Natasha Oslinger

September 16, 2014

David Corner

Problem Statement

For this problem, you must use the numbers 1, 2, 3, and 4 to create arithmetic expressions for the numbers 1 through 25. You can use addition, subtraction, radicals, exponents, factorials, or juxtaposition to make 1-2-3-4 expressions for the numbers. For example, 1+2-3*4 is 4 and 1*3-2+4 is 5.

Process Description

To start this projects, I began with the basics: addition, subtraction, and multiplication. I combined and subtracted 1, 2, 3, and 4 in different orders and variations in order to get as many numbers as I could. For example, 1+2+3+4 and 1-2-3-4 would give me two numbers I could check off of the 25 I needed to find. Then, I used a calculator to “play” with the numbers by putting in different combinations of 1, 2, 3, and 4 to get as many numbers as I could. Once I couldn’t find any more numbers using simple arithmetic, I tried using the order of operations (PEMDAS) to solve the rest. For example, I would use 3*(4+1)+2 to find the number seventeen. Using the order of operation, I would know to do what is in the parentheses before the rest of the problem, this method was able to get me several more answers between 1 and 25.

Finally, I couldn’t think of any other ways to find the few last numbers I needed, then, my math teacher Mr. Corner told me about how I could use factorials to solve the last few numbers I wasn’t able to get. I was finally able to finish the problem by making the number 21 out of 4!-3!+2+1 and the number 23 out of 3!+4^2+1. I wasn’t able to find any patterns to help me figure out the answers, it was all trial and error. By the end of the problem, I was using a lot more creative expressions, at the beginning I used mostly addition and subtraction, by the end, I was using factorials and exponents.

Solution

0= 4+1-3-2

1= (3*1)+2-4

2= 1+2+3-4

3= 4-3+1+2

4= 4-3+1+2

5= 3*1-2+4

6= 1-2+3+4

7= 13-4-2

8= 2-1+3+4

9= (3*1)+2+4

10= 1+2+3+4

11= (3*4)-2+1

12= 4^2-3-1

13= 34-21

14= 3^2+1+4

15= (4*3)+1+2

16= 4!-3^2+2

17= 3*(4+1)+2

18= 4*(3+1)+2

19= 14+3+2

20= 14+(3*2)

21= 4!-3!+2+1

22= 34-12

23= 3!+4^2+1

24= 4*3*2*1

25= 1+2(3*4)

I know that the answers I found are correct because I followed the guidelines I was given and used the rules of operation. In addition to this, I went and double checked all my answers.

Self-Assessment and Reflection

From doing this POW, I learnt about factorials and how to use different mathematical techniques to solve a set of problems. I also learned that there is always a way to solve a problem, even if I can’t find the solution at first. I think I deserve an A on this project because I spent the full amount of time we were given to make sure my work is correct and exquisitely. I didn’t procrastinate until the night it was due and I double checked to make sure my answers were correct. I feel I finished this project to the best of my ability and I am the proud of the work I got accomplished. The Habit of a Mathematician I think I used well in this project was ‘Staying Organized.” Math can get very confusing when there are numbers all over the place, work in random places on the page, and when things aren’t placed as efficiently as they could be. In this project, I kept all my work and answers separate and tidy. By doing so I was able to complete the work and be proud of what I did.

September 16, 2014

David Corner

Problem Statement

For this problem, you must use the numbers 1, 2, 3, and 4 to create arithmetic expressions for the numbers 1 through 25. You can use addition, subtraction, radicals, exponents, factorials, or juxtaposition to make 1-2-3-4 expressions for the numbers. For example, 1+2-3*4 is 4 and 1*3-2+4 is 5.

Process Description

To start this projects, I began with the basics: addition, subtraction, and multiplication. I combined and subtracted 1, 2, 3, and 4 in different orders and variations in order to get as many numbers as I could. For example, 1+2+3+4 and 1-2-3-4 would give me two numbers I could check off of the 25 I needed to find. Then, I used a calculator to “play” with the numbers by putting in different combinations of 1, 2, 3, and 4 to get as many numbers as I could. Once I couldn’t find any more numbers using simple arithmetic, I tried using the order of operations (PEMDAS) to solve the rest. For example, I would use 3*(4+1)+2 to find the number seventeen. Using the order of operation, I would know to do what is in the parentheses before the rest of the problem, this method was able to get me several more answers between 1 and 25.

Finally, I couldn’t think of any other ways to find the few last numbers I needed, then, my math teacher Mr. Corner told me about how I could use factorials to solve the last few numbers I wasn’t able to get. I was finally able to finish the problem by making the number 21 out of 4!-3!+2+1 and the number 23 out of 3!+4^2+1. I wasn’t able to find any patterns to help me figure out the answers, it was all trial and error. By the end of the problem, I was using a lot more creative expressions, at the beginning I used mostly addition and subtraction, by the end, I was using factorials and exponents.

Solution

0= 4+1-3-2

1= (3*1)+2-4

2= 1+2+3-4

3= 4-3+1+2

4= 4-3+1+2

5= 3*1-2+4

6= 1-2+3+4

7= 13-4-2

8= 2-1+3+4

9= (3*1)+2+4

10= 1+2+3+4

11= (3*4)-2+1

12= 4^2-3-1

13= 34-21

14= 3^2+1+4

15= (4*3)+1+2

16= 4!-3^2+2

17= 3*(4+1)+2

18= 4*(3+1)+2

19= 14+3+2

20= 14+(3*2)

21= 4!-3!+2+1

22= 34-12

23= 3!+4^2+1

24= 4*3*2*1

25= 1+2(3*4)

I know that the answers I found are correct because I followed the guidelines I was given and used the rules of operation. In addition to this, I went and double checked all my answers.

Self-Assessment and Reflection

From doing this POW, I learnt about factorials and how to use different mathematical techniques to solve a set of problems. I also learned that there is always a way to solve a problem, even if I can’t find the solution at first. I think I deserve an A on this project because I spent the full amount of time we were given to make sure my work is correct and exquisitely. I didn’t procrastinate until the night it was due and I double checked to make sure my answers were correct. I feel I finished this project to the best of my ability and I am the proud of the work I got accomplished. The Habit of a Mathematician I think I used well in this project was ‘Staying Organized.” Math can get very confusing when there are numbers all over the place, work in random places on the page, and when things aren’t placed as efficiently as they could be. In this project, I kept all my work and answers separate and tidy. By doing so I was able to complete the work and be proud of what I did.