3
=
35
1225
1225
6
4
F4
=
45
2025
2025
7
5
F5
=
55
3025
3025
8
6
F6
=
65
4225
4225
9
7
F7
=
75
10
8
F8
=
85
11
9
F9
=
95
12
10
F10
=
105
13
14
15
16
11
F11
=
115
17
18
15
F15
=
155
19
20
16
F17
=
165
21
124
F124
=
1245
2
1. Your number is 65.
For the square of 65,
make the last two
digits 25.
2. Add one to 6 which
equals 7. Now
multiply 6*7 = 42.
3. Place the 42 in front
of 25 . So the
square of 65 = 4225.
Work Space
A
B
C
D
E
F
An
Algorithm
1
Whole
Number
Fn
=
Fn = xxx…
(Fn) = zzz…
2
0
F0
=
5
25
25
3
1
F1
=
15
225
225
4
2
F2
=
25
625
625
5
3
F3
=
35
1225
1225
6
4
F4
=
45
2025
2025
7
5
F5
=
55
3025
3025
8
6
F6
=
65
4225
4225
9
7
F7
=
75
10
8
F8
=
85
11
9
F9
=
95
12
10
F10
=
105
13
14
15
16
11
F11
=
115
17
18
15
F15
=
155
19
20
16
F17
=
165
21
124
F124
=
1245
2
1. Build this spreadsheet
with a formula for
Column A after Cell A2
2. Fill in Column B with a
“copy mechanism “
after cell B2.
3. Fill in Column C with a
copy mechanism after
cell C2.
4. Fill in Column D by a
formula after cell D2.
5. Fill in Column E with a
formula from cell E2.
6. Find an algorithm (a
formula) to fill in
Column F starting with
cell F2.
6th Grade Benchmark: State a rule to explain a number
pattern; 7th Grade Benchmark: Describe and extend
patterns in sequences.
A number pattern
{1, 4, 7, 10, 13, 16, … , 82}.
Question? What is the rule that
describes this number pattern or
sequence?
• In terms of previous entries?
• The ultimate rule for fn (this is worth $1
to the student that discovers it first)?
n
fn
n
1
f1=1
6
2
f2=1+3
7
3
f3=1+3+3
8
4
f4=1+3+3+3
9
5
f5=1+4·3
n
• What value of n gives fn= 82?
fn
Work Space
6th Grade Benchmark: State a rule to explain a number
pattern; 7th Grade Benchmark: Describe and extend
patterns in sequences.
A number pattern .
{0, 1, 3, 6, 10, 15, 21, … }.
Question? What is the rule
that describes this number
pattern or sequence?
• In terms of previous entries?
• What is the ultimate rule for fn?
n
fn
1
f1=1
2
f2=3
3
f3=6
4
f4=10
fn
10
n
fn
5
f5
6
f6
7
f7
8
f8
9
f9
10
f10
n
fn
fn
Scratch Pad
6th Grade Benchmark: State a rule to explain a number pattern; 7th Grade
Benchmark: Describe and extend patterns in sequences.
n
fn
n
fn
1.
Fibonacci Sequence
2.
What is the n-th term?
1
f1=1
6
3.
What is the ratio of [(fn+1)/(fn)]
for n large?
2
f2=1
7
4.
What is the “Divine
Proportion” (from the DaVinci
Code) or Golden Ratio?
3
f3=2
8
5.
Let’s build a spreadsheet.
4
f4=3
9
5
f5=5
10
n
Fn
Fn
(Fn+1)/(Fn)
F1
F2
1
1
1
1
1
2
F1
F2
3
F1
+
F2 =
F3
2
1.00000000000000000000
4
F2
+
F3 =
F4
3
2.00000000000000000000
5
F3
+
F4 =
F5
5
1.50000000000000000000
6
F4
+
F5 =
F6
8
1.66666666666667000000
7
F5
+
F6 =
F7
13
1.60000000000000000000
8
F6
+
F7 =
F8
21
1.62500000000000000000
9
F7
+
F8 =
F9
34
1.61538461538462000000
10
F8
+
F9 =
F10
55
1.61904761904762000000
11
F9
+ F10 =
F11
89
1.61764705882353000000
12
F10 + F11 =
F12
144
1.61818181818182000000
13
F11 + F12 =
F13
233
1.61797752808989000000
14
F12 + F13 =
F14
377
1.61805555555556000000
15
F13 + F14 =
F15
610
1.61802575107296000000
16
F14 + F15 =
F16
987
1.61803713527851000000
17
F15 + F16 =
F17
1,597
1.61803278688525000000
18
F16 + F17 =
F18
2,584
1.61803444782168000000
19
F17 + F18 =
F19
4,181
1.61803381340013000000
20
F18 + F19 =
F20
6,765
1.61803405572755000000
21
F19 + F20 =
F21
10,946
1.61803396316671000000
22
F20 + F21 =
F22
17,711
1.61803399852180000000
23
F21 + F22 =
F23
28,657
1.61803398501736000000
24
F22 + F23 =
F24
46,368
1.61803399017560000000
25
F23 + F24 =
F25
75,025
1.61803398820532000000
26
F24 + F25 =
F26
121,393
1.61803398895790000000
27
F25 + F26 =
F27
196,418
1.61803398867044000000
28
F26 + F27 =
F28
317,811
1.61803398878024000000
29
F27 + F28 =
F29
514,229
1.61803398873830000000
30
F28 + F29 =
F30
832,040
1.61803398875432000000
31
F29 + F30 =
F31
1,346,269
1.61803398874820000000
39
F37 + F38 =
F39
63,245,986
1.61803398874989000000
40
F38 + F39 =
F40
102,334,155
1.61803398874990000000
41
F39 + F40 =
F41
165,580,141
1.61803398874989000000
42
F40 + F41 =
F42
267,914,296
1.61803398874989000000
43
F41 + F42 =
F43
433,494,437
1.61803398874989000000
44
F42 + F43 =
F44
701,408,733
1.61803398874989000000
45
F43 + F44 =
F45
1,134,903,170
1.61803398874989000000
46
F44 + F45 =
F46
1,836,311,903
1.61803398874989000000
47
F45 + F46 =
F47
2,971,215,073
1.61803398874989000000
48
F46 + F47 =
F48
4,807,526,976
1.61803398874989000000
49
F47 + F48 =
F49
7,778,742,049
1.61803398874989000000
50
F48 + F49 =
F50
12,586,269,025
1.61803398874989000000
51
F49 + F50 =
F51
20,365,011,074
1.61803398874989000000
52
F50 + F51 =
F52
32,951,280,099
1.61803398874989000000
53
F51 + F52 =
F53
53,316,291,173
1.61803398874989000000
54
F52 + F53 =
F54
86,267,571,272
1.61803398874989000000
55
F53 + F54 =
F55
139,583,862,445
1.61803398874989000000
56
F54 + F55 =
F56
225,851,433,717
1.61803398874989000000
57
F55 + F56 =
F57
365,435,296,162
1.61803398874989000000
58
F56 + F57 =
F58
591,286,729,879
1.61803398874989000000
59
F57 + F58 =
F59
956,722,026,041
1.61803398874989000000
60
F58 + F59 =
F60
1,548,008,755,920
1.61803398874989000000
61
F59 + F60 =
F61
2,504,730,781,961
1.61803398874989000000
1.61803398874989000000
From Points to Lines and
From Lines to Quadratic
Equations
Scenario: Stringer High School Beta Club wants
to sponsor a dance to raise money for convention.
To help them decide if the dance will be worth the
trouble, the members polled the students to see
just how many students would attend and how
much they would be willing to pay to attend the
dance. They estimated that the cost of the dance
would be about $200. Does the data show that
the dance will be worth it and generate a profit?
15
Poll Results
Ticket Price
$ 6.50
$ 6.00
$ 5.50
$ 5.00
$ 4.50
$ 4.00
$ 3.50
$ 3.00
$ 2.50
$ 2.00
Number of People willing to
pay the ticket price
49
58
38
64
51
36
69
51
48
72
16
Enter Data in Lists
Enter ticket price in L1 and
people willing to pay in L2.
You must set the
window before you
graph or use your
“quick” windows
under zoom.
17
What kind of
relationship
is this?
18
Same ole Data, or is it?
Ticket Price
$ 6.50
$ 6.00
$ 5.50
$ 5.00
$ 4.50
$ 4.00
$ 3.50
$ 3.00
$ 2.50
$ 2.00
Number of People who
said they would pay the
ticket price
49
58
38
64
51
36
69
51
48
72
Attendance in a
new light
49
107
145
209
260
296
365
416
464
536
19
Enter Data in Lists
Enter ticket price in L1 and
attendance in L3.
You must set the
window before you
graph or use your
“quick” windows
under zoom.
20
Now, that’s more like it!
Oh Great Calculator, help us to see the
relationship! Work your magic!!!
21
22
What ticket price will generate the
greatest Profit?
X = ticket price
Y = -105.95X + 735 (Attendance)
How do you find Profit???
(Attendance * ticket price ) – cost = profit
(-105.95X + 735)X – 200 = P(X)
23
24
25
49
49
$6.00
58
107
$5.50
38
145
$5.00
64
209
$4.50
51
$4.00
36
296
$3.50
69
365
51
Attendance
$6.50
$3.00
The Dance
Number of people
New way of looking at data
willing to pay the price
80
70
60
50
40
30
20
10
0
$0.00
$2.00
$4.00
$6.00
Price per ticket
$8.00
price/attendance
The Dance 2
260
Attendance
Price per ticket
500
450
400
350
300
250
200
150
100
50
0
$0.00
y = -103.8x + 723.88
$2.00
$4.00
$6.00
Price per ticket
$8.00
416
price/attendance
$2.50
48
464
26
Price per ticket
Profit
Income less cost
$1,200.00
$6.00
$5.50
$118.50
$442.00
$597.50
$1,000.00
Income Less Cost
$6.50
$3.50 ,
$1,077.50
$800.00
$600.00
$400.00
y = -100.71x2 + 697x - 146.66
$200.00
$5.00
$845.00
$4.50
$970.00
$4.00
$984.00
$3.50
$1,077.50
$3.00
$1,048.00
$2.50
$960.00
$0.00
$0.00
$2.00
$4.00
Ticket Price
$6.00
$8.00
27
Tessellations
What Are They?
28
Basically, a tessellation is a way to tile
a floor (that goes on forever) with
shapes so that there is no overlapping
and no gaps. Remember the last
puzzle you put together? Well, that
was a tessellation! The shapes were
just really weird.
We usually add a few more
rules to make things
interesting!
29
REGULAR TESSELLATIONS:
RULE #1: The tessellation must tile a floor (that goes on forever)
with no overlapping or gaps.
•RULE #2: The tiles must be regular polygons - and all the same.
•RULE #3: Each vertex must look the same.
What's a vertex?
where all the "corners" meet!
What can we tessellate using these
rules?
30
Triangles? Yep!
Notice what happens at each vertex!
The interior angle of each equilateral triangle is
60 degrees.....
60 + 60 + 60 + 60 + 60 + 60 = 360 degrees
31
Squares? Yep!
What happens at each vertex?
90 + 90 + 90 + 90 = 360 degrees again!
So, we need to use regular polygons that add up to 360 degrees.
Will pentagons work?
The interior angle of a pentagon is 108 degrees. . .
108 + 108 + 108 = 324 degrees . . . Nope!
32
Hexagons?
120 + 120 + 120 = 360 degrees Yep!
Heptagons?
No way!! Now we are getting overlaps!
Octagons? Nope!
They'll overlap too. In fact, all polygons with more than six sides will
overlap! So, the only regular polygons that tessellate are triangles,
squares and hexagons!
33
SEMI-REGULAR TESSELLATIONS:
RULE #1: The tessellation must tile a floor (that goes on forever) with
no overlapping or gaps.
•RULE #2: The tiles must be regular polygons (more that one has to be
used)
•RULE #3: Each vertex must look the same.
Neat web site to practice….
http://matti.usu.edu/nlvm/nav/frames_asid_163_g_4_t_3.html?open=activities
34
Another neat website……
http://www.shodor.org/interactivate/activities/tessellate/index.html
This is a great way to show transformations of regular polygons that will
tessellate!
http://mathforum.org/alejandre/students.tess.html
Take time to practice…..
35
Tessellations with ‘Paint’
• Select your favorite color from the menu.
In Paint, use the “filled Box” rectangle tool
to draw a square. If the shift key is held
down as the “rectangle” is drawn, it will
make a square.
– It is recommended that that you save your
work frequently. By saving it after each step,
you will not have to start over.
36
• Using the “Free-Form Select” tool, start
on the top edge, wiggle around as you
are going down and ext the lower edge
of the square. Once you exit, the
mouse button is released, a dotted box
will appear around the selection. Grab
the box and slide the selection to the
opposite side of the square. Make sure
you have selected the ‘no fill’ option.
• Repeat the process but this time go
from bottom to top.
37
• Look at the figure. What does it look
like to you? Use the pencil or
paintbrush with a contrasting color to
decorate your figure.
• Use the “Free-Form Select” to select
your object and copy and paste. You
can use the paint can to color your
pasted object before you move it.
• Select the pasted figure once you have
colored it and with the ‘no fill’ option,
move it to the border of your original
figure.
38
– Make sure you are saving your changes
frequently or else you will have to start
over
– This will take practice…..
39
Example:
40
652  6 10  52
652  6 102  2  6 10  5  52
652  6 2 10 2  6 10  2  5  52
652  6 10 2  6  6 10 2  52
652  6 10 2 6  1  52
652  10 2  6  6  1  52
652  10 2  6  7  52
652  100  42  52
652  4200  52  4200  25
652  4225
1252  12 10  52
1252  12 102  2  12 10  5  52
1252  122 102  12 10  2  5  52
1252  12 102 12  12 102  52
1252  12 102 12  1  52
1252  102 12  12  1  52
1252  102 12 13  52
1252  100 156  52
1252  15600  52  15600  25
1252  15625
41