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)

Divisibility Rules

A Math friend sent me info on divisibility rules. Figuring out if a number is divisible by 7 has a few steps -- 1. double the units (ones) digit
2. subtract it from the rest of the number
If the difference is divisible by 7, so is the original number.
Example:
343 -- take off the 3 and double it to 6
subtract 6 from the remaining numbers (34)
34 - 6 = 28 -- which is divisible by 7 -- therefore, 343 is divisble by 7.

Statistics and 'The Curve'

After using IQ as an example of a normally distributed data set, students wanted to clarify that the mean will not always be 100 and the standard deviation will not always be 15 -- ;)

We spoke about how the class average and standard deviation can greatly influence letter grades because how one does relative to everyone else is often how grades are determined. We used an example of

Student A: Test Grade 78 Class Average 69 Std Dev 5
Student B: Test Grade 86 Class Average 78 Std Dev 8

z score for A = (78-69)/5 = 1.8 A is 1.8 standard deviations above the mean
z score for B = (86-78)/8 = 1.625 B is 1.625 standard deviations above the mean

Therefore A did better than B relatively even though their actual numerical grade was less!

Boy, did I ever live by this in engineering school. Senior year, I got a 7 (3.5/50) on a Communications Systems Design exam that I actually studied for!!

TODOS CEO Lecture at Teachers College

Last night (Monday 2/26) was the Colloquim at TC -- as always, it was informative -- the CEO of TODOS Miriam Leira from UNC Charlotte spoke about Equity for All in Math. She had some great slides on the achievement gap and emphasized that all students need individual attention.

Exponent Rules

WARNING!!! Do NOT memorize these -- they easily get jumbled!!

Test makers know that students can easily mix these up!!
Encouraging students to write it out is the most useful!
(x^2)^3 = (x)(x) three times = (x)(x)(x)(x)(x)(x) = x^6!!

Divisbility Rules

The most important one is 3. If the digits in a number add up to 3 or a multiple of 3, then the number is divisible by 3. Some examples include 12, 51, 57, 111, 666, 1113.

If the number is divisible by 9, then the digits will add up to 9 or a multiple of 9. Plus if it is divisible by 9, then it must be divisible by 3 as 9 is composed of 3 x 3. Examples include 27, 333, 981, 1836, 111111111.

If the number is divisible by 3 and is even, then 3 and 2 must be factors and therefore the number is divisible by 6. Some examples include 666, 1836, 222, 3300.

Two Negatives Make a Positive -- I Don't Not Like Pizza

8 - - 12 = 8 + 12 = 20

8 - 12 = -4

Using a number line to illustrate these concepts is very helpful. Only put a few numbers on a number line -- include 0 and wherever the beginning point is (in this case 8). Then consider if your answer is more positive or negative than your beginning point (in this case 8).


The Negative Sign is the Most Common Mistake in Math!!

Either there is one too many negative or one too few. Check work carefully as this can be tricky -- expect to find mistakes!!

Also, the comparative can help tremendously here: -6 + 6 can not equal -6 + -6!!

What Do Students Need Help With?

Algebra
Generalizing from y = mx + b so that the result is an equation like y = 2x + 5.
Students benefit from being reminded that they are finding the equation that describes the entire line not just a point.
I like to ask them "How many points are on the line?" -- the answer I like the most is "Infinite" but "Too many to count" or "A real lot" are very acceptable answers.

It takes a while to see that (x-5)/(5-x) = -1!!


Calculus and Number Lines

The idea of limits and that inscribing an n-sided polygon in a circle, the more sides you have, the closer the area is to the area of the circle. So as n approaches infinity using a circle with a radius of 1, the area approaches pi (3.14159265....).

Secant and tangent lines have nothing to do with trig functions secant and tangent!!

A really cool problem was f(x) = [x] + [-x] which involves the greatest integer function.
The TI-84 function for greatest integer function is Int. The greatest integer function is best demonstrated with a number line and asking the student what was the last number that you passed. If you are exactly at an integer, then the answer is that number.
For example, [2.5] = 2 we passed 2 on the way to 2.5.
and [4] = 4
But [-2.5] = -3 we passed -3 on the way to -.25 (but we have not yet passed -2!!)
The cool thing about this problem is that for most x's, such as x = 2.5,
f(x) = [2.5] + [-2.5] which comes out to
f(x) = 2 + -3 = -1
The exception is at exactly an integer -- f(4) = [4] + [-4] = 4 + -4 = 0.
This graph looks like a straight line like y = -1, however at each integer value, there is a discontinuity, as y jumps to 0 at -3, -2, -1, 0, 1, 2, 3, etc.
This is best seen by setting xmin = 0.99 and xmax = 1.01 on the window menu -- at x = 1 there is a dot on the x axis!!

Trigonometry

The Law of Cosines is really the Pythagorean Theorem with a little extra.
We have always been using it except that if we can use the Pythagorean Theorem then we must have a right triangle!
If we rewrite a^2 + b^2 = c^2 as
c^2 = a^2 + b^2 - 2ab cos C (cos 90 = 0)
so we are used to seeing it as
c^2 = a^2 + b^2


General Math (Percents)

I love to work with students on 'trick questions' .

For example, you buy a stock at $100 -- it goes up 10% and then down 10%, what is its final price?

You buy a stock for $40 -- it goes up 50% and then down 50%, what is its final price?

And, you buy a stock for $60 -- it goes up 100% and down 100%, what is its final price?

Math is a metaphor for life!!

• Learning how to recover from mistakes

• Not giving up even when the going gets tough

• Problem solving is useful in any profession or endeavor

Math Education is such a hot topic. Last night, Richard Mills, NYS Comissioner on Education was on PBS on NY Learns. The teacher certification process was discussed and what they still do not realize is that the math ed requirements for a certificate leave out engineers and finance people who might actually take the punge into teaching. The Math content that is required is Math that engineers have not taken but have little to do with High School Math. Why not reverse engineer what courses teachers should take by studying the content of the Regents, SAT and AP exams?

Math Confidence has had a busy winter with conferences, private students and the College.

Thanks to my brother Adam I now have a blog. Stay tuned for entries!!