Multiplying by 12
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Better to see it than me write it. Just separate the number being multiplied by 11 in this case, also 11 so that there's room for your number in-between. That gives you The slide it in-between the 5 and 8 and you get But, that's not right, since you need to carry that one over. Go ahead and carry it over and you'll end up with This base method uses the powers of 10 tens, hundreds, thousands, etc.
Again, much better to see what I am saying then trying to read it. Here's an example of multiplying two, two-digit numbers a base of hundreds. Then he cross-adds one set of numbers either pair would work to get the first number of the answer.
- A Multiplication Table.
- What’s Wrong with Multiplying by the Square Root of Twelve (Digest Summary)!
- Math-Trick's - Multiplying by 12 shortcut - Wattpad;
So, that'd be the first part of the answer. Now multiply the two smaller numbers x to get the second part of the answer. So, that makes the answer This same method works for any base of ten.
You can see this at around the mark in the video. The cool thing about math is watching how seemingly impossible combinations seem to walk out perfectly in the end.
12 times table worksheets PDF - Multiplying by 12 activities
By doing certain operations, you can turn wildly complex equations into simple, step-by-step solutions. Using the Vertical and Crosswise Pattern, we can easily multiply large two-digit numbers like the one pictured below. First, we multiply vertically up the right side. So, 8 will be the last digit in our answer. Next, we cross-multiply. Carry over the 1 like you normally would, and you are left with 0 , which will go in front of the 8 we already have. Lastly, we vertically multiply up the left side.
Place that in the front of our answer line and we get If you're more of a visual learner that really hates numbers, you can also go all Japanese on them and substitute those digits for lines, like YouTuber kimelicious does. I'm not going to explain this one—just watch and you'll see. There are some really amazing math tricks using Vedic-style mathematics, so be sure to watch the full video to get a firmer grasp on it and start using it in your everyday life. Does that work with problems that ends with 0 at the end of the answer? Just wondering. I just want to learn the new way to do math.
Multiplying by Twelve (12) with Factors 1 to 12 ( Questions) (A)
Then add them together! Students have to remember a lot of numbers when looking at a multiplication chart. Prodigy can help you teach multiplication charts, assign specific questions and track how students are doing in real time — at no cost!
- Multiplication Tips and Tricks.
- LAmphitryon [French & English Bilingual Edition] - Paragraph by Paragraph Translation (French Edition)?
The game keeps students engaged with content through an exciting story and loads of features. Reports can then tell you which topics students are struggling with. Prodigy creates a personalized learning experience for every student, so they can practice the skills they have trouble with. This allows all students to learn at their own pace. Meanwhile, from the teacher dashboard, you can create assignments and practice tests, view statistics, and plan lessons. Prodigy is a free tool for educators and will remain free — forever!
In this game, students race against the clock to complete as many equations as possible. Supply students with a worksheet that has several columns of equations. For example, one column will only have the four multiplication table while the next has the seven multiplication table.
Once every student has a worksheet, give them a countdown of 60 seconds to answer 20 questions. Then have students tally up their answers to see how they did. This exercise is just for fun, but adding an incentive — like crowning a class champion — can get students more interested in the activity. Try to do this activity a couple of times a week. Eventually, students will have no trouble filling out entire columns of questions correctly in less than 60 seconds! Try this free table race template in your class! This game can be played as a class or in groups.
Bingo sheets will have the solutions to multiplication chart equations in a five-by-five grid. You can fill out these charts yourself, or have students create their own bingo sheets to get them a little more involved. Now, use dice or a random number generator to get your equations. Students then solve the equations to get the numbers on their bingo cards. If students are working in groups, have them write down the equations so you can check their work.
For dice, roll a pair of them and add the values to get the first number. Then roll the pair of dice again and add the values together to get the number to multiply by.
Trachtenberg method for multiplying by 12…19
For the random number generator, randomize two numbers from and use them for the equation to solve. Once students have solved the equation they can check their bingo sheet to see if the value is there. This is a great way to get students to think critically and avoid simple memorization of multiplication tables in list form.
This game creates a class competition to see who can complete their multiplication tables the quickest — and more importantly, the most accurate. Shuffle up the entire multiplication chart and have students compete to see who can answer questions the fastest. The trick is, as soon as a student answers a question wrong, their climb is over.
Create a visual board for students to look at with a picture of a mountain. Students will love competing to reach the top. Make an event of this game. Students can answer questions at the front of the class to try for the mountaintop title. Students will be excited to compete and showcase their skills. Multiplication is one of the most common forms of math that we encounter every day. Learning the basic multiplication chart is essential for young students.
It will help them throughout their educational career and their daily lives. Use our free printable multiplication chart in your classroom to teach your students how to do simple multiplication quickly and flawlessly. Our list of tips and games makes multiplication easy and fun. Ryan is a Junior Content Writer at Prodigy. Your email address will not be published. Loved by more than , teachers and 30 million students, Prodigy is the world's most engaging math game and platform.
And it's free for everyone. Keep up with our blog's research-backed advice by signing up for your Prodigy account now! Printable multiplication chart We have two multiplication charts available for your class — one for reference and one blank template for students to complete themselves.
Download them here!


Posted by Jon | Apr 24 2016
10 Free Maths Revision Resources Every Student Should Know
Whether you’re a maths student or a tutor, there are many fantastic resources online to help you. Best of all, they’re mostly free! People create these resources out of a love for maths and a genuine desire to help others.
If you’re a student, then you should find these resources a really helpful boost to your maths revision at GCSE or A-level. If you’re a tutor or a teacher, you should find these resources helpful for your lessons, and you need to get your students using them ASAP!
Here are the resources, in no particular order.
1. Mathway – Live equation editor and homework generator
Alright, so the idea of a ‘maths homework generator’ might send a chill down your spine, but let’s be honest, doing worked examples is by far the best way of revising for maths exams.
By the time of writing this, I haven’t found any other website that generates free maths homework for any topic, so you need to check it out if you want to try out more maths problems for your revision.
The main feature of Mathway though is the live equation editor and solver. Using the symbols, it uses LaTeX (a maths formatting language) and understands the input and recognises what type of problem it is.
If it’s a quadratic equation, it will recognise that and will offer to try to solve it for you. Really useful!
Mathway also has a premium option where if you pay, you can get explanations behind these solutions.
2. Paul’s Online Notes – More advanced maths revision for A-level and beyond
Professor Paul Dawkins is a professor of mathematics at Lamar University in Texas, USA, and has constructed a massive resource of advanced maths revision.
I personally used it all the time during my maths degree! It really is one of the best advanced maths revision resources available.
It is mostly used by university students, but it does have plenty of study materials appropriate for maths at A-level. He really starts from the basics and moves forward.
He also has a great list of links to other maths revision resources, both free and premium, which I recommend you go and check out.
If your plan is to study maths or a STEM subject at university, and you want a taster of university maths, I highly recommend you check out the more advanced sections.
3. Physics and Maths Tutor – London maths tutor’s website
Gorkem Cavli is a fellow maths tutor in London and has a comprehensive list of all maths past papers at GCSE, A-level, and more.
As a tutor I’m constantly using his maths past paper page to download PDFs of past papers. It’s got all the exam boards and nearly all the modules.
If you’re a student them I’m sure you’ll find it invaluable for your revision too. I get all my students to use it!
It’s really handy to have a go-to page that you can always rely on to download maths past papers from.
I tried to reach out to him via email but he hasn’t responded yet. Oh well!
4. Wolfram Alpha – Solve any maths problem in seconds
Wolfram Alpha is a website that can solve any maths problem, and provide a clear picture on any equation.
It’s powered by a supercomputer that can recognise what you type in and provide a solution. It’s almost like magic – most students’ jaws drop when they see it work for the first time.
Go ahead and type in x^2 + 2x + 1 and it’ll recognise this as a quadratic equation, solving it for x, plotting it on a graph, and more advanced stuff too.
My favourite use for it is to quickly plot a graph of a function – it can do this in 2D and 3D!
It’s not just for maths though – type in ‘population of the UK’ and it’ll show you details population figures and statistics! It’s an amazing website.
5. Mr Barton Maths – Math teacher’s resources for GCSE maths
I don’t tend to take on many GCSE maths students, but I tend to find this resource useful.
His past paper page isn’t anywhere near as comprehensive as Physics and Maths Tutor’s in my opinion, but I think the website is clearly laid out, and the resources are completely free.
His GCSE maths grade passports are a bit of fun and really useful for checking up on what grade you’re likely to get with your current knowledge. This can be a bit scary and a good motivational tool for your GCSE maths revision!
As a bonus for teachers, his new site Diagnostic Questions might prove useful for your teaching efforts. It’s not free but it looks like a useful resource for sharing notes and lesson and homework plans with other maths teachers.
6. Exam Solutions – Maths tutor’s A-level video lessons and past paper worked examples
Stuart Sidders has done a tremendous job in creating an incredible 3,600 videos. These are of him doing maths lessons from basic AS-level topics to A2 further maths, and worked examples.
The A-level past paper worked example videos are the best part by far. On every single question in past papers, he will record a video of him solving it, and link to it straight on the page.
It really is one of the best resources out there.
Even I find his worked examples videos very useful if I find myself stuck on a topic! (It happens to the best of us!)
If you’re not already using Exam Solutions’ past paper worked example videos, you definitely should be!
7. GCSE.com Maths – A resource full of GCSE maths notes
Once again, I don’t tend to take on many GCSE maths students, but I’ve been impressed by this free resource.
Whoever runs the website (I think it’s Andrew Harmsworth) writes in an informal way with very clearly-written notes and explanations.
All the core GCSE maths topics are on there, so it’s a useful addition to your GCSE maths revision.
As an aside, “GCSE.com” is a cool domain name!
8. Bitesize Revision – The BBC’s GCSE maths revision notes
This is already quite a well-known one. The BBC have done a great job keeping a comprehensive set of resources for maths (and many other GCSE subjects) up-to-date with new exam specifications in 2015.
Not only do they fill the resources with lots of helpful examples, but I think the diagrams and visual examples are fantastic too.
They’ve also started uploading some decent videos with the revision pages, and link out to other things like BBC podcasts about maths.
I love visual examples of maths, and I sometimes turn to the Bitesize website to help explain a topic!
9. OpenStudy – Ask for maths help and get answers immediately
A bit like Yahoo! Answers but for maths, OpenStudy is a great website that I’ve recently discovered.
You can get your questions seen by thousands of users at the touch of a button, and get immediate help on how to solve your question.
It has quite a big range of level too – you can get help on basic GCSE core topics all the way up to A-level maths and degree level maths.
The users there are knowledgeable and helpful. The more prolific users tend to hang around the website all day, so you can be sure of getting help with most maths problems from your revision.
Links are public and saved, so they encourage you to search the site to see if someone has asked the same question as you already.
10. TexPaste – Maths equation and revision note editor
TexPaste is a “pastebin”, which is a type of website that lets you write stuff and save it for later.
Huy Nguyen created TexPaste to allow you to write maths documents, and then save them for later and print them out for your revision.
I think tools like this are highly under-utilised by schools.
Just think about how useful this is. You can write your maths notes, with maths equations and everything, and then share them with other people and print them out.
If you’re a maths tutor or a teacher, start using tools like this right away!
It’s a fantastic addition to hand-writing notes, because with TexPaste and other tools like it, you can make the notes really neat and clear. They can be much easier to read than human handwriting.
It’s also really useful for learning the LaTeX language too. If you’re planning on doing a degree in maths or another STEM subject, you’ll be required to write your dissertation and any other papers in LaTeX, so this is good pre-university practise.
As a top tip for your UCAS university applications, they will be very impressed if you’ve taught yourself how to write LaTeX.
It’s completely free too! You should definitely create an account and use this website whether you’re a student or a teacher.
(I even made a website like this myself! – Link)
(Bonus) 11. Further Maths Tutor – My own website!
Alright it’s a bit cheeky, but I’m also putting forward my own website.
I feel really passionately about encouraging students to take up further maths at A-level, and so I made a website with study materials for further maths modules FP1, FP2, and FP3, for the Edexcel and OCR MEI exam boards.
I worked really hard making these revision materials, and thoroughly enjoyed it. I hope you find them useful, and please share them as much as possible with your further maths friends!
More from the blog:


1 + 1 = OLICO Maths Education
Our support of OLICO Maths Education just adds up (see what we did there?), the lasting impact and confidence that children develop from having a clear understanding of their schoolwork can be seen for decades to come.
OLICO Maths Education (formerly known as OLICO Foundation) facilitates Maths Clubs that are designed to improve the core strands of maths proficiency: procedural fluency, conceptual understanding, problem-solving, reasoning and disposition toward maths.
The OLICO Foundation (OLICO) began operating Maths Clubs from mid-February 2022, as they had to wait for schools to complete their own enrolment processes. Maths Clubs take place at OLICO’s centre and on-site in partnering schools. At the schools, OLICO works with learners from Grades R to 6, while at the centre, they work after school with learners from Grades 4 to 6.
The target for enrolment in 2022 was 1 000 learners; however, OLICO anticipated this target would be exceeded by March and prepared accordingly. Total enrolment has now reached 1 346 learners. To meet the increased demand, OLICO recruited a new facilitator in March and recruited two more facilitators in April. In addition to this, two of their ex-students are also volunteering in the clubs.
The current ratio is 1:15 and there are almost 100 clubs operating. Learners at the centre clubs have completed Early Grade Mathematics Assessment (EGMA) registration; this is still in progress for school-based clubs.
Name change: TIMCA Global NPC
OLICO Foundation is now registered as TIMCA Global NPC. A few years ago, OLICO Foundation decided to split its Maths and enterprise activities. Maths activities now fall under OLICO Maths Education and enterprise activities fall under TIMCA Global.
“Without mathematics, there’s nothing you can do. Everything around you is mathematics. Everything around you is numbers.”— Shakuntala Devi


Calculate a Moving Average
A Moving Average is a function that is often used to filter out the noise of random price fluctuations and get a better idea of the trend. A moving average takes into account the value of the current row of data, as well as the surrounding rows. For example, if you have weekly sales data, a 4 week moving average would take into account the current week and the previous 3 weeks of data. In Sigma, we can set this up very easily.
Moving Averages are a type of Window Function. Windows functions act on the group level they are placed in, and are affected by the order of the rows. The first step to setting up a Moving Average is therefore setting up your Groups, and the sorting for your worksheet. For an accurate Moving Average, the worksheet must be uniquely sorted. Multicolumn Sorting Criteria and Groupings with no duplicate values are both ways to ensure a unique sort order. Once your data is grouped and sorted, you can accurately apply a Moving Average.
When Calculating a Moving Average:
Create your Groups
Create a unique sort order
Enter your Moving Average formula
Looking again at the example of a four week moving average of weekly sales data, lets work through the three steps. We want to see a four week moving average broken down by store and department. We start with a Worksheet with department level weekly sales data from several stores each with several departments. Our first step is to group the data by Store, then create a second level grouping by Department. Within each Department there should only be one entry every week for Weekly Sales, allowing us to Sort Ascending for a unique sort order. To ensure that the sort order is unique, we can use multi column sorting criteria and add a second sort criteria that will be used to break ties.
Now that the set up for the window function is done, we can calculate our Moving Average. Create a new column and enter in MovingAvg([WEEKLY SALES], 3). This tells the function to look at the column [WEEKLY SALES] and average the current row with the three above it. This computes a four week moving average. If we wanted to sort our weekly data Descending, we could use MovingAvg([WEEKLY SALES], 0, 3) to average the current row with the three rows below it. The function is confined by the groupings, and will only use the rows in its own group in calculating the averages. The result is a department level four week moving average for every department in every store.


Dilution calculations are common in both chemistry
courses and in lab. The calculations are actually quite
simple to perform as they only use one equation M1V1
= M2V2 where
M1 = the initial concentration
V1 = the initial volume
M2 = the final concentration
V2 = the final volume
The concentration is usually given in M (mol/L) and the
volume can be in any unit you want, as long as you use the same units
for V1 and V2.
"What volume, in mL, of
concentrated sulfuric acid
(18.0 M H2SO4) is needed
to prepare 2.50 L of a 1.00 M
solution? How would I do this problem?"
We'll start with our equation
M1V1 = M2V2
where M1 = the initial concentration (18.0M)
V1 = the volume of that solution you will need
M2 = the final concentration you want (1.00M)
V2 = the final volume of solution you want
(2500mL, or 2.5L)
so you plug the numbers in, we're going to use 2500mL so it gives us
our answer in mL.
18.0 (V1) = 1.00 (2500mL)
V1 = 2500 / 18
V1 = 138.89 mL
So you would measure out 138.89 mL of sulfuric acid and dilute to
2500mL (Add 2361.11 mL of water)
*** VERY IMPORTANT - do not pour water into a
concentrated acid solution. The mixing of water and a
concentrated acid is an exothermic process (creates heat).
Instead, get about 1/2 to 3/4 of the water you need, add the
concentrated acid to that, then finish diluting with the rest of your
Did I miss something?


Look at a few examples:
What do absolute values do to positive numbers?
What do absolute values do to 0?
What do absolute values do to negative numbers?
MAKE THEM POSITIVE!
That's it, Baby! That's as hard as it gets!
Why are the answers to absolute value guys never negative?
Because absolute values are distances!
Check this out:
The real way to look at absolute values is that they are a number's distance from 0 on the number line:
Use a number line to show the answers to these


Roulette is a game that was originally born in France during the 17th century. Its name in fact refers to the wheel that is used to determine a winning number during each turn of the game. But before spinning the wheel, every player can bet on a number from 0 to 36 (some versions even include a 00). Players can also bet on groups of numbers, adjacent numbers, or if the number will be odd or even. Another possibility is to bet on the color of the numbers. In fact each number can be red or black. All these possibilities are represented on the common table, that is divided into sectors, where each sector represent a number or group of numbers. Obviously each sector has a different multiplier when determining how much a player wins.A legend says that François Blanc supposedly bargained with the devil to obtain the secrets of roulette. The legend is based on the fact that the sum of all the numbers on the roulette wheel (from 1 to 36) is 666, which is the “Number of the Beast”.


The International Group for the Psychology of Mathematics Education (PME), is a group of researchers, established in 1976 at the International Congress on Mathematics Education (ICME3) in Karlsruhe, Germany. The aims of the organization are:
- to promote international contacts and exchange of scientific information in the field of mathematical education;
- to promote and stimulate interdisciplinary research in the aforesaid area; and
- to further a deeper and more correct understanding of the psychology and other aspects of teaching and learning mathematics and the implications thereof.
IGPME welcomes all who are interested in how students learn mathematics, how teachers teach mathematics, and how mathematicians, teachers and students do mathematics. Please feel free to explore this website. If you would like to join us as a member, please contact IGPME via its Administrative Manager.
IGPME gathers once a year to share our work and interests at our Annual PME Conference. For more information about the conference, please refer to the conference page.
If you have questions, you can send an e-mail to our Administrative Manager.


This program will calculate the approximate weight and creep of an individual printed item (magazine, newsletter, etc.) or the total weight of the job.
If you are concerned about the unit weight of your publication for mailing purposes you may want to try our calculator below.
By choosing “1” for the quantity and entering the specifications related to your enquiry, our calculator will give you an approximate unit weight.
If you need help with how to use this calculator, please feel free to ask one of our helpful Sales team to guide you.
The above figures are approximate only and the actual may vary.


Let the crapshoot begin
So those of you who have nothing better to do than remember everything Iâ€™ve written here will of course recall my periodic forays into Bradley-Terry Ratings for baseball.Â Itâ€™s time to use math to predict the playoffs.Â Why not?Â It canâ€™t be worse than anyone elseâ€™s predictions, can it?Â It may not be better, but no one will ever be able to prove that itâ€™s worse.
A brief recap: to make Bradley-Terry Ratings, you need a dataset in which everyone has played everyone else a number of times, and you create a single number to represent the strength of that team such that you come closest to predict the actual record of that team in expectation over the season when every match between two teams is of the form:
Pr(Road Team Win) = RoadRating/(HomeRating x HomeFieldAdvantageFactor + RoadRating)
Letâ€™s start with the rankings for the playoff teams.
There are a number of things you have to make Bradley-Terry ratings.Â The first is to decide the period over which they are calculated.Â The standard method is to use that seasonâ€™s games.Â Under that criterion, here are the ratings:
But of course using the whole seasonâ€™s games can be misleading.Â Teams can change dramatically over the course of the year, as a certain team which retooled at the deadline last year might remind you.Â What if we just ranked teams based on July-October?Â That would yield the following rankings:
You have to drop all the way down to 16th place to get all 12 playoff teams.Â And the Braves rise to 2nd. (This proves that the whole season matters. Without those first two months, the Yankees would be fishing now.)
One more entirely biased method is to go from June 1 to the end of the year.Â That yields this:
Ahhâ€¦ thatâ€™s more like it. Now the right team is the best team.
Thereâ€™s an important lesson here even before we begin playoff simulations. Anyone who claims to be able to rank teams must have in their mind something that is the team and something about the timeframe they are measuring over. None of that is a guarantee of performance in the playoffs, but you wonâ€™t get any better predictions out that the strength of the team you assumed going in.
So we can start with the most pessimistic Bravesâ€™ method: the full year estimate, which is standard. I’ll go through the playoffs in some detail, and then circle back to show the change in probabilities from only using post-May data.
The next step is to assign a home field advantage.Â I am really skeptical of trying to measure a playoff home field advantage, so Iâ€™m just going to go with the sort of numbers Iâ€™ve seen in the literature and go with a 1.1 multiplier.Â This really only has a big effect in the Wild Card round, where the better team gets up to three home games.Â The following table gives the Wild Card round estimates:
|Wild Card Round|
For each team, this gives their probabilities of winning a game in the series and their chances of winning two out of three. So the Padres, for example, have a 42.9% chance of winning a game at Citifield, which gives them a 36.1% chance of winning two out of three. The last two columns are just the complement of columns (2) and (3). Note, by the way, how little a 2-out-of-three format changes the probabilities. That’s the basis for the basic insight that if you want to upset someone, your best chance is a short series.
Now we get to the LDS round.Â The lack of reseeding under the new schedule makes this pretty easy. When we look at all the possibilities, we get this:
|Game 1 Visitor||Game 1 Home||Matchup Probability||PR Vis||PR Home|
I have sorted these by the probability that the matchup occurs. Since the Mets have the highest probability of winning their Wild Card Series, it gets the first row. I list the Visitor in Game 1 and the Home Team in Game 1, the probability that matchup occurs, and then the probability that the Visiting Team (in Game 1) survives, followed by the complement, the probability that the other team survives. All the home teams are substantial favorites, which you’d expect.
Next comes the LCS.Â Following the same structure as the previous table (though now we are playing 7 game series) we get the following:
The probabilities here require both teams to survive to this round, which is just the multiplication of each teams chances of surviving the second round.
Finally, we come to the 36 possible World Series matchups, sorted by their probability of occurrence, and in the same format as the previous two tables:
One thing you can do with this table is calculate fair odds. If I wanted to bet that Seattle beats the Padres in the World series, I ought to get around 1,000:1 odds.
We can then cumulate by team across this table and get the full crapshoot probabilities.Â The chances of winning the World Series are:
|Team||BT Rating||Championship Probability|
How robust are these probabilities?Â Well, suppose we use team rankings since June 1st.Â The revised probabilities are:
|Team||BT Rating||Championship Probability|
Thatâ€™s more like it, although the Braves are barely better than 4:1 odds. But it’s hard to refute these probabilities. They don’t say the Padres won’t win, only that it’s about 150-1 against. Current Vegas odds are around 30-1 on the Padres. That’s a terrible bet.
On the other hand, the current Dodger odds of about 3.5:1, while not exactly fair. are not strikingly unfair either. But the Braves odds of 6:1 are really good if you think the last four months are who the Braves are.


Mark Fiecas received his PhD degree in Biostatistics from Brown University. His research was on connectivity measures in brain imaging. Mark is now a post-doctoral scholar in the Department of Psychiatry at UC-San Diego.
Cristina Gorrostieta received her PhD degree in Biostatistics from Brown University. Her dissertation research was on complex dependence measures in multivariate time series analysis.
Collaborator Giovanni Motta (Maastricht Univ) visited UC-Irvine from Jan - March 2012.
Hernando Ombao started his new position as Associate Professor in the Department of Statistics at UC-Irvine.
Hakmook Kang received his PhD degree in Biostatistics from Brown University. His research was on Spatio-Spectral Analysis of functional magnetic resonance imaging data. Dr. Kang is now Assistant Professor of Biostatistics at Vanderbilt University.
Devin Koestler won the Student Paper Award which was presented at the ENAR conference in Miami, FL. His paper, co-authored with H. Ombao, was on forecasting census counts using both seasonal and patients' clinical data.
Hakmook Kang is the 2011 winner of the John van Ryzin Award for Best Paper. The award was presented at the ENAR conference in Miami, FL.
Dan van Lunen received his ScB degree in Applied Mathematics from Brown University. Dan wrote a thesis on online change-point detection under the direction of H. Ombao.
Mark Fiecas received one of the Student Awards for his paper on generalized shrinkage for estimating partial coherence. The award was presented at the New England Statistics Symposium at Harvard University.
Dan van Lunen, undergraduate student in Applied Mathematics, received the summer research training award to compare diffusion tensor imaging data recorded under different scanning protocols.


From what I can see, no. I had both a Destiny Knot and Everstone on my Pokemon for both the Tyrunt and Honedge and both Pokemon hatched shiny at opposite sides of the spectrum. If yours is taking awhile just remember, the chance to hatch is the same for each egg.
For those interested, the exact probability of encountering a shiny Pokemon is calculated as follows. The game generates four random numbers ranging from 0 to 65535. These are represented as binary numbers, 16 bits in length. The bitwise operation "exclusive or" is performed on each of the numbers, resulting in a single binary number in the range 0 to 65535. If this number is less than 8, the Pokemon is shiny. Hence the probability of this occuring is 8/65536 or 1/8192.
Pokemon Shiny Gold was released in 2006 after fears that nintendo would not revisit the Jhoto region. It is a hack of Pokemon Fire Red and is a near identical remake of Pokemon Gold, with a few exceptions of course.


Math Calculation Problems – Your “Achilles Heel”experts are online now
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Upper-level math courses are tough – but you already know that, don’t you? You have been working on some calculation problems for days, the assignment is due, and you cannot figure out why the correct answers are not coming. You have either missed a step or made a simple math error somewhere, and now you face going back and starting all over again or just giving up. Don’t give up and don’t start over. There is a third solution!
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You are so ready to meet your friends for a beer – admit it! And you can do just that if you just access the order form at Essays Deluxe, request solutions to your math problems, and upload the original problems. While you are out enjoying your night, a personally assigned mathematician I hard at work on your problems. Chances are, they might even be finished when you get home – if not, certainly by whatever deadline you have given us. We really are that fast.
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Inseikai Tohoku Spring Camp 2023 -- Mathematics & Computation
My friends (Zhang Ye, and Zhao Yikai) and I organized a Math Camp to assist new Master’s/ Ph.D. students with mathematics and computation, opened on campus at Tohoku University. The camp was concluded on Friday (March 24, 2023).
For mathematics, the content covers most of the techniques commonly used in multivariable calculus and static optimization. For computation, we taught introduction to Python, and how to use them in economic modeling. The content mainly follows Essential Mathematics for Economic Analysis (2016) by Sydsaeter, Hammond, Strom & Carvajal. Although the materials are not as rigorous as other formal textbooks (such as Simon & Blume (1994), Chiang(1967)), it is sufficient for practical treatment of doing and presenting mathematics in writing papers. Our notes cannot substitute for the original book, and we greatly recommend students buying these books.
The lecture notes have been uploaded to https://github.com/thanhqtran/spring23_camp.
For the next camp, we intend to cove Differential Equations and Dynamic Optimization, whereas the Computation can extend to cover techniques in MATLAB and econometrics with R.


An Introduction Statistics is a set of tools used to organize and analyze data.
Added to Your Shopping Cart Add to cart Description A complete set of statistical tools for beginning financial analysts from a leading authority Written by one of the leading experts on the topic, An Introduction to Analysis of Financial Data with R explores basic concepts of visualization of financial data.
Through a fundamental balance between theory and applications, the book supplies readers with an accessible approach to financial econometric models and their applications to real-world empirical research.
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability – p. 1/ Outline Basic probability concepts Parameters vs. Statistics A parameter is a number that describes the population. Usually its value is unknown. In this course, you will learn R via your existing knowledge of basic statistics and gain a familiarity with R to use it to conduct statistical analysis. In this course, you will learn R via your existing knowledge of basic statistics and gain a familiarity with R to use it to conduct statistical analysis.
The author supplies a hands-on introduction to the analysis of financial data using the freely available R software package and case studies to illustrate actual implementations of the discussed methods.
The book begins with the basics of financial data, discussing their summary statistics and related visualization methods.
Subsequent chapters explore basic time series analysis and simple econometric models for business, finance, and economics as well as related topics including: Linear time series analysis, with coverage of exponential smoothing for forecasting and methods for model comparison Different approaches to calculating asset volatility and various volatility models High-frequency financial data and simple models for price changes, trading intensity, and realized volatility Quantitative methods for risk management, including value at risk and conditional value at risk Econometric and statistical methods for risk assessment based on extreme value theory and quantile regression Throughout the book, the visual nature of the topic is showcased through graphical representations in R, and two detailed case studies demonstrate the relevance of statistics in finance.
A related website features additional data sets and R scripts so readers can create their own simulations and test their comprehension of the presented techniques.
An Introduction to Analysis of Financial Data with R is an excellent book for introductory courses on time series and business statistics at the upper-undergraduate and graduate level.In this course, you will learn R via your existing knowledge of basic statistics and gain a familiarity with R to use it to conduct statistical analysis.
Installing the Data Analysis Tool One tool that we will use heavily to complete data analysis is the Data Analysis package.
This can be found under the Data tab. Analysis and interpretation of data An Introduction to Interaction Analysis Tyler J VanderWeele Departments of Epidemiology and Biostatistics Harvard School of Public Health Gerhard Bohm. click on the list of catgories on the list below You may view or hide descriptions of the CHAPTER 1: INTRODUCTION TO STATISTICS 3 student fidgeted Presenting a spreadsheet with an ad analysis .
An Introduction to Statistical Learning with Applications in R. Web based materials for teaching statistics.
Analysis tools VassarStats by Richard Lowry Text Analysis of Variance (PDF) by B. Gerstman. The power of p values¶. Statistics provides the answer. If we know the distribution of typical cold cases – roughly how many patients tend to have short colds, or long colds, or average colds – we can tell how likely it is for a random sample of cold patients to have cold lengths all shorter than average, or longer than average, or exactly average.


ВИНТНЕР (WINTNER AUREL FRIEDRICH)
08.04.1903, ВЕНГРИЯ - 15.01.1958, ?
Венгерский и американский математик. Поместил небесную механику на более надежный математический фундамент. В русском переводе вышли "Аналитические основы небесной механики" ("Наука", 1967).
Aurel Freidrich Wintner
Born: 8 April 1903 in Hungary Died: 15 Jan 1958 Aurel Wintner published the first proofs of the basic facts about Hilbert spaces in 1929. In 1930 he married HЎlder's daughter and, in the same year, took up a post in the USA where he remained until his death. In 1944 he became editor of the American Journal of Mathematics. He published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener). Wintner (along with Poincarщ and George Birkhoff) placed celestial mechanics on a more sound mathematical basis. These innovators were more concerned with the underlying theory, less concerned with quantitatively accurate prediction of celestial body motion.
JOC/EFR December 1996
Дата последнего редактирования:


• A line graph represents change over time, and a scatter plot shows the relationship between two variables. This is a fast activity, but will help you see where students … That's why I love math mazes, like this one for rational and irrational numbers. different activities you can do with task cards. Jun 24, 2017 - Explore Nanci Madsen's board "Box Plot" on Pinterest. Literacy Center Ideas. Your story begins with two students inside the school, a note from the principal and a fish bowl.
Task cards include multiple choice problems and constructed response. Math Games Middle School Probability Activities Maths Activities Middle School Plot Activities Feedback For Students Math Review Game Math Time Scatter ... and clusters. It covers topics for "Scatter plots, Correlation, and Line of Best fit" such as making predictions given an equation for a line of best fit, making scatter plots utilizing a... Nine weeks into the school year, I am still in love with foldables.
Sometime, students just need to get a lot of practice under their belt. • A dot plot shows the difference between individual groups. 4 B. reveal some websites with only one or two brief lesson ideas, which lack classroom management techniques, worksheets, and assessments. ... (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Post was not sent - check your email addresses! Displaying top 8 worksheets found for - Scatter Plot For Middle School. Scatter Plot For Middle School - Displaying top 8 worksheets found for this concept.. A scatter plot is designed to show a broad swatch of values for a given data set. These cookies will be stored in your browser only with your consent. In this scatter plot graphs game students answer question and check their answers through a QR code reader. Share skill. Some of the worksheets displayed are Scatter plot work, Scatter plots, Name period scatter plots algebra 10, Concept 20 scatterplots correlation, Tall buildings in cities building city stories height, Scatter plots, Interpreting data in graphs, Name hour date scatter plots and lines of best fit work. • A line graph represents change over time, and a scatter plot shows the relationship between two variables. Some of the worksheets for this concept are Scatter plot work, Scatter plots, Name period scatter plots algebra 10, Concept 20 scatterplots correlation, Tall buildings in cities building city stories height, Scatter plots, Interpreting data in graphs, Name hour date scatter plots and lines of best fit work. 5 Present Findings Create a presentation for your classmates in the form See more ideas about scatter plot, middle school math classroom, middle school math. • A pie chart compares different parts of a whole group. Some might consider this group of heist writing prompts very similar to bank robbery prompts and they’d be right. They can be used as bell ringers, exit tickets, homework, or just general practice. Ws teacher created date. Lessons and Ideas Books and Authors Top Teaching Blog Teacher's Tool Kit Student Activities The Teacher Store Book Clubs Book Fairs ... Students will use a scatterplot and line of best fit to determine a relationship between two sets of data, while examining how scientists and actuaries use math and data to study earthquake probability. This is absolutely crazy! Graph a scatter plot using the given data. There are differentiated versions of the worksheets that use pictures and color coding. See All Resource Types. I love thinking about plot ideas and idea starters for novels, short stories, and other fiction. Aug 4, 2016 - Explore Courtney Bratcher's board "Plot Activities", followed by 219 people on Pinterest. Interpret slope and y-intercept. M & Ms scatter plot activity from Math Equals Love. Difficulty Level: At Grade | Created by: CK-12. The best teaching strategies are the ones that make school … free scatter plot worksheets for middle school is government more than the management of the affairs of a Nation? I decided to try an interactive approach this year and selected twelve graphs and 18 verbal descriptions. I love doodle notes and the science behind them. Scatter Plots and Trend Lines Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 * Warm Up Graph each point. Modifications are considered for both struggling learners and high flyers. The scatter plot below shows their results with the line of best t. Scatter plot middle school. The graph looks like a bunch of dots, but some of the graphs are a general shape or move in a general direction. Check It Out! Topics include probability, scatter plots, linear models, dot plots, histograms, box plots, two-way frequency tables, shapes of distributions, outliers, residuals, correlation versus causation, quadratic models, exponential … Related Posts. • The mean is the average. :o) Unit 4a: Scatter Plots, Correlation and Line of Best Fit This year I only had two weeks to spend on scatter plots, correlation and line of best fit. Scatter plot middle school. Moreover, students have to read a scatter plot, calculate percent, determine probability, and find the mean of the given set of base hit data. Title: Lesson 11.4: Scatter Plots 1 Lesson 11.4 Scatter Plots. 10 Heist Writing Prompts and Mystery Story Ideas. I hope you have fun making these stories. In the past couple of months, a few of my newsletter subscribers have requested YA plot ideas. Get your students involved with practicing Interpreting Data with Scatter Plots and Line of Best Fit! Log in. This scatter plot … This can provide an additional signal as to how strong the relationship between the two variables is, and if there are any unusual points that are affecting the computation of the trend line. Classroom Forms. Have a student share a short story out loud with the class (such as what they did after school yesterday). The captain of the high school debate team does his best arguing … Author: Kathleen McCullough School/Organization: William T. Tilden Middle School. As a class, identify the beginning, middle and end. Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Free Story Plot Ideas Short Story. ... and Interpreting Scatter Plots notes and practice are differentiated based on some common needs found in the middle school math classroom.


Mortgage blog: Why being bad at maths could cost you your home
How good are your maths skills? New research has found that if you struggle with numbers then you're more likely to be in arrears with your mortgage payments and have your home repossessed.
The study by the University of Lausanne, in Switzerland, found that the risk of you missing mortgage payments is directly linked to how good your maths skills are. And the author of the report has urged lenders to offer ‘mortgage counselling' to borrowers with poor numeracy.
Borrowers with poor numeracy skills more likely to be in mortgage arrears
The study, published in the Journal Proceedings of the National Academy of Sciences of the United States of America, questioned 339 subprime borrowers in the USA who took out their home loans between 2006 and 2007. It also assessed each individual's numerical abilities in a telephone survey.
The questions that the respondents were asked involved calculating percentages and the basics of compound interest - the idea that the lump sum on which the interest is calculated changes each year. The results of the survey showed that borrowers with poor maths skills were more likely to be in default on their mortgage. Borrowers who did least well at the maths test spent about 25% of the time in arrears on their mortgage payments compared with only 12% for the top group.
Budgeting problems and an overall lack of understanding about the mortgage they had taken out were also likely to have played a part, the researchers said.
The Daily Telegraph reports that it "indicated owners' poor maths skills could have contributed to the mortgage defaults that led to the recent global financial crisis."
Keith Osborne, editor of WhatHouse.co.uk, says: "This research shows that it's not just financial skills that are needed to be able to manage your household finances, but basic maths skills as well. The researchers believe that lenders ought to do more to help people who are less mathematically minded and offer counselling services to help people avoid getting into arrears on their home loans."
Professor Lorenz Goette, the author of the report, said: "We need to think about whether we can teach people to avoid these mistakes. It's always possible for people to spend too much, regardless of their income. Recent studies have found changes in financial education curricula in high schools have important effects on financial decisions later in life, that foreclosure [repossession] counselling can reduce incidences of foreclosure and mathematical skills in general may be more malleable and less genetically driven than previously thought.
"If financial education can reduce sub-optimal financial decision making, this could have profound effects on household behaviour, as suggested by our results."
Click here to find out more about how whathouse.co.uk can help you find the right mortgage.


When I heard about Quicken Loans' $1-billion promotion in connection with the NCAA basketball championships in March, it struck me that this imaginative exercise to stimulate business could be very positive for Detroit. But it wouldn’t be possible without insurance or the advanced mathematics that created it.
Raising awareness of Quicken Loans could motivate more homebuyers to use the retail mortgage lender to attain financing, as well as more homeowners to rely on it for refinancing. That’s more work for Quicken employees, the bulk of whom work in Detroit and pay city taxes. Thanks, therefore, to Fibonacci, Pascal, Markowitz and many other math pioneers for their help to revitalize the city. (Disclosure: My son is a Quicken banker.)
By dint of its digital technology – not to mention its relentless energy – Quicken has vaulted to No. 4 among U.S. mortgage lenders, according to the latest league tables, behind Wells Fargo, JP Morgan Chase and Bank of America. It’s been as high as No. 3; and I have no doubt that Dan Gilbert, Quicken’s founder, won’t be entirely satisfied until his company is on top.
Is anyone going to win the $1 billion? Almost certainly not. The odds of a winner, roughly one in a quadrillion, are infinitesimal. Quicken hopes as many as 10 million will register for the contest, according to Jay Farner, the company’s chief marketing officer. Each will have to provide a name and email address, possibly a bit more information about plans to buy or refinance a home. At some point they should expect to read a Quicken ad or hear from a Quicken banker.
“Americans love March Madness anyway,” said Farner, “our contest will just make it more fun and exciting.”
The billion-dollar sweepstakes idea isn’t unprecedented. Pepsi, to drum up soft-drink sales, in 2003 created a contest that involved a billion codes, 100 of which would make the holders eligible to appear on a two-hour game show. Eventually, a $1 million winner would become a $1 billion winner if a six-digit number held by the winner matched one selected by a monkey. The numbers, of course, didn’t match.
Pepsi wasn’t interested in accepting even the remote risk of paying out $1 billion – and neither is Quicken. But Warren Buffett, left, and his company, Berkshire Hathaway, were prepared to guarantee the jackpot, if it was won, for the princely sum of $10 million. How Berkshire came up with that premium is a question only an actuary could answer. (If you’re interested in how human beings figured out the math that created insurance, read Peter Bernstein’s fascinating book, “Against the Gods.”)
Hole-in-one contests, in which a car or a large sum is given away to a golfer making an ace on a specified hole, usually involve an insurance policy. The probability of a hole-in-one changes for the length of the hole, whether professionals are playing and so forth, said Daniel Kaufman, an executive of Kaufman Financial in Farmington Hills. A precise assessment of the risk is determinable via equations and formulas applied to real-world data, such as how many times a golfer swings and how many holes-in-one result.
Farner declined to say how much Quicken is paying for its insurance policy. The premium probably is similar to Pepsi’s. Berkshire Hathaway can underwrite the policy because it has the financial wherewithal to be able to pay out $1 billion – it would sting, but the company could survive.
I’m no actuary, but Berkshire Hathaway’s bet sounds like a pretty safe one on a one-in-quadrillion chance of a number coming up. I’d like to take a bet like that, too, for $10 million; I’m just a bit dubious how I’d get the billion for the payoff, if my luck ran out.


Finally, the interest rate in an ordinary annuity is calculated using the equation A = P(1 + rt).
What is the formula for calculating annuity?
Based on the present value of an annuity due, effective interest rate, and numerous periods, the annuity formula assists in establishing the values for annuity payment and annuity due. As a result, the formula is based on an ordinary annuity, which is computed using the present value, effective interest rate, and various periods. The following are the annuity formulas:
The annuity formula for calculating the present value and future value of an annuity is particularly useful for quickly and easily estimating the value. The following are the Annuity Formulas for future and present value:
What is the typical interest rate on an annuity?
According to AnnuityAdvantage’s online rate database, the maximum rate for a five-year fixed-rate annuity is 3.71 percent as of December 2019. It’s 4.00 percent for a 10-year annuity and 2.70 percent for a three-year guarantee. These are terrific rates for accumulating funds in a secure manner. You don’t have to go overboard.
How do you calculate the rate of an annuity in Excel?
To determine the annuity’s periodic interest rate, type “=RATE(A2,A4,A3)” in cell A8. To compute the yearly interest rate, enter “=RATE(A2,A4,A3)*12” if you are using monthly intervals rather than annual periods.
How do you calculate interest rate example?
Example 1: If you deposit Rs.50,000 in a fixed deposit account for a year at an interest rate of 8%, the simple interest earned is as follows:
At the end of the one-year period, you will receive Rs.4,000 in interest. As a result, the FD’s maturity amount will be Rs.54,000.
Example 2: If you put Rs.8 lakh in a fixed deposit account for a period of 5 years at a 6.85 percent FD interest rate, the simple interest earned will be:
At the end of the 5-year term, you would receive Rs.2.74 lakh in interest. As a result, the FD’s maturity amount will be Rs.10.74 lakh.
How do you calculate monthly interest rate?
Divide the yearly rate by 12 to get a monthly rate that reflects the 12 months in a year. To finish these procedures, you’ll need to convert from percentage to decimal format.
Assume your annual percentage yield (APY) is 10%. What is your monthly interest rate, and how much would you pay or earn on $2,000 if you were to borrow it?
- Divide the annual rate in percent by 100 to get a decimal: 10/100 = 0.10.
- To get the monthly interest rate in decimal notation, divide that figure by 12: 0.0083 = 0.10/12
- Multiply $2,000 by the total money to find the monthly interest: 0.0083 multiplied by $2,000 equals $16.60 every month
- Return the decimal monthly rate to a percentage (by multiplying by 100): 0.0083 x 100 = 0.83 percent
Do you want a spreadsheet with this example already filled in? Make a copy of the free Monthly Interest Example spreadsheet and use it with your own numbers. The example above shows how to compute monthly interest rates and costs for a single month in the simplest method possible.
Interest can be calculated for months, days, years, or any other time period. The rate you use in calculations, regardless of the term, is known as the periodic interest rate. Typically, rates are quoted in terms of an annual rate, so you’ll need to convert to whatever periodic rate corresponds to your query or financial product.
What is annuity and how it is calculated?
Before diving into the concept of estimating the amount of annuity pay-out for your plans, it’s critical to first grasp a basic understanding of annuities and how they typically pay their beneficiaries.
An annuity plan is one that pays you regular payments over a certain length of time for the amount you pay in premiums. Your payment can be made in one lump sum or at regular intervals. The insurance company agrees to pay you the annuity either right away or at a later time. These annuity plans are retirement plans that allow you to receive regular income payouts so that you may maintain your current lifestyle once you retire.
Fixed and variable annuity programs are the two types of annuities available. Fixed plans have an interest rate that is guaranteed. Variable plans invest your premiums in other investments, thus their rate of interest is determined by the market’s performance.
This will be pre-determined between you and your insurance provider when you sign up for the plan, so there will be no surprises afterwards. You have the option of choosing from one of the following pay-outs that are frequently linked with these plans:
- The plan continues to pay the agreed-upon sum to the policyholder at the agreed-upon frequency. The balance annuities are paid to the beneficiary in the case of the policyholder’s death during the period.
- The plan pays until the policyholder dies; there is no idea of a beneficiary, thus no payments are made after the policyholder dies.
- The beneficiary will receive periodic payments from the plan for the rest of his or her life.
- The plan is only valid for a set period of time; this includes payments to the beneficiary after the policyholder’s death, but only for the agreed-upon period of time.
An annuity calculator can help you figure out how much your plan will pay you in the future. You can also use this calculator to figure out how much you’ll have to pay in capital to get a plan to run for a certain period of time.
For instance, if you want to see how much money you may take out of your annuity plan each month, input the following information into the annuity calculator India:
When you click’Calculate,’ you’ll see how much your annuity plan will pay you out each month.
You can also see how long your annuity plan will run by entering all of the above information (including the monthly withdrawals you choose) but leaving the term column blank.
This calculator will help you discover the approximate annual returns that yourprincipal will create if you enter all of the other details and leave the growth rate blank.
It is critical to have a thorough grasp of annuity plans before making a decision.
Each annuity plan is unique in terms of pay-out options, premium payment terms, death benefit specifics, and other factors. If you have any questions, you can contact your insurance carrier, and you should carefully examine the conditions of the policies to ensure complete understanding. Because annuity plans have the potential to provide lifetime income, even after you retire, you must understand them well in order to make the best use of them. Visit our main page to learn more about Aegon Life’s life insurance products, such as term insurance and other options.
What is a good rate of return for an annuity?
All genuine fixed indexed annuities in the study had an average annual return of 3.27 percent. Annuity returns ranged from 5.5 percent average annualized (highest) to 1.2 percent average annualized (lowest) (worst).
This time period includes the stock market’s roller coaster ride during the 2008 economic recession, as well as the “recovery” years.
On the surface, this doesn’t appear to be a negative situation. But it all depends on what you’re comparing them to. For example, below are the returns of a couple of no-load, low-cost index funds, as well as several blends of the two, illustrating some easy asset allocations, during the same time period:
If you’re wondering why the index fund (non-annuity) sets have n/a in the best and worst columns, it’s because there is no range of returns. The only returns would be the average, whereas annuity returns would vary greatly across the best and worst performing contracts.
This research isn’t intended to be a recommendation for or against any of the investments listed above. It’s more about grasping average annuity returns and the dangers associated with various investment strategies. However, there were a few things that caught our attention:
How much does a 100000 annuity pay per month?
If you bought a $100,000 annuity at age 65 and started receiving monthly payments in 30 days, you’d get $521 per month for the rest of your life.
How do you find total simple interest rate?
Simple interest is computed using the formula S.I. = P R T, where P is the principal, R is the annual rate of interest in percent, and T is the rate of interest in percentage r percent, expressed as r/100.
How do you calculate interest rate when not given?
Subtract the total amount of interest paid over the course of the year from the current loan balance. For instance, $3,996 divided by a $83,828 current loan balance equals 0.0476. Multiply that amount by 100 to get an estimate of the interest rate, which is 4.76 percent in this situation.


All Models of 2x4 LEGO Bricks
The answer to the question "Take six eight-stud LEGO bricks (2x4) - how many ways can they be combined?" is currently accepted as being 915,103,765. This is how many ways the bricks can be combined when we restrict them to be at straight angles, that is, as if placed on a grid. This limitation is extremely useful when finding an answer to the question as it limits all but the most basic of geometrical considerations. In this paper we find the answer without imposing the limitation. That is. We allow bricks to be connected at odd angles at their corners.
Our theoretical contribution is a natural method for distinguishing when two models of LEGO bricks are the same. We present our aswer to the original question using this method for up to six LEGO bricks. Our result is that using sic bricks, one can combine them into
Previous results in counting how many ways basic two-by-four LEGO bricks can be assembled only count combinations where the bricks are in places as in a grid. This limitation makes computation of the number of combinations easy. In this paper we wish to include all combinations where bricks are in general position, that is, are allowed to be connected at non-rectilinear angles.
Care has to be taken when deciding if two models are "the same". If we are simply interested in identifying the number of different models, the result is simply "One for a single brick and infinity for any number of bricks larger than one". This is realized by connecting two bricks at a corner and observing that the angle they are connected at is freely adjustable between the extremes.
We use the concept of homotopic equivalence: Two models are homotopically equivalent (or simply "homotopic") if one can be transformed into the other simply by rotating any free angles. If two models are not homotopic, they are homotopically distinct. We are interested in finding the number of homotopically distinct models. Notice that this definition extends to the previous results. These results count the number of homotopically distinct models where each model is homotopic to a model where all bricks are at rectilinear positions.
In the following section we present strict defitions for the concepts used in the remainder of the paper. Using strict definitions makes it easier to understand exactly what is referred to, and removes any ambiguity. Once the definitions are set, we dedicate a section to the geometry of the problem at hand, followed by a section with discussions of practical considerations when computing the results. The results and conclusion are presented in the last two sections.
In this section we present the definitions to be used for the remainder of the paper.
We only consider the standard "2 x 4" LEGO Brick, or simply brick . A brick consist of a block and a eight studs on top of the block. The exact geometry of the brick is presented in a later section.
The block of a brick is open underneath allowing a stud from another brick to be placed inside the block. There are eight such places in a brick (each corresponding to a stud above). Two bricks are connected when one of them has at least one stud inside the block of the other. With bricks being connected like this, we can limit ourselves to consider bricks as always having the studs pointing up, that is, with the block placed on the X/Y plane and the studs pointing along the Z-axis in the positive direction. See the following section for the geometry of bricks.
Two bricks are said to be connected by a corner if their shared connection consists of a single stud and this stud is at one of the four corner locations of its brick.
Two bricks are strongly connected if their shared connection consists of two or more studs. Notice that the angle of two strongly connected bricks can not be changed.
The way bricks are connected imposes a natural level on their position along the Z-axis. Consider a set of connected bricks. We say that the bricks with lowest Z-position are at level 0 . The bricks directly above are at level 1 and so forth. When two bricks are at level n and level (n+1) we say that they are at neighboring levels .
A configuration is a set of connected bricks. We only consider physically realizable configurations. This means that any stud inside a block of a brick must be at one of the eight allowed locations, and that two blocks may not intersect.
A configuration can only be moved by rotation and translation in the X/Y-plane. We say two configurations are the same (or identical) if one can become the other by simply moving it.
Notice that already for two bricks there are infinitely many different configurations as the bricks can be connected by a corner and the angle they are connected at can be turned freely between the two extremes.
A configuration in which all bricks are connected at straight angles is called rectilinear .
Given a brick a in a configuration, the strongly connected configuration of a is the rectilinear configuration obtained by greedily adding strongly connected bricks to the set of bricks initially containing a .
Notice that the division of a configuration into strongly connected configurations is unique.
A configuration can be canges using a using a simple transformation . A simple transformation is performed by rotating one or more pairs of bricks that are connected by their corners.
Two configurations are homotopically equivalent , or homotopic if one can be transformed into the other by a set of simple transformations.
Two configurations are in the same homotopy class iff they are homotopic. We use the term model to denote a homotopy class and say the model is rectilinear if it contains a rectilinear configuration.
We define the turn points of a model to be those connections for which there exist at least two configurations in the model where the angle between the bricks of the connection are different. Notice that two bricks connected by the corner are not necessarily contributing a turn point.
A turn point is free (as opposed to restricted ) if it can be turned as if there were only the two bricks of the connection in the model.
A turn point is non rectilinearly locked (or simply locked ) if it is restricted and the angle of which is can turn does not contain a straight angle.
Notice that two models can be different simply because of a turn point being locked at two different intervals of angles.
Notice also that the angle of one turn point can affect whether other turn points are locked. There are even non-rectilinear models where for each turn point there exists a configurations where the turn point isn't locked.
A circle is a set of three or more turn points in a model with the property that for each turn point of the set, each rectilinear configuration of the turn point is connected to exactly two of the turn points of the set, and that the turn points of the set form a connected component when considering the connections of the rectilinear configurations.
A circle is locked if at least one of its turn points is locked.
Counting All Models
Previous results have computed all rectilinear models.
We are interested in finding all non-rectilinear models. In order to find non-rectilinear models, all physically realizable configurations are considered. Turn points are identified in order to identify homotopy classes.
We divide the discussion of how to find non-rectilinear models into sections based on the number of bricks.
The Trivial Case: 1 brick
This case is trivial as there is only one model when using a single brick. This model is rectilinear and there are thus no non-rectilinear models.
The easy case: 2 bricks
From previous results  it is given that there are 24 models with two bricks. Four of these models have a turn point, but these turn points are obviously free, so all 24 models are rectilinear.
Still no non-rectilinear models: 3 bricks
There are 1560 rectilinear models .
In order to explore potential non-rectilinear models we consider models with either one or two turn points.
Models with one turn point
A configuration of a model with one turn point consist of a rectilinear configuration a with two bricks (without turn points) and a single brick b connected to this configuration at a corner (the turn point).
From the previous section we know there are 20 (24-4) ways a can be constructed. To connect b to a at a corner can't cause the turn point to be locked. With the low number of combinations this can even be realized quickly by trying to construct them all.
Models with two turn points
Consider a model with 3 bricks and 2 turn points. One turn point can't cause the other to be locked, so all models of this kind are rectilinear.
The first interesting case: 4 bricks
For 4 bricks there are both models with locked cycles and locked turn points.
In order to identify all non-rectilinear models without circles we have constructed a program that attempts to connect models with three or less bricks at corners of bricks and compute the intervals of possible angles of the turn points.
For models with circles there are at most one circle in a model and it has either 3 or 4 turn points.
If a circle has 4 turn points then it is also rectilinear as there is a configuration where the turn points form a square.
If a circle has 3 turn points then there are so few ways in which the turn points can be constructed that the non-rectilinear models can be found simply by investigating all such possible models. Notice! There are models with where the circle isn't locked, so these models need not to be counted.


Sometime during the early hours of Boxing Day, I fear I may have bought my ticket to Hell.
As part of a Christmas gift, the missus and I had been given a variation on the Rubik’s Cube, as pictured below.
It’s similar to the Rubik’s Revenge, in that each face consists of a 4x4 grid (instead of the usual 3x3. Addditionally, the colours of each face are supplemented by a small picture of an animal — meaning that the way each facet on a side can end up in an entirely different orientation to the others becomes really clear.
As a child, I never actually had a Rubik’s Cube (though I did have a Rubik’s Magic). So, I’d spent the last couple of hours of Christmas Day engrossed in the toy, succeding in being able to solve a single face, but progressing nowhere towards being able to solve the whole puzzle.
And then I got to thinking about ways in which the puzzle could be made even harder. And then I started thinking about SuDoKu. And then I found myself bound for Hades.
In SuDoKu, the object is to fill each 3x3 region of a 9x9 grid with the numerals 1-9, such that each numeral is placed once and only once within each region, and also occurs once and only once for any given horizontal or vertical line in the grid as a whole.
One could combine this with the traditional 3x3 Rubik’s Cube such that each face consisted of the numerals 1-9 — instead of filling in the blanks, one would have to rearrange the facets on the cube such that there isn’t a repeated numeral on any face.
This actually represents the solved cube, though it’s not at all easy to tell at a glance. On a normal Rubik’s Cube, it’s instantly obvious (colour-blindness notwithstanding) if the puzzle is completed — the real difficulty here would be that it’s no longer in any way obvious whether the cube is solved or scrambled.
Incidentally, I’m not actually at all sure whether the template represents the only solution to the puzzle. It might not, I think, given the way the cube allows facets to rotate and change position.
Anyway, here, unlike real SuDoKu, the relationship between the regions (the faces of the Cube) isn’t important — since there are four faces for each “line”, it’s not possible for each line to contain only one instance of each numeral. One could restore this constraint by having a 4×4 grid like the cube with the animals — the fiery Gehenna of hexadecimal SuDoKu, as found (so I’m told) in The Independent.
If, in the near future, a plague of boils is bestowed on London, it might not be terrorism. It might be that I’ve made a prototype and sent it to ThinkGeek.
- In Stuff


Moose Math for Apple iOS [Free for Limited Time]
Moose Math engages kids in a mathematical adventure and teaches counting, addition, subtraction, sorting, geometry and more. While playing 5 multi-level activities in the Moose Juice Store, Puck’s Pet Shop and Lost & Found, kids can earn rewards to help build their own city and decorate buildings. Moose Math introduces a new whimsical group of Duck Duck Moose characters, The Dust Funnies, who help with mastering math skills. Moose Math is aligned with Common Core State Standards for Kindergarten and 1st Grade and includes a Report Card section where parents and teachers can monitor progress and find additional skill-building activities. AGES: 3-7.
Moose Math features 5 engaging math activities:
1) MOOSE JUICE: Make smoothies while practicing counting, addition and subtraction
2) PAINT PET: Match the pets by counting the number of dots
3) PET BINGO: Solve addition, subtraction and counting problems to get BINGO
4) LOST & FOUND: Learn and sort through shapes and colors
5) DOT TO DOT: Help the Dust Funny find his way home by joining the dots
Kids will learn the following math skills based on Common Core State Standards:
– Understand the relationship between numbers and quantities
– Solve word problems and algebraic thinking
– Practice number pattern recognition
– Count by 1’s, 2’s, 5’s and 10’s
– Master counting to 100
ADDITION & SUBTRACTION:
– Add and subtract by 1’s, 2’s, 5’s and 10’s
– Add and subtract up to 20
– Learn to add and subtract with numbers, dice and rekenrek racks
– Master geometry at Kindergarten and First Grade levels
– Learn to identify and recognize shapes
– Understand and compare lengths
PARENT REPORTING: Includes a Report Card section where parents and teachers can monitor progress for each child and learn about new skill-building activities.
DEVELOPED WITH EDUCATORS:
– Developed in conjunction with Stanford University Educator, Jennifer DiBrienza, PhD in –
Early Elementary Education and former NYC public school teacher (K-Grade 2)
COMMON CORE STANDARDS: K.CC.1, K.CC.2, K.CC.4, K.CC.6, K.OA.1, K.OA.2, K.OA.5, K.MD.2, K.MD.3, K.G.2, K.G.3, K.G.4, 1.OA.1, 1.OA.2, 1.OA.6
ABOUT DUCK DUCK MOOSE
(A wholly-owned subsidiary of Khan Academy)
Duck Duck Moose, an award-winning creator of educational mobile apps for families, is a passionate team of engineers, artists, designers, and educators. Founded in 2008, the company has created 21 top-selling titles and has received 21 Parents’ Choice Awards, 18 Children’s Technology Review Awards, 12 Tech with Kids’ Best Pick App Awards, and a KAPi award for “Best Children’s App” at the International Consumer Electronics Show.
Compatibility: Requires iOS 6.0 or later. Compatible with iPhone, iPad, and iPod touch.


In my last post I discussed the topic of modularity of code in the context of generating prime numbers. In this post I am going to discuss the topic of recursion in SQL server in the context of modular code subsequently in the context of generating prime numbers.
In the prior post we showed how modularizing would allow us more opportunity to improve and correct past work without redoing everything from scratch. We created a function for testing primality called intIsPrime. We call it intIsPrime because if we really want a lot of primes (as many as we can get from SQL Server without inventing a new data type or doing some column and constraint tricks or both) we would need a bigintIsPrime function for all of those primes bigger than the integer data type which ends with the 2,147,483,647 which is prime. 2,147,483,647 is in fact the 3rd double Mersenne prime.
We then went on to query for the primes in the next 100 or so numbers using a technique called a recursive common table expression (CTE). In the spirit of modularity and reaching beyond a mere hundred tests through modular design consider this function:
CREATE FUNCTION dbo.hundredMorePrimeTests(@n int) RETURNS TABLE AS RETURN ( WITH recurseMore AS ( SELECT @n AS p, 1 AS n UNION ALL SELECT p + 1 , n + 1 FROM recurseMore WHERE n < 101 ) SELECT p, n FROM recurseMore WHERE dbo.intIsPrime(p) = 1 )
By putting our recursive CTE into a function. We can pass it the parameter we want to test the next hundred integers for primality. This allows us to call it in a stored procedure that bases the call on the last prime in our prime table. We can also have that stored procedure call its self (another type of recursive technique) which enables us to run the stored procedure once and have it execute 32 times (an error occurs is a stored procedure nests more than 32 levels).
CREATE PROCEDURE thirtyTwoHundredPrimalityTests @c INT = 1 AS BEGIN SET NOCOUNT ON; DECLARE @n INT; SELECT @n = MAX(p) FROM dbo.primes; INSERT INTO dbo.primes SELECT p FROM dbo.hundredMorePrimeTests(@n); SET @n = @c + 1; IF @n < 32 EXEC thirtyTwoHundredPrimalityTests @n ; END
There you have it, modular and recursive. If we wanted to we could call thirtyTwoHundredPrimalityTests from another stored procedure that could serve to limit the run based on how long we want to run, or how many integers we want to test. We can pass @c as any integer and the result would be that many 100s of integers tested.
Pingback: OPTION (MAXRECURSION 32767) | Thomas [W] Marshall


This week we are going to focus on non-objective art with geometric shapes! This activity is geared toward kindergarten through second grade.
Shapes are flat, two-dimensional, enclosed (has no gaps) objects. Common shapes are squares, rectangles, circles, ovals, and triangles. These shapes, along with others, are part of a group called geometric shapes.
Geometric shapes are related to math (geometry) and usually are symmetrical (equal on both sides) or irregularly shaped.
Today, you will make an abstract, non-objective artwork full of geometric shapes. Your final artwork will be called Non-Objective because it will have no objects (or things) from our real-world. Real-world objects include houses, plants, machines, people, places, and animals.
-Paper (colored or white)
-Color drawing media (crayons, colored pencils, chalk, or markers)
-(Optional) Geometrically shaped items to trace: like lids, toys, or other items in the home. And/or a straight edge/ruler to help you draw shapes.
Kindergartners and first graders
First, if you are a kindergartner or first grader let’s watch Choo-Choo!
On YouTube, watch “Learn 2D shapes with Choo-Choo Train part 1. Shapes for kids (kindergarten and students of grade 1)” at www.youtube.com/watch?v=8uIjdI0qbi8.
Then watch “Shapes for kids (preschoolers and grade 1). Learn about 2D Shapes with Choo-Choo Train – part 2” at www.youtube.com/watch?v=qny6CPtJvpo.
Second graders and older
If you are in second grade or older we will learn about Geometric Shapes by watching the Video “The Greedy Triangle by Marilyn Burns – Children’s Book Read Aloud | Storytime with Elena” at www.youtube.com/watch?v=9Xm3McQ6upw.
After watching the video, we will start out drawing geometric shapes with your pencil. If you found things in your home to trace or a straight edge/ruler, use those to help you. If not, try drawing them by hand.
To help you remember the shapes you learned from the video(s), look at Diagram 1.
Try to make some of the shapes go over the edges of other ones or inside, Overlapping them. Also, try to spread out the shapes across the page!
To start with color, follow your pencil lines by tracing them — making your pencil drawing disappear!
Now, decide if you are filling in the shapes with color, patterns or lines (maybe all three). Every time you start to color (or add lines/patterns) to a new shape or section of a shape, change the color (lines/patterns). It will make your design fun and festive!
Diagrams 3 and 4
I have two versions of my non-objective art:
-colored using crayons (Diagram 3)
-line patterns using chalk on black paper (Diagram 4)


This curriculum bundle includes a printed Textbook and a Workbook.
The Curriculum Textbook is a comprehensive math curriculum book for the Algebra 1 (8th grade math). It contains easy to understand and fun math lessons that teach various math concepts with appropriate visuals and text. It is a 598 pages printed textbook (B&W) with excellent visuals that help your students understand math concepts clearly.
The Curriculum Textbook offers the following key features:
* Quick reference guide with easy access to facts and formulas.
* Concise and "to-the-point" lessons that explain math concepts.
* Demonstration of concepts with sample problems to ensure your student fully understands how to apply concepts to real world math problems.
The Workbook (Worksheets, Chapter Tests and Answer Keys) includes 134 worksheets, 15 chapter tests and answer keys. The Workbook contains worksheets for each lesson in the curriculum book and test for each chapter along with the answer keys. Each problem provides enough working space where students can document the steps they follow to arrive at the answers. It allows your student to practice problem solving to master various math concepts that build a solid foundation in math that you can count on.
The main subject areas covered in this book are:
Chapter 1. Basics Review
Chapter 1.1 Understanding the Numbers
Chapter 1.2 Fractions
Chapter 1.3 Basic Real Number Operations
Chapter 1.4 Exponents
Chapter 1.5 Rates and Dimensional Analysis
Chapter 2. Variables and Operations with Variables
Chapter 3. Expressions, Equations and Formulas
Chapter 4. Graphing Relations and Functions
Chapter 5. Linear Equations
Chapter 6. Solving Linear Inequalities
Chapter 7. Solving Systems of Linear Equations and Inequalities
Chapter 8. Polynomials
Chapter 9. Factoring
Chapter 10. Quadratic Equations and Functions
Chapter 11. Radical Expressions and Triangles


By Marcos ChiquettoBy Marcos Chiquetto
Let’s go back in time thousands of years, to when humans began to count. Let’s imagine a primitive man wanting to count how many sheep there were in a herd, so he could check if any went missing the next day. To do this, he could maybe use little stones to represent the sheep. For each sheep, he would set a stone apart from the rest. On the next day, all he had to do would be to compare the number of stones with the number of sheep.
However, instead of stones, our shepherd had other objects that were more readily available: the fingers on his hands. For each sheep, he could stretch one finger.
But what if he ran out of fingers after using his two hands? He could start counting again and again with his fingers, making sure to remember how many times he had restarted, again associating each restart with one finger. This way, the count progresses in rounds of 10, which are also grouped in sets of 10, and so on.
Well, after counting the sheep, our ancient counter would not want to keep his fingers in the same positions until the next morning to check the quantity of sheep. Instead, he would draw a figure representing the fingers. Such figures evolved to become the Arabic numerals we came to use afterwards.
In this article, I’ll skip the long and fascinating history of the zero, one of the most important abstractions conceived by human beings. Instead, I’ll make a huge jump from the ancient shepherd to the numerical system we use today, called “the decimal system”, which uses ten symbols, including zero, to represent ten different basic quantities: 0,1,2,3,4,5,6,7,8,9.
Using numerals instead of the fingers of a hand, it’s easy to register the number of times we restart the count of 10. We just add one additional numeral in front of the first one, representing accumulated groups of ten:
1 group of 10 already counted: 10, 11, 12, 13, 14…
2 groups of 10 already counted: 20, 21, 22, 23, 24…
Someone in a far-off time associated the elements of this counting to the word “digitus” (finger, in Latin). And so, today we call decimal numerals digits.
Much more recently, around 1940, driven by the World War II effort, scientists from Germany, The United States and England focused on projects for powerful electronic machines capable of performing calculations (which the North Americans called “electronic computers”, see our article on the first computers).
Initially, they tried building machines that operated using the decimal system, but they realized this would be too complicated, since electronic circuits don’t have anything similar to hands with 10 fingers. To represent decimal numerals in an electronic machine we would have to use, for instance, 10 different levels of voltages in the circuit. Operating with this kind of data would be an engineering nightmare.
Within the scope of electronic resources, the most natural way of counting things would be something like a lamp: to register a sheep, turn on a lamp. The second level of counting would be a second lamp, and so on. Of course, one would need many lamps, but this wouldn’t make the system complex. Even with many lamps, the system remains simple. Much simpler than the decimal system.
The next figure shows how we can count sheep using a row of lamps. You may assume we have as many lamps as needed.
I will not try to explain the logic of this counting here, since the explanation would be more complicated than the logic itself. I suggest you spend some time observing the sheep and the lamps. It is likely that you will see the rule.
And so, a new system was developed for counting, in which the counting device has only two states (lamp), instead of ten (our hands). As opposed to the decimal system, this system was called “the binary system”. The corresponding set of numerals only needs two symbols. Borrowing from the decimal numerals, the symbols for the binary system are “0” and “1”.
Looking at the above figure, we can create a table relating the first decimal and binary numbers:
Although this numeric system might seem complicated to us because we are used to decimal numbers, it is actually much simpler. This invention paved the way for the development of computers.
Even though it does not derive from the use of fingers, the binary system kept the word “digit” to refer to its symbols, which are called “binary digits”. Moreover, any equipment using this system is referred to as “digital”. Thus, digital equipment is any device that represents quantities using binary digits.
Each binary digit is called a “bit” (abbreviation of binary digit).
But what physical devices are used to identify these two states in a computer? Of course, a computer would not use lamps, as the machine does not need visual information to operate. The very first computers used vacuum tubes, which are electric valves that can open or close the flow of electric current, working much like water faucets. For example, a closed valve means “0” and an open valve means “1”.
Today, bits are stored in microscopic devices, built into silicon wafers through chemical reactions, which form chips. One chip of a few square millimeters can store billions of binary digits.
However, besides numbers, computers also operate with letters and graphic symbols. How can we represent so many elements with just two symbols?
The solution was to gather symbols in groups.
In the early stages of computer development, scientists concluded that groups of 8 binary digits generated a number of combinations large enough to represent the numerals and all letters and graphic symbols. The group of 8 digits was thus adopted as the basic element of information in computers.
A group of 8 bits was called a “byte”. Here are a few examples of bytes:
Comparing this table with the previous one, you will see that the numerals are the binary numbers with a series of digits “0” added to complete groups of 8 digits. The other combinations are used for letters, graphic symbols etc.
Besides letters and numbers, screen pixels and instructions for programming in machine languages are also represented by bytes. So, letters, numbers, graphic content and programming instructions are stored in a computer in bytes.
The previous picture showing vacuum tubes depicts a logic unit which stores one byte (8 bits).
Well, returning to the initial question of this post, what does a 500GB HDD mean?
The international system of units adopts the following multiples:
k (kilo) = 1000
M (mega) = 1,000,000 (million)
G (giga) = 1,000,000,000 (billion)
T (tera) = 1,000,000,000,000 (trillion)
These multiples are powers of 10, appropriate for the decimal system of numerals. In the binary system, powers of 10 don’t have any special meaning. What’s relevant are powers of 2. However, since we are used to thinking in powers of 10, engineers adopted some powers of 2 that give very similar results to these multiples, so that we can evaluate quantities the same way we are used to.
This is a power of 2 which is very close to 1000. This quantity of bytes is called 1 kilobyte. For practical purposes, we can think of it as being one thousand bytes without much error. The same way, there are other powers of 2 which are very near to one million, one billion, etc, which are named as the other decimal multiples.
So, this is the rule:
1 KB (kilobyte) is approximately one thousand bytes
1 MB (megabyte) is approximately one million bytes
1 GB (gigabyte) is approximately one billion bytes
1 TB (terabyte) is approximately one trillion bytes
So, when you buy a 500 GB HDD, it means you can store approximately 500 billion bytes in it.
Just for comparison, the circumference of the planet Earth measures 49 thousand kilometers. Measured in millimeters, that would be 49 billion millimeters. In other words, a 500 GB HDD stores bytes in a quantity 10 times greater than the circumference of the Earth stated in millimeters.


Our VisionAt Harlow Green Primary school, we embrace the aims of the National Curriculum: fluency, accuracy, precision, reasoning and problem solving. ‘Mastery approaches’ to teaching mathematics are used to ensure deep, long-term, secure and adaptable understanding of the subject. We believe that mastery of mathematics is a tool for life and teachers reinforce an expectation that all pupils are capable of achieving high standards in mathematics. As a school, we aim to produce confident learners who are able to see and make connections between the different areas of Mathematics. All students can succeed in mathematics; a positive mind-set and strong subject knowledge are key to their success. Mathematical learning at Harlow Green will allow children to deepen their understanding and master key concepts giving them the opportunity to use their skills and confidence to contribute positively to society. Mrs R. Carr – Mathematics Lead
Purpose of studyMathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.
AimsThe national curriculum for mathematics aims to ensure that all pupils:
- become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
- can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.