Wrongful Punishment was the Best Thing

When I was in 1st grade I suffered punished from a wrongful accusation, well, kind of.  And the punishment would land people in jail today.  In front of the entire 1st grade class I was spanked, but not spontaneously.  I was paraded to the front of the room and quite a spectacle was made of the ordeal.  Then I was sent to the principal’s office where I suffered a similar fate.  And at home, once again.

And while it is true, I did knock all of the lunch boxes off of the stand where they were stored, and they did spill open and they did make a big mess, it was an honest accident, no intent involved.  And I was very willing to clean all of it up, all of it!  The opportunity was not provided.  Instead, I bent over, knees straight, palms placed flat on a chair at the front of the room while the teacher drew back her arm, equipped with a wooden paddle, and brought it forward squarely on my six-year-old hind end, with all of my peers in audience.  Thrice over it happened.

It was spring and I had been spanked at school so frequently that my parents made a bargain for me, a bribe really.  If I could manage to not get spanked for an entire week then I could go out for ice cream on Friday.  And, being that it was spring, the local ice cream shop had my favorite ice cream in the whole world … peppermint ice cream!

Somehow I mustered the will-power to keep the tornado of energy that sprung forth from my body under wraps.  I sat properly, as instructed by the teacher.  And the teacher’s name, Mrs. Fortenberry, was pronounced clearly and accurately, all week.  I raised my hand at appropriate times, passed papers forward neatly and returned from recess with punctuality.  There was no cutting in line at lunch, no teasing other kids during class and the shenanigans that transpired when the teacher would turn around ceased to occur.

The staying power of six-year-olds is questionable, but I held this superb behavior through all of Monday, Tuesday, Wednesday, Thursday, and half of Friday.  Upon returning from lunch on Friday I went to place my lunch box on the stand and it slipped.  I had not been goofing off, running around, just an honest accident.  But it was as if I threw a basketball at them, they toppled like dominos … to this day I marvel at the physics involved in knocking over all of those lunch boxes in such a fashion.  Entropy is sometimes spontaneous in the presence of six year old boys!

My protests of innocence might as well have been mute.

That evening my family went out for ice cream.  I’m sure there was plenty of the delicious peppermint ice cream eaten.  I wouldn’t know, though, because I was left home by myself.

And while this may sound tragic, I realized something from this experience. I had my first epiphany.  I realized that nobody believed me with just cause.  They should not have believed me, my reputation was well earned.

From this unjust punishment I realized the power of reputation.  I realized that if I wanted to be believed in uncertain circumstances, when light spilled over me in a questionable fashion, that I needed to have a reputation of being honest.  The only way to establish such a reputation was through being honest.

My behavior did improve after this.  Perhaps not getting the ice cream was the best thing that ever happened to me.

Sometimes the best lessons are hard earned.

 

Sets of Numbers and the Problem with Zero Chapter 1 – Section 1

1.1 3

Sets of Numbers
and the
Problem with Zero and Division

We will begin with the various types of numbers called Real Numbers. Together, these numbers can be ordered and create a solid line, without gaps.

Ø  Natural Numbers: These are counting numbers, the smallest of which is 1. There is not a largest Natural Number.

Ø  Whole Numbers: All of the natural numbers and zero. Zero is the only number that is a Whole Number but not a Natural Number.

Ø  Integers: The integers are all of the Whole Numbers and their opposites. For example, the opposite of 11 is -11.

Ø  Rational Numbers: A Rational Number is a ratio of two integers. All of the integers, whole and natural numbers are rational.

o   Decimals that terminate or repeat (have patterns) are rational as they can be written as a ratio of integers.

Ø  Irrational Numbers: A number that cannot be written as a ratio of two integers is irrational. Famous examples are π, and the square root of a prime number (which will be discussed next).

Together these make up the Real Numbers. The name, Real, is a misnomer, leading people to conclude that the word real in this context has the same definition as used in daily language. That misconception is only strengthened when the Imaginary numbers are introduced, as the word imaginary here harkens back to a day when the nature of these numbers, and their practical use, was unknown.

Is zero rational?

A rational number is a number that is the ratio of two integers. Before we tackle the issues that arise from zero, let’s reframe how we think about rational numbers (fractions) and develop a different language for these to promote greater proficiency in Algebra and allow for greater ease in understanding how zero causes real problems with rational numbers. (If you understand the nature of what follows you do not have to memorize or remember the tricks, you just understand.)

Consider the fraction 8 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aaabaGaaGOmaaaaaaa@3785@ . You were likely taught to think of this fraction as division and would also likely be taught to ask the question, “How many times does two go into eight?” That is sufficient for this level of mathematics, but the Algebra ahead is seemingly more complicated, but by simply rephrasing the language we use to talk about fractions, we can expose the seemingly more complex as being the same level of difficulty.

Instead of asking, “How many times does two go into eight,” the better question is, “Two times what is eight?”

It is true that 8 2 =4, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aaabaGaaGOmaaaacqGH9aqpcaaI0aGaaiilaaaa@39F9@ because two times four is eight. Simply answer the question “Two times what is eight,” and you’ve found the answer.

This will come into play with Algebra when we begin reducing Algebraic Fractions (also called Rational Expressions) like:

9 x 2 3x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aGaamiEamaaCaaaleqabaGaaGOmaaaaaOqaaiaaiodacaWG4baa aaaa@3A74@ .

If you ask the question, “How many times does three x going into nine x squared,” you’ll likely be stuck, especially when the expressions become more complicated.

But asking, “three x times what is nine x squared,” is a little easier to answer.

9 x 2 3x =3x, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI5aGaamiEamaaCaaaleqabaGaaGOmaaaaaOqaaiaaiodacaWG4baa aiabg2da9iaaiodacaWG4bGaaiilaaaa@3DE4@ because 3x3x=9 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hacqGHflY1caaIZaGaamiEaiabg2da9iaaiMdacaWG4bWaaWbaaSqa beaacaaIYaaaaaaa@3F64@ .

There will be much more on reducing Algebraic Expressions later in this chapter. Let’s turn our attention to zero and how it “behaves” in with rational numbers.

Zero is an integer, and again, a rational number is a ratio of two integers. Consider the following:

5 0 0 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaGimaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8+aaSaaaeaacaaIWaaabaGaaGynaaaaaaa@5AFB@

The first expression asks, “Zero times what is five?”

The second expressions asks, “Five times what is zero?” (Again, phrase the question in this fashion to provide easier insight into the math.)

The product of zero and any number is zero. So, the answer to, “zero times what is five,” is … well, there is no answer. There is no number times zero that is five. There is not a number times zero that equals anything except zero. We say this is undefined, meaning, there is no definition for such a thing.

The second expression, “five times what is zero,” is zero. Five times zero is zero.

One of these two expressions is rational, the other is not a number at all. It does not just fail to fit within the Real Numbers, it fails to fit in with any number.

5 0  Not a Number 0 5 Rational MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaGimaaaacaaMc8UaaGPaVlaaykW7cqGHsgIRcaqGGaGa aeOtaiaab+gacaqG0bGaaeiiaiaabggacaqGGaGaaeOtaiaabwhaca qGTbGaaeOyaiaabwgacaqGYbGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGa aGimaaqaaiaaiwdaaaGaeyOKH4QaaeOuaiaabggacaqG0bGaaeyAai aab+gacaqGUbGaaeyyaiaabYgaaaa@7129@

 

 

 

Repeating Decimals Written as Fractions

Consider the fraction 1 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaG4maaaacaGGUaaaaa@3831@ This is a rational number because it is the ratio of two integers, 1 and 3. Yet, the decimal approximation of one-third is 0. 3 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaiodaaaaaaa@3831@ (the bar above the three means it is repeating infinitely).

Here is how to express a repeating decimal as a fraction. Let us begin with the number 0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaikdacaaI3aaaaaaa@38F1@ .

We don’t know what number, as a fraction is 0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaikdacaaI3aaaaaaa@38F1@ , so we will write the unknown x.

 

x=0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaicdacaGGUaWaa0aaaeaacaaIYaGaaG4naaaaaaa@3AF4@

Since 0. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaikdacaaI3aaaaaaa@38F1@ is repeating after the hundredths place, we will multiply both sides of the equation by 100.

(note, for 0.333333… we would multiply by 10, since the decimal repeats after the 10ths place, but we would multiply 0.457457457457…by 1,000 since it repeats after the thousandths place.)

 

100×x=0. 27 ¯ ×100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeG+aaaaaai vzKbWdbiaaigdacaaIWaGaaGimaiabgEna0+aacaWG4bGaeyypa0Ja aGimaiaac6cadaqdaaqaaiaaikdacaaI3aaaa8qacqGHxdaTcaaIXa GaaGimaiaaicdaaaa@45C9@

 

100x=27. 27 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIWaGaamiEaiabg2da9iaaikdacaaI3aGaaiOlamaanaaabaGa aGOmaiaaiEdaaaaaaa@3DE6@

 

The following step is done by a procedure learned with solving systems of equations, which will be covered later. (In fact, this procedure would be a great topic to review when systems of equations is learned.)

 

Subtract the first equation from the second.

 

 

 

Note: 27. 27 ¯ 0. 27 ¯ =27 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiE dacaGGUaWaa0aaaeaacaaIYaGaaG4naaaacqGHsislcaaIWaGaaiOl amaanaaabaGaaGOmaiaaiEdaaaGaeyypa0JaaGOmaiaaiEdaaaa@401E@

 

100x=27. 27 ¯ ( x=0. 27 ¯ ) _ 99x=27 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaadaa abaeqabaGaaGymaiaaicdacaaIWaGaamiEaiabg2da9iaaikdacaaI 3aGaaiOlamaanaaabaGaaGOmaiaaiEdaaaaabaGaeyOeI0YaaeWaae aacaWG4bGaeyypa0JaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaI WaGaaiOlamaanaaabaGaaGOmaiaaiEdaaaaacaGLOaGaayzkaaaaaa GaaGPaVdqaaiaaykW7caaMc8UaaGPaVlaaiMdacaaI5aGaamiEaiab g2da9iaaikdacaaI3aaaaaa@585F@

 

 

 

Divide both sides by 99 to solve for x.

 

Recall that x was originally defined as the fractional equivalent of the repeating decimal.

 

x= 27 99 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9maalaaabaGaaGOmaiaaiEdaaeaacaaI5aGaaGyoaaaaaaa@3B0D@

 

 

 

 

 

Practice Problems.

1.      Change the following into rational numbers:

a.       5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@36B6@

b.      0

c.       3 0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaaabaGaaGimaiaac6cacaaI0aaaaaaa@38EE@

d.      0. 23 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiaac6 cadaqdaaqaaiaaikdacaaIZaaaaaaa@38ED@

2.      Why is a the following called undefined: a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbaabaGaaGimaaaaaaa@37A7@ ?


3.      List all of the sets of numbers to which the following numbers belong:

a.       0 b. 9 c. -5 d. 5.37 9 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaac6 cacaaIZaGaaG4namaanaaabaGaaGyoaaaaaaa@39BA@ e. 5 π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaeqiWdahaaaaa@3883@ f. 5.47281…


4.      Can a rational number also be a whole number?



5.      What number is whole but not natural?

Exponents v Rational Expressions: The Most Difficult Topic to Teach in Algebra 1?

There are many topics in math that are difficult for students to grasp because either the concept itself is elusive and maybe they don’t have the foundation required, or the procedures are complicated and riddled with traps.  Algebraic Fractions or Rational Expressions, whichever name you prefer, is a great example of a topic that hits both of those.

Algebraic Fractions are intimidating from the get-go.  Students see fractions with algebraic expressions and their heads practically explode!  Where as exponents, are inviting and seemingly easy, at first anyway.  But as students get into more complicated problems involving Algebraic Fractions, the path becomes clearer.  To find a common denominator first find the LCM of the denominators, multiply each term, reduce and then combine like terms.  To finish it up, reduce if possible by seeing if a GCF exists between all terms.  If one exists, divide it out of all terms and done!

Once students get away from some of the basic properties and have to combine several ideas involving exponents together at the same time the students are hopelessly lost.  An expression with exponents that needs to be simplified that might involve negative bases with negative exponents that are written in the denominator, then combining like-bases with the properties of exponents all while trying to traverse the muddy lines separating the order of operations (PEMDAS) when sometimes the first three (P, E, and M) are completely interchangable because it’s all multiplication, where other times you must take care of the group first and the multiplication last.  It’s over-whelming and all comes back to my favorite harping point, the ability to read mathematics by understanding the meaning behind how it is written.

While I work very hard to build a strong conceptual foundation for my students, sometimes I start doing the framework of the house before the foundation has has time to set.  This is often the case with Algebraic Fractions.  But, if their procedural fluency is sufficient, they can often repair their weaknesses in the foundation, or never need to shore it up at all and suffer little, if any, from it.

There are other topics that can be taught with concept only, for example, graphing a linear equation by finding the intercepts.  If students know what intercepts are, know how to find their coordinates, and know that linear equations are equations whose solutions form a line, they can put all of that together on their own.

Exponents are different.  The basic properties of exponents can be taught without covering the “rules” at all.  By simply knowing that exponents are repeated multiplication, and writing things out, students can get quite a long way.  I’d suggest they can get through division like the problem below without using the “Negative Exponent Rule.”

Now this is not a practical way to go about simplifying an expression.  The rules, or properties, prevent students from having to reinvest the wheel, as it were.  However, if the rules are introduced too soon, before the foundation has set and hardened, then with exponents, there will be serious troubles ahead.  And yet, visualizing this method displayed above, without using the properties to obtain the answer by some procedure, allows one to visualize that there are two more x’s in the denominator and four more y’s in the numerator.

But without a measurable level of fluency with applying the rules of mathematics, simplifying expressions involving exponents quickly becomes insurmountable. Consider a problem with multiple brackets, rational exponents, negative bases and negative exponents, like the one below. This would be overwhelming to approach with only a conceptual understanding, but no applicable short cuts in the proverbial tool box. With exponents it is imperative that a balance between the ability to approach any problem from a conceptual framework or to approach the same problem from a procedural framework with equal accuracy, must be achieved.  Students with strong conceptual understanding but little practical procedural ability will struggle with a similar level of difficulty compared to students with little conceptual understanding but that are well versed in the “rules” of exponents.

Students and teachers are very accustomed to seeking and rewarding answer-getting techniques.  This is what makes exponents a wonderful topic to spend a lot of time on.  You can show students how focus on procedure alone is insufficient, and how multiple methods of finding solutions is a powerful tool because combined with a little experience clear paths to solutions unfold when there is no proper first step.

An example of a problem without a proper first step is below:

There are many, equally efficient, and mathematically correct ways to begin simplifying such an expression. Students that rely on procedures will struggle mightly with these types of problems because the first step depends on what you recognize first (conceptual understanding).  How to respond to what is first recognized depends on the procedural fluency of the student.

It is a great exercise to take a problem like this and have one student perform some operation of their choosing, then another to perform a second operation and so on, until the problem is simplified completely.

By exposing students to problems where the initial step is highly variable and the method of arriving at correct answers vary upon what is noticed by students, they can develop a sense that math works this way.

Students should not consider math to work as follows:  Well, this problem looks like the last problem we did and I used the quadratic formula last time, so let’s do that.

It is my intent that when teaching something like exponents (and radical expressions can be similar) I am trying to plant the seed of this idea that math can be approached from many points of view, and sometimes, the right answers are not so cut and dry.

I am currently working on a series covering exponents, square roots and rational expressions where both the needs of the student are addressed through tutorial/remediation videos, and instructional coaching, lessons and practice problems for teachers.   The videos will be posted here on this website, but also on my YouTube channel.  The lesson will serve to meet two callings.  First, students have conceptual understanding of the topics and procedural fluency, but both done in a way that promotes mathematical literacy.

If you are interested in such a treatment of exponents, done in a way to encourage to view mathematics in a different light entirely, you can subscribe to my newsletter, or visit my YouTube channel and subscribe there.  Either way, I’m finishing up the series on exponents soon and will post them when done.

Thank you for reading.  And please, if you found this to be informative or useful, spread the word.

 

Philip

The Case Against Point-Slope Formula

One of the biggest problems in public education, math in particular, is answer-getting.  And this is endemic, not the fault of students, teachers or administrators, legislators or parents … alone.  Students seek answer-getting methods and are rewarded for finding answers.  Teachers are given massive amount of curriculum to be covered in a year while at the same time deeper understanding is demanded.  It’s not any one person’s fault, nor is it one party’s fault.

Leadership is trying to fulfill expectations and avoid tripping over the strings attached to funding.  For example, graduation rates are a coveted measure of success of a school.  On the flip side, what constitutes the requirements for high school graduation are not closely monitored, or incentivized.  There is a lot of pressure to promote students through the ranks of education, starting in Kindergarten, even when it is not always in the best interest of the student.

I won’t even get into how curriculum calendars and pacing guides became en vogue, but they are a massive burden on teachers.  A teacher must decide to cover material on a predetermined schedule, and if they fall behind, they must hurry to catch up or face reprimand for not being on pace … This is complete insanity!  Learning takes time and that time cannot be compressed.  It cannot be compressed when students are cramming for tests (that aren’t strictly fact based), and it can’t be compressed during the school year either.

And yet, from the same mouths of the people who bestow the virtues of standardized testing comes the call for differentiation in education!  What’s the saying about those who live in glass houses shouldn’t throw stones?  I am working on a constructive solution to these types of things, but for now, I’ll do as my mother says and engage in silence on the topic as I have nothing nice to say.

All that said, all parties involved are well-intentioned. Yet, must traverse a jungle of pitfalls that land us in poor teaching.

There is one particular topic in Algebra that sticks around because it satisfies all of these pressures that, in the end, result in low-quality education.  It is easily taught, gets quick answers and can be tested easily on a multiple choice (standardized) test:  The Point Slope Formula.  Now, I taught this formula for years.  When I first heard its usefulness being called into question, when I first heard that it was a simple answer-getting technique, I bucked at the idea.  I mean, after all, I am a good teacher who has high standards and plenty of rigor.

And that’s one of the issues with change.  We are all part of the system, and while it is easy to break apart what everybody else is doing wrong, they’re part of the same system that produced you…look at yourself first.  Back to the topic at hand…

A typical problem would say, Given the point (2, 3) and the slope of -1, find the equation of the line.

Students would use the formula below and they’d plug in 2 for x and 3 for y and -1 for m.  Then they’d distribute and then solve for y.  Dang, if they mess up distributing or the one step of inverse operations required, BOOM, there’s a chance for remediation or reteach or whatever the fancy phrase at the time is.  

But none of that, the process or the reteach would ever touch the nature of the question at hand.  None of it would connect the question to the concept of linear equations and how graphs are related to equations, which means the students could not build upon that concept in the future.

The concept at play here is the nature of the relationship between dependent (x) and independent (y), how every point on a line is a solution to the equation, and perhaps even how slope describes the relationship between the dependent and independent variables.

If a student knows the following, they can arrive at a solution in a way that addresses practical issues at hand and without having to be taught yet another thing!

  1. The equation of a line can be written y = mx + b.
  2. The values of x and y are an ordered pair, which means they’re a solution to the equation we need.

Consider the following two tables where I show the concepts (related to linear equations) at play using the two methods discussed.  Then, I’ll show the simplicity of one method versus the other.  As is often the case, the method that approaches a problem from the concept has a simpler solution.

I do believe there is benefit from teaching and learning the point-slope formula, especially if it is taught when learning order of operations or how to distribute. But as a method of answer getting when learning about linear equations, I believe, it is inappropriate.

I hope this post caused you to pause and reflect upon your own teaching.  What other things do we teach kids that circumvent the concept at hand in order for a quick grab at an answer?  Leave me a comment, let me know your thoughts.

Thanks for reading.

 

How to Make Formative Assessments Powerful Learning Tools

How To Make Formative Assessments Powerful Learning Tools

 

I’d like to share with you one way to get students to engage in metacognition.  But before I do, let me explain why I believe this is one of the most powerful learning experiences a student can experience.

It is my opinion that if a formative assessment does not provide feedback to the students it is of little use.  In educational training, teacher-evaluation and professional development too much focus is placed on the teacher.  And yet, it is clear, that a motivated student will learn without a teacher. Sometimes, groups of students learn DESPITE bad teaching.  Students that seek understanding, in any topic, are successful.  A quiz or test is more than an evaluation, and in fact, rarely does a student perform in a way that is a surprise to the teacher.  We know who will get an A and who will be middle of the road and so on.  

A quiz should be a learning opportunity. I didn’t say, “A teaching moment,” with intent because learning doesn’t always come from teaching.  We can set up an opportunity before handing back graded quizzes and tests that will be a powerful learning experience for the students.  We just need to get them to think about what they did.

The issue here is, how do you get kids to learn from mistakes on their quizzes and tests?  I mean, they have years of experience doing the following:

  1. Quizzes are handed back.
  2. Kid says to another: What did you get?
  3. Then they say:  Let me see yours … they hold them up side by side seeing if all of the teacher’s marks match.
  4. “Mister, he got one marked wrong that I got right!”
    1. And of course you could ask, How do you know you are right?  Maybe I didn’t mark yours wrong.  Maybe I marked hers wrong on accident.  Maybe you’re both wrong!

Reviewing their own work and thinking about their understanding and performance is perhaps the single biggest learning opportunity that students have at their disposal.  We are remiss, terribly so, if we do not make full use of these opportunities.

But HOW do you get someone to think about their thinking?  You can’t really make them, can you?  It can’t be coerced, tricked or done under threat of punishment.  For someone to be willing to engage in metacongition, they must be intrinsically motivated.

Here’s one way I get students to engage in this.

Before handing back a quiz or test, I review a problem that was largely misunderstood.  Then, I have the students practice a similar problem on their own (from the quiz or test, if possible).  I walk around the room to check for understanding and then after a few moments, I allow them to help each other on the problem briefly.  I continue this as long as I feel is appropriate for the level of competency displayed on the test.

It is absolutely critical that this review is done when the students do not yet have their quizzes or tests back in their possession.

When this method of review is finished, I back their tests and instruct them to find the problems we reviewed as a class.  I ask them to figure out what they could’ve done differently on the test to improve their score.  

Without such a discussion they will often fail to realize how simple sign errors wreck their grades, or how a simple conceptual misunderstanding is causing all of their work to just … well, collapse.  You could encourage students to highlight mistakes and annotate their new understanding by the mistakes, or have them use small sticky notes to write down questions that are still confusing to them.  

This is a powerful technique, but like all methods, their effectiveness diminishes if done too frequently or too infrequently.  I would not try this method on a test where the class average was high.  I believe it is best reserved for those topics students generally struggle with.  I would advise against using this method for a summative assessment (end of unit, where the class will be moving on).

 

Success Develops Confidence

Education isn’t really about the subject being learned, or the specifics of the topic being practiced.  No, education is about changing who we are for the better by learning how to get more out of ourselves.  An education should change you, change you think, how you see the world and it should change how you carry yourself, for the better.

The following is a story of how facing challenges and experiencing success did just that.

Like a typical freshman girl, Cristina was sometimes awkward, shy, and on occasion, over-reacted to situations. And like most kids, she had more going on in her life than just school.  But, she had a great desire to be successful in the honors math class I taught, Cambridge Math.  However, the challenge was great, and likelihood of success…well, not so great.

See, the previous year, the first our school participated in the Cambridge program, not a single student passed the end of course examination (IGCSE).  In fact, only 8% of students in Arizona passed that year.  To further complicate things, I was now appointed as the new teacher for Cambridge and there was a huge learning curve ahead of me.

So there we were, Cristina and I, facing a difficult situation together.  I’d never taught any honors program and know the teacher that taught Cambridge before me is a quality teacher.  And Cristina, as well as the other students, had never faced a course like this.  At the end of their sophomore year they would take a pair of hand-written tests.  To prepare for the tests the students had two school years to learn everything we teach in Algebra 1 and Geometry, most of what is taught in Algebra 2, Probability and Trigonometry, as well as large portions of Statistics and a handful of other topics not usually taught in the US.  To make it more complicated, most of the test required complex thought and application of concepts in unpredictable, unteachable ways.  To have just 8% of students, and these are honors students, pass in the state of Arizona was alarming to all of us!

Cristina did not stand out as a particularly strong math student.  In fact, when speaking with her mother one day, her mother said, “Cristina’s going to do what Cristina is going to do.  However the day strikes her will determine how the day goes.”

She passed the first year, but not without tears and heartache as she received far lower grades than she ever got in middle school.  There was a lot of frustration and the decision to stay in Cambridge her sophomore year, or move to an easier regular class, was considered at some length.

During her sophomore year she became pretty inconsistent, often sabotaging her own efforts.  I believe she saw herself as a weak math student with little to no chance of success.  Often when we see ourselves in a particular way we unknowingly take steps to fulfill that expectation.  This was unfortunate in Cristina’s case as she’d sometimes lack the discipline to complete homework, and often when it was completed, it was done so in low quality…just to get it done, not to promote learning and to practice.

On one occasion in particular Cristina was become very frustrated with her lack of progress.  Her performance had been suffering, grade dropping and agitation was on the rise!  During class that day we were working on a complex problem, the type they’d see on the difficult portion of the end of course Cambridge exam.  Cristina wasn’t participating, not even working on her own.  I asked her to work and eventually she snapped at me, “Why do I have to do this?!?!”

She’s not a bad kid, and as I mentioned, she had some things going on outside of school that added to her stress.  But, the fact remains, she was sabotaging her own efforts with inconsistent work, poor work ethic and sometimes bad attitude.

Cristina, like many students, would often say things like, “Let’s just get this over with, I’m going to fail anyway.”  I believe those are defense mechanisms, designed to take the sting out of the potential failure.

I encouraged all of the students to try, without reservation.  If they try their best and fail, it’s a win because our best is plastic…it improves or diminishes depending on what we demand of ourselves.  I believe that students that try their best and come up short will over-come, they will succeed!

When test day finally arrived Cristina was a nervous wreck.  I was a nervous wreck.  The students took the tests and we mailed them overseas to Cambridge University to be graded.

And then we waited … and waited… May to August we waited.

The night before the grades we to be released I had nightmares about having to tell students like Cristina that she did not pass.

Cristina passed the Cambridge exams … she did something that over 90% of participating honors students in the state failed to do!  Not only did she pass, she smashed it!  Maybe I’m wrong, but the experience seemed to change her. I believe the success she experienced, not just my class, but her other Cambridge classes (equally difficult) gave her the background to KNOW, not just hope, that she was capable of such difficult tasks.

Cristina just graduated High School.  I spoke with her about writing this blog post about her, and she consented, hoping that sharing her story would embolden others to try their best to achieve their goals, with reckless abandon…swing for the fences, as it were.

Wednesday’s Why

There are many things in math that are just memorized, with little or no understanding of the meaning behind the scenes.  To help promote greater understanding, promote recall and accuracy, but most importantly, to empower people to be able to glean this deeper understanding behind the math by learning how to read the math, I am starting a video series on my YouTube channel called Wednesday's Why.   Every Wednesday I'll take a topic that is either largely misunderstood or just assumed to be true without any questioning and unpack it for you, show you how to read it so that you see what factors are at play.

This last week was the first episode and we tackled a tricky property of exponents.  The video is here below.

Summer School

I am teaching Algebra 1 for summer school this year, finally.  I’ve been teaching Geometry during summer school for a number of years, but prefer to teach Algebra 1.

…but … I’m taking a big risk in summer school this year.  Yup.  And there will be consequences if I fail.

But before I explain those consequences and the risk, let me set the stage.  The first idea is this:  In high school, even good high schools like where I work, there is an enormous amount of pressure on teachers to pass students.  The unintended result is that standards are lowered.  In math, and this is well articulated in nearly every TED Talk about the state of mathematics in education today, but students are taught these disparate procedures.  Students don’t learn concepts and thus cannot connect ideas or build upon past learning.  The end result of trying to make it easier by just showing kids how to arrive at an answer is that math becomes this enormous weight with seemingly thousands of things to memorize and recall.

That is tragic because the beauty of math, to me, is that you only need to understand a few things and those seemingly thousands of things just present themselves to you!

The second idea to consider is the population of students taking Algebra 1 in summer school.  The upperclassmen will have failed many times and will be jaded.  The freshmen will likely be behavior problems.  There will those that failed due to truancy and others still that failed because they’re simply lazy.  Then there will be the truly fearful students and the self-defeating students, those who never give themselves a chance.  (It’s easier to not really try and fail then really try and have to face failure without the out, Well, I never really tried.)   All of these kids have the aptitude to be successful in math, but getting them to realize it is where the art of teaching really comes into play.  The easier group is the very few who truly lack the aptitude in math, though it’s likely all students in summer school would identify themselves as belonging in this group!

The last thing to consider is that learning takes time and the time cannot be compressed.  Yet, summer school will be 11 days per semester, 7 hours of instruction time per day.  One day will be state testing and final exams.  So 10 days of class time.

I have set a goal within those 70 hours, in an environment where it is acceptable to lower standards a little bit, and with a group that would greatly resist pushing themselves.  I want all of the students to be truly proficient in Algebra 1, first semester.  I am going to try and teach them to be aggressive learners who challenge themselves and their understanding.  I want them to be introspective and reflect upon mistakes, beliefs and thinking.

In short, I am going to try and mold their thinking about math and education like I do with my honors students who take the Cambridge IGCSE test.  I will hold the standards high, I will not be dumbing down anything we cover, though I will be selective about the specific things we learn.

At first, students are going to struggle mightly with the idea that I will not be explaining everything to them, I will not be writing out steps.  They will struggle with the idea that their notes should be things they’ve realized, not just things I’ve written. I will be writing as little as possible and guiding them, with vigilant reminders to be actively engaged and so on.

If I am successful then the students will not only learn Algebra 1, but they’ll also recast the light in which they see themselves.  They will learn how to learn.

If I fail, they’ll fail and their bad mindsets will be reinforced by yet another bad experience in math class.  I take a lot of pride in the service I provide to students and this outcome would be completely unacceptable to me!

But, I think the reward is worth the risk.

A few specifics about how I’ll execute my plan … without a plan, remember, a goal is just a dream.

  1. The expectation of active engagement will be made explicit on day one.  (I’ll share the essence of this post with them.)
  2. I will provide accessible and engaging (I hope) support materials for them that focus on concept and show procedure as a consequence of properties of the concept.
  3. Organization:  Students will know the plan for the 11 days, and I will break each day’s activities down for them so they know exactly what to expect.
  4. Remediation plan:  Quizzes will be taken daily, short and sweet.  Students will grade these check-point type quizzes themselves and will be given a small amount of participation points for correct grading.  Homework will be fixing the errors made and completing a remediation assignment.

Here’s a map of what’s going to be covered, generally speaking:

Day 1:  Sets of numbers, prime numbers, LCM/GCF, and Percent problems

Day 2: Time Problems and the calculator, Algebraic Fractions (rational expressions), Order of Operations and function notation introduction.

Day 3:  Square Roots, Cube Roots and Exponents

Day 4: Test 1, Reading and Writing in Algebra, solving simple equations

Day 5: Inequalities, solving rational equations, variation

Day 6: Functions, graphs of various functions, function arithmetic and inverse functions

Day 7: Test 2, Linear Equations introduction, t-charts

Day 8:  Slope, intercepts, graphs of vertical and horizontal lines, slope-intercept form

Day 9: writing equations of lines, parallel and perpendicular lines, linear inequalities

Day 10: Systems of equations by graphing, substitution, elimination

Day 11: Review, Final Exam, AZ Merit

Once summer school begins I’ll be posting a daily vlog on my YouTube channel about how it is going, what I’ve tried and how the students have responded.  So, stayed tuned!

 

How to Save Time Grading

 

How to Grade Efficiently

and Promote Assignment Completion

 

Grading papers is one of the most time-consuming responsibilities of teaching.  Hours upon hours can be, I argue, wasted, pouring over daily homework assignments.  This article will discuss how to integrate awarding credit for daily assignments in a way that saves hours of time while increasing your awareness of student progress, increases student completion rates and better informs students regarding their progress in the subject.

This routine described here is a daily variety, not how I grade quizzes, tests or projects.  However, there are some tips that apply to recording those grades later in this article.

Let’s begin with a question: What is the purpose of homework?  For me, it’s practice needed for students to gain proficiency.  Homework is about trying things, working out how to struggle through difficult problems and making, and learning from, mistakes.  Without effective homework, students will not integrate their learning into a body of knowledge that they can draw upon for application or just recall.

The breadth of the purpose of homework and how that purpose is best served is beyond the scope of this article, but I would like to suggest that homework is something done in their notes, whenever possible.  The reason being is that notes are a receipt of their learning, to be reviewed in the future to help remember observations and important facts.

Overview of How It Works:

At the beginning of class, often before the bell rings, I begin walking around the classroom stamping homework that deserves full credit.  (What merits full credit is up to your discretion, but it should be a clear and consistent expectation, known to students.)  As I circle the room, I look for common mistakes, ask kids if they have questions or difficulties and make small talk.

Students that didn’t do, or complete, the homework have to answer for it on the spot!

Then, I simply mark those that did not receive credit for the homework on a student roster I keep on a clip board.  (For a video of how this works, visit the link here: )

Quick Notes:  This method has students ready for class because they have their notes.  They’ve also asked me questions if they had any, so I can begin with meaningful review.  I also have forced students that are remiss to account for their actions and done so in a way that applies positive peer pressure.  The scores are recorded by leaving blanks for completion and only marking those that do not get credit (which will be very few).

Credit:  I award full credit or zero credit when checking homework.  If a student attempted all problems, with evidence of attempt demonstrated by work shown and questions written, they get full credit.  Those that fail to receive full credit have the opportunity to reclaim 80% (the percentage is arbitrary but again needs to be consistent, clear and known by all), the students must see me during tutoring time by the Friday of the week of the assignment to show that they’ve fulfilled the expectation.  Students that did not attempt the homework can also see me during tutoring time (before or after school, not between class times or lunch), and receive partial credit.

But the rule of being due the Friday of the week assigned is big.  The purpose of homework is practice.  Without proper practice skills and knowledge are not developed.  Homework is not about compliance and fulfilling an expectation with a grade as a reward.  Students that are hustling to complete homework from two months prior are likely not promoting their understanding of current materials.  Plus, by having the time requirement applied to the homework policy, students are not enabled to fall too far behind.

The added bonus is that you will not be buried with make-up work the last week before grades are due to be reported!

Work to be Turned In:  If the nature of the work is not something that can be kept and must be turned in, have the students pass their work forward by row.  As you collect each row’s stack, count them.  If a row’s stack is incomplete, ask who in the row didn’t turn in the work.

If students can NOT fulfill the expectation and only receive a bad grade from it, and that reprimand comes well after the unwanted behavior, they will quite happily go along thinking nothing bad is going to happen.  Having to answer, publically, for their lack of work, especially when the vast majority will work, is a powerful deterrent!  Just as when checking the work of students and asking those who failed to complete for an explanation, this keeps them accountable and will increase the amount of students completing their work.

When collecting the papers, alternate the direction of the stacks and do not mix them up when grading.  This will allow you to quick return the papers after you’ve been done.  If it is a daily practice type of work turned in, I’d suggest awarding full or no credit and only recording, again on the printed class roster, those that were awarded no credit.

Recording Grades:  Whether you’ve collected daily practice or are carefully grading quizzes and tests, how you record those grades can either waste your time, or greatly reduce the amount of time spent.

By recording each grade as it is calculated by hand on the student roster it is quick and easy to transfer them to the computer.  This is a huge time-saving practice.  You don’t need to hunt on the computer screen for each student, and do so for each assignment.  When they’re recorded by hand, you can simply enter the column of numbers in the computer.  When the last name lines up with the last number that you entered, you know they’re all entered correctly.

By following this method, the data entry side of grading is done in a few moments of time instead of over hours, working through those stacks of papers, again!

Final Thoughts:  By looking at, and discussing, homework with students on an individual basis, very briefly, you gain insight into their progress.  They get a chance to ask questions.  Students that need a little bit of motivation receive it as an immediate consequence for poor behavior, rather than waiting until the end of the quarter, when a lot of pressure will be placed on you to help them bring up their grades.

This routine has proven to be a cornerstone of my classroom management.  It gives me a way to set the expectation that we are here to learn and that learning is done through work and reflection.  Students that need discipline receive it immediately and in a way they find uncomfortable, but it is done so with an invitation that guides them to the desired behavior (of completing their work).

 

How to Study!

Can you think of a thing that more often praised, promoted as virtuous and imperative to success, yet largely remains undefined than studying?  Well, I have four easy steps you can follow to structure your studying and give you great use of your time and will produce excellent outcomes both in the short term (for your quiz), but also improve retention of difficult and elusive topics.

  1.  Set aside time.  You can't rush learning.  I'd suggest starting at least a week before your big test.  Pick a time every night where the only thing you're doing is studying, but the time period doesn't need to be long.  Twenty to thirty minutes should be long enough, provided you study with focus and for several days consecutively.

    The time is required because learning is developing your brain...can't compress the time needed to make learning permanent.

  2. Create a study guide.  List all topics that you've covered in class that might be tested.  When finished, notate which are most difficult for you.  Briefly review the topics you feel are mastered and research the rest.

    To research use the internet, your book, notes, friends and teacher.  But be focused in your questions.  Know what it is you don't know so that your teacher can actually help you!

  3. Practice Problems:  From old tests, quizzes, and homework, create a practice test of your own.  Perform the problems under testing conditions.  If notes are not allowed, don't take practice with notes.

    Pay particular attention to problems you missed on old quizzes and tests.  Learn them, figure them out!  That's the point of studying anyway, right, to learn what you don't know?  Keep practicing until you're solid!

  4.  Nail the test!

If you set aside time and follow through you'll be rock solid.  You can't control what grade you'll get, but you will have taken care of the things within your control.  You'll be confident on test day and things will go well for you.

Thanks for reading.