Sequences are fun to learn and not really that difficult.
There are many similarities between arithmetic and geometric sequences, so
learn both together.
Enjoy !!!!!
From Mthcounts Mini: Sequences and Series
Easier concepts:
Sequences
Arithmetic sequence/determine the nth term
Arithmetic and geometric sequences
Mathcounts strategies : review some sums
Note : Don't just memorize, but really understand the concepts.
Harder concepts:
Sum and Average of An Evenly Space
Relationship between arithmetic sequences, mean and median
Sequences, series and patterns
Some Common Sums
Sunday, December 10, 2023
Thursday, November 23, 2023
A Skill for the 21st Century: Problem Solving by Richard Rusczyk
Does our approach to teaching math fail even the smartest kids ?
Quotes from that article "According to research from the University of California, Los Angeles, as many as 60 percent of all college students who intend to study a STEM (science, technology, engineering, math) subject end up transferring out. In an era when politicians and educators are beside themselves with worry over American students’ lagging math and science scores compared to the whiz kids of Shanghai and Japan, this attrition trend so troubles experts it has spawned an entire field of research on “STEM drop-out,” citing reasons from gender and race to GPAs and peer relationships."
A Skill for the 21st Century: Problem Solving by Richard Rusczyk, founder of "Art of Problem Solving".
Top 10 Skills We Wish Were Taught at School, But Usually Aren't
from Lifehacker
Quotes from that article "According to research from the University of California, Los Angeles, as many as 60 percent of all college students who intend to study a STEM (science, technology, engineering, math) subject end up transferring out. In an era when politicians and educators are beside themselves with worry over American students’ lagging math and science scores compared to the whiz kids of Shanghai and Japan, this attrition trend so troubles experts it has spawned an entire field of research on “STEM drop-out,” citing reasons from gender and race to GPAs and peer relationships."
A Skill for the 21st Century: Problem Solving by Richard Rusczyk, founder of "Art of Problem Solving".
Top 10 Skills We Wish Were Taught at School, But Usually Aren't
from Lifehacker
Wednesday, October 11, 2023
2023/ 2024 Mathcounts, AMCs, AIMEs Competition Preparation Strategies
Hi, Thanks for visiting my blog.
Don't forget other equally interesting activities/contests, which engage your creativity and imagination.
Some also require team work. Go for those and have fun !!
Before going full throttle mode for competition math, please spend some time reading this
well- thought-out article from BOGTRO at AoPS "Learn How to Learn".
It will save you tons of time and numerous, unnecessary hours without a clear goal, better method in mind.
Less is more. My best students make steady, very satisfactory progress in much less time than those
counterparts who spent double, triple, or even more multiple times of prep with little to show.
It's all about "deliberate practices", "tenacity", and most of all, "the pleasure of finding things out on your own".
Take care and have fun problem solving.
I have been coaching students for many years. By now, I know to achieve stellar performance you need :
Grit (from TED talk), not only that but self-awareness (so you can fairly evaluate your own progress) and a nurturing-caring environment. (Parents need to be engaged as well.)
Thanks a lot !! Mrs. Lin
"Work Smart !!" , "Deliberate practices that target your weakness ", " Relax and get fully rested.", "Pace your time well", "Every point is the same so let go of some questions first; you can always go back to them if time permits."
"It's tremendous efforts preparing for a major event on top of mounting homework and if you are the ones who want to try that, not your parents and you work diligently towards your goal, good for you !!"
"Have fun, Mathcounts changes lives, because at middle school level at least, it's one of those rare occasions that the challenges are hard, especially at the state and national level."
Now, here are the links to get you started:
Of course use my blog. Whenever I have time I analyze students' errors and try to find better ways (the most elegant solutions or the Harvey method I hope) to tackle a problem. Use the search button to help you target your weakness area.
Last year's Mathcounts competition problems and answer key
Last year's handbook questions.
Near the end of the handbook, there is a page called problem index (page 7 to page 31 for 2021-2022 handbook).
For state/national prep, find your weakness and work on the problems backwards, from the hardest to the easiest.
Here are some other links/sites that are the best.
Mathcounts Mini : At the very least, finish watching and understanding most of the questions from 2010 till now and work on the follow-up sheets, since detailed solutions are provided along with some more challenging problems.
For those who are aiming for the state/national competition, you can skip the warm-up and go directly to "The Problems" used on the video as well as work on the harder problems afterward.
Art of Problem Solving
The best place to ask for help on challenging math problems. Some of the best students/coaches/teachers are there to help you better your problem solving skills.
Register for Alcumus and start using the great tool to practice deliberately.
Change the setting based on the levels of your proficiency of different topics. Do Not Rush !!
Awesome site!!
For concepts reviewing, try the following three links.
Mathcounts Toolbox
Coach Monks's Mathcounts Playbook
You really need to understand how each concept works for the review sheets to be useful.
To my exasperation, I have kids who mix up the formulas without gaining a true understanding and appreciation of how an elegant, seemingly simple formula can answer myriads of questions.
You don't need a lot of formulas, handbook questions, or test questions to excel.
You simply need to know how the concepts work and apply that knowledge to different problems/situations.
Hope this is helpful!!
E-mail me at thelinscorner@gmail.com if you want to learn with me. :) :) :)
Currently I'm running different levels of problem solving lessons, and it's lots of fun learning along with students from different states/countries.
So many students are not learning smart.
Problem solving is really fun (and a lot of the times very hard, yes).
Good questions are intriguing and delicious, so come join our vibrant community and have the pleasure of finding things out on your own.
Problem solving is really fun (and a lot of the times very hard, yes).
Good questions are intriguing and delicious, so come join our vibrant community and have the pleasure of finding things out on your own.
There is no overnight success.
My other blogs :
thelinscorner : Standardized test preps, books, links/videos for life-time learning
Take care and have fun learning.
Don't forget other equally interesting activities/contests, which engage your creativity and imagination.
Some also require team work. Go for those and have fun !!
Don't just do math.
In Chinese : 《專題報導》亞裔尖子生 職場難出頭？
Before going full throttle mode for competition math, please spend some time reading this
well- thought-out article from BOGTRO at AoPS "Learn How to Learn".
It will save you tons of time and numerous, unnecessary hours without a clear goal, better method in mind.
Less is more. My best students make steady, very satisfactory progress in much less time than those
counterparts who spent double, triple, or even more multiple times of prep with little to show.
It's all about "deliberate practices", "tenacity", and most of all, "the pleasure of finding things out on your own".
Take care and have fun problem solving.
I have been coaching students for many years. By now, I know to achieve stellar performance you need :
Grit (from TED talk), not only that but self-awareness (so you can fairly evaluate your own progress) and a nurturing-caring environment. (Parents need to be engaged as well.)
Thanks a lot !! Mrs. Lin
"Work Smart !!" , "Deliberate practices that target your weakness ", " Relax and get fully rested.", "Pace your time well", "Every point is the same so let go of some questions first; you can always go back to them if time permits."
"It's tremendous efforts preparing for a major event on top of mounting homework and if you are the ones who want to try that, not your parents and you work diligently towards your goal, good for you !!"
"Have fun, Mathcounts changes lives, because at middle school level at least, it's one of those rare occasions that the challenges are hard, especially at the state and national level."
Now, here are the links to get you started:
Of course use my blog. Whenever I have time I analyze students' errors and try to find better ways (the most elegant solutions or the Harvey method I hope) to tackle a problem. Use the search button to help you target your weakness area.
Last year's Mathcounts competition problems and answer key
Last year's handbook questions.
Near the end of the handbook, there is a page called problem index (page 7 to page 31 for 2021-2022 handbook).
For state/national prep, find your weakness and work on the problems backwards, from the hardest to the easiest.
Here are some other links/sites that are the best.
Mathcounts Mini : At the very least, finish watching and understanding most of the questions from 2010 till now and work on the follow-up sheets, since detailed solutions are provided along with some more challenging problems.
For those who are aiming for the state/national competition, you can skip the warm-up and go directly to "The Problems" used on the video as well as work on the harder problems afterward.
Art of Problem Solving
The best place to ask for help on challenging math problems. Some of the best students/coaches/teachers are there to help you better your problem solving skills.
Register for Alcumus and start using the great tool to practice deliberately.
Change the setting based on the levels of your proficiency of different topics. Do Not Rush !!
Awesome site!!
For concepts reviewing, try the following three links.
Mathcounts Toolbox
Coach Monks's Mathcounts Playbook
You really need to understand how each concept works for the review sheets to be useful.
To my exasperation, I have kids who mix up the formulas without gaining a true understanding and appreciation of how an elegant, seemingly simple formula can answer myriads of questions.
You don't need a lot of formulas, handbook questions, or test questions to excel.
You simply need to know how the concepts work and apply that knowledge to different problems/situations.
Hope this is helpful!!
Friday, August 11, 2023
Friday, August 4, 2023
V's record
V's record
weekly homework about 30 to 45 minutes
extra videos, links optional
First lesson:
2010 chapter sprint:
Hi, I got a score of 22 and got questions 16, 22, 23, 24,25,27,29, and 30 wrong.
I just guessed these questions because I didn't really find a way to do them.
Second meet:
2011-12 Mathcounts handbook (40 questions total)
Warm Up 1: 8
Warm Up 2: 19,20
Warm Up 2: 19,20
Warm Up 3: 32, 39
Workout 1: 23, 24, 30
Third meet:
2010 Mathcounts school test :
Hi, I finished trying the Sprint and Target questions:
Fifth meet : review
these are the problems that I got wrong
Sprint: I got 7/15 correct.
15,16: Attempted but answer was wrong
19,20,26,27,29: Didn't attempt
30: Couldn't find a good method to do, but was able to solve it by listing out all the possibilities.
Target: I got 6/8 correct.
The first 6 were relatively easy and I could find a clear way to do them
The last 2, I couldn't find a way to approach the problem.
Fourth meet :
2011 Mathcounts school test : last 15 sprint and last 4 target
Sprint: Out of the last 15,
I got 7 correct.
The questions that I got wrong were 20,21,22,24,27,28,29,30.
I tried 21,22, and 27 but the answers were incorrect. I wasn't able to attempt the rest.
Target: I got every question other than #5.
However, questions 1 and 2,
I got wrong at first, but when I retried them,
I was able to get them. I read the problems wrong and didn't fully understand them the first time.
Fifth meet : review
Sixth meet :
2011 chapter
Hi Mrs Lin! I was able to try both the target and the sprint round questions and here are my results:
Target: I missed 3 & 8, and I didn't attempt them.
Sprint: I got 17,21,23,24,25, 27,29,30.
I didn't attempt 27,29, and 30,
2012 Mathcounts school
Hi Mrs Lin! I was able to finish both the chapter and target, and here are my results:
Target: I got 5,6,and 8 wrong. I didn't try any of them because I didn't know how to do them.
Sprint: I got 18,23,25,26, and 30
Apr. 30, 2023
2012 Mathcounts chapter
Hi Mrs. Lin! I hope your having a very good day and week! I tried both the target and the sprint.
Target: I got only number 8 wrong, but I didn't know how to do it.
Sprint: I got number 22, 24, 26,28,29, and 30 wrong. I tried to do 22, but I got it wrong. The rest I didn't know how to do them..
May 7th, review
May 13th, 2023
2013 Mathcounts School
Hi Mrs.Lin, I was able to try both the Target and the Sprint and here are my results!
Target (5/8) I got 6,7, and 8 wrong, however, I was able to figure out the answer to problem 6 when I reviewed it.
Sprint: I got 30,29,28, and 24 wrong.
May 20th, 2023
Hi Mrs Lin!
2013 Mathcounts Chapter
I was able to try both the target and sprint round tests and here are my results:
Target: 6/8. I attempted the first 6 problems and got all of them right. I understood the problems fairly well and was able to do all of them on the first try.
sprint: I got 10/15 right. I got all problems I attempted right, and didn't attempt 30,28,25,24,21, and 19.
2014 Mathcounts school
Hi Mrs Lin, I was able to finish both the target and the sprint, and here were my results!
Sprint: 22/30, I didn't get 18,19,24,25,26,28,29, and 30. I attempted number 18, but didn't get it.
Target: I got 5 out of 8 on the target, but after reviewing my answers, I was able to figure out number 5.
2015 Mathcounts school
Hi Mrs Lin, I was able to try both the sprint and target. For the sprint, I got problems 20 and 27 wrong, and I didn't attempt any problems past 24 other than 27.
For my Target I got numbers 6,7, and 8 wrong, and I didn't know how to approach any of them.
Friday, May 5, 2023
Pathfinder
From Mathcounts Mini :
Counting/Paths Along a Grid
From Art of Problem Solving
Counting Paths on a Grid
Math Principles : Paths on a Grid : Two Approaches
Question #1: How many ways to move the dominoes on a 6 by 6 checker board if you can only move the dominoes to the right or to the bottom starting from the upper left and you can't move the dominoes diagonally?
Solution :
You can move the dominoes 5 times to the right at most and 5 down to
the bottom at most, so the answer is \(\dfrac {\left( 5+5\right) !} {5! \times 5!}\) = 252 ways
Question # 2: How many ways can you move from A to B if you can only move downward and to right?
Solution : There are \(\dfrac {\left( 4+4\right) !} {4!\times 4!}\) * 2 * \(\dfrac {\left( 4+4\right) !} {4!\times 4!}\) = 9800 ways from A to B
Counting/Paths Along a Grid
From Art of Problem Solving
Counting Paths on a Grid
Math Principles : Paths on a Grid : Two Approaches
Question #1: How many ways to move the dominoes on a 6 by 6 checker board if you can only move the dominoes to the right or to the bottom starting from the upper left and you can't move the dominoes diagonally?
Solution :
You can move the dominoes 5 times to the right at most and 5 down to
the bottom at most, so the answer is \(\dfrac {\left( 5+5\right) !} {5! \times 5!}\) = 252 ways
Question # 2: How many ways can you move from A to B if you can only move downward and to right?
Solution : There are \(\dfrac {\left( 4+4\right) !} {4!\times 4!}\) * 2 * \(\dfrac {\left( 4+4\right) !} {4!\times 4!}\) = 9800 ways from A to B
Friday, March 3, 2023
Sum of All the Possible Arrangements of Some Numbers
Check out Mathcounts, the best middle school competition math program up to the national level.
#2: Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy's list? (2004 Mathcounts Chapter Sprint #29)
Questions to ponder: (detailed solutions below)
It's extremely important for you to spend some time pondering on these questions first without peeking on the solutions.
#1:
Camy made a list of every possible distinct four-digit positive integer
that can be formed using each of the digits 1, 2 , 3 and 4 exactly once
in each integer. What is the sum of the integers on Camy's list?
#2: Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy's list? (2004 Mathcounts Chapter Sprint #29)
#3: 2020 Mathcounts state sprint #24
Solutions:
#1:
Solution I:
There are 4! = 24 ways to arrange the four digits. Since each digit appears evenly so each number will appear 24 / 4 = 6 times.
1 + 2 + 3 + 4 = 10 and 10 times 6 = 60 ; 60 (1000 +100 +10 + 1) = 60 x 1111 = 66660, which is the answer.
the four numbers.
2.5 (1000 + 100 + 10 + 1) x 24 = 66660
#2:
Solution I:
Since this time Camy wants five-digit even integer, which means that the number "4" has to be at the unit digit and only 1, 3, 5, 9 can be moved freely.
Again there are 4! = 24 ways to arrange the four numbers. 1 + 3 + 5 + 9 = 18 and 18 x 6 = 108 (Each number that can be moved freely appears 6 times evenly.)108 x 11110 + 4 x 24 = 1199976
Solution II:
Since this time Camy wants five-digit even integer, which means that the number "4" has to be at the unit digit and only 1, 3, 5, 9 can be moved freely.
There will be 4! = 24 times the even number 4 will be used so 4 x 24 = 96
As for the remaining 4 numbers, their average (or mean) is \(\dfrac{1 + 3+ 5 + 9} {4} = 4.5\)
4.5 * ( 10000 + 1000 + 100 + 10) * 24 (arrangements) + 96 = 4.5 * 11110 * 24 + 96 = 1199976
Solutions:
#1:
Solution I:
There are 4! = 24 ways to arrange the four digits. Since each digit appears evenly so each number will appear 24 / 4 = 6 times.
1 + 2 + 3 + 4 = 10 and 10 times 6 = 60 ; 60 (1000 +100 +10 + 1) = 60 x 1111 = 66660, which is the answer.
Solution II:
The median of four numbers 1, 2, 3, 4 is (1 + 2 + 3 + 4) / 4 = 2.5 and there are 4! = 24 ways to arrangethe four numbers.
2.5 (1000 + 100 + 10 + 1) x 24 = 66660
#2:
Solution I:
Since this time Camy wants five-digit even integer, which means that the number "4" has to be at the unit digit and only 1, 3, 5, 9 can be moved freely.
Again there are 4! = 24 ways to arrange the four numbers. 1 + 3 + 5 + 9 = 18 and 18 x 6 = 108 (Each number that can be moved freely appears 6 times evenly.)108 x 11110 + 4 x 24 = 1199976
Solution II:
Since this time Camy wants five-digit even integer, which means that the number "4" has to be at the unit digit and only 1, 3, 5, 9 can be moved freely.
There will be 4! = 24 times the even number 4 will be used so 4 x 24 = 96
As for the remaining 4 numbers, their average (or mean) is \(\dfrac{1 + 3+ 5 + 9} {4} = 4.5\)
4.5 * ( 10000 + 1000 + 100 + 10) * 24 (arrangements) + 96 = 4.5 * 11110 * 24 + 96 = 1199976
#3: The answer is 101.
Other applicable problems: (answers below)
Answer key:
#1: 66660
#2: \(\dfrac{1 + 3 + 5 + 7 + 8}{5} = 4.8\)
4.8 * 11111 =\(\color{red}{53332.8}\)
#3: 133320
#4: 1333272
#5: 93324
#1: What is the sum of all the four-digit positive integers that
can be written with the digits 1, 2, 3, 4 if each digit must be used
exactly once in each four-digit positive integer? (2003 Mathcounts Sprint #30)
#2: What is the average (mean) of all 5-digit numbers that
can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly
once? (You can use a calculator for this question.) (2005 AMC-10 B)
#3: What is
the sum of all the four-digit positive integers that can be written with
the digits 2, 4, 6, 8 if each digit must be used exactly once in each
four-digit positive integer?
#4: What is
the sum of all the 5-digit positive odd integers that can be written
with the digits 2, 4, 6, 8, and 3 if each digit must be used exactly
once in each five-digit positive integer?
#5:What is
the sum of all the four-digit positive integers that can be written with
the digits 2, 3, 4, 5 if each digit must be used exactly once in each
four-digit positive integer?
Answer key:
#1: 66660
#2: \(\dfrac{1 + 3 + 5 + 7 + 8}{5} = 4.8\)
4.8 * 11111 =\(\color{red}{53332.8}\)
#3: 133320
#4: 1333272
#5: 93324
Friday, January 20, 2023
2015 Mathcounts State Prep: Mathcounts State Harder Questions
Check out Mathcounts here, the best competition math program for middle school students.
Download this year's Mathcounts handbook here.
Question: 2010 Mathcounts State Team Round #10: A square and isosceles triangle of equal height are side-by-side, as shown, with both bases on the x-axis. The lower right vertex of the square and the lower left vertex of the triangle are at (10, 0). The side of the square and the base of the triangle on the x-axis each equal 10 units. A segment is drawn from the top left vertex of the square to the farthest vertex of the triangle, as shown. What is the area of the shaded region?
There are lots of similar triangles for this question, but I think this is the fastest way to find the area.
\(\Delta \)AGB is similar to \(\Delta \)DGC and their line ratio is 15 to 10 or 3 : 2.
\(\Delta \)CGF is similar to \(\Delta \)CBE.
\(\dfrac {CG} {CB}=\dfrac {GF} {BE}\)
\(\dfrac {2} {5}=\dfrac {GF} {10}\)\(\rightarrow GF=4\) From there you get the area =
\(\dfrac{10\times 4} {2}=20\)
Question: 2010 Mathcounts State #30: Point D lies on side AC of equilateral triangle ABC such that the measure of angle DBC is 45 degrees. What is the ratio of the area of triangle ADB to the area of triangle CDB? Express your answer as a common fraction in simplest radical form.
Since each side is the same for equilateral triangle ABC, once you use the 30-60-90 degree angle ratio and 45-45-90 degree angle ratio, you'll get the side.
Since area ratio stays constant, you can plug in any numbers and it's much easier to use integer first so I use 2 for \(\overline {CD}\).
From there you get the side length for each side is \(\sqrt {3}+1\).
\(\overline {AC}-C\overline {D}=\sqrt {3}+1-2\) = \(\overline {AD} = \sqrt {3}-1\)
\(\Delta ABD\) and \(\Delta CBD\) share the same vertex, so their area ratio is just the side ratio, which is \(\dfrac {\sqrt {3}-1} {2}\).
Download this year's Mathcounts handbook here.
Question: 2010 Mathcounts State Team Round #10: A square and isosceles triangle of equal height are side-by-side, as shown, with both bases on the x-axis. The lower right vertex of the square and the lower left vertex of the triangle are at (10, 0). The side of the square and the base of the triangle on the x-axis each equal 10 units. A segment is drawn from the top left vertex of the square to the farthest vertex of the triangle, as shown. What is the area of the shaded region?
There are lots of similar triangles for this question, but I think this is the fastest way to find the area.
\(\Delta \)AGB is similar to \(\Delta \)DGC and their line ratio is 15 to 10 or 3 : 2.
\(\Delta \)CGF is similar to \(\Delta \)CBE.
\(\dfrac {CG} {CB}=\dfrac {GF} {BE}\)
\(\dfrac {2} {5}=\dfrac {GF} {10}\)\(\rightarrow GF=4\) From there you get the area =
\(\dfrac{10\times 4} {2}=20\)
Question: 2010 Mathcounts State #30: Point D lies on side AC of equilateral triangle ABC such that the measure of angle DBC is 45 degrees. What is the ratio of the area of triangle ADB to the area of triangle CDB? Express your answer as a common fraction in simplest radical form.
Since area ratio stays constant, you can plug in any numbers and it's much easier to use integer first so I use 2 for \(\overline {CD}\).
From there you get the side length for each side is \(\sqrt {3}+1\).
\(\overline {AC}-C\overline {D}=\sqrt {3}+1-2\) = \(\overline {AD} = \sqrt {3}-1\)
\(\Delta ABD\) and \(\Delta CBD\) share the same vertex, so their area ratio is just the side ratio, which is \(\dfrac {\sqrt {3}-1} {2}\).
Subscribe to:
Posts (Atom)
