Proof of Euclid's Number Proposistion 26

Proof of Euclid's Number Proposition 26

 

I loved writing this proof because I am able to clearly understand a theory of Euclid's.  Many famous mathematics come up with such complex theorems that as an undergraduate mathematics student I can barely understand the theorems, much less the proofs of those theorems.  I appreciate that this proposition is simple and the proof is simple, but the implications are vast. 

In the future I would look into proving more complicated number theory propositions by Euclid.  This type of mathematics appeals to me and number theory would be an interesting topic for me to explore more.

A Review of Mathematician’s Lament by Paul Lockhart

    Paul Lockhart makes many interesting points in his book Mathematician’s Lament.  He argues that mathematics education in schools today is far from where it should be.  The author thinks that mathematics is an art, cheapened and stripped down to formulas and definitions.  Although I think reform is needed in mathematics classrooms today, Lockhart’s idea of reform is not practical.
    Lockhart’s main theme of the book is that mathematics is an art form and should be treated as such.  Just as art is taught with a blank canvas, students should be given a blank canvas to explore mathematics as they please.  Although I agree that mathematics is so much more than procedure and “plug and chug” formulas, can anyone really expect a middle schooler to work on mathematics for an hour every day given no guidance or instruction?  There has to be some structure in a mathematics classroom especially for younger students. 
    The author makes it clear that we stress notation too much in our classrooms.  What I don’t understand is how he expects students to create and explore mathematics with no base to start from.  You have to crawl before you can walk and you have to be able to add numbers before you can create mathematics.
    In “Mathematician’s Lament” the author states that there should be no mathematics curriculum, no standardized test, and no regulation of mathematics teachers cross country because teachers are being too bogged down by scores to actually teach mathematics.  There are several alarming implications to this.  First, if there were no regulations then teacher performance would probably decline because there is no way to keep them accountable.  Second, there is a sad truth that many teachers do not have enough mathematical understanding to guide students through any and every mathematics the students will explore.  Teachers teach procedures mostly because they were taught procedures.  There are ways to encourage teachers to branch out, but taking away all standards and expectations is not one of them. Finally, students who can achieve high level mathematics will never be taught the mathematical background needed to succeed.
    Lockhart writes that mathematics is a fun art form that has only a few real life applications.  This point breaks my heart, because I believe mathematics is fun, but I also believe that mathematics is important.  He makes it seem like the only purpose of mathematics fun, making it essentially unimportant to teach in schools.  In his argument to reform, he goes so far as to make mathematics a trivial subject.  This seems far from accurate.  Where would our world be without mathematics? Computers, medicine, science, technology, business and more wouldn’t even exist without mathematics.  So yes, Lockhart, mathematics is fun, but it is also vital to the world we live in.

Unit Plan: Honors Senior Project

7th Grade Mathematics Unit Plan Outline: 

Number Lines and Negative Integers

 

Common Core State Standards: 7.NS.1

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
1. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
2. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
3. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
4. Apply properties of operations as strategies to add and subtract rational numbers.

I Can Statements:

1. I can name real world examples of two numbers that combine to make zero.
2. I can tell you the distance between two rational numbers p and q on a number line.  
3. I can associate directions on the number line with subtraction and addition.
4. I can use integers and rational numbers in all operations.
5. I can tell you why adding a negative number is the same as subtracting a positive.
6. I can define the vocabulary words integer and rational number.
7. I can tell you how far away a is from b and where a+b is on a number line.

Activities 

Score Keepers

   a. Standard alignment: NS.1.a, NS.1.c, NS.1.d

   b. Use a jeopardy game with negative and positive score values to introduce adding and subtracting negative numbers  This activity is meant for the whole class to do together.  The teacher will guide the score keepers in keeping score.  Point out times where the score is zero because negative points balanced with positive.  

   c. 1-2 Class periods

   d. Materials: Projector

Lesson Plan

Chip Board

   a. Standard alignment: NS.1.b, NS.1.c, NS.1.d

   b. Use red and black checkers chips to represent positive and negative numbers.  Have students practice subtracting and adding negative numbers using their checkers.

   c. 2-3 Class periods

   d. Materials: Black and red checkers, student page

Lesson Plan

Student Page

Debt and Negative Numbers

   a. Standard alignment: NS.1.a, NS.1.c

   b. Use debt as an example of real world negative numbers.  Use national debt as an example.  Ask question like, if x amount is the US debt, how much would it take to break even?  Use a number line to describe debt.  Also have students explore adding and subtracting debt.   

   c. 2-3 Class periods

   d. Materials: Debt Power point, at home assignment

Lesson Plan

Slide Presentation

Student Page

Number Line Introduction: Thermometers

   a. Standard alignment: NS.1.b, NS.1.c, NS.1.d

   b. Use an activity with traditional thermometers to introduce number lines.  This will give students a real world examples of number lines.

   c. 1-2 Class periods

   d. Materials: Student page

Lesson Plan

Student Page

Performance task: Create your own Timeline

   a. Standard alignment: NS.1.b, NS.1.c, NS.1.d

   b. Students will work on creating their own timeline.  They will use personal life events and write about the mathematics involved

   c. 2-3 Class periods

   d. Materials: Student pages, rubric, student peer review, markers/colored pencils, glue/tape, poster board/graph paper/11x17 paper.  

Lesson Plan Day 1

Student Page Day 1
Lesson Plan Day 2
Student Page 2

Who Dun It?

   a. Standard alignment: NS.1.a, NS.1.b, NS.1.c, NS.1.d

   b. Set up 4 stations, one for each role.  Students will work on solving the mystery, the undoing of -7.  This activity will help review the unit while letting the students have a fun mystery activity.

   c. 4 class periods

   d. Materials: Caution tape, graph paper, poster paper, coloring utensils, rulers, student pages

Lesson Plan 

Rainy Day Game

(Thanks to Dr. John Golden)