Download Polygon Patterns

Polygon Patterns
1. From any vertex of a 4 sided-polygon, one diagonal can be drawn.
Draw this shape.
2. From any vertex of a 5 sided-polygon, two diagonals can be drawn.
Draw this shape.
3. From any vertex of a 6 sided-polygon, three diagonals can be drawn.
Draw this shape.
4. From any vertex of a 7 sided-polygon, four diagonals can be drawn.
Draw this shape.
5. Create an equation that will generate the number of diagonals for nsided polygons.
6. State the slope and interpret it in the context of this problem.
7. State the y-intercept and interpret it in the context of this problem.
8. How many diagonals can be drawn from any vertex of a 20-sided
polygon?
9. How many sides does a polygon have if you can draw 42 diagonals?
Source: Algebra I NCSCOS Indicator
Polygon Patterns
(These were the responses to the 3 prompts from the group completing this task).
What’s the important mathematics?
o
Objective 1.02 patterns
o
Objective 4.01 tables/charts
o
To make linear functions
What scaffolding could make the task accessible to all?
o
Manipulative for figures
o
Suggest students set up a table
o
Help them label the independent and dependent variables
How would you extend?
o
Discuss domain/range
o
Verification methods
Source: Algebra I NCSCOS Indicator
The Honeycomb Project
Three friends, Clay, May, and Jay, are using a drawing program to create a project
in art class. Their class has been studying various tilings and their job is to create
an example of tiling that occurs in the real world. They’ve decided to draw a
honeycomb and found that the easiest way to do this with their software requires
using rotations. The bad news is they must know the angle measures in a regular
hexagon and unfortunately, they were all doodling that day in geometry class and
cannot remember the formula they need to use.
Each of them started scribbling on paper, trying to come up with a technique to
find the measure of a hexagon’s interior angle. Surprisingly, they all arrived at the
same answer but used slightly different methods. Each of their methods is shown
below. Use each method below to see if you can determine the interior angle sum of
a hexagon.
Clay's method
May's method
Jay's method
Clay, May, and Jay were so impressed with their problem solving ability that they
wondered if they could find a formula that would always produce the angle sum for
any regular polygon. They agreed that a good strategy would be to consider other
regular polygons, organize the information in a table, and look for a pattern. They
knew that a triangle’s angles summed to 180º so they began with a polygon having 4
sides.
[Hint: Write the expanded form of each computation and then the total.]
No. of sides
4
5
6
7
8
9
10
n
Clay’s method
May’s method
Jay’s method
Do their methods always produce the correct sum? Why or why not?
Adjust their formulas to find the measure of one angle of any regular polygon.
SITE Geometry - Summer 2007
Authored by Eleanor Pusey
The Honeycomb Project
(These were the responses to the 3 prompts from the group completing this task).
What’s the important mathematics?
o
Patterns
o
Angle sum of a triangle
o
Polygon Angle Sums
o
360º in a circle
o
180º in a straight line/straight angle
o
Radii are congruent
o
Inductive Reasoning
What scaffolding could make the task accessible to all?
o
Read the first paragraph aloud to set up the context of the problem
How would you extend?
o
Verify all 3 formulas are equivalent
o
Ask them for another method to write the formula
o
Draw a picture to go with it
o
Do the methods extend to non-regular polygons? Why?
o
Note the formula for any angle of a regular polygon using May’s method
results in:
180n ! 360
360
. How is this related to the exterior angles
= 180 !
n
n
of the polygon?
SITE Geometry - Summer 2007
Authored by Eleanor Pusey
Heart Rates
Suppose a patient is given a 250mg injection of a therapeutic drug that has a side
effect of raising the Heart rate. Table A gives the relationship between Q, the
quantity of drug in the body (in milligrams, or mg) and r, the person’s heart rate (in
beats per minute).
Table A. Heart rate r as a function of drug level Q
Q, drug level (mg)
0
50
100
150
R, heart rate (beats per min) 60
70
80
90
200
100
250
110
1. How does the heart respond to higher drug levels?
Over time, the patient’s body metabolizes the drug, and the level of the drug in the
body falls. Table B gives the drug level, Q, as a function of t, the number of hours
since the injection was given.
Table B. Drug level Q, as a function of time, t
T, time (hours)
0
1
2
3
4
Q, drug level (mg)
250 200 160
128
102
5
82
6
66
7
52
8
42
Since the patient’s heart rate depends on the drug level, and the drug level depends
on the time, then the heart rate depends, via the drug level, on time.
2. What is the heart rate 1 hour after the drug is administered?
3. Estimate the heart rate for each time in Table C below.
Table C. Heart rate r, as a function of time, t
T, time (hours)
0
1
2
3
R, heart rate (beats/min)
4
5
6
7
8
4. Verify your estimates by using a graphing calculator to evaluate a composition of
functions formed by the first two tables. What is the rate of the heart after
24 hours?
Source: adapted from Functions Modeling Change, Wiley & Sons, 1998
Heart Rates (Solutions)
1. How does the heart respond to higher drug levels?
The Heart rate increases as the drug level increases. The actual equation for the model
in Table A is linear and given by: R = 15 Q + 60
2. What is the heart rate 1 hour after the drug is administered?
From table B, we observe that one hour after the drug is administered the remaining
amount of the drug is 200 mg. From table A we see that the heart rate with 200 mg of
the drug would be 100 beats per minute. The equation for the model in Table B is
" 4%
exponential and can be approximated by: Q = 250 ! $ '
# 5&
T
3. Estimate the heart rate for each time in Table C.
Look up time in Table B to find the amount of drug. Find the heart rate in Table A for
the given amount of drug. Fill in the values for Table C.
Time 0 → 250 mg of the drug remains → heart rate is 110 beats per minute → (0, 110)
Time 1 → 200 mg of the drug remains → heart rate is 100 beats per minute → (1, 100)
Time 2 → 160 mg of the drug remains → heart rate is ??? beats per minute → (2, ? )
Since 160 mg is not in Table A, we must approximate. We can use a proportion (linear
interpolation) to make the approximation. Find the two values so that 160 is between
them: Drug Level
Heart rate
200
100
160
???
150
90
The slope of a line is constant so the slope calculated using points 2 and 3 must be the
same as the slope calculated with points 1 and 3. Solve the proportion D/ (160 – 150) =
(100-90)/ (200-150) or D/10 = 10/50. Hence D =2.
Note D is the difference in the y-values (Heart Rate).
The Heart Rate would be approximately 90 + 2 or 92 beats per minute.
Time = 3, Drug Level = 128
Time = 4, Drug level = 102
Drug Level
Heart rate
Drug Level
Heart rate
150
90
150
90
128
???
102
???
100
80
100
80
D/ (128-100) = (90-80)/ (150-100) D/ (102-100) = (90-80)/ (150-100)
D = 28(10/50)
D = 2(10/50)
D = 5.6
D = 0.4
Heart Rate ≈ 80 + 6
Heart Rate ≈ 80 + 0
Source: adapted from Functions Modeling Change, Wiley & Sons, 1998
Table C. Heart rate r, as a function of time, t
T, time (hours)
0
1
2
3
R, heart rate
110
100
92
86
(beats/min)
4
80
5
76
6
73
7
70
8
68
4. Verify your estimates by using a graphing calculator to evaluate a composition of
functions formed by the first two tables. What is the rate of the heart after 24
hours?
Find a best fit function for Table A.
Enter Q into List1 and R into List2.
Linear regression shows a perfect fit: Y1 = .2X + 60
Find a best fit function for Table B.
Enter T into List3 and Q into List4.
An exponential function fits very well (R squared is 0.999955):
Y2 = 249.99(0.79995) ^X or Y2 = 250(0.8) ^X
Form the composition so time is the input and heart rate is the output. In other
words, compute ( R ! Q ) (T ) = R(Q(T )) = 50( 45 ) + 60
T
Y3 = Y1 (Y2) and look at the tables to compare values.
This composition can also be done at the home screen.
Manual calculation:
X → 250(0.8) ^X → .2(X) + 60
4 → 250(0.8) ^4 or 102.4 → .2(102.4) + 60 or 80.48
The rate after 24 hours is: 250(0.8) ^24 ≈ 1.2 → .2(1.2) + 60 ≈ 60.2
(These were the responses to the 3 prompts from the group completing this task).
What’s the important mathematics?
o
Interpret the table data
o
Modeling linear, exponential functions
o
Graphical and algebraic representations of data
o
Use data to predict
What scaffolding could make the task accessible to all?
o
Handout for putting the data into calculator to get regression out.
o
Choosing the independent and dependent variables, appropriate model
How would you extend?
o
Poster display of the mathematics
o
Algebraic composition of the functions
o
Make a new problem with new Tables A, B
Source: adapted from Functions Modeling Change, Wiley & Sons, 1998