Just Count the Pegs Write-up
Lizzy Young
Math
Ms.Vee
11~5~14
Just Count the Pegs
Problem Statement:
For this problem, there are three kids with geoboards, and each of them comes up with a rule that goes with shapes on the geoboard. Freddie’s rule is that if you have a shape with no pegs on the inside of it, then you can find out the area based on the amount of pegs in the boundary. Sally’s rule says that as long as you know the amount of pegs that are inside the shape, you can create any shape, as long as it only has four pegs on it’s boundary, on the geoboard and figure out the area. Frashy’s rule says that you could have any shape and figure out the area as long as you know the amount of pegs on the boundary and the amount of pegs on the inside. Now, I had to create shapes on geopaper that fit these rules, make in-out tables for them, and create a formula for each rule.
Process:
First I made shapes on geopaper that matched Sally’s, Freddie’s and Frashy’s requirements. Then, I created the following tables based on the shapes I had drawn.
X
Y
2
3
0
1
1
2
X
Y
4
1
8
3
6
2
7
2 1/2
3
1/2
5
1 1/2
X
Y
A
1
8
4
9
12
14
0
4
1
5
8
8
2
4
3
1
4
2
Then, I took the data that I had collected and put into tables, and I started to look for similarities in the differences between the X and Y’s of the tables. This way, I started to form possible formulas. When I came up with one I thought might work, I plugged in the numbers from the table to see if it fit into the equation.
Solution:
Sally’s Formula:Y=1+x
Freddie’s Formula: Y=(x/2)-1
Frashy’s Formula: A=(y/2)-1+x
Reflection (habits of a mathematician):
The habit of a mathematician I used in solving this problem was conjecture and test. I used this habit when I was trying to find a formula to fit the rules of the different people’s shapes and when I thought I had a formula that might work, I plugged it into the equation to see if it worked. Sometimes it did and sometimes it didn’t but when it didn’t, I would just change it a little and plug in another number.
Math
Ms.Vee
11~5~14
Just Count the Pegs
Problem Statement:
For this problem, there are three kids with geoboards, and each of them comes up with a rule that goes with shapes on the geoboard. Freddie’s rule is that if you have a shape with no pegs on the inside of it, then you can find out the area based on the amount of pegs in the boundary. Sally’s rule says that as long as you know the amount of pegs that are inside the shape, you can create any shape, as long as it only has four pegs on it’s boundary, on the geoboard and figure out the area. Frashy’s rule says that you could have any shape and figure out the area as long as you know the amount of pegs on the boundary and the amount of pegs on the inside. Now, I had to create shapes on geopaper that fit these rules, make in-out tables for them, and create a formula for each rule.
Process:
First I made shapes on geopaper that matched Sally’s, Freddie’s and Frashy’s requirements. Then, I created the following tables based on the shapes I had drawn.
X
Y
2
3
0
1
1
2
X
Y
4
1
8
3
6
2
7
2 1/2
3
1/2
5
1 1/2
X
Y
A
1
8
4
9
12
14
0
4
1
5
8
8
2
4
3
1
4
2
Then, I took the data that I had collected and put into tables, and I started to look for similarities in the differences between the X and Y’s of the tables. This way, I started to form possible formulas. When I came up with one I thought might work, I plugged in the numbers from the table to see if it fit into the equation.
Solution:
Sally’s Formula:Y=1+x
Freddie’s Formula: Y=(x/2)-1
Frashy’s Formula: A=(y/2)-1+x
Reflection (habits of a mathematician):
The habit of a mathematician I used in solving this problem was conjecture and test. I used this habit when I was trying to find a formula to fit the rules of the different people’s shapes and when I thought I had a formula that might work, I plugged it into the equation to see if it worked. Sometimes it did and sometimes it didn’t but when it didn’t, I would just change it a little and plug in another number.
Rug Games Write-up
Lizzy Young
Math
Ms.Vee
10~3~14
Rug Games Write Up
Problem Statement:In this problem, there is a hypothetical “magic trap door” which releases a dart at random at a patterned rug. On the rug(s), there are two or three colors (gray, white, or black). The colors are divided into sections on the rug(s) so there are several sections of different colors. If the trap door releases a dart at the rug at random, what is the the theoretical probability of the dart landing on white, gray or black?
Process:Describe what you did in attempting to solve the problem. Do this part even if you did not solve the problem.
For each rug, I would first examine the rug and the sections and what colors they were. Then, I would see how the sections were divided by lines, and how I could continue the lines to create a grid, with even sections, so I could figure out the total amount of gray, black, or white sections. By then, I would have a rug with even spaces on it colored in one of the three colors, for example, the rug on the left.This rug obviously only has two colors, black and white. As you can see, there are 30 spaces, 15 white and 15 black. Now that I know how many of each color there are, and the total amount of spaces, I would calculate the theoretical probability. The theoretical probability for black would be 15/30 because out of 30 total spaces, 15 of them are black. We can simplify this fraction to 1/2. The theoretical probability for white would be 15/30 because out of 30 total spaces, 15 of them are white. We can simplify this fraction to 1/2.
Solution:State the solution clearly as you can. Explain how you know your answer is correct.
There is no one solution or answer for this problem, but I will discuss the solution for the example rug above. The theoretical probability of the dart hitting a white space on the rug is 15 in 30 or 1/2 (if you prefer). The theoretical probability of the dart hitting a black space on the rug is 15 in 30 or 1/2 (if you prefer). So the “solution”, you might say, for this problem is that there is a 50% chance of the dart hitting black or white.
Extension:
Invent some extensions or variations to the problem. You can solve these or ask another classmate to do so.
An extension to this problem could be adding a value to the squares. Let’s say that the black squares are worth 5 points (meaning every time the dart hits a black square, you get 5 points) and the white squares are worth 6 points. In this case, you would want to “bet on” white because even though the theoretical probability is not any greater than that of hitting a black square, you would benefit more from it.
Evaluation:Discuss your personal reaction to the problem.
Math
Ms.Vee
10~3~14
Rug Games Write Up
Problem Statement:In this problem, there is a hypothetical “magic trap door” which releases a dart at random at a patterned rug. On the rug(s), there are two or three colors (gray, white, or black). The colors are divided into sections on the rug(s) so there are several sections of different colors. If the trap door releases a dart at the rug at random, what is the the theoretical probability of the dart landing on white, gray or black?
Process:Describe what you did in attempting to solve the problem. Do this part even if you did not solve the problem.
For each rug, I would first examine the rug and the sections and what colors they were. Then, I would see how the sections were divided by lines, and how I could continue the lines to create a grid, with even sections, so I could figure out the total amount of gray, black, or white sections. By then, I would have a rug with even spaces on it colored in one of the three colors, for example, the rug on the left.This rug obviously only has two colors, black and white. As you can see, there are 30 spaces, 15 white and 15 black. Now that I know how many of each color there are, and the total amount of spaces, I would calculate the theoretical probability. The theoretical probability for black would be 15/30 because out of 30 total spaces, 15 of them are black. We can simplify this fraction to 1/2. The theoretical probability for white would be 15/30 because out of 30 total spaces, 15 of them are white. We can simplify this fraction to 1/2.
Solution:State the solution clearly as you can. Explain how you know your answer is correct.
There is no one solution or answer for this problem, but I will discuss the solution for the example rug above. The theoretical probability of the dart hitting a white space on the rug is 15 in 30 or 1/2 (if you prefer). The theoretical probability of the dart hitting a black space on the rug is 15 in 30 or 1/2 (if you prefer). So the “solution”, you might say, for this problem is that there is a 50% chance of the dart hitting black or white.
Extension:
Invent some extensions or variations to the problem. You can solve these or ask another classmate to do so.
An extension to this problem could be adding a value to the squares. Let’s say that the black squares are worth 5 points (meaning every time the dart hits a black square, you get 5 points) and the white squares are worth 6 points. In this case, you would want to “bet on” white because even though the theoretical probability is not any greater than that of hitting a black square, you would benefit more from it.
Evaluation:Discuss your personal reaction to the problem.
- What did you learn from it?
- Describe one Habit of a Mathematician that you used?
- How would you change the problem to make it better?
- Did you enjoy working on it?
- Was it too hard or too easy?
Aussie Fir Tree Write-up
Lizzy Young
Math
Ms.Vee
9/3/2014
Problem Statement:
In the first of the stages, the Aussie Fir tree consists of just two blocks stacked on top of each other. These blocks are referred to as “unit squares”, but I like to call them blocks. In the second stage of the Fir Tree growth, there are four blocks stacked on top of each other and on the third block down, there is an additional block on either side of it resulting in six total blocks. The tree continues to grow this way for four stages (listed below).
Stage 1 : Stage 2: Stage 3: Stage 4:
How many blocks will there be in stage five of the Fir Tree’s growth?
Process:
To solve this problem, I first figured out how many blocks the tree grew vertically in each stage. Then I used that information to figure out how many blocks tall the tree would be in the next stage. After that, I figured out the “branches”. I looked at the pattern of growth of the blocks going horizontally and realized that each branch grew by one block on each side every time. I used that information as well as the growth of the tree vertically to figure out what the tree would look like in the next stage of growth.
Solution: By the fifth stage, the fir tree was 10 blocks long, and had four branches. The first (shortest) branch was three blocks long, the second branch was five blocks long, the third branch was seven blocks long and the fourth (longest) branch was 9 blocks long. In total, the fir tree was made up of 30 blocks in the fifth stage.
Stage 5:
Evaluation:
From this activity, I learned that when patterns are not numerical, it can be difficult to explain them with words, but if you think of the patterns as numbers even if they are not, you can accomplish the same thing using different wording. To make this problem “better” to me would be to make it a little more challenging and testing my formulation skills as well as understanding patterns. If I were to change this problem, I would probably add some kind of formulation factor where you would have to come up with a formula for the growth of the tree or solve the way the tree grows by using a given formula. I wouldn't necessarily say I “enjoyed” working on this problem, but it was definitely a good brain exercise to get me back in “school mode”. I think that there are areas where this problem could grow to meet my standards because there were areas that were really easy for me and there were others that were a bit more difficult, but I think the easy parts and the more difficult parts cancelled each other out to make it just right.
Math
Ms.Vee
9/3/2014
Problem Statement:
In the first of the stages, the Aussie Fir tree consists of just two blocks stacked on top of each other. These blocks are referred to as “unit squares”, but I like to call them blocks. In the second stage of the Fir Tree growth, there are four blocks stacked on top of each other and on the third block down, there is an additional block on either side of it resulting in six total blocks. The tree continues to grow this way for four stages (listed below).
Stage 1 : Stage 2: Stage 3: Stage 4:
How many blocks will there be in stage five of the Fir Tree’s growth?
Process:
To solve this problem, I first figured out how many blocks the tree grew vertically in each stage. Then I used that information to figure out how many blocks tall the tree would be in the next stage. After that, I figured out the “branches”. I looked at the pattern of growth of the blocks going horizontally and realized that each branch grew by one block on each side every time. I used that information as well as the growth of the tree vertically to figure out what the tree would look like in the next stage of growth.
Solution: By the fifth stage, the fir tree was 10 blocks long, and had four branches. The first (shortest) branch was three blocks long, the second branch was five blocks long, the third branch was seven blocks long and the fourth (longest) branch was 9 blocks long. In total, the fir tree was made up of 30 blocks in the fifth stage.
Stage 5:
Evaluation:
From this activity, I learned that when patterns are not numerical, it can be difficult to explain them with words, but if you think of the patterns as numbers even if they are not, you can accomplish the same thing using different wording. To make this problem “better” to me would be to make it a little more challenging and testing my formulation skills as well as understanding patterns. If I were to change this problem, I would probably add some kind of formulation factor where you would have to come up with a formula for the growth of the tree or solve the way the tree grows by using a given formula. I wouldn't necessarily say I “enjoyed” working on this problem, but it was definitely a good brain exercise to get me back in “school mode”. I think that there are areas where this problem could grow to meet my standards because there were areas that were really easy for me and there were others that were a bit more difficult, but I think the easy parts and the more difficult parts cancelled each other out to make it just right.