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Make sure you read all parts 1-3!

“I made a scatter plot of the 5,675 teachers. On the x-axis is that teacher’s language arts score for 2010. On the y-axis is that same teacher’s math score for 2010. There is almost no correlation.

Rather than report about these obvious ways to check how invalid these metrics are and how shameful it is that these scores have already been used in tenure decisions, or about how a similarly flawed formula will be used in the future to determine who to fire or who to give a bonus to, newspapers are treating these scores like they are meaningful. The New York Post searched for the teacher with the lowest score and wrote an article about ‘the worst teacher in the city’ with her picture attached. The New York Times must have felt they were taking the high-road when they did a similar thing but, instead, found the ‘best’ teachers based on these ratings.

I hope that these two experiments I ran, particularly the second one where many teachers got drastically different results teaching different grades of the same subject, will bring to life the realities of these horrible formulas. Though error rates have been reported, the absurdity of these results should help everyone understand that we need to spread the word since calculations like these will soon be used in nearly every state.”

World Maths Day 2012 is underway.

The competition is designed for ages 4-18 and all ability levels. Teachers, parents and media can also register and play.

Last year, more than five million students from 218 countries combined to correctly answer 428,598,214 World Maths Day questions.

Each game lasts for 60 seconds, and students can play up to 50 games, earning points for their personal tally. Students can play beyond 50 games during the event, but points will only count to the World Maths Day Mathometer, not their personal point score. The students who answer the most questions correctly appear on the Hall of Fame. There are 5 different levels of play, 10 challenges on each level.

The official World Maths Day competition begins with the first second of March 7 at midnight in Apia, Samoa, and continues as long as it is March 7 anywhere in the world. The total competition time is 48 hours.

For further information, visit the World Maths Day site and check out the Resources section.

The 47th edition of the Math Teachers at Play Blog Carnival is now up at Math Hombre. Check it out!

Activities, games, lessons, hands-on fun — the Math Teachers at Play blog carnival would love to feature your article about mathematics from preschool to precollege level (through the first year of calculus). You can submit your article online, or email John directly to make sure he gets your submission.

The carnival will be published Friday at Math Hombre.

I wrote this note in 2008 to give a down-to-earth example to help in the understanding of negative numbers [2]. It is posted for the Math Teachers at Play blog carnival.

— 1. Introduction —

First, lets get all the formalities out of the way by observing a few algebraic rules regarding positive and negative real numbers. We will consider the familiar arithmetic operations of addition and multiplication [3]. Feel free to skip over this section and continue to section.

— 1.1. Rules Addition (Sum) —

Rules for adding any two real numbers ${a}$, ${b}$

1. The result of addition is called the sum.
2. The sum between ${a}$ and ${b}$ is indicated as:

$\displaystyle a+b$

3. If ${a}$ and ${b}$ are both positive, then the sum is positive.

$\displaystyle 4+3=7$

4. If ${a}$ and ${b}$ are both negative, then the sum is negative.

$\displaystyle -4+(-3)=-7$

5. If ${a}$ is positive and ${b}$ is negative and ${|a|>|b|}$, then the sum is positive.

$\displaystyle 4+(-3)=1$

6. If ${a}$ is positive and ${b}$ is negative and ${|a|<|b|}$, then the sum is negative.

$\displaystyle 3+(-4)=-1$

7. Addition is commutative: ${a+b=b+a}$.

$\displaystyle 4+3=3+4=7$

8. Addition is associative: ${a+(b+c)=(a+b)+c}$.

$\displaystyle 1+(2+4)=(1+2)+4=7$

9. ${a+(-b)=a-b}$.
10. ${a+(-a)=0}$; ${a+0=a}$.

— 1.2. Rules Multiplication (Product) —

Rules for multiplying any two real numbers ${a}$, ${b}$

1. The result of multiplication is called the product.
2. The product between ${a}$ and ${b}$ is indicated as:

$\displaystyle a\times b=a\cdot b=a(b)=(a)b=(a)(b)=ab$

3. If a and b have the same sign, then the product is positive.

$\displaystyle (4)(3)=12$

$\displaystyle (-4)(-3)=12$

4. If a and b have opposite signs, then the product is negative.

$\displaystyle (-4)(3)=-12$

$\displaystyle (4)(-3)=-12$

5. Distributive property: ${a(b+c)=ab+ac}$; ${a(b-c)=ab-ac}$.

$\displaystyle 4(2+1)=8+4=12$

$\displaystyle 4(2-1)=8-4=4$

6. Multiplication is commutative: ${ab=ba}$.

$\displaystyle (4)(3)=(3)(4)=12$

7. Multiplication is associative: ${a(bc)=(ab)c}$.

$\displaystyle 1(2\cdot6)=(1\cdot2)6=12$

8. The product of any real number and -1 is the additive inverse of the real number.

$\displaystyle (-1)a=-a$

$\displaystyle (-1)(-1)=-(-1)=1$

$\displaystyle (-a)(-b)=(-1\cdot a)(-1\cdot b)=(-1)(-1)(a)(b)=1(ab)=ab$

9. ${a\cdot1=1\cdot a=a}$; ${a\cdot0=0\cdot a=0}$.

— 2. Example —

Why is a negative times a negative a positive? Many people have trouble answering this question. While, most will not have an issue memorizing the rule. Understanding the reasoning for the rule is another matter. Many people try to come up with a visualization to help picture what is going on [1]. We shall use money to illustrate the concept. When going over the example try to think of a negative as a loss and a positive as a gain.

— 2.1. The Offer You Can’t Refuse —

Suppose your are bankrupt. You now own ${\0}$ dollars. You approach Mr. Mob Boss for a loan of ${\4,000}$ dollars. He does so with the condition that you pay him back 4 times that amount. You now have a debt of

$\displaystyle 4\cdot-\4,000=-\16,000$

See rule 1.2.4 for reference.

In other words, you owe Mr. Mob Boss ${4}$ debts of ${\4,000}$. You have lost ${\16,000}$ dollars. You take the ${\4000}$ and invest it all in starting an Italian restaurant. The problem is all your customers are members of La Cosa Nostra, who expect nothing less than a free meal with the finest wine.

Eventually realizing you will never break even, you approach Mr. Mob Boss to tell him the bad news. Unexpectedly, he is so happy about the service you provide that he decides to forgive your ${4}$ debts of ${\4,000}$!

$\displaystyle -4\cdot-\4,000=\16,000$

See rule 1.2.3 for reference.

You have gained ${\16,000}$ dollars, and, thus, you are back where you started…absolutely broke! However, try not to be so "negative". At least you have a job for the rest of your life.

References

[1] Dr. Math, Negative x negative = positive, available at http://mathforum.org/dr.math/faq/faq.negxneg.html.
[2] Ian Stewart, Letters to a young mathematician, first ed., Basic Books, 2006.
[3] Rong Yang, A-plus notes for algebra: With trigonometry and probability, third ed., A-Plus Notes Learning Center, 2000.