Math 251-Assignment Pick\'s Theorem PDF

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Name: Jillian Barber

Assignment: Using Manipulatives Geoboard

Part I: In this lesson, you will find areas of triangles on a geobooard. For all of this activity, assume that the pegs on your geoboard are 1 unit apart, so that the square that looks like this has an area of 1 square unit. The triangle that look

like this would have an area of

square unit.

1 of a 2

What do you think the area of a traingle that looks like this is? Answer: 1

One way to think about its area is rectangle

to enclose it in a

1. The area of the red rectangle is? Answer: 2

That means that the area of the traingle that is half of that rectangle is? Answer: 1 For problems 2 thru 4, make the figures on your geoboard. Then enclose the triangles in rectangles to help you figure out the ares.

Answer: 1

Area of 2 = 3

Area of 3 = 2

Area of 4 = 3

Part II: On your virtual geoboard A polygon that you can make on a geoboard is called a lattice polygon. In this activity you will learn a way to find the area of a lattice polygon. 1. Make a polygon that looks like this If we assume that the pinsare 1 unit apart, what is the the area of the polygons you made? Answer: 5

Attach a picture of both polygons. Or draw your shape below

Answer: 2

Area of Polygon 1 = 2

Area of Polygon 2 = 2

3.) Now change one of your polygons to have B=7 and I=0. Attach a picture of the polygons. Or draw your shape below

Answer:

Area of Polygon = 2.5 4.) Make a table that describes the number of boundary pins, number of interior pins, and the area. Attach a picture of the polygons. Or draw your shape below

B (boundary pins)

I (interior pins)

Area

6

0

2

7

0

2.5

8

0

3

9

0

3.5 3

5.) Predict: What is the area of a polygon with B=12 and I=0?

Answer: 5 6.) Check: Make a polygon with B=12 and I=0 and see if your predictions is true. Attach a picture of the polygons. Or draw your shape below

Answer:

Area of Polygon = 5 7.) Extend: Do you see a pattern in what happens to the area as the number of boundary pins increase? What do you think the rule is for the aea of a polygon with I=0?

Answer: Yes, the shape’s area will increase by .5 with every extra boundary pin added. 4

Part III: In the last lesson, you had a formula for the area of a polygon with I = 0. Next we will start to figure out what happens when I ≠ 0. 1)

Complete the following table by creating the appropriate polygons on your geoboard and calculating the area of each shape. Make sure to pay attention to the number of boundary pins (B) and interior pins (I).

B (Boundary Pins)

I (Interior Pins)

Area

6

0

2

6

1

3

6

2

4

6

3

5

7

0

2.5

7

1

3.5

7

2

4.5

8

0

3

8

2

5

8

4

7

2)

For each line in the table, did it matter what the polygon was or did it just matter what the numbers B and I were?

5

Answer: Only the B and I mattered- their limitations created the shape.

Part 4: Calculating Pick’s Theorem Look at the table you made in Part 3 to figure out the answers to the questions in this part.

Note: If most of the rows in the table make a particular pattern, but one row doesn’t, look again at that row to see if you made a mistake in the number of pins or in the area. 1. A polygon with B = 10 and I = 3 has area 7. What do you think will be the area if B = 10 and I = 4?

Answer: 4+(10/2)-1=8 2. Predict: As you add interior pins to a lattice polygon, how does the area change? Write down a rule about the number of interior pins and the area of a polygon.

Answer Rule: area increases by 1 as interior pins are added 3. Check :Make a new polygon that you haven’t made already. Check to see if your rule works with this new polygon. Did the rule Work? Show your work.

Answer: B (Boundary Pins) I (Interior Pins)

Rule

10

A(P)= I+(b/2)-1 9 5+(10/2)-1=9

5

Area

The rule you just discovered is called Pick’s Theorem, named after Georg Alexander Pick.

Finally State what Pick Theorem is Answer: A method used to prove (give mathematical evidence) the area equals the interior points plus boundary points divided in half minus 1.

A good biography of Pick can be found at https://www.math.hmc.edu/funfacts/ffiles/10002.2.shtml http://www.mathedpage.org/geometry-labs/pick/ https://matthewdaws.github.io/mathematics/files/pick_proof.pdf

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