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.
Tuesday, February 27, 2007
Friday, February 23, 2007
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!!
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!!
Thursday, February 22, 2007
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?
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 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?
Subscribe to:
Posts (Atom)