Middle School Math is the Glue That Holds Everything Together

The middle school merry-go-round: Fractions, Decimlas, Percents

Which is bigger? .02 OR .059

As soon as the third decimal place is added, the difficulty goes up by a lot! When students compare and contrast these numbers (as opposed to learning them in isolation), they can find order and beauty in decimals.

Undoing Common Student Mistakes and Leading by Example

This is where I spend most of my time with students. When students know where the traps are they learn so much especially after they have made the mistake a few times!!

How many times did it take me?? When I worked as an engineer, it took me a while to learn that if the oscilloscope shows no sign of life it is likely the wrong channel has been chosen or the wrong probe has been picked up. I probably made this mistake about once a month in the first year I worked as an engineer. It stumped me enough times that eventually I learned to check it!!

This is a valuable lesson for Math, for life and for engineers ;)

After discussing the slope of a linear line, one Math 102 student asked “How do you find the slope of a parabola?” – led me right to where I was planning to go ;)
I showed them the x y table for a linear and a quadratic and they could totally see how they differed. Next, we will discuss the algebra involved in finding the slope – the first derivative – and apply it to the parabola.

Worked with a student last night on the unit circle, cos goes with x and sin goes with y – it is alphabetical!!

Have also seen lots of logarithms lately including using logs to solve radioactivity and interest/financial problems. Using log 100 = 2 simplifies the procedure (and therefore concept) to the student. Teaching by example (rather than by concept) gets student to understand and remember the material!!

The TI-84 -- always something new to learn

At the Ten County Math conference last week, I learned how to input piecewise functions into the TI-84 in a precalculus session.

This week, worked with 2 students taking the first derivative of sqrt x -- this requires a leap of faith that delta x (or h) really does go to zero. The calculus is so algebra-dependent -- expressing sqrt x as x to the one-half is necessary to differentiate it. Interpreting the power of the derivative of sqrt x (which has x to the -ve 1/2 in it) again requires that algebra skill!!

TI is introducing a new product called N-Spire which I will get to see next month at the ATMNYC's minconference.

Related Rates

These calculus problems can be a lot of fun -- ok, I am the Math Lady.

Found an awesome web site:
http://www2.scc-fl.com/lvosbury/CalculusI_Folder/RelatedRateProblems.htm
The best demonstration of the concept is the balloon problem which shows that the rate of change in volume changes depending on how full the balloon is -- the two rates are related ;)

Strong algebra skills are a true asset here as some of the solving involves negative and fractional exponents and dividing by fractions.

How Do You Say "1/4"?

1/4 to me is "one-quarter" and to most students, it seems to be "one-fourth".
A fifth-grader that I work with made me realize that I say "one-quarter" -- maybe from working on Wall Street. Parents and teachers should keep this in mind ;)


Borrowing with Fractions = Rewriting

Fractions are often seen as having their own rules -- which is true, however, subtracting one mixed number from another involves the concept of 'borrowing'. Because borrowing seems so mechanical, students may not analyze the process. If students see borrowing as 'rewriting', then the mixed fraction rewriting will make more sense. For example, 8 - 3 1/2 -- student will often make a mistake and get an answer of 5 1/2.
If they rewrite the 8 as
7+1
and then again as 7 + 2/2, then can then perform this operation with much greater ease and confidence.

Non-Intuitive Answers

Multiplication of Fractions: Fractions get smaller and students are used to multiplication answers (products) being larger than whatever they started with. Mathematics Teaching in the Middle School has an excellent article update ("The Future of Fractions" was first published in 1978) by Zalman Usiskin from the University of Chicago -- the driving force behind the K-6 Everyday Math curriculum.

One of the features in the article was a NAEP (National Assessment of Educational Progress) items from 1978 for thirteen-year-olds.

Estimate the answer to: 12/13 + 7/8
You will not have time to solve the problem using paper and pencil (let alone a calculator!!)

The choices given -- and percents responding were

1 (7%)
2 (24%)
19 (28%)
21 (27%)
"I don't know" (14%)

Fractions have always been a sore spot for many and it is that foundational upper elementary/lower middle school Math that enables students to succeed in high school and beyond. It also contributes greatly to Math confidence or lack thereof.

The Unit Circle and "All Students Take Calculus"

Trigonometry is like a workout for the brain ;)
Some trig problems are posed like this:
sin A = 8/17 and cos A <0, find tan A.
How to start?

Where is sin positive and cos negative? Sin is +ve (positive) in Quadrants I and II. Cos is positive in Quadrant I so we must be in Quadrant II. The Quadrants are labelled with Roman numerals in a counterclockwise fashion starting with the most popular and +ve Quadrant #1.

In Quadrant II, sin is +ve but cos is -ve -- this results in a -ve tan since tan = sin/cos.

Tan = opp/adj

In Quadrant 2, we draw the right triangle of 8, 15, 17. Since sin A is 8/17 and the angle is in Quadrant II, we know that the y value (vertical part of the triangle) is positive. This makes the x value have to be -15. We also know this because in Quadrant II on a regular x, y coordinate plane, x is -ve while y is +ve.

From the angle, since tan = opp/adj and opp = 8 and adj = -15, then tan A = -8/15.

Note: We do not need to use a calculator for this!! (nor should we as the calculator will find the Quadrant I angle (less than 90) that pertains to sin A = 8/17 rather than the second angle value (between 90 and 180) where sin A also is equal to 8/17 but cos A <0.

Law of Sines and The Law of Cosines

The word best associated with the Law of Sines is PROPORTIONALITY

a b c
___ = ______ = ______

sin A sin B sin C


It can also be flipped upside down (or right side up if you prefer) to read:

sin A sin B sin C
_____ = ______ = ______

a b c


By the way, you can use just two out of the three at any time ;)



Law of Cosines

Please see the posting from Thursday 2/22/07 re: Trigonometry

The Main Thing About Logs -- Use Examples Not Formulas!!

Logarithms, like most topics in Math, are best learned and taught by example rather than formula. The log rules work like exponent rules so be wicked careful!!

If students know that log10100 = 2, they can see that 10^2 = 100. They can then apply this knowledge to any other log problem that comes up. Log 'rules' do not make sense!!
Just use thinking and analysis to remember the rules:

log101000 = 3
log10100 = 2
log1010 = 1

This is because logs are related to exponents!!

log10 (10^3) = 3

and

log10(10^2) = 2

and

log10(10^1) = 1

Therefore log 1000 = log 100 + log 10
(here you can build and see that 100*10 = 1000 and that their logs are related additively rather than multiplicatively)