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Almost every online casino has some kind of welcome bonus in order to help try and persuade players to sign up and use their website rather than their competitors. With a little thought and cleverness, we can take these new-player casino bonuses and convert them into withdrawable cash.
However, before delving into casino bonuses, let’s talk about sportsbook bonuses. Earning money from them is known as matched betting and is a little easy and may be where you get started?
But right now due to the COVID-19 pandemic, most sports aren’t running so there are not many matched betting offers available. But to try and keep players, casinos are offering better bonuses than usual. It is a great time now to try some casino bonus bagging – particularly as we all need to earn some more money!
Casino Bonus Bagging Isn’t Risk Free
Unlike matched betting, earning money from casino bonuses is not risk-free. But that doesn’t mean that it can’t be profitable and worth doing.
We call this notion expected value (EV). By using maths, we can find out how much money we will be able to make on average from the bonus. So why is that not risk-free? Due to variance. Each time we make a slightly different amount, but throughout the long term, it averages to the EV.
If you understand what I mean then feel free to skip straight to the how casino bonuses work section.
The More We Play, The Closer To Expected We Get
Now we get a bit more complicated. The longer we play, the more likely we are to lose money.
I am not going to explain the maths here, but over 1 coin flip my opportunity of walking away with a profit is 50%.
Over 100 coin flips my opportunity of making a profit has dropped to 31.7%, that’s almost a 1/3rd opportunity which still is quite high. I may decide that I’m feeling lucky and take that 1/3rd opportunity. But over 6,000 coin flips my opportunity of leaving with a profit drops only 1%.
That is the way how a casino makes money. If there are 60 people playing at your casino with each person playing the coin flip game 100 times, then each has a 31.7% chance of leaving with a profit. But since the casino is playing 6,000 total flips, it only has a 1% chance of not making a profit overall.


Math Department Overview
The Norwalk High School Mathematics Department is comprised of 16 teachers. The department chairperson is currently Maureen Ireland.
Courses taught range from Mathematics for Multi Language Learners to AP Statistics and AP Calculus, including all courses in between. The department is home to the Computer Science curriculum where AP Computer Science Principles and AP Computer Science A are taught.
The goal of the Norwalk High School mathematics department is that all students will persevere when faced with challenging content, approach problems from different perspectives, explain solutions accurately and coherently, and develop a curiosity about the nature of mathematics.
Suggestions from the department for students preparing for their assessments in Math class.


My teaching career began as a high school mathematics teacher in Levittown, NY. As my family moved around the country, I held teaching positions in Enterprise, AL, Cheyenne, WY and Fairfield, CT. I also worked as a Cost Accountant for Boeing Aircraft and as a Statistician for the WY Department of Labor. I taught my first classes at North Central in 1986. Over the years, I have witnessed many changes at the college in curriculum, staff and facilities.Download CV
- Girls and Math Workshop: 9am-1pm, July 10-14, 2017.
- "The New SAT" The 2017 Annual Meeting of the Illinois Section of the MAA, on April 1, 2017, College of Dupage in Glen Ellyn, Illinois. The SAT test from the College Board underwent a number of significant changes recently (they were put into effect in spring 2016). In this session we will look at the new SAT Math test, its makeup, format, and structure. In Illinois, the State of Illinois is now paying for high school juniors to take the SAT test (previously, they paid for the ACT). As a result, more college applicants applying to Illinois colleges and universities are using SAT scores in their applications. In this session we will raise some of the math placement issues surrounding use of the new SAT scores and share ideas. We may have more questions than answers at this point, so input will be welcome. This talk will be of interest to those teaching teacher education courses, but is also of general interest and to those who are involved with math placement. Co-presented with Jim Olsen, Western Illinois University
- "Coding and CAD in the Classroom: 21st Century Education", Metropolitan Mathematics Club of Chicago Conference Of Workshops,Saturday January 28th, 2017,at Evanston Township High School, in Evanston, IL with Jocelyn Wilcox, Irving Middle School.
- Girls and Math Workshop: 9am-1pm, July 11-15, 2016.
- Girls and Math Workshop: 9am-1pm, July 24-28, 2015.
- “Coloring, Fish Tanks and Scheduling” for K-12 teachers at the Metropolitan Mathematics Club Annual Conference of Workshops, Stevenson HS, January 24, 2015.
- Girls and Math Workshop: 9am-1pm, July 25-29, 2014.
- “Lesson Study: A Mathematics Tool for Teacher Collaboration in the Midst of the Common Core Standards” to K-12 teachers at the Associated Colleges of Illinois Summer Symposium, held at North Central College, June 27, 2014 .
- “Finding Success and Satisfaction in Realizing the Common Core in Mathematics” Single Paper with Multiple Presenters, Association of Teacher Educators 2014 Annual Meeting, St. Louis, Missouri, February 17, 2014.
- Girls and Math Workshop: 9am-1pm, July 22-26, 2013.
- “MATHEMATICAL MODELING: WHAT IS IT? WHY DOES IT MATTER? to 6-12 teachers at the Associated Colleges of Illinois Teacher Induction Academy on Friday, April 19th, 2013 at Dominican University, River Forest, IL


10monkeys Math World Giveaway
Welcome to the 10monkeys Math World Giveaway
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10monkeys Math World is an online Math Program geared for younger elementary aged children around grades K-3. It has fun, eye-catching graphics kids love! 10monkeys Math World takes a progressive approach to learning Math. It uses what is referred to as the “high pedagogic approach,” developed together with top professionals and schools. As they progress, their skills are improving as well – and they’re having fun!
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Math 561 is a graduate-level course on algebraic and geometric methods in statistics. Course homepage is here, containing detailed info on the material, grading, etc.
This page keeps track of course notes, material covered, homework schedule, and other relevant links. Unlike the static course homepage, this one is updated weekly.
- In the next iterations of the course, the order of topics will likely remain the same, but the speed at which we cover material may change.
- Most lectures use slides in the beginning, but I always also write on the board to supplement examples.
- It is impossible to follow the course only by reading these slides; students are expected to attend the lectures as well.
- All course communication, updates, announcements, logistics, and grading takes place on our Campuswire course page. Link and permission is provided to all enrolled students.
Week 1 - invitation to algebraic statistics
Homework 1 assigned. due 1/18/2023->1/24/2023
Week 2 - limiting distributions, why worry about which model fits the data, and intro to conditional independence
(This was a one-lecture week due to MLK holiday.)
Week 3 - the algebra of conditional independence
The plan is to cover the remainder of section 4.1 in the book. We will talk about the algebra / geometry behind conditional independence models.
It is important to be familiar with the independence concepts so that you can better understand the lecture.
Day 1: discrete random variables
- lecture 4 slides PDF
- there is a typo on slide 8. can you spot it? (hint: which variable is non-binary…)
- These slides contain basic information about ideals and varieties; they will serve as good reminders.
- in-class worksheet
Day 2: Gaussian random variables
For students, here are questions to verify:
- What is a positive definite matrix? That is, \(\Sigma\in PD_m\) means what, exactly?
- What is the density of a Gaussian random vector?
- How do you compute covariance (that is, entries in the matrix $\Sigma$)?
Things to review:
Understanding some of the material in here, which is still in section 4.1 of the textbook, requires digesting the normal distribution notation from Chapter 2. ( These are standard concepts you most likely have seen in some statistics course before.) Specific results you need to review are stated in the slides at the beginning of the lecture.
Homework 2 assigned. due 2/10/2023. Here is the source .Rmd file. Deadline moved to 2/13/2023
Important note regarding code submissions: If you are solving the coding problem, you should submit: clean code that I can run, ensuring that proper documentation is provided (e.g. what does each function/method do?), and a printout of at least one example illustrating how the code works.
Week 4: Maximum likelihood estimation and exponential families
Day 1: Parametric models (recap), statistics vs. parameters, parameter estimation problem
- lecture 6 slides PDF
- Material from: Statistics primer (bits of chapter 5)
- (Note: we didn’t get to exponential families on Monday.)
Day 2: Exponential families
- lecture 7 slides PDF
- Material from: exponential families (bits of chapter 6), and likelihood inference (chapter 7). In particular, section 7.1. talks about score equations and the following sections a bit on how to solve them using geometric viewpoint. We will refer to this material as needed.
- We reviewed an example on maximum likelihood estimation, but did not finish discussing all of the likelihood inference topics today.
Week 5: Exponential families, and wrap-up on likelihood inference. The sufficiency principle.
- Material from: exponential families (bits of chapter 6), and likelihood inference (chapter 7).
Day 1: How to find the vanishing ideal of a discrete exponential family. Log-linear models and binomial ideals.
Day 2: Likelihood inference & outlook: why do we even estimate parameters?
- lecture 9 slides PDF
- Last part of slides in lecture 9 are preview of the next topic, which will be core to several projects: the question of model/data fit.
Homework 3 assigned. Due 2/24/2023 hw3.pdf, hw3.Rmd.
Day 1: in-class work. See Campuswire post #39 for announcements.
Note: Homework 2 is due 2/13/2023, extended from Friday week 5.
Please note the following special assignment:
Homework X assigned. hwX.pdf Attend either of the following two talks: 1. Aida Maraj, Friday 3 March 11:30-12:30, or 2. Felix Almendra Hernandez, Friday 7 April 11:30-12:30. Write a one-page summary (one page typed in Markdown, for example) of the talk. Details are in the PDF. If your write-up is satisfactory, this grade can contribute to your your “Participation & worksheets: 10%” grade (see grading scale) in the course.
Day 2: Exact testing: motivation
- Lecture 10 slides PDF.
- Please note that there is hands-on work included in this lecture, although I did not separate it out as a worksheet!
- Review p-value definition in section 5.4 “hypothesis testing” from the book. Also, we will be working through the Markov chain example in Chapter 1, to find a way to do a goodness-of-fit test.
Project topics are to be posted this week based on the input received from students; please stay tuned. pushed to week 7 due to illness.
Week 7: GoF (goodness of fit) for log-linear models
Day 1: exact tests part 2
- lecture 11 outline PDF
- This worksheet with its Rmd source should help you walk through the questions of HW3. I explained the significance of the two parts of the assignment to understanding the GoF problem. For example, who cares about likelihood? You need the MLE
We worked through the main ingredients of an exact conditional goodness of fit test, starting with the motivation. Here are the main points, which I summarize for your review:
- Began by reviewing critical interpretation:
- What does the hypothesis $H_0: p\in\mathcal M_A$ mean?
- Hint: $p$ is a probability vector or table.
- What is “the best explanation the model $\mathcal M_A$ can give for the $p$, the best “guess”?
- We use $\hat p$ to make a decision based on statistics. It sounds like a proof by contradiction.
- Defined an asymptotic test of independence based on the chi-squared statistic which has a chi-square distribution (degrees of freedom computed using number of row/columns of the table).
- Defined an exact test that would be necessary in some cases, for example when some of the table cell counts are small. This is an example of what we call “sparse data”.
- Identified an issue with the exact computation: evaluating the generalized hypergeometric distribution in the setting of larger samples/tables.
- The alternative is to find another exact distribution to reference:
- Find a new score for each table (not the actual probability of that table, but a statistic such as the $X^2$);
- Compute $X^2(v)$ for all $v$ whose sufficient statistics are the same as that of the observed $u$;
- The sampling distribution of this statistic $X^2$ is the reference distribution to define the $p$-value.
- We did not yet define this $p$-value—we will begin Lecture 12 with it—but we did talk about all the ingredients needed to motivate why and how it will be constructed.
Announcement: Project topics and more information can be found on this page.
Day 2: exact tests part 3
Today’s lecture will finish the last definition in section 9.1 in the book, and continue covering section 9.2 in the book. The book has an example with another data table and code in
R, which you are invited to try out. I will focus on explaining the method and the algorithm on the board. For your convenience, here is a quick summary of what we will cover first:
- lecture 12 outline PDF
- Definition of the p-value of the exact conditional test of goodness of fit, using the chi-square as the GoF statistics of choice, and using the fiber as the reference set for computing the exact distribution of the statistic.
Homework 4 assigned. Update: THIS HAS NOT YET BEEN ASSIGNED. it will be assigned 2/28, early week 8.
Week 8: Bounds on cell entries or design of experiments [tentative]
Project topics selection period closes. All teams should have a topic to work on and develop a plan of action this week.
Day 1: exact test, part 3 continued
We spent a large part of week 7, day 2 on project topics. So, today we will finish the topics from slide 7 from lecture 12 outline PDF.
- Related reading: section 9.2. of the textbook.
- Here are additional slides start on slide 26 which summarize book section 9.2., corresponding to today’s lecture topic.
- At start of lecture, reminder: definition of fiber, definition of exact conditional p-value
- what are we conditioning on and why? (suff.stat.)
- what is “exact” about this test? (distribution of GoF statistic.)
- what is the main problem we have to solve next? (enumerate the fiber so we can compute the reference distribution exactly.)
- Proposition (proof) conditional distribution on the fiber does not depend on A or p.
- Formal statement of Fisher’s exact test generalization to any log-linear model.
- Markov basis [definition, example]
- Random walk on the fiber [definition]
- Metropolis Hastings algorithm [definition]
- Theorem 9.2.3: M-H outputs an approximation to the exact conditional p-value.
Homework 4 assigned. Due 3/13/2023 hw4.pdf, hw4.Rmd.
Day 2: how to compute Markov bases?
lecture 14 PDF
- The fundamental theorem
- Examples and a sampler of results
- Some issues with nonexistent MLE that may impact the GoF tests.
Day 1: Cell bounds on contingency tables
lecture 15 PDF
- Chapter 10.1 plus additional references; see slides
The lecture will be on the board and follow these notes, summarizing our book’s Chapter 14.
Week 10: Graphical models
Day 1: intro to graphical models
Lecture by Miles Bakenhus. Notes forthcoming.
Day 2: continuation.
Week 11: Graphical models [tentative]
Lecture 18 PDF.
- Here are additional notes.
Lecture 19 PDF.
Homework 5 assigned. Due 4/14/2023 hw5.pdf, hw5.Rmd.
Day 1: MLE for graphical models
Lecture 20 PDF
- Insert from Carlos Amendola: slides
Day 2: Project check-ins.
Status updates and help on projects. In person, during class time, in office RE208.
See post #82 on Campuswire:
Here are the steps to complete the project assignment:
- sign up your team for a day to present (see below for dates);
- prepare the slides (or notes to write on board if that’s what you prefer) for your presentation day;
- prepare the project report draft, and either: (a) post your project report draft here on Campuswire so that others can comment (anonymously) about it; or (b) print at least 6 copies so you can get feedback from other students in the class on the day of your presentation. Deadline for posting/sharing drafts is your presentation day.
- after your presentation, take the feedback you get from others to improve your final project report as necessary. Final report is due on Friday, May 5th, at 5pm. You are more than welcome to submit your final project report to me before this deadline, as soon as you are done.
Lecture 21 will be written on the white board.
Material: section 14.2 from the textbook; it revolves around a running example of relating hidden variables to instrumental variables and mixed graphs.
Lecture 22 will also be written on the white board.
Material: section 14.2, and then continuing into selection of several examples/results from chapter 16.
Homework 6 assigned. Due 4/27/2023 hw6.pdf.
Week 14: Project focus and hw6 work
Day 1: project check-ins
Monday, April 17, will be used again for project check-ins, which will this time take place over Zoom. The Meeting ID information will be sent separately, and each team will sign up for 10 minutes (similar to this week in person), you can use the time to share a draft of your slides, report, any issues that come up, etc.
Day 2: Final notes on identifiability.
- Note This week, meetings/lecture time will be on zoom as I am traveling to BIRS.
- The asynchronous activities for this lecture were shared via video on blackboard (zoom recording).
Week 15: Overflow & wrap-up; final presentations, part 1
Day 1: on to causal discovery with graphical models!
Since this lecture needs to be virtual, we will make it asynchronous. In lieu of the live lecture, I invite you to please watch this seminar lecture.
It is aimed at a student audience.
The lecture is 1 hour long.
It is a nice wrap-up for causal discovery, relating directly to and building on our most recent lectures on graphical models and conditional independence.
There is both a YouTube video link and slides that you can look at.
- Speaker: Liam Solus (KTH)
- Title: On the Geometry of Causal Discovery
- Abstract: The basic problem of causal discovery is to learn, in the form of a directed acyclic graph (DAG), the cause-effect relations that hold within a system of random variables based solely on data drawn from their underlying joint distribution. While a variety of approaches to this problem have been explored, algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful. In this talk, we will observe that the standard greedy approaches to causal discovery are all one and the same from a geometric perspective: they are each a different type of greedy edge-walk along a specific convex polytope. We will see how the method of proof for this result offers natural extensions of classical greedy algorithms, extensions that tend to perform better on simulated data. We will end with a discussion of the doors opened by this result: a new direction of research in discrete geometry for better causal discovery methods.
Day 2: project presentations on Wednesday, April 26, 11:25-12:40pm
Confirmed team presentations
- Team 5 (DAGs)
- Team 3 (Conditional Independence of Stocks given US Dollar index)
- Team 1
Finals week: final presentations, part 2
Thursday, May 4th, 2-3 pm, project presentations confirmed
- Team 2 (data analytics)
- Team 4 (How good are algebraic Markov Chains)
- Team 6 (uncovering proximity of chromosome territories)
finals week: Thursday, May 4, 2-4pm project presentations for 2-3 projects. Important: If nobody wants to show up during the finals week, then we need at least 2 projects to be presented on Wednesday, April 19, 11:25-12:40. If there are teams willing to do this, please just claim the date in the comments!
Sample files: how to create your first .Rmd file! All of your HW can be submitted using markdown and html/pdf. Here are some templates I created for another course, just so you know what to expect:
Types of files
- html - course handouts.
- PDF - slides for short illustrations used within a live lecture
- Rmd - raw R Markdown code. This has two purposes:
1) for students to see how the basic output files are generated, and
2) to enable collaborators to make updates to the materials in future semesters.


In this article, we’ll discuss calculating the amount of material for a tile and mixed media installation.
We’ll begin with the glass tile installation in Photo 1. This layout consists of a four-inch-by-four-inch tile with a four-by-eight pattern of one-inch-by-one-inch tiles to create the pattern. To begin the calculations for this type of installation, you need to find the repeat of the pattern. This one happens to be a four-inch-by-twelve-inch repeat (Photo 2).
Let’s use a room that is ten feet by ten feet with the tile pictured as an example. You will take the length times the width to get this number, 10x10 equals 100. Now, make sure that you add your usage/waste factor. You will need to add anywhere from 5% to 15%, depending on the layout, the more intricate the layout, the more you will add for usage/waste. We’ll add 7% to our area which now equals 107 square feet. The next step is to determine the amount of square footage of the pattern repeat. Multiply the 4 inch x12 inch, which equals 48. Take 48 and divide it by 144; there are 144 square inches to a square foot. 48 divided by 144 equals 0.33. The pattern repeat is .33 square feet. Take the 107 square feet that we need to cover and divide by .33. 107 divided by .33 equals 324.24. It will take 324.24 pattern repeats to cover the floor. We know that it will require 324.24 of the four inch by four inch tiles but we also need to determine the square footage necessary.
To determine the square footage of the four inch by four inch tile, take four times four, which equals 16 square inches. Divide 16 by 144 which equals 0.11, the square footage number. Multiply the square footage of the four inch tile to the number of pattern repeats. 0.11 x 324.24, this equals 35.66 square feet of 4 inch tile needed
To determine the square footage of the one-inch tiles, take four times eight which equals 32 square inches. Divide 32 by 144 which equals 0.22 square feet. Multiply the square footage of the one inch tile to the number of pattern repeats. 0.22 x 324.24, this equals 71.33 square feet of 1 inch tile needed. 35.66 plus 71.33 equals 106.99 square feet.
Let’s put all this into a mathematical equation:
Size of the room 10’x10’ = 100’
Add 7% usage 100 x7% = 107 square feet
Square footage of pattern 4”x12” = 48 square inches
Convert to square footage 48 / 144 = 0.33 sq. feet
Divide total square footage needed
By pattern repeat square footage 107/ 0.33 = 324.24 pattern repeats
Calculate sq. footage of 4”x4” tile 4x4= 16 square inches
Convert to square footage by dividing 16/144= 0.11 square feet
Multiply 4 in. tile square footage into
number of pattern repeats 0.11x 324.24= 35.66 square feet of 4 inch tile
Calculate sq. footage of one inch tiles 4x8= 32 square inches
Convert to square footage by dividing 32/144= 0.22 square feet
Multiply 1 in. tile square footage into
Number of pattern repeats 0.22x 324.24= 71.33 square feet of 1 inch tile
Add the two totals 35.66 + 71.33 = 106.99 square feet
Now you could just subtract the square footage of the 4 inch tile that is required from the total footage to get the square footage for the one inch tile, but I didn’t want to take a short cut on the explanation.
Make sure that you know how many square feet come in a carton with each type of tile that you will be using, as you may only be able to purchase full cartons and you will need to make sure you have this covered in your costs.
Now let’s take a look at a mixed media installation (Photo 3). In this photo, we have tile with wood to create a basketweave pattern on a diagonal. This one gets just a bit more involved as each tile shares a wood border. Let’s take another ten by ten foot room and this time let’s add 15% usage/waste factor to equal 115 square feet needed for this installation. The tile is a true 13-inch tile and the wood is measured at 5 inches in width. In order to determine the amount of wood required, it will be necessary to split the width of the wood plank for the basketweave in half; remember each tile is sharing the design; this gives us a full pattern repeat for each tile so that we can calculate. The 13-inch tile and 2.5 inches of wood (half the width of the wood) around the entire tile equals 18 inches.
Take 18 x 18, which equals 324 square inches, and divide by 144; this equals 2.25 square feet per pattern. Divide the 2.25 square feet into 115, the total amount needed to cover the floor. This number equals 51.1 pattern repeats. We now know that it will require 51.1 or 51 pieces of tile to construct this design. Now to break down the pattern, the tile is 13x13 so let’s take 13x13 which equals 169 square inches and divide by 144, this equals 1.17 square feet per tile. Subtract the tile 1.17 square feet from the full pattern repeat 2.25 square feet, this equals 1.08 of wood at 2.5 inches around each tile. Take the square footage number of the tile and times it by the number of pattern repeats required 1.17 x 51, this equals 59.67 square feet. Multiply the square footage of wood 1.08 to the total number of pattern repeats 51.1 this equals 55.18 square feet of wood needed.
Here is a mathematical equation:
Size of room 10’x10’= 100 square feet
Add 15% usage/waste equals 100’x15% = 115 square feet
Tile plus 1/2 width of wood
For pattern repeat calculation 13 + 2.5 around entire tile = 18x18 pattern
Calculate square footage of pattern 18x18= 324
Convert to square footage by dividing 324/ 144 = 2.25 square feet per pattern
Divide total square footage needed by pattern repeat square footage 115/ 2.25 = 51.1 pattern repeats
Break down of tile and wood square
Footage, tile is 13”x13” 13x13= 169 square inches
Convert tile to square foot 169/144= 1.17 square feet
Deduct tile square footage from pattern square footage to get amount of square feet of wood required for each pattern repeat 2.25- 1.17 = 1.08 square feet wood
Multiply square footage of tile, 1.17 into number of pattern repeats 1.17 x 51= 59.6 square feet tile needed
Multiply square footage of wood to pattern repeats required 1.08 x 51.1= 55.18 square feet of 5 in. wood
Add the two totals 59.6 + 55.18 = 114.78
Remember, we just made it 51 tiles to round off, instead of 51.1; that’s why the slight difference in total.
The short cut, take the total square footage required 115, and subtract tile square footage 59.67, which equals 55.33 square feet; the amount of wood required for this installation.
Keep in mind that this calculation is using random length boards. If you are going to be using a specific length of one piece for each portion of the basketweave, it may require more wood to achieve this as you may need to cut to length.


As kids start going to school they start with their first problems solving tasks. Math is one of the classes where it’s very important that the children understand what they are doing as it will be hard for them to catch up if they start falling behind the class. To achieve a perfect balance in the classes that they take, there will have to be some holes filed as every kid has one or more things where they excel and one or more where they just need a bit of help.
Helping your child can be done in two ways, the first one is to hire them a private tutor that will help them and focus on their special needs. This is the professional route and it will guarantee that your child learns how to solve math problems. If you don’t have a lot of time this can be both beneficial for your kid and for you, however, it can sometimes be costly depending on the amount of help your child might need. The second one is a lot cheaper but requires you to invest a lot of your free time, and that’s by teaching them healthy learning habits and problem-solving skills at home.
We will be focusing on the part where you can help your child at home, and in our other math tutoring miami blogs, we will explain how child tutors can specifically help your child with math problems.
There are several things that you can do to help your kid with their math problems and some of them are:
1) Start with the small stuff
Math is always surrounding us and if you start teaching your kid to count than you might as well start with everyday things. For instance, during the setup of the table, you can count forks, knives, and spoons together. This will help your kid with the basics and can be started early to give him a boost before school. This can later extend with questions if we took one fork and we had three forks and spoons in the beginning how many forks and how many spoons will we have now?
2) Set them up with their private piggy bank
As your child is slowly getting exposed to math, give them a task that they will do even when they grow up. Teach them to keep track of their coins that you can start with them in their own piggy bank. Coin counting and them keeping track of their coins will make their brains practice with numbers. This is an excellent exercise for the brain.
3) Incorporate math into games
Playing with your child is an important bonding part of their growing up period. If you want to make it playful but still teach them something you can slowly ask them questions that involves counting or numbers. This is a fun way for them to learn about math and how to think when solving math problems.
4) Ask the right questions
Asking questions is important in every part of our lives. Especially when solving problems and math problems have a beginning middle and end like every problem. Ask them where they would begin in a logical way and explain how they can approach the task at hand. This will give them analyzing skills that will help their reason in the long run when it comes to math.


In equation form: If Ea and Eb are each 100 volts,
at figure 3-27.) For example, in view A, the positive
maximum voltage generated in coils A and B act in the
direction of the arrows, and B leads A by 120. This
arrangement may be obtained by assuming coils A and
B to be two armature windings located 120 apart. If
The value of Er may be derived as follows:
each voltage has an effective value of 100 volts, the total
voltage is Er = 100 volts, as shown by the polar vectors
1. Erect a perpendicular to Er divides the isosceles
in view B.
triangle into two equal right triangles.
If the connections of coil B are reversed (view C)
2. Each right triangle has a hypotenuse of 100 volts
with respect to their original connections, the two
and abase of 100 cos 30, or 86.6 volts.
voltages are in opposition. You can see this by tracing
3. The total length of Er is 2 x 86.6, or 173.2 volts.
the circuit in the direction of the arrow in coil A.
To construct the line voltage vectors E1,2, E2,3, and
1. The positive direction of the voltage in coil B is
E3,1, in figure 3-26, it is first necessary to trace a path
opposite to the direction of the trace
around the closed circuit that includes the line wires,
armature windings, and one of the three voltmeters. For
2. The positive direction of the voltage generated
example, in figure 3-21, consider the circuit that
in coil A is the same as that of the trace.
includes the upper and middle wires, the voltmeter
Therefore, the two voltages are in opposition.
connected across them, and the ac generator phases a
3. This effect is the same as though the positive
and b. The circuit trace is started at the center of the
maximum value of Eb were 60 out of phase
wye, proceeds through phase a of the ac generator, out
with that of Ea, and Eb acted in the same
line 1, down through the voltmeter from line 1 to line 2,
direction as& when the circuit trace was made
and through phase b of the ac generator back to the
(view D) to vector.
starting point. Voltage drops along line wires are
disregarded. The voltmeter indicates an effective value
4. Ea is accomplished by reversing the position of
equal to the vector sum of the effective value of voltage
Eb from that shown in view B, to the position
in phases a and b. This value is the line voltage, E1,2.
shown in view D, which completes the
According to Kirchhoff's law, the source voltage
Figure 3-27.--Vector analysis of voltage in series aiding and opposing.


We have to start with inequality which is the difference between two values in a way that one value is bigger than the other. For example, there are two people named “A” and “B”. And person A has a salary of Rs.1000, and person B has a wage of Rs.500.
This system consists of inequalities that relate to each other. The inequalities are called the inequalities because they are not equal.
Let’s start by defining what inequality is.
An inequality is a mathematical relationship where two values are not equal. An imbalance is a statement that two values do not add up to equal a third value.
There are three types of inequalities.
The first type of inequality is an absolute inequality. This means that disparity has a solution. Absolute inequalities have answers, and they are not always unique.
The second type of inequality is relative inequality. This means there is not necessarily a solution. Relative disparities do not necessarily have an answer and may not have any solution.
The last type of inequality is mixed inequality. This means there are two types of imbalances. A mixed inequality has solutions, and it may not be unique.
This is a very serious and important issue in the world today. The inequalities that exist between people are often hidden or ignored. This means that the poorest and most vulnerable members of society are unequally deprived of their basic human rights and basic needs. A lot of people are being treated in a way that they deserve. People born into a privileged position are given advantages and privileges they do not earn simply because of their birth. It is not fair that a child should grow up in poverty when they have a wealthy parent. So there must be a way of fixing this injustice.
Definition of System Of Inequalities
The System of Inequalities is a set of mathematical equations. This system consists of inequalities that relate to each other.
The inequalities are called inequalities because they are not equal to each other. The System of Inequalities is a set of mathematical equations. In other words, the System of Inequalities is a set of mathematical equations.
Examples of system inequalities in the world
There are many types of inequalities, and they are usually represented as a fraction. For example, in the following fraction, a>b, the whole part is larger than the half.
The fraction also shows that the inequality holds for all values of x and y. However, the fraction doesn’t tell us how much a>b is bigger than b>a. This is where the notation comes in handy.
Solving system inequalities
Systems of inequalities are a set of mathematical relationships that describe some system.
For example, the system of inequalities, 1 < 2 < 3, is an example of a system of inequalities. It is a system because it describes a relationship between 1, 2, and 3.
It is an inequality because it does not say 1 = 2 or 1 = 3. It only says that one is less than 2 and 2 is less than 3.
A system of inequalities is a set of statements where the variables are greater than or less than each other.
For example, the system of inequalities, 5 > 4 > 3, is a set of disparities where the variables are greater than each other.
Solving a system of inequalities is finding all the solutions to the system.
Solving a system of inequalities can be done with the method of inequalities.
Solving the system of inequalities with the method of inequalities works like this.
Start with the first statement.
Find the first variable that is greater than the other two.
Add the first variable to the other two variables.
Now find the second variable that is greater than the other two.
Repeat steps 1–3 until you’ve found all the variables greater than the other two.
After you’ve found all the variables greater than the other two, you can repeat step 2 with all the variables that are less than the other two.
Now you’ve found all the variables less than the other two.
If you want to, add the variables equal to the other two.
Types Of Inequalities
There are two types of inequalities. One is inequality, and the other is strict inequality.
A non-strict inequality is when the equation is not satisfied exactly. For example, 3 > 2. This means that if we add 3 to 2, we get 5, greater than 4.
Strict inequalities are more rigid than non-strict inequalities. This means that if the imbalance is not satisfied, the difference between the numbers is not equal to zero. For example, if we add 5 to 3, we get 8, greater than 6.
Frequently asked questions about System.
Q: What is inequality, and what are its different types?
A: An inequality is where you have two or more numbers, and you need to find out which one is bigger, smaller, or equal to the other.
Q: What is a system of inequalities?
A: A system of inequalities is when you have an imbalance and then another disparity. You can’t just solve the disparity.
Q: How do you know which inequality to solve?
A: The easiest way to figure it out is to subtract them. If the first one is bigger, remove one from both sides. If the first one is smaller, add 1 to both sides. There’s no difference if the first one is equal to the second one.
Q: What if there’s a negative number in the inequality?
A: Then, you use the absolute value sign before the inequality. Example: |x + y| = |
Top Myths about System
- I do not know how to solve a system of inequalities.
- It will take me a long time to solve a system of inequalities.
- I am afraid that if I start solving a system of inequalities,
We’ve been trying to solve these problems for centuries. But still, we’re just scratching the surface.
For example, did you know that in 2000, there were more than 200,000 unsolved inequalities? That’s a lot of problems!
I’m sure other people are out there trying to solve these problems. Perhaps you are one of them.
If you are, then I recommend downloading my free ebook. It’s full of information on how to solve systems of inequalities.


By Tuncer Cebeci
Modelling and Computation of Turbulent Flows has been written by way of essentially the most prolific authors within the box of CFD. Professor of aerodynamics at SUPAERO and director of DMAE at ONERA, the writer calls on either his educational and commercial event whilst providing this paintings. the sector of CFD is strongly represented via the subsequent company businesses; Boeing; Airbus; Thales; United applied sciences and common electrical, executive our bodies and educational associations even have a powerful curiosity during this intriguing box. every one bankruptcy has additionally been in particular built to represent as a complicated textbook for PhD applicants operating within the box of CFD, making this e-book crucial interpreting for researchers, practitioners in and MSc and MEng scholars. * A vast assessment of the improvement and alertness of Computational Fluid Dynamics (CFD), with actual functions to undefined * A loose CD-Rom which incorporates computing device program's appropriate for fixing non-linear equations which come up in modeling turbulent flows * Professor Cebeci has released over 2 hundred technical papers and 14 books, a global authority within the box of CFD
Read Online or Download Analysis of turbulent flows PDF
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This type of correction to the ideal gas EOS is said to have first been suggested by Clausius. 49). Sometimes, this equation is also called the Noble-Abel EOS. In the study of propulsion systems, gaseous combustion products at very high densities are reasonably well 14 1. The Equations of Fluid Dynamics described by the covolume EOS. In its simplest version the covolume b is a constant and is determined experimentally or from equilibrium thermochemical calculations. 1 X w- 3 . 49) with b constant.
The shallow water equations are an approximation to the full free~ surface problem and result from the assumption that the vertical component of the acceleration Dv I Dt can be neglected. 138) gives p=pg(l)-y). 141) This is called the hydrostatic pressure relation. e. 138) are independent of y. Thus, the x and z components of velocity are also independent of y for all t if they were at a given time, t = 0, say. 145) Wt +uwx +wwz = -g'f/z. 137) with respect toy (vertical direction) to obtain + 1'1/ '1/ Ux dy 1 -h Wz dy -h + v\~h = 0.
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Granite Estimate Tutorial
How Much Will Your Stone Cost?
Need a granite estimate for new countertops? Want to get a "ballpark idea" of the cost? Your contractor will measure but it's important for you to know. Why?
Stone is priced by the square foot. Know your square footage, know your stones price per square footage and you'll have an idea of what your stone will cost.
Grab a tape measure, pen and paper and follow the example below filling your own numbers in the blanks. In my example, everything in italics, I have two countertop sections.
STEP 1.) Measure the length of your countertop in feet.
_____feet x 12 =______inches
STEP 2.) STANDARD countertop width = 26 inches
_____inches(from Step 1) x 26(countertop width) =________ total inches
STEP 3.) Divide total inches by 144.
____total inches(from step 2) /144= _______ Square Feet
DO STEPS 1, 2, and 3 FOR ALL AREAS OF COUNTERTOP NEEDED.
STEP 4.) Add together the total square feet needed.
___________ From Step 3 (22.75 Total Square Feet Countertop One(1))
STEP 5.) Total square feet x cost per square foot
_______ Total square feet (from step 4) (29.25 Total Square Feet)x _____ cost per square foot ($60.00 per square foot)
$______Base Cost ($1755.00 Base Cost)
LET'S BALLPARK ESTIMATE THE ENTIRE GRANITE COUNTERTOP JOB
$________ BASE COST(from step 5) ($1755.00 Base Cost)
$________ SPECIALIZED EDGING (0 None)
$________ SINK CUT OUT(S) ($250 One Undermount Sink)
$________ COUNTERTOP TEAR OUT ($200 Tear out)
$________ ARCHES (0 No arches)
$________ SINK (0 have my own sink)
$________ GRANITE ESTIMATE ($2205 GRANITE ESTIMATE)
This tutorial is for you to use to get a rough estimate of your granite countertop costs. Prices vary depending on your location AND on the specific granite or stone you choose. Make sure to confirm the average cost per square foot of granite in your area.
Hints & Tips
You'll find More Help On Choosing Stone Below:
Stone Blog. Subscribe via RSS!
Copyright © 2005 - 2015 Natural-Stone-Interiors.com - All Rights Reserved.


I first wrote about teachers using QR Codes almost exactly 4 years ago when Jarrod Robinson a physical education teacher in Victoria, Australia invented QR Code orienteering and later helped his students learn the major bones of the human skeleton with the aid of QR Codes. Since then there have been many imaginative uses of QR Codes in the classroom by inspirational teachers but I particularly like Tabitha Carro’s recent use of Jenga blocks to teach fourth graders fractions.
She created labels for the 54 blocks each with a simple question on fractions accompanied by a QR Code. The students work in small groups building a tower one block at a time. Students take it in turns to select a block from the pool, solve the problem and tell the answer to the group. Scanning the QR Code provides the correct answer and if the student has got it right the block can be used. If they have got it wrong the block is returned to the pool. The game continues until all 54 blocks are incorporated into a tower. Full details and a link to print the labels are on Tabitha’s blog.


This tutorial covers the formula for calculation of the Net Present Value (“NPV”) in Excel. It assumes the reader is familiar with the concept of net present values and concentrates on how to calculate an NPV in Excel and the relevan formulas and considerations to be aware of when performing this calculation.
The NPV formula can be defined as the sum of present value (PVs) of future cashflows after netting out the initial cashflow / investment made. The NPV is commonly used for capital budgeting and widely throughout many mathematics common in economics and the finance industry. NPV can be formulated as:
What is the best way to calculate an NPV of a set of cashflows?
We will discuss in this tutorial the possible method to calculate the NPV:
- Calculate from first principles without Excel function
- Calculate NPV using Excel function NPV()
- Calculate NPV using (X)NPV
- Checking the NPV calculation
An excel workbook has been prepared to demonstrate the NPV calculation, it would be best to download the workbook whilst reading this document.
Calculate NPV in Excel from first principles
This approach requires just a small amount more code than using the Excel functions but is more transparent. Calculating the NPV from first principles allows the user to sense check every aspect of NPV calculation.
From the NPV formula written above, we could lay out the calculation in the spreadsheet using following steps.
- CFi – Bring in the sets of future cash flows to be assessed
- r – Calculate the discount rate
- PVi – Calculate the PV of cashflow in each period
- NPV – Sum up all the PVi, remember to net out any initial cash / investment that has not been accounted in.
Screenshot 1: Manual calculation of NPV
Refer to Screenshot 1. Let us work through an example to calculate NPV of project cashflows in order to calculate the project NPV.
- CFi (Line 104) – This is the cashflow before funding that we would like to calculate the NPV
- Annual r (Line 106) – The discount rate used is 10% p.a.
- Periodic r (Line 107) – The periodic discount rate is calculated as shown above.
- Discount factor – We would like to discount back the cashflows to the model start date (31-Dec-08), thus calculate the discount factor as shown above.
- PVi (Line 111) – Calculate the PV of each periodic cashflow in the model by dividing the CFi with the discount factor
- NPV (Line 112) – Calculate the NPV by summing up the PVi. In this example, the NPV of project cashflows as of model start date is $20.8 million
Using the NPV() function in Excel to calculate Net Present Value
NPV(r, CF1, CF2, …)
We could calculate NPV using Excel function NPV() but it has some limitations:
- The CFi values in NPV calculation must be equally spaced in time and occur at the end of each period.
- The CFi must be entered in the correct sequence.
- The NPV calculation is based on future cash flows. If the first cashflow occurs at the beginning of the first period, the first value must be added to the NPV result.
Using the XNPV() formula in Excel to calculate the NPV
Due to its limitation, the NPV function (without the X) is best avoided. Cashflow models such as in project finance transactions are often presented in more detailed during construction period as opposed to during operations.
For example we often find many project finance models have monthly calculations during construction and perhaps semi-annual / annual during operations.
The more robust function would be XNPV(). It returns the net present value for a schedule of cash flows that are not necessarily periodic. XNPV() is an added-in function in Excel and the syntax is:
XNPV(r, CFi, dates)
Screenshot 2: Using XNPV function
As demonstrated in Screenshot 2, calculate NPV using the Excel XNPV() function yields the same result as manual calculation previous explained in Screenshot 1.
NPV calculation considerations
- Considerations to be aware of when performing the NPV calculation:
- Have you got time to do calculate NPV ‘properly’ from first principles?
- Be aware of the limitations of NPV() function in Excel
- Be aware of the assumptions when using the XNPV() function
- Clearly show the units of cashflows and the discount rate in the NPV formula
- Checking and recheck the NPV calculations at every stage
- Clearly show if the NPV is pre or post tax
- Show if the NPV is calculated as real or nominal
- Select the appropriate discount rate for the risk profile of the project as it is a key variable in the NPV calculation
- Presenting the NPV results clearly, maybe at a range of discount rates
- Communicate the shape of the NPV (discount rate) profile, maybe there is more than one root to NPV(x)=0
- Identify if it is a Project NPV or an Equity NPV (for geared projects)
A substantial benefit of using the XNPV function is that it can be copied and used to calculate the NPV at different discount rates. However this is not a good enough reason for using a black box formula as a quick 1-dimension data table with discount rates as the vertical parameter will provide the same output and can be easily tested.
Screenshot 3: Data table of NPV at various discount rates
Checking the NPV formula and calculation in Excel
NPV is related to the IRR function (internal rate of return). IRR is the rate for which NPV equals zero. We could double-check NPV calculation by firstly calculating the IRR and then feed the IRR back into the NPV calculation as a discount rate.
XNPV(XIRR(…), …) = 0
Screenshot 4: NPV and IRR relationship
Common applications of NPV calculations
NPV is an indicator of how much value an investment or project adds. Amongst other factors, theoretically a project with a positive NPV should probably be accepted. Or in financial theory, if there is a choice between two mutually exclusive projects then the one has the higher NPV should be selected.
Calculate project’s Loan Life Cover Ratio (LLCR) using the NPV
LLCR is one of the most commonly used debt metrics in project finance. It provides a measure of the number of times the NPV of projected cashflows over life of the loan can repay the outstanding debt balance.
LLCR = NPV (CFADS over loan life) / Debt balance b/f
Determine debt capacity of certain project
Related to LLCR above, the borrowing capacity is usually worked out by deciding the initial LLCR. For example if the required LLCR is 3.0 to syndicate a loan of a particular project, then the debt capacity of that project is one-third of the NPV of the project’s available cashflows.


Article by Daniel Sitaru,Claudia Nănuți-Romania IDENTITIES AND INEQUALITIES IN CYCLIC QUADRILATERALS
Proposed by Nguyen Van Nho-Nghe An-Vietnam CYCLIC INEQUALITY-452 Solution 1 by Christos Eythimiou-Greece, Solution 2 by Catinca Alexandru-Romania, Solution 3…Read More
Romanian Mathematical Magazine
The Romanian Mathematical Magazine appears in four editions: Spring, Summer, Autumn and Winter. Each issue contains proposed problems for juniors, seniors, undergraduates and math articles.
We are glad to publish in RMM author’s problems and articles from all over the world.
Bld. Carol, nr. 6, Drobeta Turnu Severin, România.


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The Math Board Games Book www.math-board-games.com Jungle Vines 2 - 4 players (Three simple steps and you're ready to play!) 1. Print the Jungle Vines Board. 2. Print the instruction cards. 3. Choose which game you want the kids to play. This is a manual which shows how to play games using the Everyday Mathematics program. Read more. Previous page. Language. English. Publisher. McGraw-Hill Wright Group. Publication date. January 1, 2003. Dimensions. 9 x 0.96 x 10.8 inches. ISBN-10. 0075727609. ISBN-13. 978-0075727606. See all details. Next page.
Edu games has a great collection of free educational games. Create crosswords puzzles, bingo cards, word wheel challenges sudokus and many more games. Fun Games for ESL Teachers and Students, Powerpoint Games, Games for Kids, Printable Board Games and Poker Card Games, Interactive ESL, Math, Science : Games, Worksheets, Videos.
Here for every moment. Homework help. Study with 55+ million step-by-step explanations, Expert Q&As & math support.
Math Whiteboard is a FREE collaborative whiteboard designed specifically for Math teaching and learning. bring math to life! Just a few of the great things you can do with Math whiteboard.
The Games Kits include all the materials needed for easy setup and play, including Gameboards, Game Pieces, and a Pre-K-6 Games Kit Teacher's Guide Updated Edition with instructions for every game. The Games Kit box helps keep it all organized. The Grades 4-6 Games Kit Update includes: Dot Dice, 1 package of 12.
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Maths & spelling personalised practice and low-stakes tests that pupils love. Games-based learning. Virtual rewards and instant feedback engage and motivate your pupils to enjoy their learning.
In this interactive game, children will listen to your commands and show their understanding of positional language by going on a scavenger hunt. What you need: 1 toy. What you do: Ask the children to hide their eyes. Place the toy in an easy-to-find location. What you say: Use easy-to-follow commands that include positional language.
2022. 7. 29. · Each grade-specific kit includes all materials needed for easy setup, game-play and storage. A Teacher’s Guide to Games is included, providing differentiation options, directions, and masters for games. Grade-level-appropriate gameboards to practice basic facts and build thinking skills. Vibrant gameboards that mirror the design and style of.
Math and Games for Kids, Teachers and Parents. Math lessons and fun games for kindergarten to sixth grade, plus quizzes, brain teasers and more.
4th grade math games for free. Multiplication, division, fractions, and logic games that boost fourth grade math skills.
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Everyday Mathematics Online provides many useful resources for students and parents, including games and an interactive reference book to help with assignments. To use it, you'll need to know your username and password, which you can get from your teacher if your school has subscribed. Interactive Arithmetic.
2020. 10. 28. · 6. Math Brain Booster Games. A variety of mind-stimulating math problems and quizzes provided in this game can bring a positive change in your memory, attention span, and logical reasoning abilities. Mental math problems.
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PRAXIS, a C library which minimizes a scalar function of a vector argument, without needing derivative information, by Richard Brent.
PRAXIS seeks an M-dimensional point X which minimizes a given scalar function F(X). The code is a refinement of Powell's method of conjugate search directions. The user does not need to supply the partial derivatives of the function F(X). In fact, the function F(X) need not be smoothly differentiable.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
PRAXIS is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
BRENT, a C library which contains Richard Brent's routines for finding the zero, local minimizer, or global minimizer of a scalar function of a scalar argument, without the use of derivative information.
COMPASS_SEARCH, a C library which seeks the minimizer of a scalar function of several variables using compass search, a direct search algorithm that does not use derivatives.
TEST_OPT_CON, a C library which defines test problems for the minimization of a scalar function of several variables, with the search constrained to lie within a specified hyper-rectangle.
TEST_OPTIMIZATION, a C library which defines test problems for the minimization of a scalar function of several variables, as described by Molga and Smutnicki.
Original FORTRAN77 version by Richard Brent. C translation by John Burkardt.


In order to determine a person's creditworthiness, a credit score is calculated by applying a mathematical formula. A credit score ranges from 300 to 850. A higher credit score increases your chances of getting a loan. Loan approval is less likely if you have a low credit score. Low credit scores will result in higher interest rates if you are approved for credit with a low credit score. Therefore, if you have a high credit score, you will be able to save thousands of dollars in interest.
Based on a mathematical formula, a credit score predicts a person's creditworthiness. The range of credit scores is 300 to 850. Your chances of getting a loan increase with a higher credit score. Loans are less likely to be approved if your credit score is low. A low credit score and approval for credit will result in a much higher interest rate than a good credit score and approval. A high credit score can save you thousands of dollars over the course of a mortgage, auto loan, or credit card.


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Welcome to my homepage!
I am a third-year graduate student in Applied Mathematics at University of North Carolina – Chapel Hill. I am interested in Mathematical Medicine/Physiology and Computational Sciences. I received my undergraduate degree in Applied Mathematics at UC Berkeley in 2013 and interned at Lawrence Berkeley National Laboratory – NERSC Division for a year as a research assistant. In my spare time, I like to play basketball, football, work out, or play guitar and sing.
If you’d like to find out more about me, please check out my CV.


AS a former Maths teacher I like to set mathematical problems. However they are at High School level and they are not too difficult.
Simply be the first to email me the correct answer to any mathematical problem below together with your name and address and I’ll send you a free signed copy of one of the books!
I have 210 meters of fencing wire and I wish to enclose a rectangular paddock on three sides with a natural boundary such as a river on the fourth side which does not need to be fenced. What should be the dimensions of the sides so as to enclose the maximum area and what is this maximum area?
I have $1000 to invest at the current rate of 5.35% per year. If interest is added to the account at the end of each year how long will it take the account to reach $10000?
For any circle the ratio of the circumference to the diameter is a number known as PI.
The first ten digits of PI are 3.141592654. What are the second twenty digits of PI, known as PI-2-20?
What is the sum of all the even numbers from 2 to 1000?
That is Sum = 2 + 4 +6 +8 +……………..+ 1000.
If men work at a constant rate and it takes 3 men 6 hours to dig 6 holes how long does it take 6 men to dig 4 holes?
In poker there are 4 suits of 13 cards each (23456789 10 J Q K A). A hand of five cards is called a ‘Full House’ if it consists of three of a kind and two of a kind. An example would be three aces and two kings. What is the probability of a poker hand of five cards being a ‘Full House’ ?
A river is 180 metres wide and the speed of the river current is 8 metres/second. There are two boats A and B each travelling at 10 m/s relative to the river. Boat A travels from point O on the bank to a point 180 m up stream against the current and then travels back down stream to point O with the current. Boat B travels across the current to a point 180 m away on the opposite bank and then back again to point O. (This boat obviously heads into the current both ways in order for the combination of its own speed and that of the current to enable it to reach a point on the opposite bank opposite to point O.) How long does it take each boat to complete its journey? Note that they are NOT the same.
NOTE: This led to one of the most important experiments in all of science namely the Michelson-Morley experiments conducted from 1881 to 1887. Instead of a boat travelling with and across a river it was a light ray travelling with and across the supposed ‘ether’. Despite the apparatus being sensitive enough NO difference in time has ever been recorded for the two light rays! This has far reaching effects and leads directly to Einstein’s Theory of Relativity. There is no ether and therefore no absolute frame of reference. The speed of light in a vacuum is always the same no matter how the observer moves!
Find P(not equal to 0) such that (X + 1/X)^2 + P^2 is a perfect square.


Often, when experimental results are collected, the plotted x- and y-pairs will not all lie exactly along one straight line. Random sampling error accounts for some of the variation, since the data collected during an experiment is only a sample of the possible sets of data that could be collected. The line of best fit to such data can be calculated by eye (acceptable in some cases) or can be calculated by the least squares method.
Examine the plot below. Three different individuals have drawn, by eye, three different lines of best fit through the data. Which one of these is really the "best" fit to the data?


These reviews can be verified at Tutorfair.
Awesome Maths Tutor
Written by Sophie U
Tom has been teaching my daughter, 9, for almost a year now. She is an August child at an academic prep school and whilst bright she has struggled with the very challenging maths curriculum. Her relationship with maths had deteriorated so far that she refused to engage with lessons, homework was fraught with tears and tantrums and she was showing signs of significant anxiety.. Tom has brought enormous emotional intelligence to his tutoring of her. His kind manner coupled with firm boundaries is a winning combination. His patience is extraordinary and he is teaching her the tools she needs to approach independent learning and working as well as simply the maths itself.
I cannot recommend him highly enough.
Outcome: We will continue to have Tom for as long as needed.
Highly recommended 11 plus and maths tutor
Written by Abigail S
I highly recommend Tom as a tutor. He has taught my son for almost two years now, both in KS2 maths and in prep for the 11 plus exams. My son really struggled with maths at the outset of his lessons with Tom (he has mild dyslexia), but the work he has done with him has been transformative. He now says he enjoys maths and has made significant progresss in his understanding. He also secured a place at a selective school which felt like an unreachable goal when he first started with Tom. Tom has a very caring style and my son says he explains concepts really clearly. From my perspective I found Tom very easy to communicate with and he has made me feel confident that my son has real support in his learning. I wish I had had a maths tutor like Tom!
Outcome: Secured a place at The Harrodian school.
Exceptional Math Tutor
Written by Susie A
Tom is my 8-year-old daughter’s math tutor. Tom is a friendly and professional tutor, who was always well prepared for his lessons, and impeccably on time. Tom took a full responsibility of tutoring my daughter, and showed incredible knowledge in teaching math, and is willing to go the extra milestone. My daughter is top math pupil, and she greatly enjoys her lessons with Tom.
I highly recommend Tom for anyone who is looking for help in Math.
Subject: Maths (Primary)
Outcome: My daughter completed Y3 as one of the top math pupil.
A fantastic tutor!
Written by Wendy Wong C
Maths has gone from being my son’s least favourite subject to his favourite subject at school!
We first approached Tom to prepare my son to sit an 8+ entrance exam for a top London school. Tom is very knowledgeable and very quickly grasped my sons weaknesses and focussed on strengthening his basic understanding of the principles of maths before progressing to other topics in the curriculum. My son has greatly benefitted from the lessons with Tom and has gone from being in the bottom set to being in the top set in his year. As we approached the exams, Tom focussed more on exam techniques and my son felt prepared on the day of the exams. Tom has always been prepared for each lesson and has made maths fun! We would highly recommend Tom to anyone!
Subject: King’s College Wimbledon School Admissions
Excellent Maths Tutor
Written by Naomi D
Tom is professional, approachable, thoughtful and knowledgeable. His teaching style enabled our son, in just a few weeks, to learn to conquer exam nerves and timing issues and acquire the necessary tools to tackle tricky questions. This in turn, led to him achieving the top grade at IGCSE.
Outcome: Level 9 at IGCSE
Written by Camille M
Tom tutored my daughter for about a year preparing her for 11+ exams (South West London independant schools). She used to attend a state primary school and therefore needed some support in studying the curriculum in greater depth and giving her the confidence and “technique” to solve more complicated problems and puzzles that often come up in exam papers. She passed her exams with flying colours, including London’s most selective girls’ school and its “notoriously challenging” 3-parts maths paper. She even scored 100% in one maths paper at another school which was mentioned in the offer letter. Once she had mastered the curriculum, Tom focused the work on the most challenging papers and workbooks. This, added to his very methodic approach to problems, helped her achieve those results. Last, Tom is very reliable and good at giving feedback, positive but also negative. I would recommend him highly.
Subject: 11 Plus
Outcome: Being offered the (very selective) independent schools she wanted.
Tom is a great tutor!
Written by Vanessa S
We needed a tutor for our 10 year daughter, Yasmin as she struggles in Maths. In the past we had a few different tutors but they didn’t help or made any difference at all.
When Tom started tutoring Yasmin we noticed the difference straight away – even her school teachers noticed too. Tom has also helped with Yasmin’s attitude towards maths – she is much more confident and happy to do her maths homework, whereas before she would panic and not do it at all. Since the tutoring she has moved up a level already – she understands a lot more in maths now. We put this down to Tom’s great tutoring. He has a firm and fun approach and Yasmin looks forward to her lesson with Tom. The difference with Tom to the other tutors is that he really cares and is very passionate about his teaching, he will think of different ways to teach Yasmin. He has sometimes called or sent emails with different ideas for teaching or just giving me advice how to help her.
Subject: Maths (11+)
Outcome: We are so lucky as he really is a great tutor.
Primary maths tutoring
Written by Emma E
Tom has been a wonderful tutor for our son. Seb was struggling with some aspects of maths so we decided to see if a tutor could help. Tom was amazing. He made our son feel completely at ease and much more confident. Seb actually looked forwards to the lessons and he has now completely caught up at school.
I could not recommend Tom highly enough.
Subject: Maths (Primary)
Outcome: Sebastian is now really confident in maths and doing really well
Written by Irina A
Tom has been teaching my son the 11+ onwards maths curriculum for the past four months. My son’s maths performance has gone from the lower end of the class (averaging 40-odd% in year 5) to the top (class average in tests 88%, son’s personal average 99%)! He finished year 6 a couple of marks off level 6b for this subject. Tom is calm, steady and well prepared at every lesson. Furthermore, my son not only loves his lessons but has even come to see Tom as his friend, which I never imagined could happen when someone is teaching you maths. He is now encouraging his younger brother to have some tutoring with Tom too. Thank you Tom. We will be continuing our sessions despite the fact that my son is starting at a highly regarded selective school. Our reasoning is that given our son’s progress and love of lessons with Tom, why rock the boat?
Subject: Maths (11+)
Outcome: excellent rating
Written by Irina A
We have used Tom for our son’s maths tutoring, who is currently in Year 7 of a top-50 selective school, for the past year. This is the most recent school report from our son’s maths teacher:
What a fabulous year so far! Early in Michaelmas term he was prone to rushing through exercises and he made a few too many silly mistakes. His first test score of 62% was respectable enough, but indicated that some of the basics needed work. However, since then he has really kicked on, and largely due to an exemplary attitude in class he has climbed right to the top of the set. Recent test results of 83% and 86% reflect the progress he has made. As well as written work of a very high quality, I am even more impressed by the excellent contribution he makes within lessons, where he is always willing to offer answers or ask questions. He also works hard on bonus material and challenge preps. This has been a successful year – congratulations!
Our son continues to enjoy his maths lessons with Tom and looks forward to them. I suspect that his “exemplary attitude” in class has been a result of Tom’s teaching and dedication in creating the spark of interest in this subject. We have since also added our younger son and older daughter to Tom’s list of tutees, and they are also going from strength to strength.
Outcome: Very pleased
Maths and coaching for common entrance/ 11 plus
Written by Charlotte F
Tom is an excellent teacher and I would not hesitate to recommend him to anyone – whether they excel at maths or whether they find it difficult . He has been tutoring my children, now aged 10 and 13, for more than three years. He first came to us to help my daughter who struggles with maths and has been unfailingly patient and helped increase her confidence as well as her understanding. My son enjoys maths and Tom has been able to set him challenging work to keep him interested for the last few years. He has also supported him in preparing for his 11 plus deferred entry.
Outcome: My son has just been offered a place at Kings College School, Wimbledon. He also has been invited for interview at St Paul’s. We don’t yet know the outcome.
Very good tutor
Written by N D
Tom is a very patient and pleasant tutor. He has a very good and positive attitude with kids. He is reliable and always on time .He always comes prepared with a tailored plan. My daughter improved her math level tremendously and ended up at the end of the academic year, at the top of her class. Tom has an ability to teach Math in a fun and efficient way. Highly recommended.
Subject: Maths (Primary)
Outcome: great math improvement .
Written by Ayesha A
I’m a master’s student studying Business Intelligence and one of my modules involved statistics which I’ve never done before. I was having a lot of difficulties trying to understand what to do. Tom helped me so much within 2-3 weeks before my exam. He went through every topic I was having difficulty with step by step. I was extremely impressed with the quality of teaching. Also, Tom was very organised and conscientious. I would recommend him to anyone who needs helps in statistics or any other subject. Tom’s tutoring is reflected through my results from the statistics exam in which I got 94%. Thank you so much Tom!
Outcome: 94% in my Masters exam for statistics – top mark in the class.
An Amazing Tutor
Written by Lara B
I called on Tom to help me study for a statistics module I was taking at university level. I was having serious trouble understanding the logic of the questions and how to approach each problem and Tom was able to really simplify the questions and help me tackle them easily.
Tom is very punctual, organized and patient. I was impressed that he was always prepared, having gone through the material i sent him before he came to our sessions.
I would highly recommend Tom, not only for statistics, but for any other area of maths as well.
Thanks so much for your help Tom!
Subject: Maths (Degree)
Outcome: 77% in bachelors level statistics exam
Special Needs/ Asperger’s
Written by Sarah T
My son has Asperger’s and suffers a great deal of anxiety about all aspects of school. Tom has taught him the whole of the AS maths syllabus in super quick time and has given much a great level of confidence in the subject. Highly recommended
Outcome: Awaiting AS results but more confidence.