In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ ABCD.
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A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted △ ABC.Read More


EXERCISE 8 STATISTICAL TECHNIQUE IN REVIEW Mean, median, and mode are the three measures of central tendency used to describe study variables. These statistical techniques are calculated to determine the center of a distribution of data, and the central tendency that is calculated is determined by the level of measurement of the data (nominal, ordinal, interval, or ratio; see Exercise 1 ). The mode is a category or score that occurs with the greatest frequency in a distribution of scores in a data set. The mode is the only acceptable measure of central tendency for analyzing nominal-level data, which are not continuous and cannot be ranked, compared, or sub-jected to mathematical operations. If a distribution has two scores that occur more fre-quently than others (two modes), the distribution is called bimodal . A distribution with more than two modes is multimodal ( Grove, Burns, & Gray, 2013 ). The median ( MD ) is a score that lies in the middle of a rank-ordered list of values of a distribution. If a distribution consists of an odd number of scores, the MD is the middle score that divides the rest of the distribution into two equal parts, with half of the values falling above the middle score and half of the values falling below this score. In a distribu-tion with an even number of scores, the MD is half of the sum of the two middle numbers of that distribution. If several scores in a distribution are of the same value, then the MD will be the value of the middle score. The MD is the most precise measure of central ten-dency for ordinal-level data and for nonnormally distributed or skewed interval- or ratio-level data. The following formula can be used to calculate a median in a distribution of scores. Median()()MDN=+÷12 N is the number of scores ExampleMedianscoreth_N==+=÷=31311232216 ExampleMedianscoreth:.N==+=÷=404012412205 Thus in the second example, the median is halfway between the 20 th and the 21 st scores.
1. The primary objective of financial accounting is: To serve the decision-making needs of internal users To provide financial statements to help external users analyze


I am a mathematician and (currently) Senior Developer Experience Engineer based in Florianópolis, Brazil. I have an 8-year-old son and enjoy crocheting, knitting, cross-stitch embroidery and all sorts of crafts. I am interested in math, programming, free/open-source software and communities, crafts, science fiction and parenting. My pronouns are she/her.
I work at Quansight, and dedicate my time to NumPy, SciPy and other open-source projects. I care deeply about teaching, mentoring, and have been involved in the Python community for some time. You can find most of my talk slides here or on GitHub; feel free to use those according to the licenses stated in each repo.
I have a B.S. in Mathematics and Scientific Computing by the Federal University of Santa Catarina (Brazil), M.S. in Mathematics by the same university and Ph.D. in Applied Math by the Facultés Universitaires Notre Dame de la Paix, in Namur (Belgium) where I was advised by Professor Philippe Toint.
After my Ph.D., I spent 9 years as a professor at the Federal University of Santa Catarina (Brazil), teaching, doing some research and other administrative work at the Math Department. My research interests include smooth optimization (mainly nonlinear programming), numerical analysis, the numerical solution of large-scale linear systems and the implementation of software for scientific computing.


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EducationI am currently enrolled in the Masters of Architecture program at The University of Hong Kong (HKU).
I graduated from the same university for my bachelor’s degree of architecture in 2017, participated in an exchange program in UCLA in 2016, and have previously studied at Sha Tin College for my GCSEs and IB, graduating in 2013.
ExperienceI’ve had experience in tutoring both primary and secondary school students for 2-3 years, and I’m familiar with the GCSE, IB and KS3 curriculums as those are what I studied myself. To this day, two of my most memorable experiences were receiving news of my Year 13 student achieving a level 7 in HL Economics in IB (who is now studying at UCL), and my P3 student achieving an A in his english class that he was struggling with.
HobbiesAs I've studied in Hong Kong for most of my life, I've really learned to appreciate what I have around me, and really enjoy the outdoor countryside that the city has to offer.
In my spare time, I enjoy going hiking, or spending time relaxing under the sun either on the beach, or by the park. Painting and sketching is also a hobby I used to do much more often, and one I'd be willing to pick up once again if I had the chance!
Architecture, Economics, Drawing, Digital Media, Contemporary Architecture, Chinese, Cantonese, Biology, Art, Arithmetic, Essay Editing, Architectural Design, Algebra 3/4, Algebra 2, Algebra 1, 9th Grade Writing, 9th Grade Reading, 9th Grade math, 8th Grade Writing, IB Extended Essay, Summer Tutoring, Study Skills And Organization, Study Skills, Kindergarten - 8th Grade, IB Visual Arts SL, IB Visual Arts HL, IB Mathematics SL, IB Language A: Literature SL, 8th Grade Science, IB Economics SL, IB Economics HL, IB BIology SL, IB BIology HL, IB, History of Architecture, Graphic Design, General Biology, 1st Grade math, 3rd Grade Writing, 3rd Grade Science, 3rd Grade Reading, 3rd Grade math, 2nd Grade Writing, 2nd Grade Reading, 2nd Grade math, 1st Grade Writing, 1st Grade Reading, 4th Grade math, 12th Grade Writing, 12th Grade Reading, 12th Grade math, 11th Grade Writing, 11th Grade Reading, 11th Grade math, 10th Grade Writing, 10th Grade Reading, 6th Grade Reading, 8th Grade Reading, 8th Grade math, 7th Grade Writing, 7th Grade Science, 7th Grade Reading, 7th Grade math, 6th Grade Writing, 6th Grade Science, 10th Grade math, 6th Grade math, 5th Grade Writing, 5th Grade Science, 5th Grade Reading, 5th Grade math, 4th Grade Writing, 4th Grade Science, 4th Grade Reading
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Use place value to add tenth; use written addition to add pairs of numbers with 1, 2 decimal places; multiply , 1; round numbers with 2 decimal places to the nearest whole , 1 , subtract numbers with 1 , 2 , divide by 10 2 decimal places; use rounding to estimate totals; begin to add three numbers with 2 decimal places. This problem would make a good challenge activity for the end of maths a block of work on time. Multiplication facts worksheets with facts to 12 × 12 = 144 including individual facts worksheets. Showing top 8 worksheets in the category - Class 3 Maths Multiplication. Great resource for lesson plans , quizzes, homework just practice different multiplication topics. There are some fun math worksheets with make- your- own- maths calendar templates that work perfectly for this.
One more ( within 10) workskeets online interactive activities , mad maths minutes other educational resources. Click on the ' change' button and watch the numbers change. There are some fun maths worksheets with make- your- own- calendar templates that work perfectly for this. Don' t forget to let your child work through maths problems with fun maths multiplication worksheets for kids. Multiplication – Add and Multiply – Repeated Addition - One Worksheet. The following worksheets are intended for multiplication fact practice or assessment after students have learned all of the multiplication facts. Click on the ' print' button to print a pupil worksheet and an answer sheet. Free dynamically created math multiplication worksheets for teachers students, parents.
Math explained in easy language videos , games, quizzes, plus puzzles worksheets. | Assessment Package 1 | Assessment Package 2 ' Build- a- Mathematician' ( BAM) Progress Tracking ( NC) Our new ' assessing without ( NC) levels' suite of assessment materials - based on the Kangaroo Maths Mastery Indicators - is intended to be used alongside our schemes of work. For K- 12 kids maths teachers parents. Multiplication Worksheets – Printable PDFs. Using math multiplication worksheets lets your child practice the subject at his own pace. Basketball Multiplication # 4 Math maniacs, sneak in some multiplication practice with a fun basketball math sheet.They might also be used as a set of questions for manipulative practice. Content filed under the Multiplication category. Maths multiplication sheet. Your child will get a great review of times tables.
The clock is counting down. Using the six numbers, create an equation that gets you as close to the target number as possible. You can use addition, subtraction, multiplication and division to. Multiplication worksheets for grade 3 Make an unlimited supply of worksheets for grade 3 multiplication topics, including skip- counting, multiplication tables, and missing factors.
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The worksheets can be made in html or PDF format ( both are easy to print). Multiplication Worksheets ( counting) Multiplication Circle Worksheets 2 Digit Multiplication Worksheets 3 Digit Multiplication Worksheets Theme based multiplication worksheets Decimal Multiplication Worksheets Lattice Multiplication Blank Multiplication Table Multiplication And Division Puzzle Worksheets Multiplication Activities.


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I do group tutoring for the following: Pre-Algebra, Algebra I, Geometry, and Algebra II middle and high school students. Monday thru Thursday from 5pm to 9pm. Just $10 per session if you pay for 2 session per week for the month or $15 for walk-ins.
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“Mom. Can I do another 24?”
This request was music my math mom ears. It could be that the 9-year-old was only asking as a ruse to prolong bedtime. Frankly, if he was, then I will take that bait every time. Who can say no to a kid who wants to do more math??
24 is a card game. The cards look like this…
The goal is to make the number 24 using all 4 numbers on the face of one card. You must use each number, and you may use each number only once. So, for the top-left card above, one possible answer is 7 – 1 = 6, 2 + 2 = 4, and then 6 x 4 = 24. There are often multiple solutions on a single card, though I haven’t found another one for that particular card. If you find another way to use 7, 1, 2, and 2 to make 24, let me know!
I’ve had this set of cards for many years. Many years. A teacher colleague introduced me to them. Though the game is appropriate for kids as young as 9 (so says the box…really the main prerequisite is that you know your basic addition, subtraction, multiplication, and division facts within the 24 range), it can be challenging for anyone because the key to success is mental flexibility and persistence. I use this game frequently with my tutoring clients. I had not thought to use it with my own son until one night when he was just a little too rambunctious before bedtime and I wanted him to calm down.
“Hey,” I said, “Check this out.” And I tossed a card in his direction. I had tutored earlier in the day, and the cards were still out on the coffee table. As he picked it up, I pulled out one of our big whiteboards and a handful of colorful Expo markers. At the top of the whiteboard I wrote: 2 x 12 = 24, 3 x 8 = 24, and 4 x 6 = 24.
“Can you make the number 24 out of those 4 numbers on the card? You can add, subtract, multiply or divide to make it happen.”
His first inclination was to add them all together. But that didn’t work. So I let him marinate in his thoughts for a bit. Then I pointed out the equations that I had written and suggested that he look for ways to make numbers that could then multiply to 24. Then I left the room, claiming that I had to check on his brother. (Parents have ruses, too…this one was so that he didn’t have an audience for any drama that might unfold as he worked through this problem.) He came up with: 8 – 4 = 4, 3 + 3 = 6, and 4 x 6 = 24.
He enjoyed the game enough to try a few more cards before (happily!) heading to bed. Yay!
(Mathematically…at least 3 things are happening in this game that are great for you and your child. First, your child is getting practice not just with math facts, but also with being flexible with her math facts. She is working her numerical manipulation and mental flexibility muscles. Second, you are observing and learning about your child’s thought processes and learning style. Your observations can then become references in the conversations you have with your child’s teacher and inform the ways you interact effectively with your child. Third, working with the white board (or any recording medium) opens up the conversation for other mathematical topics and discussion. I introduced my son to notation with parentheses (remember order of operations and PEMDAS??) and we discussed whether or not an equation was allowed to have more than one equal sign. He said no; I said maybe. It’s an ongoing discussion.)
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“This Boxset includes the following titles:1. Objective Mathematics Volume 1 for Engineering Entrances 2. Objective Mathematics Volume 2 for Engineering Entrances This book has been revised and designed by Arihant for the aspirants preparing for the upcoming popular engineering entrances like JEE Main & Advanced, BITSAT, MH CET,EAMCET, Karnataka CET, Kerala CEE, VIT, WBJEE, UPSEE, etc.
This book has been designed to help aspirants master the concepts of mathematics as thorough and comprehensive knowledge of the concepts is required for scoring high in such engineering entrances.The present Box set contains Objective Mathematics Volume 1 & Volume 2 for preparation of JEE Main & Advanced 2018 and various other engineering entrances. This combination has been designed in sync with Class 11 th and Class 12 th Mathematics NCERT textbook to help aspirants prepare for the competitions along with their school studies.
The books contain more than 5000 objective questions of all types like single option correct, matching, assertion-reason and statement based questions. Ample number of solved and unsolved questions have been provided in the books to help clear their concepts and equip them in solving different types of problems. The questions provided in the books incorporate the syllabi of almost all engineering entrances in India. Each chapter contains objective questions presented in two levels, Level I covering the basic questions and Level II covering higher difficulty order questions. The questions have been divided into two levels to help aspirants practice the concepts step by step.Separate collection a of questions asked in previous years’ national and regional engineering entrances has been provided in the books.
The questions of practice exercises have been solved in detail to help students understand the concepts better and effectively. Also JEE Main& Advanced 2016 & 2015 Solved Papers have been provided in the books to help aspirants getan insight into the recent examination pattern and the types of questions asked therein.As the combination contains ample number of questions that can be asked in the upcoming engineering entrances as well as questions asked in previous years’ engineering entrance examinations, it for sure will help aspirants master the concepts of mathematics and also the skills required to tackle the questions asked in different formats in various popular engineering entrances.”
On your way to success, Arihant Publications aim to be your companion guiding you through the hardships and helping you in achieving your goals. After a decade of successful journey, the brand Arihant is widely known as the leading force in the educational books market.
Their books were conceptualized in such a way that no one had thought about, the set of books was the first of its type in which separate textbooks were available for different topics of Physics, Chemistry & Maths, like Diffierential Calculus, Algebra, Mechanics, Electricity & Magnetism, Inorganic Chemistry, Physical Chemistry etc.


Draw Arabic numbers with matatalab
Activity Lesson Key Information
Project Name: Draw Arabic numbers with Matatalab!
Big Idea: Matatalab could be used in a math class. Let us draw Arabic numbers with Matatalab!
Concepts: Matatalab tangible coding language，Arabic numbers.
Objective: Draw Arabic numbers 0-9 with Matatalab Coding Set.
l Compare the difference between handwritten Arabic numbers and those drawn by MatataBot.
l Apply the motion blocks, including move forward, move backward, turn left, and turn right, to draw simple Arabic numbers (0, 1 and 7) .
l Draw the other Arabic numbers (2, 3, 4, 5, 6, 8 and 9) based on the provided programs.
l Explore the usage of the Loop blocks.
l Challenge to code the date of your birthday without the provided programs.
MatataBot: A Robot
Command: instructions given to a computer or robot
Move forward: the MatataBot move forward 10cm
Move backward: the MatataBot move backward 10cm
Turn right : 90º turn to the robot’s right side
Turn left: 90º turn to the robot’s left side
Loop: repeat with the specified action
Students know how to use Matatalab Coding Set to program and control the movement of the MatataBot.
Standards:(ISTE, CSTA, Common Core, NGSS, etc. )
CSTA(K-2): 1A-CS-01, 1A-DA-05, 1A-DA-06, 1A-AP-10, 1A-AP-11, 1A-AP-14
ISTE：1a, 3a, 3c, 4a, 4c, 5c
Matatalab Products & Supplementary Materials
Animation Add-On Sensor Add-On□
l Supplementary Materials
White papers, erasable water color pens
Detailed Activity/Lesson Plans
Lead in & Guided Activity
Compare the difference between handwritten Arabic numbers and those drawn by MatataBot.
The teacher demonstrates how to draw Arabic number 1 with Matatalab.
Students need to:
Figure out the program to draw the Arabic numbers 0 and 7 by themselves.
Code to draw the Arabic numbers 2, 3, 4, 5, 6, 8 and 9 based on the programs for Arabic numbers 2 and 3 provided by the teacher.
Summary the usage of Loop blocks.
Challenge to code the date of their birthday without the provided programs.
Feedback & Extension
Review the usage of Loop blocks.
Share your work with other students.
l I have all the programs for the ten Arabic numbers here. Which program could be used to draw number 0/1/2/3……?
l If I need to use the Loop blocks to draw a square, which program is the right one?
l Have you finished coding your birthday date? Please take them home and share with your parents.


Two weeks ago we began our French language classes together. Three nights a week for about 90min sessions. It's not too bad for me, but for My Man, it's straight after work, and it's a big ask.
So far, we have learnt to:
|Maybe French will help with the sharks back home?|
- introduce ourselves,
- say what nationality we are,
- what our profession is,
- the language we speak,
- where we live,
- our phone number,
- email address...
- - as well as the obvious - like greetings.
We can also ask others these same questions.
Now I know why I needed basic Arithmetic!
It was because one day I would need to learn to count in French. My challenge this week is not stringing together a sentence in French, it is learning to count above 70.
Who does this?
70 = 60 +10 : 71 = 60 + 11 : 72 = 60 + 12 ... etc.
82 = 4 x 20 + 2 : 85 = 4 x 20 + 5
Then there's the 90's... (faint!)
93 = 4 x 20 +13 : 99 = 4 x 20 + 19
Then to say it..... 99 = quatre-vingt-dix-neuf.
Damn man. If you made this equation look like this: a = 4 x 20 + 19 I would have a better chance of success at solving the equation than counting and it and saying it!
Anyway, enough of that but thank God for someone who made audio flashcards / games /tutorial / tests! (I love it when teachers use tech! Oh ja, that used to be me!)


Welcome to Our Website
Welcome to the Oscar J. Pope Elementary Web site! We are very excited that you visited! We are proud of each and every Stallion!
We are OSCAR!
- O: On task
- S: Safe
- C: Caring
- A: Accepting Responsibility
- R: Respectful
New Bell Schedule for 2014-2015
For the 2014-2015 academic year, the first bell will ring at 8:00 A.M. and the dismissal bell will ring at 3:00 P.M. This will provide an additional fifteen minutes of instructional time to our academic day.
Register New Students Now
If you are moving into the Oscar J. Pope Elementary School attendance zone, don't wait to register your children for school. School staff members are available to assist you with registration paperwork and requirements. Our office is open Monday through Thursday from 7:00 A.M. to 5:00 P.M. The office is closed on Fridays.
Free Online Math Resource
Summer learning loss in math is an issue that impacts students at all grade levels. Students lose an average of more than two months of math comprehension every summer. Polk County Public Schools is offering a free online math resource this summer for Polk County Public School students rising to first grade through Algebra 2.
To access this free resource for your child, go to www.tenmarks.com, click on "Parents-Sign Up", and enter your information with S14P3351 as the "Access Code From Your School" code. Complete the general grade information for your child and he/she will be given access to the grade specific math material.
|August 11||Teachers Return||Welcome Back #1 Teachers!|
|August 11||Teacher Workday|
|August 12||Staff Development Day|
|August 13||Teacher Workday (1/2 Day)|
|August 13||Staff Development Day (1/2 Day)|
|August 14||Student Orientation Day||3:00 - 6:00 P.M.|
|August 14||Non-Instructional Staff Breakfast and Meeting||8:00 A.M.
Jay W. Erwin Media Center
|August 14||Teacher Workday|
|August 15||Teacher Workday (1/2 Day)|
|August 15||Staff Development Day (1/2 Day)|
|August 15||OSCARbration Planning Meeting||1:00 P.M.
|August 18||First Day of School||Welcome Back #1 Students!|
|August 28||Open House||6:00 - 7:00 P.M.|
|August 29||OSCARbration: Outdoor Games||No horseshoes are required to attend. All students are invited to participate.|
|September 1||Labor Day Holiday||School Closed|


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Phone: (02) 4722 3039
Mobile: 0416 018 716
Numeracy is one of the most important and useful skills a person can possess. Just imagine how it would be if you never developed good numeracy skills. There would be so many situations where you would be dependent on other people, or on the use of calculators. It would be like being disabled! If you lack numeracy skills, you can still get by, but so many things become very much harder to do and things that should be fun are instead hard work.
At The Tuition Place we recognise that there are many levels of skills and methods that will lead you to mastery of numeracy. Probably the most important of these is the ability to visualise numbers, to be able to picture them coming apart and being reassembled in the new form that an operation requires. There is also the understanding and memory of many different methods, little tricks that can make even some very involved operations fun and easy. Finally, there is the development of understanding of how the different numeracy skills fit in with all the areas of mathematics and with problem solving.
At The Tuition Place, numeracy can be taught by itself, or as part of our Mathematics tuition program. The first step is to assess thoroughly and accurately where you are now in your numeracy skills. Then, as with all Tuition Place tuition, you can proceed at your pace, developing all the knowledge and visualisation skills that you require to master numeracy at each level, as well as laying a solid foundation for confident mastery of future levels. Tuition Place resources and teaching methods are thorough and comprehensive in covering all of the numeracy skills. The resources are also extensive, so that that a tutor has a wide choice of which resource to use in any instance.
Everything is designed to improve your knowledge and skills, develop your self-confidence and encourage you to enjoy the learning process.
If you would like a no-obligation, free assessment of your numeracy skills, contact The Tuition Place now. The assessment can be arranged at a time to suit you.


Sam is one wonderful lady. Her energy and enthusiasm and generosity of spirit is extraordinary and we're so lucky at Bushbells to have met her. She kindly organised for some of our teachers from Bluebells School to go along to Kivukoni School in Kilifi, for a teacher training day at the end of September.
The workshop was on 'Creative Numeracy in Early Years' and focused on looking creative ways to encourage young children to explore mathematical concepts through free play, numeracy learning areas and using the natural environment. They focused a lot on thinking broadly about resources and how they can be adapted and used across learning areas for different subjects to still promote mathematical awareness and development.
Sections included low cost materials to be used, loose parts, natural resources, messy maths and playdough!! How to use the natural world around them in an imaginative way was key for our staff to learn, as they work with scant resources. They learnt how learning can still be fun and creative - and outside of the classroom too! The teachers were given lots of new ideas for using playdough, cup cakes, birthday candles and numbers to promote counting and number value in an engaging and fun way.
Our huge thanks to Lucy and Kivukoni School for hosting this teacher training session, and credit and more thanks to Sian for being the teacher trainer. And thank you Sam, for organising all this - Angels all ! xx


Can you solve these 5 matchstick puzzles riddles? Genius Matchstick Puzzle Riddles with answer.
Move only one matchstick and make the equation correct. Take the challenge and solve these best matchstick puzzles. You will have to move just one stick and fix the equation. Let see, how you are going to make these equation true! Comment you answer or suggestion...
Brain Teasers & Mind Puzzles – Find Diamonds limits from image - Only for Genius Puzzles with solution;
Find how many diamonds or squares are there in this picture. Solve this brain teasers puzzles and comments your answer in the comment box before check the answer.
Brain Teasers Question:
Q) How many diamonds can you find here?
Hope you find all diamonds, let...


Brenden didn’t travel very far for his placement. In fact, he’s teaching at the very secondary school he had attended.
Christie had inspiring maths teachers back when she was in school, who had encouraged her to pursue further study – even though she didn’t really think she had the knack for maths herself.
Aaran was at a crossroads when he left university. He had a Maths degree, and wanted to do something meaningful.
Dan spent twenty-five years in the wine industry – and he loved every minute of it.
Celia had an established career as a horticulturist, but now she’s a science teacher.
After nearly nine years in the corporate world, Kath realised that she wasn’t doing what she loved.


(Complete the example. Practice practice practice. Quiz. Quiz. Done until end-of-year review time.)
That formula, with the combination and the n's and p's and r's and q's, is like a bouillon cube. Very useful, but not intended to be consumed raw. (To rip off an Ellen Kaplan analogy.) As awful as that slide is, this is what is going on in classrooms across America, because the sobering volume and ill-conceived spiraling of content means we are rushed beyond reason. I know Algebra 2 teachers whose ulcers live in fear of a snow day for the loss of instruction time.
So here's my idea and how I intend to deploy it in class. I was indeed fortuitously inspired. I didn't think of this just because Bernoulli trials are coming up.
Please bear in mind this anticipated dialog is boiled down, like bouillon, and there will be much waiting, hemming, hawing, frowning, clarifying questions, etc etc. And, it will go differently in all three classes.
"When I go visit my mom, it's my job to set the table for dinner."
"I was visiting for five days last week, and on Thursday, this happened:"
"My mom said, 'Hey, did you do that on purpose?'
'No' I said. 'That's weird.'
Why do you think we were surprised?"
(The bowls and plates match.)
"Now, my mom thought this was a miracle. A once in a lifetime Fiestaware occurrence. So, what do you think is my question?"
(What are the chances you choose matching plates and bowls.)
Here, someone will assert that I subconsciously tried to make them match, skewing the chances of a matched set. I will acknowledge this but ask if we can, for the sake of learning, pretend I was choosing plates and bowls randomly.
"So, listen. I wasn't so sure this was so improbable, because remember, I set the table for five days."
(It would be more likely the more times you did it.)
"Right. So in order to convince my mom this wasn't so unlikely, what would I need to calculate?"
(The probability of choosing two matched sets one time in five days.)
Write this on the board.
"I'd like you to discuss possible approaches with your partner. You may not be able to solve the problem yet. That's ok. I'm going to give you four minutes to discuss. I want you to make progress. What would we need to know to make progress? Write it down. And if you can't, what questions do you need answered that would help you make progress? Write them down. Is there an easier problem, that you can solve? Write it down. Go."
I don't think I can script any more after this. It depends on what they come up with. They are likely to be able to, in order of increasing difficulty/decreasing likelihood:
- calculate the probability of choosing one matched set
- calculate the probability of choosing two matched sets, performing the experiment once
- calculate the probability of choosing two matched sets on a particular day during the five days
After the four minutes, I would ask each pair what was their finding, or what is their question, and write all these on the board. And then I would go from there. With the intention of deriving the formula for Bernoulli trials that day or the next day, with copies of some practice problems at the ready to hand out, once we got this one.
Suggestions? Better ideas? I readily admit this question kind of sucks, because there's no intrinsic buy-in. Nobody really cares about the dishes at my mom's house. Or maybe it is the best I can do in my classroom, because I thought of it, so I can do it justice. What do you think?


This page is here to collect our thoughts about a joint project around dynamic geometry, proofs and constructions. This is a draft under construction which collects ideas for cooperation initiated at Castro Urdiales in February 2010.
Technological tools are widely used to teach mathematics in schools. Dynamic Geometry Software (DGS) and Computer Algebra Software (CAS) are the two families of software that are well represented. These tools are widely used to explore, experiment, visualize, calculate, measure, find counter examples, conjecture... but most of them can not be used directly to build or check a proof.
Proof is a crucial aspect of mathematics and therefore should be integrated more into the mathematical tools. However the status of proof in the education and the impact of dynamic geometry software on the teaching of geometry and more precisely on the proving activity is a well addressed issue in the literature (See for example Furinghetti, F. & Paoloa Domingo (2003), de Villiers (1990)). In Furinghetti, F. & Paoloa Domingo (2003), the authors “confirm the interlacement of exploration and proof”. We think that these two activities could better interlaced if they were both conducted using the computer. The idea collected here are driven by the motivation to bring together dynamic geometry systems and various aspects of computer-aided theorem proving. We think that both worlds have their individual strengths (appealing visualisation and rigorisity respectively) which could benefit to each other.
So these webpages address software eliciting interest in and satisfying curiosity for the principles of mathematics and their beauty, and since anyone knows that essential insights cannot be scheduled for a certain lesson at high school, we can conclude:
- The envisaged software shall cover a wide range of functionality.
In order to gain wide usage for such software, it also should support mathematics as is at schools today.
State of the art
|DGS||Dynamic Geometry System|
|CAS||Computer Algebra System|
|ATP||Automatic Theorem Proving|
|ITP||Interactive Theorem Proving|
|CTP||Computer Theorem proving (ATP+ITP)|
|IDE||Integrated Development Environment|
|MMA||Mechanised Mathematics Assistant|
|RP||Repository of Theorems|
comment (PQ): shouldn't be RP - Repository of Problems instead of Repository of Theorems? for once the RP will match the words (or we should change it to RT), and the repository of problems will strees the fact that we may have open problems also.
DGS Dynamic Geometry Software
Dynamic Geometry Software are tools which allow to create and then manipulate geometric constructions.
ATP Automated Theorem Proving
Automated theorem provers a software system which given a statement can tell if the statement is true or not. Some ATP can produce a proof of the statement.
ITP Interactive Theorem Proving
A proof assistant is a software which can be used to interactively build a formal proof. This object is then mechanically checked by the machine. In each of these systems the user has to first define what he wants to prove and then guide the software until it accepts the proof as a valid proof. Proof assistants are different from theorem provers: proof assistants are designed for an interactive use whereas theorem provers a build to find proofs automatically. However also proof assistants usually include automation: decision procedures for particular problems as well as heuristic proof search methods.
With respect to educational use ITP's emphasis for comprehensibility by humans is important: in order to have human readable proofs, also the underlying knowledge, axioms and definitions, need to be human readable. This is accomplished best by LCF-style provers like Coq or Isabelle.
Mechanised Mathematics Assistant
Regarding the variety of concepts and technologies discussed in this wiki one might expect a conglomeration of different pieces of software. Quite in contrary, our goal is unifying the technologies in one Mathematics Assistant (MA), which is general enough to include algebra as well --- general as, for instance, underlying tools like Isabelle or Coq.
Existing systems and their capabilities
The vision is that we integrate existing systems rather than starting from scratch. For convenience we give here a list of systems which could be involved. This list is not supposed to be a final decision, just an enumeration of candidates which we deem suitable according to our current knowledge and are represented to a certain extent among the authors; other suitable systems may be added accordingly, while not all systems listed here may turn out to be really suitable in the end.
- GCLC Sect.
Automatic deduction in geometry
The main methods for automated deduction in geometry are:
- Gröbner basis
- Wu's method
- Area Method
- Full-angle method
- Geometric algebra
Roughly speaking, geometric constraints solving consists in producing geometric figures satisfying a specification. This specification can be given under the form of a geometric statement, in the education domain, or a dimensioned sketch, in the CAD domain.
Example of a geometric statement:
Let two parallel lines, D1 and D2 be given, A and B be two points on D1 and D2 respectively, and O be another point. Construct two points, Z1 and Z2 respectively on D1 and D2 such that AZ1+ AZ2 = k where k is a given non negative number.
Example of a dimensioned sketch:
From a logical point of view, such a statement is a <math>\forall \exists</math>-formula and solving a geometric constraint system consists in yielding
- either a model of the statement, also called a witness or a solution, for this formula (such a model usually is a numerical figure),
- or, in education domain, a way to construct all the solutions in all the cases, that is, a geometric construction program.
In the following, we focus on the second aspect.
Automatic generation of geometric construction programs
Progé by Pascal Schreck, ...
Geometric construction language
Tutoring in stepwise construction
The introduction stresses the role of proof as 'a crucial aspect of mathematics', and addresses the importance of 'interlacements of exploration and proof'.
However, an educational context makes clear, that proof is a highly elaborated concept comprehensible only togehter with concepts like axiom, theorem etc. So there are good reasons, that high schools precede proving with another entry point to geometry:
Stepwise construction of figures is how children first encounter geometry, using paper and pencil or a DGS. Such construction problems provide the possibility of active learning, of learning by trial and error, while intuitively convincing feedback by the figure is guaranteed.
Thus we stress the demand of software, which supports stepwise constructions in geometry and proving.
Support for construction implies efficient and highly mechanised user guidance -- such kind of tutoring has not yet been realised by any system, it poses a challenging issue for our considerations here. Mechanised generation of user guidance might start from programs in a high-level language addressed above, coded by hand or generated automatically.
The experimental tutoring system ISAC employs a new technology  for user guidance in stepwise construction of calculations in applied mathematics --- together with a discussion, how this technology could be transferred from algebra to geometry.
Remarkable in the context of these webpages is, that the design of such a tutoring system raises challenges for ATP and ITP, which are discussed below.
Interactive proof in geometry
- Geometry Tutor, Déﬁ (Desmoulins 94) (Anderson et al. 85) (Baulac et Giorgiutti 91)
- Cabri-Euclide, Chypre (Laborde 95,Sutherland 63,Luengo 97,Bernat 94).
- Mentoniezh (Dominique Py) is based on an expert module i.e. an automated theorem prover which calculates beforehand many different proofs for the statement.
- Baghera is a multi-agent system for learning in geometry.
A very well develloped software for interactive proof in geometry is Geometrix
It allows both:
- contruction exercises and
- proof exercises
Interactive formal proof in geometry
Interactive formal proof in geometry has been studied by Frédérique Guilhot who developped a large amount of highschool geometry using the Coq proof assistant.
Automated formal proofs in geometry
There has been a lot of efforts invested in automated theorem provers in geometry and also in developing collections of formalized geometry knowledge. But, it is still to develop systems that automatically generate formal proofs. Also, it would be welcome to develop systems that automatically generate synthetic (hopefully readable) proofs of geometry conjectures. One option is to use coherent logic as a framework for that (Janicic 2010).
Some ADG methods have been formalized in ITP.
- the Area Method in Coq (Narboux 2004, Narboux 2010)
- the Gröbner basis method in Coq (Pottier 2008)
- to be completed
The recently emerged connection of ITPs to ATPs for proof search under the brand sledgehammer  could also turn out helpful.
Human-readable interactive formal proofs
In the early days of interactive theorem proving, the task of writing »proofs« meant to write down so-called tactic scripts which consisted of a programmatic combination of so-called tactics of which each would implement a particular proof strategy. This was successful but turned out to be inconvenient: tactic scripts seldom resemble the abstract human-invented ideas behind a proof, are brittle, and always refer to an implicit goal state which is not represented in the script itself, making maintenance difficult. To overcome this, declarative proof languages were developed,
- starting with the Mizar system
- and later on the proof languages Isar for Isabelle
- and C-zar for Coq.
The core idea is that instead of writing a programmatic description how to accomplish a proof, the proof writer gives a proof text describing what to prove in each step, where each step needs to be justified by either a subproof or a terminal tactical proof. This means a proof writer is free to express his ideas about the proof in his text such that a reader can follow his thoughts accordingly.
This is interesting in our context because such proof languages allow for fine-granular control while still be able to integrate automatic proof procedures. For example, in the Isar framework, the fundamental single proof step is resolution with natural deduction rules. Any proven theorem can again be used as natural deduction rule. Consequently, if domain-specific proof steps can be encoded into natural deduction rules, they can be applied directly within Isar. Beside that the system provides generic proof tools (called methods) which can instantiated for particular domains: induction (for higher-order natural deduction rules), simplification (equational rewriting with equalities), classical reasoning. Especially the two latter help to avoid spelling out proof parts explicitly which are considered trivial by the human reader.
Database of geometry problems
Geothms - GeoThms is a web-based framework for exploring geometrical knowledge that integrates Dynamic Geometry Software (DGS), Automatic Theorem Provers (ATP), and a repository of geometrical constructions, ﬁgures and proofs. The GeoThms users can easily use/browse through existing geometrical content and build new contents. In this paper we describe GeoThms functionalities, focusing on the interface solutions required for a system aimed at supporting studying and teaching geometry via Internet. GeoThms is a publicly accessible system with a growing body of geometrical constructions and formally proven geometrical theorems .
TGTP (Thousands of Geometric problems for geometric Theorem Provers) is a system aiming to support the testing and evaluation of geometric automated theorem proving (GATP) systems, to help ensure that performance results accurately reflect the capabilities of the GATP system being considered .
Proof document model
A recent developement in ITP user interfaces could have beneficial impact on our aims: a model for proof documents . It is motivated by the observation that typical ITP systems barely provide support for contemporary editing technology and it would be waste of resources to build editors from scratch. Therefore it is proposed to design ITP systems in a way that they can interact with existing editor frameworks. One should refrain from the temptation to model arbitrary details of the ITP system in the user interface since this would be bare duplication with all negative effects like keeping versions consistent etc. Instead the interaction between ITP system and user interface is kept to a simple document model which supports basic activities like open new document, insert document fragment, drop document fragment, conclude document etc. The ITP system judges which parts of the document are to be considered valid or not and reports on that to the user interface which visualizes this to the end user accordingly (e.g. red underlining), thus mimicking the look-and-feel of contemporary IDEs for programming languages. Hence the user interface need only the knowledge whether something is valid or not, but not why.
A concrete instance of this document model is presented using Isabelle/HOL and the JVM-based jEdit editor framework, but the ideas can be transferred to similar components.
In our scenario the visual user interface would be a DGS. It would map actions in a geometry scenario (draw circle, connect dots, construct triangle, ...) onto changes of a underlying document; the checking is done within the ITP which gives feedback which steps are valid.
Goals and means
- proof checking
- produce a system which checks geometry conjectures (this would need integration of DGS and ATP)
- proof discovery
- produce a system which finds geometry conjectures
- exercise correction
- produce a system which can check if the answer to a construction exercise is correct (this can involve ATP methods embedded in ITP or not); this can be regarded a special case of the following:
- exercise hints
- produce a system which can help the student to stepwise build a construction, see above; this includes developing techniques to adapt the hints to the curriculum of the students.
- interactive geometric proof
- produce a system to build proofs in geometry interactively (this requires having ATP to ease the interactive proof, ways to visualize statements in DGS, a readable proof language, …)
- problem collection
- produce a database of geometry theorems with their proof and figures (starting with GeoThms).; conjectures should be accompanied by (when possible):
- formal proofs (not neccesssarily automatic)
- automatically generated proofs (not neccessarily formal)
- dynamic figures (for different DGS)
- solution collection
- produce an electronic handbook (not necessarily online) about geometry integrating DGS, ITP and ATP
- integrated system
- design one system providing support from high-school to university: pupils doing an elementary construction should be able to ask the system for underlying knowledge, for instance the axiom system the MMA is using in this construction.
Interoperability between proof system and DGS
Automated deduction in geometry
Dealing with degenerated cases
Research issues raised by tutoring
The tutoring technology of Lucas-Interpretation (LucIn) combines computation and deduction such that this section is closely related to many other sections here.
Examples: tutoring by Lucas-interpreter
These examples shall provide concrete material for discussion. The collection of examples might be extended during the course of discussion.
The construction of a circle, which contains all points of a given triangle, is one of the first constructions a learners encounters --- and DGS convincingly show the invariant, when dragging one of the points on the screen. Leet's discuss two appoaches:
- A simple set-theoretic approach which reflects the kind of comprehension general to learners at the age of about 10 years. This example is simple such that a program for the LucIn can cope with the combinatorial explosion of similar cases, which however cannot be expected for other examples.
- An approach based on the area method, which is strong enough to cope with a large class of geometry problems. This approach follows  and raises questions with respect to ATP.
Since the area method is not always close to human readable proofs, other approaches might emerge during discussions here, probably using more 'traditional axioms'  restricted to Euclidean plane.
Formalize ruler and compass construction in ITP
Define and formalize the notion of ruler and compass construction in a proof assistant. ITP systems could be used to ensure that construction steps given by the user are applicable and consistent.
Check constructions correct by ATP
Check that the constructions given by an oracle (a student or a software) are correct. Check by ATP (and possibly by ITP, following links from ?) that the construction meets the given specification.
Presentation of geometry knowledge
- T. Nipkow and L. C. Paulson and M. Wenzel, Isabelle/HOL – A Proof Assistant for Higher-Order Logic, Springer LNCS 2283 pdf
- Remark: ISAC is located here under ITP, because is neither belongs to DGS nor to RP. A summary on the relations to ITP, ATP and DGS is given in #Tutoring_in_stepwise_construction
- Walther Neuper, Lucas-interpretation mechanises tutoring by combining computation and deduction. preprint
- Loïc Pottier, Connecting Gröbner Bases Programs with Coq to do Proofs in Algebra, Geometry and Arithmetics, KEAPPA 2008 pdf
- Sascha Böhme and Tobias Nipkow, Sledgehammer: Judgement Day, IJCAR 2010 pdf
- P. Rudnicki, An Overview of the MIZAR Project, 1992 Workshop on Types for Proofs and Programs
- Markus Wenzel, Isar – a Generic Interpretative Approach to Readable Formal Proof Documents, TPHOLs '99 pdf
- Pierre Corbineau, A Declarative Language for the Coq Proof Assistant, TYPES 2007 pdf
- Pedro Quaresma and Predrag Janicic, GeoThms --- a Web System for Euclidean Constructive Geometry, UITP 2006
- Pedro Quaresma, TGTP - Thousand of Geometric problems for Theorem Provers, ADG 2010,
- Makarius Wenzel, Asynchronous Proof Processing with Isabelle/Scala and Isabelle/jEdit, UITP 2010 pdf
- P. Janicic, J. Narboux, P. Quaresma, The Area Method. A Recapitulation. in: J.of Automated Reasoning. 2011 to appear preprint
- Nicolas Magaud, Julien Narboux, Pascal Schreck, A Case Study in Formalizing Projective Geometry in Coq: Desargues Theorem. In: Computational Geometry: Theory and Applications (2010) preprint
People currently involved into discussion
- Florian Haftmann, TU München, email@example.com
- Markus Hohenwarter, Univ. Linz, firstname.lastname@example.org
- Predrag Janicic, Univ. Belgrade, email@example.com
- Julien Narboux, Univ. Strasbourg, firstname.lastname@example.org
- Walther Neuper, TU Graz, email@example.com
- Tomas Recio, Universidade de Cantabria, firstname.lastname@example.org
- Pascal Schreck, Univ. Strasbourg, email@example.com
- Pedro Quaresma, Universidade de Coimbra, firstname.lastname@example.org


The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich’s model) and the limit of small width (Sadowksky’s model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Frénet-Serret’s frame, the present work unifies the description of thin ribbons and thin rods.
Microorganisms are rarely found in Nature swimming freely in an unbounded fluid. Instead, they typically encounter other organisms, hard walls, or deformable boundaries such as free interfaces or membranes. Hydrodynamic interactions between the swimmer and nearby objects lead to many interesting phenomena, such as changes in swimming speed, tendencies to accumulate or turn, and coordinated flagellar beating. Inspired by this class of problems, we investigate locomotion of microorganisms near deformable boundaries. We calculate the speed of an infinitely long swimmer close to a flexible surface separating two fluids; we also calculate the deformation and swimming speed of the flexible surface. When the viscosities on either side of the flexible interface differ, we find that fluid is pumped along or against the swimming direction, depending on which viscosity is greater.
We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin- walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations.
We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed.In the case of no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures, we derive continuum equations, valid in the limit of vanishing spacing between folds, to describe the smooth surface intersecting all the mountain folds. We find that this surface has negative Gaussian curvature with magnitude equal to the square of the fold’s torsion. A series of open folds with constant fold angle generate a helicoid.
Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold, and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in creases with constant curvature and oscillatory torsion. However, relatively softer folds inherit the broken symmetry of the buckled shape with oscillatory curvature and torsion. Our asymptotic analysis of the isometrically deformed state is corroborated by numerical simulations that allow us to generalize our analysis to study structures with multiple curved creases.
Recent experiments have imposed controlled swelling patterns on thin polymer films, which subsequently buckle into three-dimensional shapes. We develop a solution to the design problem suggested by such systems, namely, if and how one can generate particular three-dimensional shapes from thin elastic sheets by mere imposition of a two-dimensional pattern of locally isotropic growth. Not every shape is possible. Several types of obstruction can arise, some of which depend on the sheet thickness. We provide some examples using the axisymmetric form of the problem, which is analytically tractable.


How to Determine Your Golf Handicap
Calculating your golf handicap can be one of the more difficult aspects of playing the game. While most golfers choose to ignore this score, if you intend to participate in tournament play, knowing and understanding golf handicaps and their calculation is incredibly important. Today, there are generally two ways to calculate your golf handicap.
Method 1: Do the Math
The first method of calculating your golf handicap is to simply use the correct formula to calculate your handicap. This is done using at least five golf scores. You’ll need to use your adjusted gross scores, and it will be necessary for you to know the USGA course and slope ratings for the courses you’ve played. Then, you’ll need to determine your handicap differential for every score that you’re planning to enter by subtracting your score from the course rating, and then multiplying that result by 113. You will then divide that result by the slope rating of the course.
Next, you’ll need to determine the number of differentials you’ll be using. You won’t use each differential you’ve calculated; if you’re only entering five rounds, you’ll only use the lowest. If you choose to enter 20 rounds, you’ll use your 10 lowest, and so on. Finally, in the last step you’ll average your differentials and multiply that result by 96 percent, dropping all the digits following tenths. This will be your handicap score.
Method 2: Use a Handicap Calculator
Of course, there are easier ways to find out your golf handicap than using the golf handicap formula yourself. Our golf handicap calculator will help you determine your score with just the press of a single button. There is no need to sit down and try to do all of the mental legwork; just enter your five scores, and our program will calculate your golf handicap for you in a matter of seconds. No pen and paper required! You may have seen other sites offering golf handicap lookup, but they can be unreliable at best. Our golf handicap calculator uses the correct formula to automatically turn your scores into a useable golf handicap for tournament play across the country and around the world. Why not use our easy shortcut?


Introduction to Matrix Algorithms
Matrix algorithms form the backbone of numerous computational tasks, ranging from solving systems of linear equations to advanced data analysis techniques. Immediate mastery of these algorithms is essential for unlocking their full potential and harnessing their power in various applications.
Fundamental Matrix Algorithms
Gauss Elimination Method
The Gauss elimination method is a fundamental technique for solving systems of linear equations by transforming the coefficient matrix into row-echelon form.
LU decomposition decomposes a matrix into the product of a lower triangular matrix and an upper triangular matrix, facilitating efficient solution of linear systems and matrix inversion.
Cholesky decomposition is specifically designed for symmetric positive definite matrices, providing a computationally efficient way to solve linear equations and compute determinants.
Advanced Matrix Algorithms
QR decomposition decomposes a Immediate Matrix matrix into the product of an orthogonal matrix and an upper triangular matrix, offering insights into matrix properties and enabling various applications.
Singular Value Decomposition (SVD)
SVD factorizes a matrix into three matrices, revealing its singular values and singular vectors, and finding applications in image processing, data compression, and dimensionality reduction.
Eigenvalue decomposition decomposes a square matrix into its eigenvalues and eigenvectors, playing a crucial role in numerous mathematical and scientific computations.
Strategies for Rapid Algorithmic Mastery
Practice with Basic Algorithms
Regular practice with fundamental matrix algorithms builds proficiency and lays the foundation for understanding more advanced techniques.
Leveraging Online Resources and Tutorials
Utilizing online resources, tutorials, and interactive platforms provides additional guidance and facilitates self-paced learning of matrix algorithms.
Exploring Real-World Applications
Engaging in real-world applications of matrix algorithms enhances understanding and highlights their relevance in diverse fields such as engineering, finance, and computer science.
Overcoming Algorithmic Challenges
Addressing Computational Complexity
Efficient algorithm design and implementation are essential for managing the computational complexity associated with large-scale matrix operations.
Handling Numerical Instability
Numerical stability issues, such as round-off errors and ill-conditioned matrices, require careful consideration and appropriate numerical techniques for mitigation.
Optimizing Algorithmic Efficiency
Optimizing algorithmic efficiency through parallelization, algorithmic improvements, and hardware acceleration enhances performance and scalability.
Applications of Matrix Algorithms
Matrix algorithms find applications in various domains, including:
- Data Analysis and Machine Learning: Matrix algorithms underpin essential tasks such as regression analysis, clustering, and dimensionality reduction in data science and machine learning.
- Image Processing and Computer Vision: Techniques like convolution and matrix transformations are used extensively in image processing for tasks such as filtering, edge detection, and feature extraction.
- Cryptography and Security: Matrix-based encryption schemes, such as the Hill cipher and RSA algorithm, are used to secure communications and protect sensitive information.
Case Studies: Algorithmic Success Stories
Examples of Algorithmic Breakthroughs
From the development of the Fast Fourier Transform (FFT) algorithm to the breakthroughs in deep learning, matrix algorithms have played a pivotal role in driving innovation and technological advancements.
Impact on Scientific and Technological Advancements
Matrix algorithms have revolutionized fields such as scientific computing, finance, and telecommunications, enabling rapid progress and opening new frontiers for exploration.
In conclusion, immediate mastery of matrix algorithms is paramount for effectively tackling complex computational tasks and leveraging their potential across various domains. By understanding the fundamental principles, exploring advanced techniques, and applying strategies for rapid learning, individuals can unlock the power of matrix algorithms and drive innovation in science, engineering, and technology.


Nonograms are logic puzzles in which you fill in the boxes to reveal an image using the numerical clues on the sides. You’ll adore nonograms if you enjoy solving sudoku puzzles!
But since Nonogram puzzle is a logic game, you can easily get stuck while playing it. Fortunately, there are tips to help you play these interesting and brain-teasing puzzle games. Here are these amazing tips.
- Utilize the mark function.
Knowing what to do lets you solve puzzles without relying on guesswork. Knowing the definition of the mark function will make solving nonograms more enjoyable. You can use this feature to circle every potential place. You can eliminate areas where blocks aren’t by doing this. Mark the areas where blocks should be but are still missing with an x if you have circled them. You can move if you get trapped by doing this and separating the connecting lines of blocks.
- Start with large numbers.
Experts advise beginning with large numbers in a puzzle before moving on to smaller ones. As you fill in the largest numbers, you can work with the horizontal lines or cross-check the vertical lines where they meet to obtain the answer. For instance, you can locate 8 particular square positions if a horizontal line of 10 blocks includes a 9 in it.
Where orientation squares terminate can be easily determined with bold line markers. You can solve every block line as fast as you can with this information. For example, a line of 10 squares has a five-block marker line in the center. If you plan to position a 6, you will know it should probably go on the square to the left or right of the divider. However, understanding patterns is essential to know this.
- Add numbers in single lines
You may quickly determine the solution points for a line by multiplying multiple values in a single line. For instance, adding a line that contains the numbers 4 and 5 will result in the number 9. So, if you’re filling a line with a 10, you’ll know it can be solved.
When three numbers add up to 8, four numbers add up to 7, five add up to 6, and six add up to 5, other solutions might be possible.
- Try counting loudly
Several modern nonogram puzzles have automated counting elements built in to make it simple for you to count squares. If a nonogram you’re playing doesn’t have this option, you can still count by hand and indicate the potential spots. For instance, if the numbers 3 and 4 are next to the number 10, it is better to begin counting on the left and mark 1, 2, and 3, then skip and mark 1, 2, 3, and 4. Repeat the process on the right side. Mark any choices that cross your marked squares with a permanent circle.
Wrapping it up
You won’t ever need to guess the answers if you’ve learned the logic required to solve nonogram puzzles. While some logic puzzles could seem challenging, following these few simple tips will reduce the hassle of searching for solutions.


Comparing Areas of Parallelograms in Mathematics Education
What is a Parallelogram?
A parallelogram is a geometric shape that has two pairs of parallel sides. It is similar to a rectangle or a square, but the opposite sides are not necessarily perpendicular. The opposite sides of a parallelogram are equal in length, and the opposite angles are also equal.
Parallelograms are important because they are used in many real-world applications such as engineering, architecture, and construction. They are also commonly used in mathematics to illustrate concepts such as vectors and transformations. Comparing the areas of parallelograms is an important aspect of studying them, as it helps us understand their relationships with other shapes and their geometry.
How to Compare the Areas of Parallelograms
The formula for calculating the area of a parallelogram is base x height, where the base is one of the parallel sides and the height is the perpendicular distance between the two parallel sides. If two parallelograms have the same base length but different heights, then the one with the greater height will have the greater area. Similarly, if two parallelograms have the same height but different base lengths, then the one with the greater base length will have the greater area.
Another way to compare the areas of parallelograms is to use the formula for the area of a triangle, which is 1/2 base x height. If we draw a diagonal line through a parallelogram, we can split it into two equal triangles. The area of each triangle is half the area of the parallelogram. So, we can compare the areas of two parallelograms by comparing the areas of the triangles that they are split into.
Examples of Comparing Areas of Parallelograms
Let’s take two parallelograms with the same height but different base lengths. The first parallelogram has a base length of 6 cm and the second parallelogram has a base length of 8 cm. Since they have the same height, the second parallelogram will have the greater area because it has the greater base length.
Now, let’s take two parallelograms with the same base length but different heights. The first parallelogram has a base length of 5 cm and a height of 4 cm. The second parallelogram has a base length of 5 cm and a height of 6 cm. Since they have the same base length, the second parallelogram will have the greater area because it has the greater height.
Comparing the areas of parallelograms is an important aspect of geometry and is used in many practical applications. To compare the areas of two parallelograms, we can compare their base lengths and heights or use the formula for the area of a triangle. By understanding how the areas of parallelograms compare, we can better understand their properties and geometry.
What is a Parallelogram?
A parallelogram is a two-dimensional shape that has four sides and four vertices. A parallelogram has two pairs of parallel sides, which means that the opposite sides of the parallelogram are both parallel and congruent. This means that if you were to draw a line across the parallel sides of the parallelogram, you would create two identical triangles. The properties of a parallelogram include opposite angles being congruent and opposite sides being congruent, the diagonals bisecting each other, and the sum of the interior angles equaling 360 degrees.
How do the areas of parallelograms compare?
The area of a parallelogram can be found by multiplying the base (one of its sides) by its height (the perpendicular distance between the two parallel sides). The formula for finding the area of a parallelogram is:
Area = base x height
The base and height of the parallelogram can be any pair of opposite sides, as long as the height is perpendicular to the base. Therefore, the area of parallelograms with the same base and height will be equal to each other, regardless of the congruency of its sides and angles.
However, if the parallelograms do not have the same base and height, their areas will differ. This means that if two parallelograms have the same base, but different heights, the one with the longer perpendicular distance will have a larger area. Similarly, if two parallelograms have the same height, but different bases, the one with the longer base will have a larger area.
It is also worth noting that the area of a parallelogram is equal to the area of a rectangle with the same base and height. This is because a rectangle is simply a special type of parallelogram where all angles are right angles, which makes calculating its area easier since it only requires multiplying its length and width.
In conclusion, when comparing the areas of parallelograms, it is essential to consider the length of its base and height. The area of two parallelograms with the same base and height will be equal, but if either of those measures differ between the parallelograms, then so will their area.
Area of a Parallelogram
Parallelograms are geometric shapes that have two pairs of parallel sides. The area of a parallelogram is the amount of space inside the parallelogram. To calculate the area of a parallelogram, you need to know the length of the base (b) and the height (h). By multiplying the base by the height, you can determine the area of the parallelogram.
The formula for the area of a parallelogram is:
Area = base x height
Area = b x h
For example, if the base of a parallelogram is 6 units and the height is 8 units, the formula would be:
Area = 6 x 8 = 48 square units
The area of the parallelogram would be 48 square units.
It is important to note that the unit of measurement used for the base and height must be the same. If the base is measured in feet, then the height should also be measured in feet. Otherwise, the formula will not produce the correct answer.
You can also use trigonometry to calculate the area of a parallelogram if the lengths of two sides and the angle between them are known. The formula for the area using trigonometry is
Area = a*b*sin(θ)
where a and b are the lengths of two sides and θ is the angle between them.
Overall, calculating the area of a parallelogram is a simple process once you have the length of the base and the height. By using the formula, you can determine the amount of space inside the parallelogram and compare areas of different parallelograms.
Comparison of Areas
Parallelograms are geometric shapes that have two pairs of parallel sides. They come in different shapes and sizes and can be classified based on their properties. One of the most important properties of a parallelogram is its area. The area of a parallelogram is the amount of space it takes up, measured in square units.
When comparing the areas of different parallelograms, it’s important to consider the dimensions of each shape. The formula for finding the area of a parallelogram is base x height. Therefore, the area of a parallelogram will increase or decrease depending on the length of its base and height.
Example 1: Let’s consider two parallelograms: parallelogram A and parallelogram B. Parallelogram A has a base of 5 cm and a height of 8 cm. Parallelogram B has a base of 10 cm and a height of 4 cm. To find the area of each parallelogram, we can use the formula base x height:
Area of parallelogram A = 5 cm x 8 cm = 40 cm²
Area of parallelogram B = 10 cm x 4 cm = 40 cm²
In this example, we can see that even though parallelogram A and B have different dimensions, they both have the same area. Therefore, we can say that the areas of parallelogram A and B are equal.
Example 2: Let’s consider two more parallelograms: parallelogram C and parallelogram D. Parallelogram C has a base of 6 cm and a height of 9 cm. Parallelogram D has a base of 4 cm and a height of 12 cm. To find the area of each parallelogram, we can use the formula base x height:
Area of parallelogram C = 6 cm x 9 cm = 54 cm²
Area of parallelogram D = 4 cm x 12 cm = 48 cm²
In this example, we can see that parallelogram C has a greater area than parallelogram D. Therefore, we can say that the area of parallelogram C is greater than the area of parallelogram D.
It’s important to note that the area of a parallelogram can also be expressed in terms of its diagonals. The formula for finding the area of a parallelogram using its diagonals is ½d₁d₂, where d₁ and d₂ are the lengths of the diagonals.
Example 3: Let’s consider two parallelograms with the same base and height, but different diagonals: parallelogram E and parallelogram F. Parallelogram E has diagonals of 10 cm and 6 cm. Parallelogram F has diagonals of 8 cm and 4 cm. To find the area of each parallelogram, we can use the formula ½d₁d₂:
Area of parallelogram E = ½ x 10 cm x 6 cm = 30 cm²
Area of parallelogram F = ½ x 8 cm x 4 cm = 16 cm²
In this example, we can see that even though parallelogram E and F have the same base and height, their areas are different because their diagonals are different. Therefore, we can say that the area of parallelogram E is greater than the area of parallelogram F.
In conclusion, the areas of parallelograms can be compared using their base, height, and diagonals. By understanding and applying the formulas for finding the area of parallelograms, we can determine which shapes have greater or lesser areas.
Why the areas of parallelograms are comparable?
When we talk about geometry, one of the basic shapes that come to mind is the parallelogram. You may have learned that the area of a parallelogram equals the base times the height, and while this is true, have you ever wondered why this formula works? The answer lies in the fact that all parallelograms are similar to each other.
Similarity means that two shapes have the same shape but may differ in size. Two parallelograms are similar if they have the same shape and all angles are congruent, but not necessarily the same size. This is important because when we compare the areas of parallelograms, we can use proportions to easily calculate them.
Relationship between parallelograms and triangles
Another important relationship to note is the one between parallelograms and triangles. Both shapes have similar features, such as having bases and heights that can be used to find their areas. Additionally, any parallelogram can be divided into two congruent triangles, which helps us understand the similarities between these shapes even further.
One way to find the area of a triangle is to use the same formula as a parallelogram and divide the result by 2. For example, if you have a parallelogram with a base of 6 and a height of 4, the area would be 24. If you divide 24 by 2, you get 12, which is the area of the triangle formed by half of the parallelogram.
Parallelograms and rectangles
Another shape that is closely related to parallelograms is a rectangle. While a parallelogram can have any angle measure, a rectangle has four right angles, making it a more specific type of parallelogram. One way to think about a parallelogram is as half of a rectangle, where the height is not perpendicular to the base.
To find the area of a rectangle, we use a simpler formula: length times width. This formula can also be used to find the area of a parallelogram if we know the height and the length of the side that the height intersects. Essentially, we can think of the height as the width of the parallelogram, and the base as the length, which makes it easier to compare the areas between these shapes.
Parallelograms and trapezoids
Lastly, we can see the relationship between parallelograms and trapezoids. A trapezoid is a shape with one pair of parallel sides and one pair of non-parallel sides. While the area formula for a trapezoid is a bit more complicated than a parallelogram, they share the same base concept of using the height and the length of the side that the height intersects to find the area.
A parallelogram with a right angle is a special type of trapezoid called a right trapezoid. In this case, we can use the formula for finding the area of a rectangle to find the area of our right trapezoid since one of the legs is perpendicular to the base.
The relationships between parallelograms and other geometric shapes are important to understand. Parallelograms are similar to each other, allowing us to compare their areas easily using proportions. Additionally, we can see the similarities between parallelograms and triangles, rectangles, and trapezoids, which helps us better understand these shapes as a whole. Overall, geometry is a fascinating subject full of connections and relationships between shapes that can help us better appreciate the world around us.
Real Life Applications
The concept of comparing areas of parallelograms is not just a math problem that students learn in school but has many applications in everyday life. One of the fields where this concept is widely used is architecture. Architects and engineers use parallelogram-shaped structures extensively in their designs, and they should have good knowledge of finding the correct areas of those structures.
When designing a building, architects have to work with measurements related to floor areas, wall areas, and roof areas. A building with a larger area often requires a stronger foundation, more support, and higher costs for construction. This is where the concept of comparing areas of parallelograms comes in handy as it allows architects to efficiently determine the size and shape of a building that would fulfill the requirements within a budget. For example, in a commercial building, an architect may need to calculate the area of a parallelogram-shaped roof for installation of solar panels or HVAC units.
In civil engineering, finding the area of a parallelogram is also essential for calculating the amount of material required for road construction, pavement, and drainage systems. Engineers use the formula to estimate the amount of asphalt required to pave a parallelogram-shaped stretch of road. It is also useful for stormwater management, where the engineers need to calculate the areas of the parallelogram-shaped drainage basin that can hold a specific amount of water to ensure that the water is channeled appropriately.
The concept of comparing areas of parallelograms finds extensive use in product design. Designers visualize and create a product in a 3D design software before manufacturing it. The software helps the designers to estimate the material required for manufacturing the product.
For example, if a designer wants to create a parallelogram-shaped phone case, it is essential to know the area to determine the amount of material required for manufacturing the case. In this case, the length and width of the parallelogram and the perpendicular height are necessary for calculating the area of the shape. Moreover, artists and sculptors use the concept of comparing areas of parallelograms to create various art pieces such as paintings, pottery, and sculptures. They may use this concept to calculate the area required for creating a specific shape or to create different ratios of areas to make their artwork more visually appealing.
Land surveying refers to the measurement and mapping of land and the related natural features. It finds application in the construction industry, environmental management, and agriculture. The concept of comparing areas of parallelograms is essential in land surveying as the surveyor needs to determine the exact size and shape of plots of land.
Land surveyors use specialized tools and equipment to measure the length and width of the land parcel to calculate its area. The land parcels are sometimes in the shape of parallelograms. For example, a farmer can use the measurements to determine the area of a farm and decide how much fertilizer would be required to accommodate the crop. A land surveyor also uses this concept in construction projects to calculate the land area required for the construction of buildings, roads, or bridges.
The concept of comparing areas of parallelograms is essential in everyday life, and it finds extensive application in architecture, engineering, product design, and land surveying. By understanding this concept, people can make more informed decisions before constructing buildings or purchasing land. The importance of this concept cannot be overstated as it plays a significant role in various industries and has practical implications.
Importance of Understanding and Comparing the Areas of Parallelograms
Parallelograms are one of the most common shapes in geometry, and understanding their areas is essential in various real-life applications. The formula for finding the area of a parallelogram is simple; it is the product of base length and height. The straightforwardness of this formula is exactly what makes it such a useful tool for measuring surface areas in various fields and disciplines. Here are some of the reasons why understanding and comparing the areas of Parallelograms is essential:
1. Geometric Calculations
In geometry and trigonometry, the area of a parallelogram plays a significant role in calculations related to shapes and angles. For example, understanding and comparing the areas of parallelograms is essential when calculating the area and perimeter of a polygon. It is also used in determining the angle between vectors in physics and engineering.
2. Construction and Architecture
Parallelograms are very useful shapes in the fields of architecture and construction. Architects and builders use parallelograms to create a variety of structures and designs. For instance, they can use parallelograms to create roofs, gables, and other shapes. By understanding the areas of parallelograms, they can determine the amount of material required to build these structures, which helps them estimate project costs.
3. Transportation and Logistics
The transportation and logistics industry uses parallelograms in a variety of ways, such as calculating the volume of cargo required for shipment. By understanding the areas of parallelograms, transportation and logistics professionals can determine the size of the containers required to ship products efficiently. This knowledge helps them optimize space in storage and shipping containers, resulting in fewer trips and lower costs.
4. Land Surveying
Land surveyors use parallelograms in their work, determining the size of plots and land ownership boundaries. They can also use the area of parallelograms to calculate the location of roads, buildings, and other structures. Without an understanding of parallelograms, accurate land surveying would be impossible.
5. Agriculture and Farming
The cultivation of crops often requires area measurement since it is crucial in determining how many plants or seeds are needed to fill a given field. Farmers and agriculturalists use the area of parallelograms to estimate the potential yield of their crops and create accurate planting plans. This knowledge helps them maximize their harvests, resulting in better crop productivity.
6. Everyday Life
Parallelograms are prevalent in our everyday lives, such as in packaging and home décor. Understanding the areas of parallelograms can help us measure and calculate the amount of material required for packaging products, such as gift wrapping paper. Measuring the areas of floor spaces, wall surfaces, and ceiling spaces in home decor is also possible by using parallelograms.
7. Education and Learning
Understanding and comparing the areas of parallelograms is essential in education and learning, specifically in geometry and trigonometry. A clear understanding of the formula for finding the area of parallelgrams, as well as other common geometric shapes, is required for students to progress through math courses and understand various real-life applications. Additionally, knowledge of the areas of parallelograms is required when studying more advanced mathematical concepts, such as calculating the surface area and volume of three-dimensional shapes.
In conclusion, understanding and comparing the areas of parallelograms are essential in various fields, disciplines, and everyday life applications. As highlighted in this article, different sectors rely on the formula for finding the area of parallelgrams to calculate surface areas, optimize spaces, improve productivity, and estimate project costs, among others. Additionally, knowing the areas of parallelograms is a basic requirement for learning geometry, trigonometry, and more advanced mathematical concepts.


You may find yourself wondering, sometimes, how certain mathematical computations are relevant to your everyday life.
One helpful example is algebra; many students wonder whether they would ever use what they learn in this subject in the real world.
The reality is that algebra has a wide variety of applications in many different areas of Life.
For instance, firms might use it to determine whether they made a profit or sustained losses.
Even if there is no visual representation of X and Z, this does not imply that algebra has no use in the actual world.
There are quite a few situations where algebra comes in handy in everyday situations.
If you better understand the practical applications of algebra that you would run across daily, you might begin to see it in a new light.
Because algebra covers such a broad domain of mathematics, many mathematicians spend much of their time concentrating on how to manipulate abstract symbols.
However, algebra is a valuable tool in various contexts.
To give one illustration, you should be able to solve any mathematical problem by applying algebraic thinking.
To do this, a high level of mathematical expertise is not required.
Unless you are a programmer, you will be required to solve increasingly tricky equations daily.
Using Algebra Everyday
This article contains some of the fantastic aspects of life in which algebra plays an important role:
Algebra is part of your morning routine.
Setting your alarm clock is an algebra problem.
You use math to estimate how long it takes to get ready or commute.
Calculating profits often requires the application of algebraic operations for business owners and investors.
In terms of intelligence and analytics, it constitutes a very significant component.
Most systems are powered by software or house data centers that heavily depend on mathematical calculations.
Company owners and investors use it to analyze and price stocks in their companies.
It is sometimes beneficial for businesses to get rid of things before such products lose value while being stored in their warehouses.
In addition, those who work in finance need algebra symbols to compute things like exchange and interest rates.
Raising of the Child
It may surprise me, but even very young children can understand and apply algebraic concepts.
As an example, a newborn that is seven months old may already be able to track the paths taken by moving objects.
They can readily compute trajectories of approaching the item, which allows them to determine whether or not they will land close to the thing.
In addition to this ability, kids can estimate the distances between themselves and other things, like toys.
Even if the infants have no idea what these operations are, algebra is nevertheless responsible for making these instances feasible.
Calculation of Health Parameters
Numerous applications of algebraic concepts may be found in physical health and well-being.
For example, those told to reduce their body mass should figure out their body mass index.
Because of this, they can keep an eye on the amount of food they consume.
Calculating body mass index (BMI) requires mathematics in a roundabout way.
Without it, calculating other amounts like the percentage of fat found in the body or the weight lifting at the gym would be impossible.
Estimation and Calculation in Sports
Algebra is used in sports without players realizing it.
For example, basketball players can calculate the power needed for a shot.
It is also used for trajectory calculations in other sports and by commentators to understand ratios.
Algebra was necessary to develop almost all of the technology we use today, including televisions, mobile phones, and computers.
All of these widgets are available due to the efforts of computer wizards who are skilled in mathematics and responsible for making them a reality.
In addition, search algorithms are developed by working through complex algebraic problems.
Delicious Meal Prep
Algebra is useful in the kitchen when making meals like cake and soup.
It helps determine ingredient quantities for different serving sizes, ensuring the dish is delicious and accessible.
It’s an essential tool in the culinary arts.s.
Construction and Renovation
Because algebra deals with numbers, it is essential for anyone interested in DIY house remodeling.
Knowledge of elementary algebra is necessary for improving or estimating the number of resources, such as the cost and the required supplies.
A plumber, for instance, may do a few simple calculations to determine.
Electricians perform multiple steps to ensure proper electrical circuit construction in homes.
Learning algebra in middle or high school can benefit you for life.
It is crucial for academic success and job opportunities.
When you decide to settle down and have a family, algebra may greatly assist you.
It can assist you in house hunting, budget planning, and family insurance selection.
Algebra covers almost all aspects of your life and benefits you in every stage.


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What Is Half Of 1 4 Cup In Tablespoons?
Half of 1/4 cup in tablespoons is equal to 2 tablespoons. An easy way to remember this is to think of 1/4 cup as equal to 4 tablespoons, so half of that is 2 tablespoons. To make this even easier, you can also remember that 1/2 cup is equal to 8 tablespoons, so half of that is 4 tablespoons, which is the same as 2 tablespoons.
What is half of 1/4 cup in TSP?
It is important to note that the measurements of teaspoons and tablespoons used in the United States are different from those used in other countries. Thus, the conversion factor for other countries may differ.
What is half of 1/4 cup tbsp?
In conclusion, half of 1/4 cup tbsp is equal to 2 tablespoons (1/8 cup). This is an important conversion to remember when measuring ingredients for baking and cooking, and it is important to keep in mind the size of the measuring spoon when measuring out ingredients.
What is 1/4th of a cup in tablespoons?
When measuring ingredients, it is important to be precise. This is especially true when baking, as even a slight variation in measurement can have a big impact on the finished product. A good trick for measuring ingredients accurately is to use measuring spoons and cups that are labeled with measurements in both cups and tablespoons. This makes it easy to quickly determine how many tablespoons a given measurement is, such as 1/4th of a cup.
Is 3 tablespoons equal to 1 4 cup?
One tablespoon is equal to 3 teaspoons, so it is very easy to convert tablespoons to cups. For example, if you need to measure one cup, simply use 16 tablespoons or 48 teaspoons. Similarly, if you need to convert from cups to tablespoons, simply divide the number of cups by 16, and you will have the number of tablespoons.
This can be useful when you are measuring ingredients for a recipe and need to convert from one unit of measurement to another. Just remember that 3 tablespoons is equal to 1 4 cup, and you can easily convert between the two.
The answer to the question of what is half of 1/4 cup in tablespoons is 2 tablespoons. This is an easy conversion to remember, as 1/4 cup is equal to 4 tablespoons, so half of that would be 2 tablespoons. This is helpful when measuring out ingredients for baking or cooking, as it can be difficult to accurately measure out 1/4 cup of a dry ingredient. Knowing this conversion can make the process much easier and more accurate.