Because I'm never sure if the 400 or so subscribers to the MathNotations feed ever see revisions I've made to a post (which I do frequently!), I'm letting everyone know that the latest post displaying the calculus videos regarding the 'boring a hole through a solid sphere' problem has been updated a few times. I've restated the problem originally posted in the beginning of 2008 and I have annotated the video clips so that you can decide which ones you want to view. It is very long and may be at times as 'boring' as the title of the problem!! Hey, folks, it's a learning process for me and eventually I'll just post links on my blog to these videos which I will upload to YouTube.
Look for fully developed K-14 math investigations, math challenges, and standardized test practice both for SATs and Common Core assessments. The emphasis will always be on maintaining a balance between skills proficiency and developing conceptual understanding in mathematics. There will also be dialogue on key issues in mathematics education. Use Blogger Contact Form at right for personal communication. See top sidebar to subscribe to solutions of Daily Twitter Problems.
Contact Form
Welcome to MathNotations
I hope you enjoy the math challenges and discussion on this site. If you have any questions, you can contact me using the new contact form in the sidebar.
Proper Attribution
When using or adapting any of the materials on this site which I have personally developed, please include proper attribution by indicating that "This investigation (activity, problem) was authored by Dave Marain." Please read and adhere to the Creative Commons License below. Thank you.
Posts
Translate
MathNotations Ranking...
Link here to see MathNotations' ranking and other math sites/blogs as well.
Feedjit
Top Ten Edu-Blog 2007 - WaPo
SAT Math Tips
READ/LEARN ALL OF THE FOLLOWING!
ZERO IS A 'WEIRDO'!
(W)hole
(E)ven
(I)nteger
(R)Rational/Real
(DO) Cannot Divide by O!
BUT Zero is NOT Positive and NOT Negative!
POSITIVE INTEGERS start from 1
PRIMES start from 2 (not 1)
INTEGERS can be NEGative (and zero!) as well as positive
MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.
Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?
Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.
Step 2: Identify the values which are given
First term or a(1) = 3
Common difference or d = 4
Number of terms or position of desired term or n = 100
Step 3: Substitute into formula and solve
a(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399
Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!
Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!
ZERO IS A 'WEIRDO'!
(W)hole
(E)ven
(I)nteger
(R)Rational/Real
(DO) Cannot Divide by O!
BUT Zero is NOT Positive and NOT Negative!
POSITIVE INTEGERS start from 1
PRIMES start from 2 (not 1)
INTEGERS can be NEGative (and zero!) as well as positive
MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.
Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?
Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.
Step 2: Identify the values which are given
First term or a(1) = 3
Common difference or d = 4
Number of terms or position of desired term or n = 100
Step 3: Substitute into formula and solve
a(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399
Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!
Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!
Popular Posts
-
The English language has many confusing phrases but " as many as " IMO has blighted the youth of many an algebra student. Perhap...
-
Don't forget to submit the name of our Mystery Mathematician. Contest ends around 4-15-08. Thus far, only one correct submission (emaile... -
OVERVIEW Hundreds of math posts over the last 6+ years and my most popular post is still by far "There are twice as many girls as boy...
-
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided ...
-
Update: (1) See the visualization for the difference of squares posted on 1-9-08. (2) Read the comments in this post for considerable clari...
-
Many readers of this post objected to the scientific fallacy in asking for per cent change in temperature even if the arithmetic is correct....
-
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided ... -
Use new contact form at top of right sidebar to contact me directly. Note: If you'd like detailed solutions and Common Core strategies ...
-
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided ...
-
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided ...
Global Visitors
Click on Globe for enlargement.
NEW!! MathNotations Fan (Like) Page on Facebook
A Useful and Fascinating Link
Excellent Site for Finding The Data
Highly recommended extensive data comparison of academic institutions' libraries. Many other comparisons of all types are available too, including blogs.
Academic Libraries
Academic Libraries
Search This Blog
Followers
Ranking by Blogged.Com
MathNotations Interviews
Amazing Fraction-Decimal Calculator!
1/9801 = 0.00010203040506... has to start repeating at some point but how far will the pattern continue? Link here to find out and much more!
ADP/Achieve Links (Algebra I & II EOC Exams)
THIRD MATHNOTATIONS ONLINE MATH CONTEST IS OVER
RESULTS OF SECOND MATHNOTATIONS ONLINE CONTEST!
CLICK HERE FOR DETAILS!
Results of MathNotations First Math Contest!
Here's the link!
MathNEXUS: Mathematics Portal
Click here for a weekly update of this fabulous site!
Blogs I Read! (Latest Updates)
BlogRoll
- Ars Mathematica
- Carnival of Mathematics
- Continuities
- Drat These Greeks!
- dy/dan
- e's ponderings
- Geomblog
- Good Math, Bad Math
- Homeschool Math Blog
- jd2718
- Joanne Jacobs
- john kemeny
- kitchen table math
- Let's play math!
- Math Concepts Explained
- Mathematics Weblog
- MathMom
- MathNEXUS
- Meeyauw
- Polymathematics
- squareCircleZ
- Text Savvy
- Unapologetic Mathematician
- Vlorbik on Math Ed
- Wild About Math
Blog Archive
-
▼
2008
(155)
-
▼
December
(21)
- 'CHARMED ERA' = RADEMACHER - Our MathAnagram final...
- A Holiday Riddle: What do you call solving an equa...
- A Different 'View' of Sums of Cubes? An Algebraic ...
- Updates...
- Boring Hole in Sphere Calc Video - Finally!!
- Get Ready for Happy 41*7^2
- The Number Warrior and the Mysterious Minds of Stu...
- New Feature! Blogs I Read Daily (Latest Updates)
- The Passing of Kyosito Ito - Farewell to a Modest ...
- A Holiday Geometry Gift -- All Rolled Up For You!
- A Traditional Algebra 2 Rational Equation for the ...
- Santa Knows Linear Programming!
- WSJ Article Calls for National Standards, Complete...
- Teaching Probabilities and Strategies Via Games!
- Instructional Strategy Series for Middle School an...
- A Different "Approach" to 0.99999999...??
- Odds and Evens - FEATURED BLOGS and Update for Dec...
- FRACTIONS, FRACTIONS, FRACTIONS EVERYWHERE - A MID...
- Geometry Problem Requiring Critical Reading & Thin...
- The Product of a 40-Digit Integer and a 60-Digit I...
- Just Another SAT-Type Combinatorial Problem
-
▼
December
(21)
Labels
- 0.999...=1 (1)
- 1-significant digit estimation (1)
- 3-4-5 triangles (3)
- 3-dimensional geometry (2)
- 3-dimensional graphing (1)
- 30-60-90 (3)
- 6/pi^2 (1)
- A Class Apart (3)
- absolute value (2)
- ac-method of factoring (2)
- achieve (10)
- activities (3)
- activity (1)
- ADP (1)
- ADP Algebra 1/2 questions (1)
- ADP Algebra 2 questions (2)
- advanced algebra (27)
- advice for parents (1)
- age problem (1)
- Alec Klein (4)
- algebra (38)
- Algebra 1 end of course test (2)
- algebra 1 standards (1)
- algebra 2 (33)
- algebra 2 end of course test (6)
- algebra 2 standards (4)
- algebra for all (1)
- Algebra I Standards (1)
- algebra sense (1)
- algebra standards (1)
- algorithms (4)
- alphametic (1)
- altitude on hypotenuse (1)
- AM-GM Inequality (2)
- AMC (1)
- american diploma project (8)
- amortization (2)
- anagram (2)
- AP Calculus (1)
- AP Courses/Policies (1)
- application (1)
- approximation (2)
- archimedes (1)
- area (9)
- arithmetic sequence (5)
- arithmetic-geometric mean inequality (1)
- arrangements (1)
- assessment (3)
- asymptotes (1)
- average rates (2)
- averages (4)
- balanced view (3)
- binary representation (1)
- Bingo (1)
- binomial probability (2)
- book review (4)
- calculus (18)
- careless errors (1)
- Carnegie Report (1)
- carnival of mathematics (14)
- Cauchy (1)
- CCSS (1)
- CCSSM (1)
- chain rule (2)
- Challenge Problem (1)
- chaos (1)
- chess (1)
- children's anecdotes (1)
- circle problems (1)
- circles (19)
- clock problems (2)
- coin problem (1)
- coincidence (1)
- combinatorial math (18)
- Common Core (9)
- common math errors (1)
- completing the square (1)
- complex numbers (3)
- compound interest (4)
- compound probability (2)
- concept (2)
- conceptual understanding (3)
- cone (3)
- cone in sphere (2)
- congruences (1)
- conjecture (1)
- conjugates (1)
- connections (2)
- contest (16)
- coomon core (1)
- coordinate problems (13)
- core curriculum standards (2)
- counting problems (8)
- critical thinking (2)
- cubes (2)
- curriculum (6)
- cylinders (1)
- dedication (2)
- deductive reasoning (2)
- Desmos (3)
- developing thinking skills (1)
- differentiation (1)
- digit problems (5)
- Diophantine equation (2)
- direct instruction (1)
- direct variation (1)
- discovery learning (2)
- discrete math (5)
- dissection (1)
- distribution (1)
- distributive property (1)
- divisibility (4)
- division algorithm (2)
- division concept (1)
- dodecagon (1)
- early years (1)
- education (1)
- educational philosophy (1)
- egyptian fractions (1)
- elections (1)
- ellipse (1)
- Escalante (2)
- estimation (2)
- Euler (1)
- Experimental mathematics (1)
- explain (1)
- Exploration (1)
- explorations (4)
- exponential function (4)
- exponents (10)
- factoring (3)
- factoring by grouping (1)
- factoring quadratic trinomials (1)
- factors (8)
- fallacies (1)
- famous mathematicians (6)
- featured blogs (2)
- financial math (2)
- fractions (7)
- Friday the 13th (1)
- fun problems (1)
- functional notation (1)
- functional relationships (2)
- functions (8)
- games (3)
- Gauss Circle Problem (1)
- gcf (1)
- generalization (2)
- generalized means (1)
- geom (1)
- geometric sequence (7)
- geometry (79)
- gifted education (6)
- Goldbach Conjecture (1)
- golden ratio (1)
- golden triangle (1)
- grandson (1)
- graphing calculator (2)
- graphs (1)
- growth rate (1)
- harmonic series (1)
- hens laying eggs (1)
- heuristics (1)
- higher-order questions (1)
- higher-order thinking (2)
- history of mathematics (15)
- hole in sphere problem (1)
- humor (2)
- hyperreals (1)
- implicit differentiation (2)
- improper fractions (1)
- improper integral (1)
- indirect proof (3)
- inequalities (1)
- infinite geometric series (1)
- infinite repeating decimal (2)
- infinite series (1)
- inscribed (3)
- inscribed rectangle (2)
- inspiration (2)
- instructional strategies (14)
- Instructional Strategy Series (5)
- integers (1)
- integration (1)
- interview (7)
- inverse variation (1)
- Inverses (1)
- Investigation (1)
- investigations (67)
- issues in education (3)
- issues in math education (5)
- iteration (1)
- jokes (1)
- lattice points (2)
- law of cosines (1)
- lcm (1)
- learning (1)
- learning styles (1)
- limitations of technology (1)
- limits (4)
- linear algebra (1)
- linear equation (1)
- linear function (1)
- linear inequality (1)
- linear programming (1)
- lists (1)
- locus (1)
- logic (12)
- Lynn Arthur Steen (2)
- MAA (1)
- mandelbrot (1)
- Martin Gardner (1)
- Math (1)
- math anxiety (1)
- math challenge (23)
- math club (1)
- math contest (3)
- math contest problems (18)
- Math Education (1)
- math humor (7)
- math instruction (1)
- math mnemonics (3)
- math signs (1)
- math software (1)
- math strategies (1)
- math teachers (1)
- math tricks (2)
- math videos (6)
- math wars (5)
- mathanagram (4)
- mathematical practices (2)
- mathematicians (2)
- MathNotations Contest (12)
- mathnotationsvids (4)
- matrices (1)
- maxima/minima problem (4)
- mean (4)
- median (1)
- median (geometry) (1)
- mental math (3)
- method of shells (1)
- middle school (49)
- middle school math (10)
- mixed numerals (2)
- modular arithmetic (1)
- more (28)
- mortgage (3)
- multidigt multiplication (2)
- multiple representation (2)
- multiples (1)
- multiplication concept (1)
- multiplication of probabilities (1)
- multiplication principle (8)
- multiplication tables (1)
- mystery mathematician (19)
- Name that Mathematician (3)
- national math curriculum (4)
- national math curriculum england (1)
- national math panel (8)
- national standards (6)
- nested radicals (1)
- new year (1)
- nine (1)
- NJ Math standards (1)
- number sense (2)
- number theory (14)
- number tricks (1)
- numerical analysis (1)
- odd/even problems (1)
- odds and evens (2)
- online math contest (4)
- open-ended (4)
- optimization (4)
- organized list (1)
- overlapping circles (1)
- packing (1)
- packing problems (1)
- paper-folding (origami) (1)
- parabolas (8)
- paradox (2)
- parallelogram (1)
- parameter (2)
- parametric (2)
- partitions (1)
- patterns (17)
- Paul Cohen (1)
- pearson (2)
- pedagogy (10)
- percent (10)
- percent change (1)
- percent discount (1)
- percent increase problem (3)
- percent word problem (10)
- percent word problems (1)
- performance (1)
- periodic behavior (1)
- perpendicular lines (1)
- personal philosophy (5)
- Philosophy of Mathematics (1)
- pi (10)
- pi day (10)
- pi poem (1)
- pi^2/6 (1)
- Pick's Theorem (1)
- piecewise function (1)
- place value (1)
- placement test (2)
- podcast (1)
- poll (2)
- popularizations of math (1)
- Power function (1)
- power of a point (1)
- power series (1)
- powers of 2 (1)
- prealgebra (3)
- precalculus (12)
- pressure (1)
- prime factorization (1)
- primes (16)
- probability (13)
- problem of the day (1)
- problem-solving (4)
- Problems of the Day (1)
- Prof. Lynn Arthur Steen (1)
- projectile motion (1)
- proof (10)
- proof without words (1)
- proportions (2)
- PSAT (3)
- purpose of blog (1)
- puzzle (8)
- puzzles (1)
- pyramid (2)
- quadratic (3)
- quadratic function (5)
- quadratic systems (1)
- quadratic trinomials (1)
- quadrilaterals (4)
- quotes (1)
- radical equations (2)
- radicals (5)
- random triangle (1)
- rate-time-distance problems (2)
- rational equation (1)
- ratios (15)
- ratios in geometry (1)
- real number system (1)
- real-world applications (1)
- reasoning (9)
- rectangles (1)
- recursion (4)
- recursive functions (4)
- recursively defined sequences (1)
- reflection (1)
- reform (2)
- regular polygons (1)
- related rates (1)
- relatively prime (2)
- released items (1)
- remainders (5)
- research (1)
- rhombus (1)
- riddle (2)
- right triangles (2)
- rotations (1)
- Rule of Four (3)
- SAT strategies (21)
- SAT-type problems (87)
- scientific notation (2)
- secant-tangent power theorem (1)
- sequences (4)
- series (3)
- sieve (1)
- similar (1)
- similar triangles (8)
- similarity (1)
- Singapore Math (4)
- skills practice (1)
- slope (2)
- slopes (1)
- software review (1)
- spatial sense (10)
- sphere (1)
- square dissection (1)
- squares (6)
- standardized assessment (3)
- standardized tests (6)
- standards (1)
- starting the year (1)
- statistics (2)
- Stuyvesant HS (3)
- subtraction (1)
- sum-difference identities (2)
- sums of consecutive positive integers (2)
- Super Bowl (1)
- survey (3)
- susan ohanian (1)
- symmetry (2)
- systematic counting (2)
- systems of equations (1)
- tangent circles (2)
- tangents (11)
- Taylor Mali (1)
- tc'sTotal Challenges (2)
- teaching (3)
- teaching for understanding (1)
- technology (1)
- test construction (1)
- testing (2)
- Texify (1)
- textbooks (1)
- Thanksgiving message (1)
- theory of equations (2)
- TI program (1)
- tiling patterns (1)
- totally clueless challenge (1)
- triangle inequality (1)
- triangles (3)
- triangular numbers (1)
- tribute (2)
- tribute to teachers (1)
- trigonometry (5)
- twitter (2)
- twitter problem of the day (2)
- two trains in the tunnel classic (1)
- u-substitution (1)
- Unions (1)
- unit fractions (1)
- update (40)
- variation (1)
- Vedic Maths (1)
- verbal phrases (1)
- video lesson (8)
- video review (1)
- volume (3)
- warmup (14)
- weighted averages (1)
- winning strategies (2)
- word problems (2)
- work problems (1)
- writing in math (1)
- XMAS problem (1)
About Me
- Dave Marain
- NJ, United States
- Recently retired math educator and Supervisor of Mathematics; 30 years experience as an Advanced Placement Calculus (BC) teacher; Former Author of Math Teachers of New Jersey Annual HS Math Contest; Former K-5 Chair of New Jersey Math Content Standards and Curriculum Frameworks; Former member of Math Item Review Committee for New Jersey High School Proficiency Assessment; Experienced SAT Math Instructor and author of SAT materials; speaker at many regional and national math conferences
No comments:
Post a Comment