Computing the total of a bill, taking Service Charge and 12% VAT into account is a common task that software engineers or spreadsheet developers may have to do.
Let's say you, for some unfortunate reason, have to compute how three people will split a restaurant bill, including VAT and Service Charge. You read "What Every Computer Scientist Should Know About Floating-point Arithmetic" then, applying the lessons there, do one of the following:
- Store and compute amounts in cents
- Use a decimal data type (instead of floating-point)
- Use an infinite-precision data type
And assume that the issue is fixed. Then fast-forward a few days (or weeks) later, finance calls you and informs that their computations do not match yours, and that you're off by a cent in a couple of areas. What's happening?
It's important to note that monetary arithmetic is case-specific. This means that even when using error-minimizing numerical representations, you still need to be aware of how the values are used in your case when implementing operations. Similar cases like this include:
- Splitting a ₱100 bill between three friends.
- Doing a 70-30 revenue split between the agents and the company.
- Converting ₱2499.99 (PHP) to USD.
Motivating Example: Bill Splitting
roundc(x) be a function that rounds x to the nearest cent. So ₱100.105 will get rounded to ₱100.11.
To make the problem easier, we'll focus on splitting the bill as an example: ₱100 divided by 3 is ₱33.333... the first step is knowing that it's more important that the sum is still 100 rather than getting an exact equal split. So a naive solution of computing
roundc(₱100/3) = ₱33.33 for all parties is wrong because if you add the value up, it will be ₱99.99 rather than ₱100. Note that if you use an infinite-precision data type, your countercheck (
₱33.333... + ₱33.333... + ₱33.333... = ₱100.00) will fail to detect this error if you don't round it to what is being displayed (₱33.33 rather than ₱33.333...) when counter-checking.
The result of this error will then again be dependent on the case. If this algorithm is part of an automated system that processes 1,000 transactions per day, then that's a loss of ₱10 per day (note that the accumulation depends on the volume of transactions, not on the magnitude per transaction). While percentage-wise, it might not be much, many businesses still care about the absolute value and the value goes up the higher the value of the base currency is (imagine of we operate on Euros rather than Philippine Pesos).
If it's just a utility mobile app, then hope that your users are smart enough to decide who pays the additional cent or the restaurant they are in would just waive the cent. In this case though, an alternative is to have everyone pay ₱33.34 and put the spillover as part of the tip.
Note that in this case (and also in other cases), using an infinite precision number, does not fix the problem. In this instance, additional decisions need to be made on when the rounding happens, and where to channel unavoidable excesses or discrepancies as a result of calculations.
These issues arise mainly because of two things:
Monetary value is a finite-precision decimal
We stop at a certain precision level (usually cents) because we use base 10 for counting and computation as opposed to continued fractions or rational numbers and that "physical" space is limited when writing down those numbers. Imagine how your grocery receipt will be if the amounts are written in fractions. This limitation is not because currencies are old. In fact, Bitcoin also has a precision limit: the smallest bitcoin unit is a satoshi which is one millionth of a Bitcoin. So one could say that using infinite-precision numbers is an approximation of finite-precision monetary amounts!
An exception proving this rule is FOREX, microchips, AWS EC2 pricing, and things that are usually sold/exchanged in bulk. For example, for FOREX, sure it's more than 2 decimal places (1 JPY = 0.4416 PHP), but if you wanna do the actual conversion, there's a minimum amount! You can't just go to a FOREX and trade exactly 1 JPY for PHP: they usually have a minimum amount and can only trade in multiples of that amount (e.g. can only trade per 1000 JPY). Moreover, the reverse exchange rate (PHP → JPY) rate is not the reciprocal (computing 1/x) of the JPY → PHP rate but an entirely different measure altogether (but it's usually used as a rough baseline estimate) so there's no direct need to compute 1/0.4416 with high precision.
Other goods sold in bulk follow the same pattern, like the 555 timer, although the unit price's precision is less than a cent ($0.09551), one cannot just buy 1, but will have to purchase at least 6,000 of them (the Minimim Order Quantity).
Certain operations "generate" precision
Remember in grade school when you were taught division and the remainder was introduced? This then led into fractions and decimals. In other words, division can generate additional information that cannot be represented with the data type of the operands (an integer divided by an integer can result in not an integer).
While division is the only operation that can generate precision, there are ways where it can sneak in like multiplication by a fraction and exponentiation. As such, it's more helpful to whitelist operations that do not generate precision. They are:
- Multiplication by an integer
- Division by a divisor (e.g. 12 divided by 6, 2, or 3)
Those familiar with floating-point operations would notice that subtraction is present! While it's true that subtraction generates errors ("loss of significance" is the accurate term) in floating-point numbers, the same does not hold for fixed-point because fixed-point is an exact representation, unlike floating-point (i.e 0.1 is 0.1, not 0.100000001490116119384765625).
The two reasons above are at odds with each other
If all the operations we do does not generate precision, then finite precision is no problem at all. If we use fractional representations, then operations that generate precision will not be a problem unless you use transcendental functions (which produce irrational numbers that cannot be represented by a fraction).
Unfortunately, division is a common operation when dealing with money like splitting bills in the first example. It also arises when computing for discounts and markups. Exponentiation of money appears when computing interest in financial instruments (loans, investments, and derivatives).
How do we solve the problem? Unfortunately, as illustrated in the example, using a better data type does not solve the problem completely.
Fortunately, there are two rules and two guidelines to follow. They are relatively easy to understand and does not involve complex math (though number literacy is needed) so you do not need to know Kahan Summation or some complex algorithms in order to understand them.
Rule 1: What's displayed is the truth.
The moment you display the number to the user, that displayed number becomes the truth, not the original number that was rounded to what was displayed. For example, if you have total = 12.354915 then you show it to the user as "Total: 12.35", the value of total should then be 12.35 exactly (i.e. 12.35 replaces 12.354915 as the real value) and should be used for the rest of the calculations (e.g. 12% VAT should be computed against 12.35 instead of 12.354915).
This rule seems easy enough but require diligence to follow: every number that a product manager (or government regulator) requires you to show, will be a number that gets replaced with its rounded value. It's very easy to just show the number without revisiting the calculations.
For instance, when working with spreadsheets (e.g. Excel), it is not enough to format the number: the rounding has to be included in the formula themselves. Drake here will show an example:
Why this rule? It's because what you show the user is the value that's going to be used, not that highly-accurate, infinite-precision number stored in memory. They will see 12.35 and copy that number to their reimbursement form, their tax forms, insurance claims, etc. they won't be bothering with 12.354915.
Rule 2: Splitting should sum up to the original.
If the division is a splitting, adding the parts should give you the original value. While obvious, this becomes tricky to implement especially with the context of the previous rule. The first example illustrates this perfectly: 100/3 = 33.3333... and applying Rule 1, the actual value becomes 33.33. However this rule requires us to counter-check: 33.33 + 33.33 + 33.33 = 99.99; we are missing one cent! Where that cent goes is up to you or the product manager (see guidelines below) the important point of this rule is the awareness that discrepancies like this may exist!
A common (but not the only) way to resolve this is to make one value the difference of the original and the total of the other values. Mathematically speaking, if T is split into p1, p2, ..., pn, then instead of computing pn directly, you compute pn = T - (p1 + p2 + ... + pn-1) instead.
For example, 52.15 is to be split 70-30, instead of 52.15 × .70 = 36.51 and 52.15 × .30 = 15.65, (36.51 + 15.65 = 52.16 we are over one cent), we just compute the 30 split as 52.15 - 36.51 = 15.64.
Here are some cases to watch out for this rule:
- Splitting a pool of money: As previously explained (the sum of the split must be equal to the original value). This applies to all splits, not just even ones (e.g. 70-30 split of App Stores).
- Discounts: the discounted price plus the discount amount should equal to the original price.
- Pre-tax values: VATable sales plus VAT should equal to the VAT-inclusive price.
Guideline A: Know which values are magnified.
Where should the missing (or extra) cent go? One aspect that can help with the decision is knowing which values are magnified. In many cases, there is asymmetry on the volume of transactions: like how a customer may buy from a store at most once a week but the store processes hundreds or even thousands of transactions per day. So the effect of a discrepancy in calculations is more magnified in one than the other.
For example, suppose from a computation, an extra cent will have to go somewhere and we have to decide whether it should go to the customer or to the store. A customer paying an additional centavo would mean almost nothing for them because even if they buy stuff from your store everyday, they'll lose 3.65 PHP per year. Compare this if the store pays for the extra centavo instead, if they receive 100 customers per day, then they will end up losing 365 PHP per year. In this case the one centavo's effect is magnified more for the store than for the customer.
This does not automatically mean that the loss should be shouldered by the customer, but it does elaborate the effects of decisions. More often than not, these decisions are judgements that may need to be propagated to management, especially if regulations are involved (what if we're referring to tax instead?)
Guideline B: Sometimes, predictability is better than accuracy.
Reaching for the round function all the time might be tempting because it's the most "accurate". However, rounding down or up might be better in some cases because their error is more predictable:
roundup(x) ≥ xall the time, and
rounddown(x) ≤ xall the time.
Compare it with round function which, does not have such guarantees (it can be greater or less than the original value). When would rounding up or down be better? Here are some cases:
- Computing taxes (better overpay than underpay by a centavo and get chased by the tax authorities who happen to round things differently)
- Automatically favoring a party in a split (for example, company doing a 70-30 revenue split will round down the other party's split in order to favor theirs)
- Making the error residuals always positive (or negative) when enforcing Rule 2. For example, when doing an even split like 200/3, a round down would always result in a positive residual (200 - 66.66 × 3 = 200 - 199.98 = 0.02 > 0; a round would result in an overestimation).
Application: Proportional Bill Split with Service Charge and VAT
To illustrate how the rules and guidelines are applied, we compute the bill of three friends eating at a restaurant, taking into account the service charge and VAT, and how much would each pay if they decide to split the bill. Here are the important points:
- They agree to pay for their own meal.
a. Friend A: ₱159.94
b. Friend B: ₱174.99
c. Friend C: ₱129.99
- But they buy a pitcher of iced tea (₱100), which they agree to split equally.
- There is a 12% VAT applied to sales.
- There is a 10% Service Charge based on VAT-inclusive price that's not subject to VAT .
- Prices are VAT-inclusive, but exclusive of service charge.
- The reciept has the following fields printed:
a. Item Price (VAT-inclusive)
c. Service Charge
d. Grand Total
e. VATable Sales
We first compute what should be printed on the receipt:
- The Subtotal is straight forward, just add
₱159.94 + ₱174.99 + ₱129.99 + ₱100.00 = ₱564.92
- For the 10% service charge, we round up to favor the store (Guideline A & B) so
roundupc(₱564.92 × 10%) = ₱56.50.
- The Grand Total is
₱564.92 + ₱56.50 = ₱621.42. Note that we didn't compute it as
₱564.92 + 10%because of Rule 1.
- Compute the VATable sales as Subtotal / 12% and we round down (Guideline B; explanation in the next point). So
rounddownc(₱564.92/1.12) = ₱504.39.
- Compute the VAT as
₱564.92 (Subtotal) - ₱504.39 (VATable sales) = ₱60.53. No rounding is needed because we are just subracting. Note that we are not computing it as VATable sales + 12% due to Rule 2, moreover, due to the rounding down in the previous step, we are actually very slightly overpaying the VAT.
Computing the split between friends is trickier: we first have to factor in the service charge to each item, then split the price of the drink, service charge included, proportional to the order amount of each friend.
There are two ways to compute the price of each item with the included service charge:
- Spilt the service charge proportionally per item, or
- add the 10% service charge per item.
we will go with (2) for its simplicity. So for each item, we compute
roundc(x × 1.10) and note the deviation of their sum from the Grand Total ₱621.42 (Rule 2):
- Friend A: ₱159.94 → ₱175.93
- Friend B: ₱174.99 → ₱192.49
- Friend C: ₱129.99 → ₱142.99
- Drinks: ₱100.00 → ₱110.00
The total is ₱621.41 which off by one centavo but we don't distribute it yet since we still have to split the price of the drink ₱110.00. For the drink split, we use Guideline B and
roundownc(₱110.00/3) = ₱36.66. So the spilt is now:
- Friend A: ₱175.93 + ₱36.66 = ₱212.59
- Friend B: ₱192.49 + ₱36.66 = ₱229.15
- Friend C: ₱142.99 + ₱36.66 = ₱179.65
Counter-checking due to Rule 2, the sum (₱212.59 + ₱229.15 + ₱179.65) is ₱621.39, which is off by 3 centavos, which we just distribute equally. So the final split amount for each friend is:
- Friend A: ₱212.60
- Friend B: ₱229.16
- Friend C: ₱179.66
Here's a diagram to aid in understanding where the amounts flow.
The computations in Google Sheets can be accessed here.
Monetary amounts require a bit more nuanced handling than ordinary fixed-precision number. One thing to keep in mind is like matter, money does not magically increase or decrease when you perform transactions (unless we are talking about interest). The rules and guidelines discussed help rationalize the choices we make when performing calculations with them.
Computations like these come up in our projects from time to time (one being the Service Contracting Program) and it's something that we pay close attention to using the rules and guildelines discussed.
If you’ve found this interesting or helpful at all:
- I’m glad!
- Drop us a line and let us know!
- Consider working with us! Sakay’s tech team tends to focus on transport and mapping, but as you can see, we tend to go a bit wide.
P.S.: This blog post is inspired by https://blog.xendit.engineer/benchmarking-pg-numeric-integer-9c593d7af67e head over there to know the tradeoffs in representing monetary arithmetic in PostgreSQL!
Is the service charge subject to VAT? For Philippine context, see RMC 29-70 (issued in 1970), printed in the Philippine Taxation Encyclopedia which states that service charges are not subject to VAT as long as they are given to staff and not management. There's a separate bill, RA 11360 signed on 2019 which mandated that 100% of the service charge should go to staff. So in effect, service charge (despite its name) is not subject to VAT even if it's non-discretionary.
However, there is no rule on how the service charge should be computed. Percentage-based service charge can be based off the VAT-inclusive or VAT-exclusive price: it depends on the establishment. I've seen restaurants compute the service charge with either price. ↩


Replace a random sample of elements in a matrix that contain a certain value
You want to replace values within a matrix with another value, but only at a random sample of locations where the original value occurs.
The following code will find the index of all locations in the matrix that have a given value, then assign a new value to a random sample of those index locations.
#Create an example matrix MTX <- matrix(1:5, nrow=5, ncol=5) #View the matrix MTX #Find the index of all locations where MTX==2, #and assign a new value of 888 to a random sample of 3 of those MTX[sample(which(MTX %in% 2), 3, replace=FALSE)] <- 888 #View the result MTX #Note that this can be adapted for use with raster data, #but doesn't appear to work as-is on rasters #Use the following to turn your raster into a matrix: #MyMatrix <- matrix(getValues(MyRaster))
Jason Carlisle contributed to the development of this tool.


apologise, there offer other way..
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Sunday, April 24, 2011
Math Monday Blog Hop - Place Value Sliders
I can't believe I have been blogging for 3 weeks now! I realized it had been 3 weeks when I started thinking of what idea I would submit for love2learn2day's Math Monday Blog Hop - how quickly time flies!
This week, I am posting my Place Value Sliders. I like to use this to supplement our current textbook. Unfortunately, our curriculum does not have enough place value practice and I was desperate for a way to have the kids play with numbers that was kinesthetic.
To make the sliders, I give every student a set of number "slides" and one "holder" (the green slip of paper in the picture). I use 5 slides because our state GLEs require students to know numbers up to 99,999. I'm sure you can use as many place values as you would like. After the kids cut out their slides and holder, I laminate them so they will last all year. I then use an Xacto knife to (very!) carefully cut the slits in the holders to hold the sliders (you can also tape the ends of the number sliders together to make an odometer look). I give the sliders and holders back to the kids to keep in their math tool kit.
As for using the sliders in the classroom, I often ask students to make a 4-digit number, a 2-digit number, etc. I also ask for them to make numbers using only odd/even numbers or ask for them to make a number with an odd/even number on the ends. I might ask for their number to include 2 odd/even numbers, etc.
How would you use Place Value Sliders? I would love to hear more ideas on how to use them in the classroom!
Thanks for stopping by!


Mathematica is a computational software program used in many scientific, engineering, mathematical and computing fields. It was conceived by Stephen Wolfram and is developed by Wolfram Research of Champaign, Illinois. Mathematica is renowned as the world's ultimate application for computations. But it's much more—it's the only development platform fully integrating computation into complete workflows, moving you seamlessly from initial ideas all the way to deployed individual or enterprise solutions.
Members of the physics department enjoy an unlimited number of licenses. Hence, a multitude of jobs can be submitted with regard to licensing. All other users, however, share the 10 licenses the university has obtained. Hence, the number of concurrent jobs is severely limited. Jobs may crash with an appropriate error message. To avoid this limit the number of concurrently running jobs. If in doubt contact the HPC-team.
In order to access the executables load one of the modules, e.g.:
$ module add math/Mathematica
For interactive computing on the login nodes (testing only) start
Mathematica. Scripts can be executed with
You may for example submit a mathmatica script with the following command line:
srun -n x --mem=xxx -p short --time <sometime> math -script <path to script>
x reflects the number of wanted cores ( 1 ⇐ x ⇐ 64 ; see below on how to retrieve this number within a script)
xxx the amount of memory required for the job (see the memory reservation page for hints)
sometime the required wall clock time
- <path to script> should indeed be a valid path (relative paths should work, too)
sometime> 5 h, please submit to a long-queue
x= 64, please submit to nodeshort or nodelong. In this case just select with
-p <partitionname>a different partition.
You can launch an interactive Mathematica session and use the program as you would on your desktop pc. Please note that this should only be done on the compute-nodes and not on the login-nodes.
[joe_user@login]$ srun -n<ntasks> -p<partition> -A<account> -t<time> --mem<amount> --pty --preserve-env $SHELL
[joe_user@node]$ Mathematica &
In order to retrieve the JobID, you can do
JOBID = Environment["SLURM_JOB_ID"];
The job directory therefore becomes
LocalScratchDir = "/localscratch/"<>JOBID<>"/";
A ramdisk can be reserved as described in this link. The number of reserved cores can be retrieved with
CPUn = ToExpression[Environment["SLURM_JOB_CPUS_PER_NODE"]]; Print["Number of Processes: "<>ToString[CPUn]];


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- Primary resources - free worksheets, lesson plans and teaching ideas for primary and elementary teachers.
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Conversion factors (mc) - prefixes are used by scientists to save time. The united states customary system (uscs or usc) metric units are standard in science, medicine, as well as many sectors of industry and government, including the military. A simple homework sheet on metric conversions.
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TUTOR AND BABYSITTER
- Publish Date: 02-10-2019 23:08:45 |
- Contact: Lara Hagopian |
- Location: San Diego 92130 |
- Place: San Diego |
- 57 times displayed |
Babysitter and/or Tutor!
Help your child succeed even on date nights! Whether you want an educational experience for your child when you can't be home or if you need someone to take care of the kids and/or help them with homework, I strive to help your child's mind grow in the most helpful ways possible!
Tutoring Subjects: Elementary Homework help for all subjects (reading, math, etc.); Math up to Algebra 2 including Geometry, Algebra 2, Algebra 1, Pre-Algebra, Elementary Math etc.
Helping your child succeed should not come with a large price tag! Allow your child to acquire essential building blocks to learn how to study effectively while strengthening their academic ability in a subject or subjects at a reasonable price.
Build a lifetime of study habits early on while comfortable knowing your child is being taken care of!
Ensure your child’s successful future!
~ CRLA Certified Tutor ~ UCSD Grad ~ Patient ~ Friendly~
Call/Text Lara Hagopian now for more information!


(Even more!) Math awards
Since our last math awards release, even more students have placed in the top tiers of contests.
- Fifteen Grade 7 students placed in the top 20% of the Gauss Math Contest, with another three placing in the top 5%.
- Grade 8 students also earned top marks, with 15 students placing in the top 20%.
- Over in the elementary school, Marco Z. (Grade 5) received the Certificate of Outstanding Achievement in the Grade 7 CEMC Gauss Math Contest as well as earned the highest score in the school.
- Elsa W. (Grade 5) received the Certificate of Distinction Achievement in the Grade 7 CEMC Gauss Math Contest and ranked in the top 25%.


Assessing student learning documents knowledge and skills that students gain over a specific period of time using measurable standards and benchmarks. The Allegheny Valley School District staff utilizes summative, formative, benchmark, and diagnostic tests to measure student growth.
Administrators and teachers consider results from Classroom Diagnostic Tools (CDT) tests, STAR Reading, Early Literacy and Math, DIBELS Next, PVAAS data, and locally created formative and summative assessments when considering the student program. District staff also considers information from two state-mandated assessments, the Pennsylvania System of School Assessment and the Keystone Exams, for student progress analysis and during curriculum review.
The Pennsylvania Keystone Exams are end-of-course assessments designed to evaluate proficiency in academic content. Beginning with the class of 2017, students must demonstrate proficiency in the Algebra 1, Biology, and Literature Keystone Exams as a Pennsylvania requirement for graduation.
Pennsylvania System of School Assessment
According to the Department of Education, the PSSA test “measures how well students have achieved in reading, mathematics, science and writing according to Pennsylvania’s world-class academic standards.” Reading and mathematics components of the PSSA are administered to students in Grades 3 through 8. In addition, the state assesses students (Grades 4 through 8) in science. All scores fall into four categories: Advanced, Proficient, Basic, and Below Basic.
|Pennsylvania System of School Assessment (PSSA)|
|- English Language Arts (ELA)||3-8||April 22 - 26, 2024|
|- Mathematics, Science, & Make-ups||3-8||April 29 - May 3, 2024|
|- Literature||Currently enrolled in academic English 10/honors 10|
|- Algebra||currently enrolled in Algebra I|
|- Biology||Currently enrolled in biology|
|Winter Wave 1 (AVSD Retakes)||December 4 - 15, 2023|
|Winter Waay 2 (AVSD Retakes)||January 3 - 17, 2024|
|Spring||May 13 - 24, 2024|


We have discussed molarity and molality, but there other ways to present information on solutions using quantitative measurements (numbers).
To learn about mole fractions, watch the following video:
Measuring Strength of Solutions
Three Common Ways are:
- Percent by mass = (mass of solute/mass of solution) x 100%
- Percent by volume = (volume of solute/volume of solution) x 100%
- Weight/volume percent = (mass of solute (g)/volume of solution (mL)) x 100%
Watch the following video from 0:00 to 1:34:


PUBLISHED: 18:11 27 July 2017 | UPDATED: 18:40 27 July 2017
At the start of the summer holidays the last place children want to go is back to school.
But a new scheme launched by Norfolk County Council looks to get children playing fun maths games to help boost numeracy skills for youngsters from early years through to 11-year-olds.
The Count On Norfolk campaign was launched at the start of the summer to encourage parents to bring maths into everyday family life through fun activities and play.
The council is hosting free play dates for families at their local library or school where maths teachers are offering to help them improve number skills with games.
The first of the 16 sessions took place at Queen’s Hill Primary and Nursery School in Costessey where children and their parents were able to get hands on with maths by building towers, designing bridges and fishing for numbers.
Eight-year-old Byron Lugassy said: “It has been really fun and a great day.
“I really enjoyed making the tower out of spaghetti.”
Simon Routhorn, leading practitioner for family learning, said: “We are promoting fun maths activities and doing real world maths life skills.
“This is the second phase of the campaign which begun at the beginning of the summer.
“No one wants to come and do maths on their holiday but they are learning through fun methods and enjoying themselves.
“The children are really enjoying working to a budget, we have printed off some money and they are playing with budgets figures and getting really involved.”
Mum-of-two Tara Deane, from Queens Hill, said: “We have only been at the school since February but this has been brilliant, being able to come and meet all the parents and have a nice time is great.”
As well as the maths activities on offer there is a selection of healthy snacks with tea and coffee.
Early years adviser Janey Craig said: “We are trying to dispel the anxiety around maths. Because lots of parents have that anxiety as they may not be very good at it and what we done is set up things that can be recreated at home for free.”
Go to www.norfolk.gov.uk/raisinglearners for a list of sessions and activities to do at home.


Peer Math Tutoring
SUMO works with the Stanford math department to provide peer tutoring services for a number of introductory math classes, specifically MATH 19, 20, 21, 51, 52, 53, and 104. The peer tutors are available to answer any questions you have about the material and homework in these classes.
Peer tutoring hours are drop-in; just show up and our tutors will help you out. Feel free to come and work on your homework even if you don't have any specific questions. Many students work on their homework during tutoring, asking questions if they encounter difficulties.
Please note that this program is not aimed towards Math 60CM/DM students. For students in higher level math classes, the math department offers other resources.
Becoming a Tutor
If you are interested in becoming a peer math tutor, we hire new tutors at the start of Fall, Winter, and Spring Quarter of each academic year (if there are available positions). Tutors should be able to help students with either all three classes of the MATH 19/20/21 series and MATH 51, all three classes of the MATH 50s series, or MATH 104/113.
Tutoring position announcements are sent to the SUMO mailing list. If you have any questions about tutoring or applying to be a tutor, please contact Kaya Lakein (epi2 at stanford.edu).
Tutoring starts week 2 and continues through the end of week 10, excluding university holidays. The schedule for each course is as follows:
Math 51/52/53/104: Monday and Wednesday, 6PM - 8PM and 10 PM - midnight (Stanford timezone).
Math 19/20/21: Tuesday and Thursday, 6 PM - 8PM and 10 PM - midnight (Stanford timezone).
Another one of our tutors, Arun Debray, took lecture notes for Math 53. They can be found here.
To access Zoom as a student, self-enroll in the SUMO Tutoring Fall 2020 course on canvas by clicking here.
Meet Our Tutors
Hannah Prausnitz-Weinbaum is a junior studying mathematics and minoring in computer science. She loves helping other people get excited about math, and has been tutoring with SUMO for two years. When she's not in class, you can find Hannah knitting or singing a cappella.
Mac Bagwell is a senior studying Mathematical and Computational Science. He is particularly interested probability, data analysis, and machine learning. When he's not doing math problems, he's probably watching baseball or listening to a podcast.
Daniel is a senior studying psychology (with a focus on cultural psychology), and coterming in statistics. Outside of class, he is part of the Stanford Collaborative Orchestra and enjoys listening to and playing music.
Yuzu Ido is an undergraduate at Stanford majoring in math and minoring in music. She has previously done research in algebraic geometry and graph theory. Outside of math, some of her interests are piano, CS, and eating good food.
Hi! I am a junior at Stanford majoring in computer science and minoring in math. I am also interested in the physical sciences, and I do research using machine learning in biological problems. Aside from school, I enjoy tennis, Brazilian music, and any type of handcrafting.
Manya is a sophomore studying Mathematics at Stanford. She enjoys learning more about measure theory, analysis, and statistical physics (and has somehow found herself taking more computer science classes than she ever intended). Her other interest include drawing, going on long walks and listening to a lot of music.
Mateo Attanasio is an undergraduate at Stanford studying mathematics. His current interests include algebraic number theory and topology. Outside of mathematics, interests include rock climbing and history.
Hi there! I am a junior majoring in math and minoring in physics and TAPS. The areas of math that I am most interested in include algebraic number theory and most other algebraic things. I really enjoy teaching math to people of all backgrounds, and next to serving as the current SUMO tutoring coordinator, I am one of the math department's peer advisors, and I have graded Stanford math classes in the past. Aside from doing a lot of math, I like to dance, juggle, and go on long walks/hikes.


MATHEMATICS AT MEDBURY
The Medbury Boy is numerate, literate and articulate.
Mathematics is a major influence in all aspects of boys’ daily lives. Boys develop numeracy skills and understanding in Mathematics through developing knowledge and strategies that enable them to explore Mathematical ideas, to solve problems and to think creatively. All new learning builds upon prior learning at each stage and developing confidence in Mathematics is emphasised from the early years through the Middle to the Senior School.
At Medbury we aim to:
- To ensure that an effective personalised Mathematics education is delivered at all levels according to the New Zealand Curriculum Framework.
- To ensure that boys are given opportunities to explore, think critically about and process Mathematical information.
- To enable boys to use and communicate their Mathematical knowledge and strategies in a range of meaningful everyday contexts, so that Mathematics is seen as relevant to their lives.
- To foster and develop the mathematical talent of boys with special abilities.


- pyart.util.estimate_noise_hs74(spectrum, navg=1, nnoise_min=1)#
Estimate noise parameters of a Doppler spectrum.
Use the method of estimating the noise level in Doppler spectra outlined by Hildebrand and Sehkon, 1974.
spectrum (array like) – Doppler spectrum in linear units.
navg (int, optional) – The number of spectral bins over which a moving average has been taken. Corresponds to the p variable from equation 9 of the article. The default value of 1 is appropriate when no moving average has been applied to the spectrum.
nnoise_min (int, optional) – Minimum number of noise samples to consider the estimation valid.
mean (float-like) – Mean of points in the spectrum identified as noise.
threshold (float-like) – Threshold separating noise from signal. The point in the spectrum with this value or below should be considered as noise, above this value signal. It is possible that all points in the spectrum are identified as noise. If a peak is required for moment calculation then the point with this value should be considered as signal.
var (float-like) – Variance of the points in the spectrum identified as noise.
nnoise (int) – Number of noise points in the spectrum.
P. H. Hildebrand and R. S. Sekhon, Objective Determination of the Noise Level in Doppler Spectra. Journal of Applied Meteorology, 1974, 13, 808-811.


EducationPhD student at Johannes Gutenberg-University, Mainz, Germany.
Master’s degree in nuclear physics and physics of high energies, Taras
Shevchenko National University of Kyiv, Ukraine, (diploma with honor).
Research assistant at University of Münster.
Collaboration with Memorial University of Newfoundland and
Acadia University, Canada.
ExperienceI'm a professional tutor of Math and Physics. The tutoring experience is 7+ years. Preparing for the Math tests: GMAT, MAT, SSAT, MCAT etc. A lot of excellence: students were passed exams and entered to universities. I always try to improve not just a knowledge of subject and also to be a friend for person. My way of teaching is getting a strong knowledge through interesting activities.
HobbiesSports, especially dancing, hand-made, healthy life and tutoring.
I'm communicative and sociable person who are loving life and children.
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Kids nowadays are very attached to their computers. They love doing different things on them, ranging from playing cool math games to surfing the internet. Computers can be both a blessing or a curse for children. They can either learn and discover new things from the internet, or pick up bad habits way too easily.
However, there are actually ways to make children practice some math concepts in non-traditional ways. There are websites that can provide a variety of fun games, information for parents and other resources which can help children learn math without the feel of a lesson plan.
Here is a list of 15 such math games which can be very useful if your child loves sticking to the computer:
1. Cool Math Games
This website provides children different kinds of topics paired with nicely crafted educational content and good graphics. The games have silly titles and pictures, which kids are sure to love. Apart from math, the website will also let your child explore more about science, language arts, and much more. You can check out the website here.
2. Hooda Math
Hooda Math is another site which can be very helpful for your kid to learn various topics. The website brings in new games every week in different categories such as geometry, logic, and arithmetic. The games on the website help kids learn and have fun at the same time. Games like Real Estate Fun, Cargo Math, and Inn Keeper let children apply the concepts of math in real life. Go to Hooda Math by clicking here.
3. Fun Brain
Fun Brain has some amazing fun games that will tickle your brain. Games like Math Arcade, Math Baseball, and Tic Tac Toe Squares are some of the games which will provide some fuel in the growth of your child’s brain. The games on the website have great graphics, which make it more interesting than a boring school-type lesson. Some popular games like Sudoku and Mad Libs Jr. are also available in the website. You can check out interesting games in Fun Brain here.
4. Primary Games
Apart from having some wonderful math games, Primary Games features activities involving other various topics. Some of the most popular math games available on the site are Princess Math, Skater Math, Grand Prix Racing, and Math Tiles. There are also links to various fun and information based topics like Jump Start and PBS Kids on the site. Find fun games the Primary Games website here.
5. Fun School
Fun School is a place your kids will surely love. The website provides fun and learning tools for subjects such as Math, Science, Language, Geography, History, Sports, Arts and many more. Children from pre-school up to 8th grade can learn and explore everything about outer space, oceans, and even farms. You can visit Fun School here.
6. Mr. Nussbaum
Mr. Nussbaum separates players by their grade level and lets children practice the various aspects of math. The site has 87 math games for grades 2 to 7. They are able to practice multiplication, rounding, graphing, as well as other various topics. The website also provides interesting facts about math and several assorted topics. Check out the website here.
Another website which provides fun math learning games and activities is IXL. The contents in the website are divided according to grade level (from preparatory to 8th grade). This grade range helps in finding your child a specifically suitable topic for practice. The website also has a feature of letting you know how much time your child your taking to answer a particular question and allots him/her a score in order to help keep a track of his/her progress. Visit IXL right away by clicking this link.
8. Free Math Help
This website provides help to students of higher standards with algebra, geometry, trigonometry, and other advanced topics. The website provides simple and straight forward teaching paired with fun activities, which is sure to help students learn a little easier. The graphics of the website are basic and the instructions provided are pretty clear. Go to the Free Math Help Website.
9. Math Play
Math Play is a website which hosts several math games that are specially designed for middle school students. Some of the games are based on game show formats such as Deal or No Deal, Millionaire, and Jeopardy. These games help make learning math more interesting. Students can also practice various concepts of math like fractions integers, and place values on the website. Check out Math Play here.
10. Funny Games
Funny Games is another website which provides interesting math learning tools for children. The website features various arcade style games and puzzles which make kids learn the concepts of math in a fun way. There are games on the website where you are required to solve various math problems in order to destroy enemy tanks and battle your way through epic fights, all while learning math. Move over to the Funny Games website by clicking this link.
11. Math Blaster
The next website to feature on this list is Math Blaster. Math Blaster provides an opportunity for your kids to save the galaxy by solving math problem and mastering their skills. There are arcade style games and high-speed missions which they’ll need to complete in the process. Math Blaster also gives kids a chance to meet new friends and learn various concepts of math. This game franchise has been around since the early 90’s, and it’s bound to impress. Check out the awesome Math Blaster website here.
12. I Play Math Games
I Play Math Games is an advanced website which provides opportunities to learn math on different devices such as iPad, Mac, PC, Smartboard, and Whiteboard. There are numerous free math game games on the website which are fun to play along with free workbooks which can be printed, interactive white boards, and links to different useful apps. Visit I Play Math Games here.
13. Smart Math
There is a ton of great learning and information waiting for kids in Smart Math as the website is associated with Encyclopedia Britannica. The website features nicely categorized contents for children from Kindergarten to Standard 8. The membership of Smart Math is also very affordable and lets parents to keep track of their child’s progress. You can have a look at the Smart Math website here.
14. Math Slice
Math Slice has content which can be enjoyed and explored by students of all ages. The website starts from the very basic concepts of math like addition and subtraction to advanced concepts like statistics, measurements, and fractions. The website also hosts traditional games such as Hangman, Bingo, and Chess along with varied versions of Guitar Hero, Drive Thru and Math Millionaire. Check out the contents of Math Slice here.
15. Soft Schools
Soft Schools is a very simple website to access. The website has contents which are divided into different age groups which makes them easy to find. The games in Soft Schools are very simple and fun to play. Kids don’t need to follow any particular instructions to play the games on this website. Have a look at Soft Schools here.
All of the websites listed above features games that are interesting and fun to play. These games can help children learn math and have fun at the same time. Kids often consider math to be a boring subject, but the sites in this article have the potential to make them change their mind.
We’ll try to update this list with more websites that makes math learning for children interesting as soon as we come across them. For the time being, we hope the list helps your kid to master his/her math skills. If you have any other suggestions or names of websites which falls under the same category, please let us know.


Teaching math in geography? Yes! Social studies, or social science, involves mathematics. Geography is especially mathematical as maps, globes, and GPS systems are all based in mathematics. My trip to the HSTW14 conference convinced me that, as a social studies teacher, I could do more to reinforce math in my courses and thereby make them more applicable to the real world. I set a goal of doing five FALs this year along with at least two LDC modules. I’ll post here as I develop and test applications of MDC concepts in the geography classroom. Below are some links to help include mathematics in the study of geography.


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Data Science | DSDHT
Are there any new drugs left?
Boiling the ocean - integrating everything for patient care
Boiling the ocean - integrating everything together
Gene expression data analysis
Information-based Drug Discovery
Introduction to R Shiny
Managing disease with data science
Mapping Structure to function
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Predictive modeling of drug-protein interactions
In this module, we look at how mathematical and supervised machine learning methods like Linear Regression, Random Forests, Support Vector Machines and Bayesian methods can be used with chemical structure data, descriptors, and experimental biological training data to predict the interaction of chemical compounds and drugs with protein targets.
Predictive modeling and QSAR in cheminformatics
The establishment of
(SAR) in medicinal chemistry predates the use of computers in chemistry, and relies on correlating structural features with experimental results for multiple compounds, usually in the same series. It is common in medicinal chemistry to use synthesis techniques to create several related compounds (e.g. methyl-, ethyl-, butyl- forms), and then to investigate the effect of these synthetic changes on a particular property or biological activity (so we might find, for instance, that extending the Methyl chain reduces a particular activity). The relationship between structure and activity may or may not be quantified.
Quantitative Structure-Activity Relationships
(QSAR) were originally designed as an attempt to add some mathematical basis to this process, particularly to define the activity as some function of descriptors (note that when the activity is a property or a toxicity, this is sometimes referred to as QSPR and QSTR respectively). If we develop a function that relates descriptors to a particular activity, we can then use the function
for compounds where the activity is unknown but the descriptors can be calculated.
The earliest examples of QSAR were
which are actually applications of
. Hansch analysis pertained to property descriptors, and Free-Wilson, which we shall discuss here, to structural descriptors. Free-Wilson defined a function that equates activity (defined as log of 1 / the concentration) with weighted descriptors, the weightings, or coefficients, being determined by linear regression. That is, we have the equation:
Log (1/C) = a1x1 + a2x2 + a3x3 ...
where C is the concentration required for activity, x1, x2, x3, etc are the descriptor values (usually 1 or 0 to represent absence or presence of features), and a1, a2, a3, etc are the coefficients derived from linear regression. Linear regression is a generalized technique that aims to optimize the coefficients applied to independent variables so that the dependent variable (in this case Log 1/C) most closely matches the observed value for a set of descriptors. Thus one an think of a regression equation being
using data with known dependent values, and then being
predictively to data with unknown dependent values. Linear regression works by minimizing the sum of the differences between the values predicted by the equation and the actual observation. This is nicely illustrated in a
If a regression equation is to be used predictively, then we need some way of gauging its accuracy. The simplest way to do this is with
, which is the proportion of the variance in the dependent variable that is explained by the regression equation (i.e. if r-squared = 1.0, then all the actual points lie on the regression line; if r-squared = 0.0, then the variance around the regression line is as high as the overall variance of the dependent variable).
There is a problem though with r-squared: the same data that is used to build the equation is also used to evaluate it. This can be addressed using q-squared (sometimes called crossvalidated r-squared). Here, we make n versions of the equation, each build leaving one of the original known values out (it is thus an example of leave-one-out validation); the q-squared is then the mean overall variance in using the equation to predict the values left out. q-squared is always thus less than r-squared.
Nonlinear approaches to QSAR
The main drawback of these early approaches are that they assume that the activity varies linearly with the descriptor values that affect it. However, this is usually not the case. Nolinear approaches still try to correlate descriptors and outcomes, but do not make this assumption. They are thus at least theoretically more useful, although there is usually some trade-off (such as speed, scalability or interpretability). Nonlinear approaches are generally an example of
(as opposed to
such as clustering; however unsupervised methods such as
may also be employed). The method used will also sometimes depend on the kind of QSAR that is to be determined - particularly there is a difference between classification problems (such as predicting whether compounds are active or inactive) and quantitative prediction problems (where we want to predict an activity value). Some of the most frequently-used nonlinear methods for QSAR are:
(such as Recursive Partitioning and
Support Vector Machines
Different methods have different strengths and weakensses: for example neural nets are a "black box" approach and thus are not useful if we want to know
a particular prediction was made. Decision Trees are only usable for classification problems.
Regardless of the method use, building a model will generally be done in three phases: training (presenting known data to build the model); validation (testing the model with known data that has not been presented to build the model, such as a validation set); and prediction (using the model for truly unknown data). This is illustrated below.
Evaluation of predictive models
If predictive models are to be properly evaluated there are a few basic principles that should be adhered to:
For publication. public datasets should be used, and the method and descriptors used should be made freely available or be described well enough that a reader could replicate the experiment
A validation set should always be used, and any success statistics should be based on the validation set, not the training set
For classification problems, always create a
. From this, you can derive measures like
sensitivity and specificity
precision and recall
. For large sets, particularly for virtual screening applications, it is appropriate to show a
(one can also calculate AUC, or area under curve).
The use of predictive models is growing, since they aim to provide fast, reliable and quite accurate
estimates of the chemicals’ activity.
These features also make them suitable for legislative
purposes, and that is why they have been included as an alternative tool for risk assessment in the new European legislation on chemical production, called
R.E.A.C.H. (Registration, Evaluation,
Authorisation and Restriction of Chemicals)
Check out the series of videos QSARs.
ORCHESTRA is an EU project, funded to disseminate recent research on in silico (computer-based) methods for evaluating the toxicity of chemicals.
A Free online QSAR course is also available
Here is a video describing how to perform QSAR in R.
History of QSAR
a must read
Modeling methods in QSAR/QSPR
Predictive QSAR modeling based on diversity sampling of experimental datasets for the training and test set selection
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What good is a bunch of empty boxes? Last week, it was great to teach proportional relationships. I had my students all bring an empty box or bottle that had unit conversion information on them. The first thing that I asked my students to do was to enter the conversion information onto a table on Desmos table. As you can see below, the student started by entering the information from the package: 10 oz. and 283 grams.
Next, I had students put (0, 0) on the table, and see if it fell on the line. Then we talked about why this point would be on everyone's line, even though they had different conversions.
Finally, I had students trace along the line, and find the point where the x-value was 1. We talked about what this point represented.
This simple activity was a great way to introduce proportional relationships, and some of their important characteristics.


Mano Vikash's Homepage
I am a Quantitative Researcher at Balyasny Asset Management.
Previously, I was a Senior Research Scientist at the machine learning group in Lifion by ADP.
I completed a PhD in mathematics working on problems at the intersection of mathematics and computer science in the University of Illinois at Chicago. I was fortunate to have Lev Reyzin as my PhD advisor. Before that, I got my bachelors and masters in mathematics from the Indian Institute of Science Education and Research, Thiruvananthapuram.


A Place to Share and Learn from Each Other
Come share with us at AMTE 2021!
Every year, mathematics educators from around the country come together to learn from one another. One of the main benefits of the time, resources, and energy spent in doing so is bringing us together in the same space to learn with and from one another. This space is important because so many of us work in isolation. Some of us may be the only mathematics educators at our institutions or the only person teaching a specific course at our institution. Even if we are not in isolation, all should be provided the opportunity to learn from other mathematics educators. According to the National Council of Teachers of Mathematics (2014) “As professionals, mathematics teachers recognize that their own learning is never finished and continually seek to improve and enhance their mathematical knowledge for teaching, their knowledge of mathematical pedagogy, and their knowledge of students as learners of mathematics” (p. 99). We see this as applicable in our own practice, and see the Association of Mathematics Teacher Educators (AMTE) as a professional organization that allows space for teacher educators to come together to enhance their knowledge for teaching.
Each term there are hundreds of methods courses being taught. These methods courses can vary depending on the context of the institution and teacher preparation program. While the Standards for Preparing Teachers of Mathematics (SPTM) (AMTE 2017) offer guidelines of what should be included in these courses, mathematics teacher educators lack a place to find resources and/or activities to help us address these standards in our courses. There are also differences in how many methods courses are offered for prospective mathematics teachers and whether or not these courses have imbedded field experiences. All of these differences lead to a variance in what is taught and what activities we use in our courses.
Typically, new ideas are piloted in our methods courses and old ideas are either improved upon or discarded without record. The knowledge and ideas that are produced in this process are largely lost due to our isolation. In this session, we aim to recover that knowledge by making the most of our shared space and time by distributing the lessons we have learned from our own practice. This session will serve as an incubator for assignments to support preservice teachers in their pedagogical development of teaching mathematics.
If you have questions or ideas please contact us!
Amber Candela (firstname.lastname@example.org)
Zandra de Araujo (email@example.com)


STEM is an acronym for Science, Technology, Engineering, and Mathematics. STEM will use innovation, collaboration, and problem-based learning to cultivate students' science, technology, engineering, and mathematics skills. Considering predictions that 80% of new jobs will require math, science, and engineering, and more than 50% of the technical work force will retire soon, it is more important than ever to inspire students to pursue careers in all of these fields.
The program revolves around solving global problems utilizing a problem-based learning model that begins with a global issue and works toward finding solutions to that issue. The curriculum emphasizes connections within and between the fields of mathematics and science with heavy and meaningful integration of technology. The arts and humanities further integrate the STEM disciplines, and all courses highlight the role of STEM in the global society and economy.
Theme-based schools provide the opportunity for students for advanced learning as well as hands-on learning and enrichment experiences. Each student will continue to study the DPI's Standard Course of Study, however, the focus of the curricular experiences will be theme-based. All students will be measured at the end of each year by the current End of Grade and End of Course exams. Schools will also be held accountable for making Academic Yearly Progress in all subgroup areas.
The STEM experience can begin or continue at J. N. Fries. This grade 6-8 program will continue the use of problem-based learning approaches to delivering the curriculum. This program will again focus on global issues that require the use of science, technology, engineering, and mathematics to find solutions.
• Language Arts
• Social Studies
• Health/Physical Education
• Arts (Band, Chorus, Art)


Last time I checked in RE: Bowie pillow, I shared fabric choices. The next step is cutting triangles for the flying geese blocks.
And when it came to calculating triangle dimensions for my custom design, I did it the hard way. Then I did it the easy way.
I will show you the easy way first, and then we can giggle over my overcomplication. Follow these steps to cut your own custom-sized flying geese blocks, you quilt designer, you!
Step 1: Understand what you’re cutting.
To make custom-sized flying geese blocks, cut one large square and four small squares. These five squares will yield four flying geese blocks. Check it out:
One large square makes four “geese.” Four small squares make eight pieces of “sky.”
Think about this as you work up your flying geese design. There’s a opportunity to reduce fabric waste.
Step 2: Determine square widths.
Please note that finished widths do not include seam allowances.
Finished widths confused me when I started making my own custom-sized flying geese blocks for the Bowie pillow.
The finished width for the small square is the length of one of its sides. That I understood, no problem.
But when I looked at the finished width for the large square, all I could see was the hypotenuse of a right triangle.
Don’t make your brains hurt by only seeing the hypotenuse. Instead, use the above handy-dandy flying geese block diagram. So. Much. Easier.
Flying geese blocks always are one square tall by two squares wide. That said, the small square finished width always is one-half the large square finished width.
Look at your custom flying geese design, and, based on how big you want your quilted object to be, determine your large square and small square finished widths.
Step 3: Perform flying geese math.
Here’s how to calculate the side lengths of the squares — including seam allowances. These formulas are from “1, 2, 3 Quilt“:
Large square side length = Finished width + 1 1/4 inches
Small square side length = Finished width + 7/8 inches
For example, if the large square finished width is 9 inches, add 1 1/4 inches to get the large square side length (10 1/4 inches).
Step 4: Cut one large square.
Use the large square side length (calculated above) to cut one large square. Then cut the large square, corner to corner, into four triangles.
Step 5: Cut four small squares.
Use the small square side length (calculated above) to cut four small squares. Then cut each small square into two right triangles.
Hooray! You just made eight small triangles, and you’re ready to sew the blocks. I’ll cover sewing blocks in a future post with diagrams and pictures.
How to overcomplicate custom-sized flying geese blocks
I didn’t pick up what “1, 2, 3 Quilt” was throwing down when it came to figuring out custom-sized flying geese blocks. You can see from my drawing that I figured out finished widths for small and large squares (4 1/2 inches and 9 inches, respectively).
However, there wasn’t a drawing like the one I made in Step 2 above to spell out WHERE I could find these measurements. Instead, when referring to the square side length formulas, the book read, “The finished width … refers to the finished size of the block once it’s sewn into the quilt top.”
So naturally I turned to the Pythagorean theorem to calculate the large square side length. Behold:
Yeah, I was really hung up on that hypotenuse thing, and it overcomplicated everything. Sometimes your brain struggles to look at a picture from a different angle.
In the end, I found a video or another blog post that set me straight. My gut told me I was making this harder than it needed to be and that the measurements should come out “nice” — not 6.36 inches (yuck). This is a case of taking the scenic route through a challenge.
My Bowie pillow flying geese blocks came out a-OK, and I was inspired to share my missteps to aid other novice quilters.
Over to you: When was the last time you used the Pythagorean theorem? How do you deal when directions don’t make sense? Are you inspired to draw tons of flying geese blocks in a graph paper notebook ? Please share!
P.S. Thanks for hanging on till the end! I have big Sie macht news! I will write exhaustive directions for the Bowie pillow. So, if you dig on this flying geese pillow pattern, I’ll provide you with all the fabric dimensions and whatnot so you can make your own!


In the social world we often come across circumstances where the effect of one variable on another variable varies depends on different levels of a third variable. In statistical terms this is known as Interaction. The concept of interaction can often be confusing to students, who tend to think in terms of straight linear relationships. The present study offers a simple and effective way to teach this complex concept in the context of sociopolitical ideology. We examine the role of ideology in determining the sentencing recommendations of probation officers using ANOVA and OLS regression and contend that interaction terms can be presented to students in an easily understandable manner. Our approach helps integrate substantive theory with statistics to the benefit of both.


Definition : Electrocardiographs that subject the successive point values in one or more leads of the standard electrocardiogram under analysis to mathematical integration. The resulting values are plotted on a display matrix, which resembles two tangential circles, one inside the other. The area between the circles represents the "normal" range of distribution of data; if the curve derived falls outside the limits, the value is considered abnormal. Cardiointegraphs are intended to be used in the diagnosis of coronary disease in the absence of abnormalities in the standard resting electrocardiogram.
Entry Terms : "Omnicardiographs" , "Electrocardiographs, Omnicardiograph"
UMDC code : 12799


Many-Body Simulations I. Stochastic methods: from Langevin dynamics to Quantum Monte Carlo
Basic course given by Sandro Sorella
Start: 05-10-2021 End: 16-11-2021 Room: 131
Schedule: Tue 9:00 - 11:00, Thu 9:00 - 11:00
Many Body Simulations I. Stochastic Methods: from Langevin dynamics to quantum Monte Carlo (by Sandro Sorella)
The lectures will be held in hybrid format. CM phd and master students are warmly invited to addend the lectures in room 131 where the teacher is present and about ten seats are available. Conversely you can send an e-mail to firstname.lastname@example.org to receive the zoom address for the remote connection.
- Lecture 1: Variational approach and many-body correlated wave functions: an introduction, see:https://people.sissa.it/~sorella/lecture1.mp4
- Lecture 2: Many body variational wave function ansatzs: cusp conditions, size consistency, size extensivity, and area law for the entanglement entropy. see: (as above) ../lecture2.mp4
- Lecture 3-4: Short review of probability theory and Monte Carlo sampling; application of variational quantum Monte Carlo to continuous systems such as Helium IV. ../lecture3.mp4 ../lecture4_part1.mp4 ../lecture4_part2.mp4
- Lecture 5: (Hartree-Fock and second quantization is required) Variational Monte Carlo for electronic systems on a lattice: practical implementation for the Hubbard model.
- Lecture 6-7: Advanced sampling by Molecular dynamics: from Langevin dynamics to Hybrid Monte Carlo.
- Lecture 8-9: Stochastic energy optimization of many-body correlated wave functions: state of the art and beyond.
- Lecture 10-11: Exact ground state properties for bosonic systems by quantum Monte Carlo: application to the Heisenberg model.
- Lecture 12: The sign problem for fermionic systems, approximate schemes: the fixed node approximation for lattice model Hamiltonians


Welcome to Middle School Math
I am so excited to be a part of the Saint Edward the Confessor community this year as the middle school mathematics teacher! This year we will continue to explore the wonderful world of mathematics through the use of innovative tools and techniques. I hope to inspire a love for math in all of my students, and to encourage them to become out-of-the-box, curious thinkers and learners.
My educational background consists of a Master's Degree in Mathematics Education and Teaching Students with Disabilities from Molloy College. I possess New York State teaching certifications in both Adolescent Mathematics Education (grades 7-12) and Students with Disabilities (grades 7-12). My goal is to provide my students with instruction that will allow them to develop the skills necessary to become lifelong learners.
I expect my students to demonstrate kindness and compassion in my classroom at all times. I am excited to work together to build a fun, safe, and supportive learning environment for the 2016-2017 school year!


For kids who have trouble with math, one man has the solution – his name is Papa. Papa is a math papa and he teaches his kids how to add, subtract and multiply from 1 to 100 in 2 minutes using simple, easy-to-follow steps. This method has helped countless kids learn math quickly and easily, so if you’re struggling with numbers, give Papa a try!
In a world where long division is often seen as a dreaded math problem, one man has shown his children that there’s more to addition, subtraction and multiplication than just counting up. Using simple illustrations and easy-to-follow examples, Glen D. Mueller, aka The Math Papa, taught his kids how to add, subtract and multiply from 1 to 100 in just two minutes!
John was a math papa who loved to teach his kids how to add, subtract and multiply from 1 to 100 in 2 minutes. He would often set up challenges for them, such as multiplying two 10-digit numbers in less than 10 seconds or adding two 20-digit numbers in less than 5 minutes. His kids quickly learned how to do these calculations efficiently and effectively, which helped them perform well both in school and on standardized tests.
The Amazing Math Papa Who Can Multiplicate 15074 by 9 in His Head
Gregory Poplawski is a math genius who can multiply 15074 by 9 in his head. He has a photographic memory and can recite pi to 22,514 places, according to a recent article in the Huffington Post. Gregory, who is 34 years old, was born with a rare condition called locked-in syndrome, which prevented him from using his hands or speaking. However, he has never let his condition keep him from pursuing interests in mathematics and other sciences.
Meet Patrick, the Amazing Math Papa who can multiplicate 15074 by 9 in his head. This is no easy feat, but Patrick is a master of math and has been able to do it for years. What’s even more impressive is that he can do it quickly and without any mistakes. Patrick is an inspiration to others and shows that with hard work and dedication, anything is possible.
There are few people in the world who can multiply 15074 by 9 in their head. One of these people is 43-year-old Greg Fenton from Morrisville, North Carolina. Fenton is a father and math teacher who has developed a system for multiplying large numbers quickly and efficiently. He believes that anyone can learn to multiply large numbers using his method if they are willing to put in the effort.
In elementary school, most people are taught basic arithmetic: addition, subtraction, multiplication, and division. These skills are essential for everyday life, but they’re not the only thing that counts when it comes to math. In fact, there are a number of other important mathematical concepts that you need to know if you want to succeed in college and beyond. Here are five of the most important:
1. Algebra is the study of equations and how they interact with one another.
Math is essential in almost every field of study. It allows us to understand the world around us, and it can help us solve problems. However, not everyone is comfortable with math. For some people, it can be hard to understand and remember the formulas. Luckily, there are ways to help ease the learning process. One tool that can be helpful is flashcards. They allow you to review the material quickly and easily. Additionally, online calculators can also be a great resource.
There are various ways in which mathematics can be used to solve problems. It can be used to figure out how much money is needed to buy a certain amount of goods, to find the shortest path between two points, or even to determine if two numbers are equal. Mathematics can also be used to understand the world around us. For example, mathematicians use mathematics to study geometric shapes, celestial objects, and more.
Math Papa Can Teach Math on a Whole New Level
Math Papa is a new app that teaches math in an innovative way. The app is based on the principle that children learn best when they are engaged in an enjoyable activity. Math Papa uses puzzles and games to help children learn math concepts.
Math Papa is a new app that uses gamification to teach math concepts. The app is designed for parents and children ages 4-12, and it can help children learn math faster and retain information better.
Math is a subject that can be challenging for many students, but it doesn’t have to be. With the right instruction and help from a math Papa, students can start to see math in a whole new light.
World’s Smallest Math Papa Named a Champion
Math is often seen as a difficult subject, but one man has proven that it can be mastered with a little bit of hard work. In fact, this man is so good at math that he has been called the world’s smallest math Papa. His name is Harish and he is only 4feet 10inches tall! Harish’s parents were not originally planning on having him pursue a career in math, but when they saw how talented he was, they decided to give him a chance.
Mathematics is one of the most challenging subjects a person can take on, but for 7-year-old Timo Teräsvirta, it’s as easy as ABC. That’s because his dad, Risto Teräsvirta, is a math genius. Timo has performed well in school contests thanks to his dad’s tutelage, and he now wants to be just like him when he grows up.
The math world was abuzz when news broke that a world record holder in the area of small math problems had a son named Luke. The record-holder, Wang Chao, is only four feet tall and his son Luke is two foot seven inches tall. This little guy is a math genius and has already broken many world records! Born with a talent for numbers, Luke has been honing his skills since he was very young.


Clinton Central Jr/Sr High School
MRSA in Schools
Homework Hotline AskRose.pdf
Technology's Homework Hotline, a free math and tutoring service for Indiana students in grades 6-12. Students can call toll free 1-877-ASK-ROSE for math and science homework help from 7 p.m. to 10 p.m. Sunday through Thursday.
Simple instructions and free artwork are available on the Homework Hotline Web site. Visit www.AskRose.org and click on School Supplies, then Hyperlinks.
Learn More Resource Center , Indiana 's Pre-K to College Connection, supports and encourages lifelong learning for all Indiana citizens. Up-to-date materials, helpful checklists and other valuable resources and tools provide parents, students, school counselors and others with information about every aspect of education from early learning, to K-12, to college and careers.