Yr 5 – Mrs Croydon & Mr Biggs – Kinder Scout
Our Topic this term is Groovy Greeks. Please encourage your child to investigate our history topic. Homework for this term is shown below and will be shared with the class weekly.
Weekly spellings focus on high frequency words as listed in the National Curriculum. Children will continue investigating and learning the year 5 and 6 words. Spellings will be given on a Monday and tested the following Monday. The spellings to be learned will be highlighted in their diary and then the children will √ the ones they spelled accurately.
Please continue to encourage your child to read at home. In their diary are a list of other activities which will develop your child’s love of reading and the reading experience, try to complete at least 4 of them weekly. We always welcome book reviews and recommendations to encourage other readers in class, book review sheets are available in class. When completed come and display the review for other children to read.
This will be given out on Friday to be completed and returned any day but by the following Friday at the latest. Please encourage your child to complete this homework as it supports the grammar, punctuation and spelling learning, which takes place within the classroom. Your child will complete this task independently weekly.
In Numeracy this term we will be concentrating on multiplication and division, fractions and geometry.
A quick recall of times tables facts are an essential skill to support understanding in mathematics. Encourage your child to practise, as all children in year 5 need to know their times tables to 12 x 12 and the related division facts. Times Table activities are completed weekly.
Maths Skills Check
Homework will be given weekly to support and consolidate the learning in class. This is given out on Friday to be completed and returned any day but by the following Friday at the latest. Your child should use the ‘Prompt Sheets’ to support them. They will complete this task independently.
PE will be on Thursday and Friday afternoons. Please ensure that your child has both an indoor and outdoor PE kit on these days, as we will be venturing outside as much as we can.
FOR YOUR CONTINUED SUPPORT
Times tables – A quick recall of times tables facts are an essential skill to support understanding in mathematics.
School closes for the Summer on 20.07.2018 and re-opens on Tuesday 4th September 2018.
14/07/2018 - Hady Primary School is 50 this year and to celebrate we are holding a Celebration Fun Day on 14th July from 2.00pm to 7pm come and join us for a great afternoon of fun and activities.


Mrs. Sepulveda's 6th Grade Math and Science Class
Welcome to the 2022-2023 school year at Mountain View Middle School! Our classroom is not only in C-7 but we have a Google Classroom as well. Within our Google Classroom, students and parents will be able to view the daily agenda, assignments, videos, presentations, and additional materials. When students are absent, Google Classroom should be accessed to find out what they missed. Be sure to check out the links on this page to help with assignments and projects as well as fun sites for practicing various math skills.
Please look at the links to the units for math and/or the science links.
If you have any questions, please feel free to email me at any time!
Mrs. Ana Maria Sepulveda


Origami 6 : I. Mathematics
A unique collection of papers illustrating the connections between origami and a wide range of fields. The papers compiled in this two-part set were presented at the 6th International Meeting on Origami in Science, Mathematics and Education (10-13 August 2014, Tokyo, Japan). They display the creative melding of origami (or, more broadly, folding) with fields ranging from cell biology to space exploration, from education to kinematics, from abstract mathematical laws to the artistic and aesthetics of sculptural design.
This two-part book contains papers accessible to a wide audience, including those interested in art, design, history, and education and researchers interested in the connections between origami and science, technology, engineering, and mathematics. This Part 1 contains papers on various aspects of mathematics of origami: coloring, constructability, rigid foldability, and design algorithms.
Download Origami 6 : I. Mathematics (9781470418755).pdf, available at pdfstock.com for free.
- Koryo Miura, Toshikazu Kawaskai, Tomohiro Tachi, Ryuhei Uehara, Robert J. Lang
- Paperback | 368 pages
- 178 x 254 x 25.4mm | 725.75g
- Publication date
- 03 Jan 2016
- American Mathematical Society
- Publication City/Country
- Providence, United States
- Bestsellers rank


Have you ever wondered just how much your online sales could improve? I’ve developed a nifty calculator that lets you see how conversion rates impact online sales.
HOW TO USE THE CALCULATOR
It’s fairly simple to use. All you need to do is enter a handful of variables to begin to see the dramatic impact that conversion rates have on online sales.
If your store carries multiple products, simplify this exercise for 1 of your best selling items. You’ll enter variables into the calculator such as:
- website traffic,
- average number of sales,
- product price, and
- current conversion rate.
Your visitor to customer conversion rate is a percentage that’s found by dividing (# of buyers by / # of website visitors). Let’s say you have 100 buyers and 1,000 website visitors. In that scenario, 100/1,000 = 10% conversion rate.
Keep in mind, conversion rates tend to be fairly low. Many websites operate at a conversion rate under 3%. This is a mountain of missed opportunities. It means that the vast majority –as much as 97% simply visit and leave the website without making a purchase.
CONVERSION RATE IMPACT ON SALES
Here’s an example, let’s say your 50,000 people visit your website on a monthly basis. Your product sells for $50.00. At a 1% conversion rate (buyers/visitors %), your product sales are around $42,500.
What would happen if your conversion rate rose from 1% to 2%. At 2% conversion rate, your sales could jump to $85,000.
This leap in monthly revenue is achieved using existing website traffic resources, marketing assets, and existing tools. By improving your conversion rates, you are maximizing the potential of your existing marketing budget.


A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model. Unstable domain walls can also appear if spontaneously broken discrete symmetry is approximate and there is the metastable vacuum.
A domain (hyper volume) is extended in three spatial dimensions and one time dimension. A domain wall is the boundary between two neighboring domains. Thus a domain wall is extended in two spatial dimensions and one time dimension.
Important examples are:
Besides these important cases similar solitons appear in wide spectrum of the models. Here are other examples:


GraphPad InStat 3.10
Provides a fast way to learn and analyze the data through statistical tests. Features a built-in wizard for a step-by-step guide. Supports loading the data from the following files: CSV, DAT, PRN or plain text file.
GraphPad InStat is a less cumbersome alternative to typical heavy-duty statistical programs. With InStat, even a statistical novice can analyze data in just a few minutes.
InStat conquers the learning curve by escorting you through statistical analyses.InStat provides a unique analysis checklist. Double-check that your data have not violated any assumptions of the test, and that you have picked a test that matches your experimental design and really answers the question you had in mind.


Keno, as a game, has managed to attain incredible popularity. It has become even more popular since its inclusion in online casinos. The game is fairly simple and does not involve much strategic thinking to enjoy. It also offers great payouts, which makes it incredibly worthwhile to play.
Keno is played by many players in live casinos. It allows players ample of time to crunch their numbers so that they can be used wisely on the Keno cards. The game is usually played for leisure, but there are substantial monetary gains to be achieved as well.
Rules for playing Keno online
The inclusion of the game in online casinos has somewhat changed the game. Such online casinos allow players to play Keno at a pace that is quicker than any real casino. Players can play more number of games within short time duration. They can easily control the intervals between two successive games, as well as the time needed to play a single game. All of it is at the discretion of the players involved.
The game involves a Keno “board” that accommodates numbers between 1 and 80. Players are allowed to mark their individual card with up to a set of fifteen different numbers of their choice. Once the players have compiled their numbers, the program on the board generates a set of twenty different numbers randomly. The objective of the game is to match as many numbers as possible from the compiled set of numbers on a Keno card with the numbers drawn by the board. The payouts are made according to the quantity of numbers matched.
The board accompanies a pay table that informs each player about the amount of money they have won. The table is available ti players in plain sight so that they can see how many numbers they have been able to match and how much money they have been able to win.
Strategy of playing the Keno game
The game involves little to no strategic angle at all. Hence, it becomes very interesting to figure out how to play the game better. Since there is very little mathematical calculation can be done to predict the winning numbers, players have to depend solely on their chances of luck. They have to track numbers and past results, and then guess their number choices on the basis of the likelihood of them appearing in the winning combination. Honestly, it is all chance, which is as good an approach to a casino game as any.
Alternatively, players can also manage their money and budgets wisely in order to get the best out of playing this game. This will largely depend on the number of games that one intends to play as well as the total money that they intend to wager in all of these games. This will help them to determine the amount of money to wager on each game or bet. The aim here would be to recover the total money in wagers by the end of the game. Any winnings would automatically be a bonus!
Helpful tips to playing Keno in online casinos
Discussed here are some useful tips that will help you to enjoy the game thoroughly in an online casino. They will also help you to improve you chances of winning more money.
- It is always recommended that you become well-accustomed to the functioning of the game. You must understand that you are given the option of choosing up to fifteen different numbers. This means that you have to place fifteen different bets on a single game. At the same time, this also allows you the opportunity to win up to fifteen different bets in a single game if the numbers match.
- Although you have the option of playing at a fast pace, you should avoid playing any faster than you actually want to. Take your time in compiling the numbers of your choice, and only go forward with the game when you are good and ready.
- If you have the habit of tracking past winning numbers for the compilation of your own set of numbers, locate the place where those numbers are displayed. This will help you with your tracking during a game.
- Budget you wagers appropriately. Figure out how many games you wish to play and ensure that you have adequate money to wager on those games. This will not only help you place your bets wisely, but also ensure that you do not exceed your budget when playing the game. After all, money management is of prime importance when wagering money on live online casinos.


PTE Academic Test Centers issue a score report for every candidate who finishes the test successfully. Usually, this report card will be ready within 2 to 72 hours after the exam is finished.
1. Score report information
The PTE score report provides an overall score which is then broken down into two main categories: communicative skills and enabling skills. Your communicative skills consist of four contributors namely Listening, Reading, Speaking and Writing. Your enabling skills composed of Grammar, Oral fluency, Pronunciation, Spelling, Vocabulary and Written Discourse. Watch our youtube video on how to calculate your PTE overall score.
2. How to calculate the overall
In order to get 20 points for immigration to Australia, you need to have an average score of 79 with no skills less than 79. Let’s see how a computer calculates the overall score of your PTE test score.
First, they add up all the scores of enabling skills. Then, the sum is divided by 6 which is the number of added items. This number which is the average sum of the enabling skills becomes the fifth contributing factor along with the other four communicative skills in calculating the overall of the PTE score. In other words, your enabling skills can indirectly help you boost your average. For instance, I had a student who had no skills less than 79, but his average was 78. This meant that he could only claim 10 points instead of 20 because his average was not 79. Therefore, enabling skills can be decisive if your communicative skills scores are on the borderline.
The overall scores are usually rounded up, and once the average is above 88, it will be rounded up to 90 and the average of the enabling skills is automatically disregarded as you can see here in my PTE report card.
3. Sample calculations
So let’s put this hypothesis to test by calculating the average of a few score reports. So first up is this one: So I add up the enabling skills and then divide it by 6 and the result is 37.5. This will be the fifth contributor for calculating the overall score and the sum of the communicative skills plus the enabling skills average will be 189.5. This sum divided by 5 will be 37.9 which will be rounded up to 38.
4. Baffling ones
Now some baffling cases that really don’t add up and I have found no explanations for them.
So here as you can see even without using a calculator you can see that the average of the communicative skills is 74ish and if you add up the 6 enabling skills, you would have 453 and divide it by six you would get 75.5. Now if you add this figure to the above sum and divide it by 5, you will get an overage of 74.5. But as you can see here the overall score by the computer is 72. Hello! You make me think twice about asking for a rescore of my PTE test result Mr computer.
Need more information?
For learning the tricks of how to ace the PTE academic test, and for doing a free scored PTE mock test and practicing real PTE materials on our PTE practice platform, visit our website at www.thepte.com, where you can also book a free online PTE coaching with one of our Melbourne-based expert PTE trainers via the zoom app.


Gifted and talented maths and science club.
Read how busy our gifted and talented maths and science club have been.
During the last few weeks, 12 pupils from years 3 and 4 have been invited to come to the gifted and talented maths and science club, every Wednesday after school. This club encourages children to improve their maths and science skills through taking part in challenging maths and science investigations.
On November 12th, we attempted to produce successful, electrical circuits using a variety of equipment. We investigated the effect on the brightness of the bulbs of inserting 1, 2 or 3 bulbs into a circuit. After that, we experimented to find out which materials act as electrical conductors and which act as electrical insulators. We found out that all metals conduct electricity. Interestingly, we also found out that a material called graphite conducts electricity.
Maths and science club has encouraged children to reach high limits in these particular subjects. We all enjoy this club as it is fun and challenging.
Written by Sam Moreau and Jay Stevenson 4M


Tunxis students who are enrolled in Prealgebra, Elementary Algebra, and Intermediate Algebra are invited to stop by the Academic Success Center for “Math Mania,” a marathon of studying, on Friday, May 9 from 10 a.m.-2 p.m. This is specifically for students working on sample finals. Tutors will be available to review concepts and answer questions. Light refreshments will be provided. For more information, call 860.255.3570 or email email@example.com.
May 9, 2014 James Revillini


Amy Roth McDuffie conducts research on the professional development of prospective and practicing teachers in mathematics education. Specifically, she focuses on supporting teachers learning in and from practice in the areas of teachers’ use of curriculum resources and culturally relevant pedagogies as part of developing equitable instructional practices.
Dr. Roth McDuffie regularly teaches methods of teaching mathematics courses for undergraduate prospective teachers. She also teaches masters and doctoral level courses in mathematics education for our EdM program and for our PhD in Mathematics and Science Education.
- Roth McDuffie, A., Foote, M. Q., Bolson, C., Turner, E.E., Aguirre, J. M., Bartell, T. G., Drake, C., & Land, T. (in press). Using video analysis to support prospective K-8 teachers’ noticing of students’ multiple mathematical knowledge bases. Journal of Mathematics Teacher Education.
- Roth McDuffie, A., Foote, M. Q., Drake, C., Turner, E., Aguirre, J. M., Bartell, T. G., Bolson, C. (in press). Mathematics teacher educators' use of video analysis to support prospective K-8 teachers' noticing. Mathematics Teacher Educator.
- Estes, L., Roth McDuffie, A., & Tate, C. (in press). Lesson planning with the Common Core. Mathematics Teacher.
- Aguirre, J., Turner, E., Bartell, T. G., Kalinec-Craig, C., Foote, M. Q., Roth McDuffie, A., & Drake, C., (2013). Making connections in practice: How prospective elementary teachers connect to children’s mathematical thinking and community funds of knowledge in mathematics instruction. Journal of Teacher Education, 64 (2), 178-192. doi: 10.1177/0022487112466900.
- Foote, M. Q., Roth McDuffie, A., Turner, E. E., Aguirre, J. M., Bartell, T. G., & Drake, C. (2013). Orientations of prospective teachers towards students' families and communities. Teaching and Teacher Education, 35(126-136).
- Slavit, D. & Roth McDuffie, A. (2013). Self-directed teacher learning in collaborative contexts. School Science and Mathematics, 113 (2), 94-105.
- Turner, E., Drake, C., Roth McDuffie, A., Aguirre, J., Bartell, T., & Foote, M. (2012). Promoting equity in mathematics teacher preparation: A framework for advancing teacher learning of children’s multiple mathematics knowledge bases. Journal of Mathematics Teacher Education, 15 (1), 67-82. doi: 10.1007/s01857-011-9196-6.
- Roth McDuffie, A., Wohlhuter, K., & Breyfogle, L. (2011). Tailoring tasks to meet students’ needs. Mathematics Teaching in the Middle School, 16 (9), 550-555.
- Wohlhuter, K., Breyfogle, L., & Roth McDuffie, A. (2010). Strengthening your mathematical muscles. Teaching Children Mathematics, 17 (3), 178-183.
- Roth McDuffie, A., & Eve, N. (2009). Developing students’ understanding of area: The work of a professional learning team. Teaching Children Mathematics, 16 (1), 18-27.
- Roth McDuffie, A. (2009). Mathematics Curriculum Implementation via Collaborative Inquiry: Focusing on Facilitating Mathematics Classroom Discourse. In D. Slavit, T. Holmlund Nelson, and A. Kennedy (Eds.), Perspectives on Supported Collaborative Teacher Inquiry (pp. 46 – 71). Oxford, UK: Routledge.
- Roth McDuffie, A. & Mather, M.* (2009). Middle school teachers use of curricular reasoning in a collaborative professional development project. In J. Remillard, G. Lloyd, & B. Herbel-Eisenmann (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 302-320). Oxford, UK: Routledge.
Recent grants and awards
- Co-PI on National Science Foundation grant, Developing Principles for Mathematics Curriculum Design and Use in the Common Core Era (2012-2016), funded through the DR K-12 with Choppin, J., Davis, J., & Drake, C.; $2,200,000.
- Co-PI on National Science Foundation grant, Teachers Empowered to Advance CHange in Mathematics (TEACH MATH): Preparing preK-8 teachers to connect children’s mathematical thinking and community based funds of knowledge. (2010-2015), funded through DR K-12 Drake, C., Turner, E., Aguirre, J., Bartell, T., Civil, M., & Foote, M.; $3,497,467.
- Ben and Nancy Ellison Faculty Fellowship for WSU College of Education for research project, Examining Mathematics Curriculum Innovations at Delta (2011-2013); $22,991.
- Co-PI on Paul G. Allen Family Foundation grant, Delta High School research and evaluation (2010-2013) with Nagel, E., Morrison, J., Trevisan, M., & French, B.; $195,000.
Selected recent national service
- National Council of Teachers of Mathematics, Series Editor for Annual Perspectives in Mathematics Education, 2011- present.
- Association of Mathematics Teacher Educators, At-Large Board Member, 2010 – 2013.
- Association of Mathematics Teacher Educators, Research in Mathematics Teacher Education Advisory Committee, 2010 – present.
- Ph.D Mathematics Education, University of Maryland, 1998.
- M.S. Technology for Education, Johns Hopkins University, 1991.
- B.A. Mathematics, Franklin and Marshall College, 1987.
- Maryland Advanced Professional Certificate for Secondary Mathematics and Computer Science, 1989.


I had a lawn in my front yard. Emphasis on “had”. I took it out after realizing how much water it needed.
Grab a tape measure and measure your lawn. Start at one end and lay the tap measure down, if you run out of tape measure, stop in that spot, coil up the tape and start measuring again from that spot. Then add all the lengths together for that run. Then measure perpendicular from the run you just had so you end with a final Length x Width. Multiply the two numbers together to get your lawn area.
If you don’t have a tape measure, walk out the lawn area by using the length of your foot step either heel to toe or your normal walking pace. If your feet are 12″ long then each step toe to heel is a foot. Otherwise measure your pace, usually between 2-3 feet.
If you have a weird shaped lawn, measure at the widest point, and then at about an average width spot for the other dimension. It will give you a ballpark of about how big your lawn is.
You’ve got this.
I had 546 square feet of lawn in my front yard.
Lawn Area: 21 feet wide x 26 feet long = 546 square feet
Through searching online for “recommended watering for lawns“, I need to water my grass between 1″ – 2.5” of water a week. How much water is that?
1 foot X 1 foot X 1 foot = 1 cubic foot 7.48 gallons = 1 cubic foot Recommended Weekly Watering: 1 inch to 2.5 inches 546 square feet X (1 inch a week/12 inches depth of water) X 7.48 gallons = 340.34 gallons OR 546 square feet X (2.5 inches a week/12 inches depth of water) X 7.48 gallons = 850.85 gallons
1″ weekly watering is 340.34 gallons while 2.5″ weekly watering is 850.9 gallons.
Remember, this assumes 100% of the water coming out of your sprinklers goes into the ground. Here is the catch, if you water during the day when it is warm or windy, as much as 30% of the water will evaporate before hitting the ground. Turn on your sprinklers and watch the mist never hit the ground.
To calculate how much MORE water is needed, I continue using my numbers from above and get this:
340.34 gallons needed for 1 inch weekly watering 30% lost of evaporation (or 70% lands on the ground) Total Water Needed = Total Water Applied X 70% Simply Put: Total Water Needed / 70% = Total Water Applied 340.34 gallons / 70% = 485.71 gallons
For 1″ water a week, I’ll need 485.7 gallons of water.
850.85 gallons needed for 2.5 inches weekly watering 30% lost of evaporation (or 70% lands on the ground) Total Water Needed = Total Water Applied X 70% Simply Put: Total Water Needed / 70% = Total Water Applied 850.85 gallons / 70% = 1215.5 gallons
For 2.5″ water a week, I’ll need 1215.5 gallons!
To put this into perspective, with my 150 gallon tank, if I filled it all the way, I’d need to do 8 trips to the fill station a week to meet this demand. I’m sorry, but this is ridiculous. The lawn has got to go!
If you’d like to download this math in a Microsoft Excel format: http://1drv.ms/1zbpM3B Clicking the link will open in Microsoft Excel Online. Note: you are free to use as you please.


Sum of angles identities
6th September 2020
Recently whilst revisiting trigonometry I came across a neat way of proving the sum of angles identities shown below:
I have used these formulas a number of times in the past to solve problems but I had not thought much about how they came to be. It was not until I learnt the proof I will describe on this post that I finally acquired a solid intuition about what these equations mean.
We will only need two of the right-angled triangle definitions for this proof to work: sine and cosine of an angle.
From the image we can derive the two definitions:
- or written in terms of the opposite side:
- or written in terms of the adjacent side:
Draw a rectangle with a right triangle inside as shown in the image.
Assume the hypotenuse of the triangle is of length 1. We name the angle on the bottom left as and work out the lengths of the opposites and adjacent sides of the triangle using the definitions.
We now shift our attention to the triangle that emerged at the top left of the rectangle. We name the angle on the bottom left as and use the definitions to find the length of the opposite and adjacent sides of the triangle, having a hypotenuse of length .
To find the length of the sides of the triangle in the top right, we need to use the angles we currently know and the properties of right triangles. If we focus our attention on the corner of the red triangle touching the top of the rectangle, we can see that we have 3 angles that add up to 180°.
We can find the angle on the left by computing . Then we have a angle from the red triangle. And finally, as all these angles add up to 180° we can find the angle on the right by computing .
With and the hypotenuse of the right-angled triangle on the top right, we can calculate the length of the opposite and adjacent sides using the definitions of sine and cosine with the angle and having a hypotenuse of length
There are only two steps remaining to reach the goal of obtaining the lengths of all sides of the rectangle. Moreover, The right-angled triangle at the bottom will give use the remaining lengths.
Given that the bottom left corner of the triangle forms an angle of 90°, we can define the angle corresponding to the corner of the bottom triangle as .
Having the right angle and the angle described above, the remaining angle of the bottom triangle is equal to .
Now we can find the remaining lengths of the sides of the rectangle by calculating the opposite and adjacent sides of the bottom triangle using the definitions of sine and cosine with the angle .
Having found the total lengths of all four sides of the rectangle, we can use the fact that parallel sides have the same length to draw the following equivalences.
Equating the top and bottom sides:
Equating the left and right sides:
The equations above can be rewritten to reveal the sum of angles identities as shown earlier:
Trigonometry is a fundamental topic on mathematics as well as a prerequisite to subjects like calculus, physics or linear algebra. Having a good understanding of the trigonometric identities allows us to identify when these relations can be used in scenarios where angles are involved, such as when working with vectors in space. Luckily it is relatively easy to visualise trigonometric relations as we can draw triangles and other basic geometric shapes which allows to obtain intuition by using visual proofs.
- I highly recommend watching this excellent video from the YouTube account blackpenredpen where the instructor goes through this proof step by step on a whiteboard.
- Additionally the Wikipedia article about trigonometric identities is a good place to learn in more detail about the sum of angles and other identities.


Our three-year-old loves to pretend to shop. She’ll often get out her shopping cart, purse, coins, and even a (voided) credit card, and ask, “Mommy, will you play grocery store with me and check me out?”
Since she’s just three (well, technically three and a half), I haven’t worried too much about making sure she’s getting the academic basics that I’m sure would be covered in a formal preschool. (We’re homeschooling, for now). But since she’s so interested in coins, bills, and money, I wanted to capitalize on this interest.
So I decided to set up a little restaurant for my daughter.
Math is Fun! Math Money Games for Preschoolers
Affiliate links included where appropriate.
We started by raiding her piggy bank. In addition to the usual pennies, nickels, dimes, and quarters, we also found a few half dollar coins, and two dollar coins. The six types of coins fit nicely in these plastic serving trays.
I made menus and price lists using picmonkey (a free photo editing program) and pictures of our favorite food items (Creativecommons.org has lots of pictures). I plan to laminate the menus eventually or at least put them in plastic sleeves.
The first day we tried this, we had egg salad sandwiches, blackberries, pretzels, and hummus. Our little one LOVED paying for each and every blackberry!
Side note: prices change quickly at our house, so buy fast! I originally listed blackberries for one penny a piece, but when that menu needed a re-do, I changed the price to a more true-to-life price of one nickel each.
When my kiddo ran out of one type of coin (e.g. she had no more dollar coins, and she wanted to buy a $1 glass of milk), she had to figure out how to combine coins to reach the total she needed. Had I just sat her down at the dining room table and drilled her with info about the different values of coins, I guarantee we would have ended up with an epic power struggle, a tired mama, and a frustrated kid.
But she was determined to find the right combination of coins so she could buy her lunch. No fussing, no reluctance, no power struggles. Pretend play for the win!
Now every day around lunch time I hear, “Mommy, can I buy my lunch today?”
Sure, kiddo! Today lunch is on you. 🙂
For more fun preschool play ideas, check out our preschool play board on pinterest!


Numerical Integration Utility: Definite Integral of square-root(x) * sin(x)
This page contains a routine that calculates the definite integral of √ x sin(x) on an interval specified by the user.
Author: David K. Kahaner. Scientific Computing Division, NBS
From the book "Numerical Methods and Software" by
D. Kahaner, C. Moler, and S. Nash
Prentice Hall, 1988
To use this utility, the user must enter two values: b and c, which specify the range [b, c], over which
∫ √ x sin(x) dx will be computed. b and c must be entered as radians, NOT degrees, and they must both be greater than or equal to zero. If a negative value is entered for either of them, an error message appears asking the user to try again.
Note that over intervals for which the function is negative the integral is negative too.
IMPORTANT: Note the Error Code returned.
Error Code = 0: Normal Completion. e < eps (= 1.0e-12) and e < eps*abs(I).
Error Code = 1: Normal Completion. e < eps but e > eps*abs(I).
Error Code = 2: Normal Completion. e < eps*abs(I) but e > eps.
Error Code = 3: Normal completion but eps was too small to satisfy absolute or relative error request.
Error Code = 4: Aborted calculation because of serious rounding error. Probably e and I are consistent.
Error Code = 5: Aborted calculation because of insufficient storage. I and e are consistent.
Error Code = 6: Aborted calculation because of serious difficulties meeting error request.
Error Code = 7: More than 2*NMAX (= 100) iterations of main loop. Subroutine aborted.


There have been many tutorials floating around the ‘net for a while, detailing how to calculate the distance between an entry in your database, and a set of arbitrary GPS points. Not many of these tutorials will allow you to find the distance between two random points you might have.
In order to solve this (and remove the use of the rather large clumsy SQL calculation), I have re-written this MySQL calculation into a function, for easier reference.
DELIMITER $$ CREATE FUNCTION `CALCULATE_DISTANCE`(`@oLat` DECIMAL(10,7), `@oLon` DECIMAL(10,7), `@dLat` DECIMAL(10,7), `@dLon` DECIMAL(10,7)) RETURNS DECIMAL(10,7) NO SQL BEGIN RETURN ( ( ACOS( SIN(`@dLat` * PI() / 180) * SIN(`@oLat` * PI() / 180) + COS(`@dLat` * PI() / 180) * COS(`@oLat` * PI() / 180) * COS((`@dLon` - `@oLon`) * PI() / 180) ) * 180 / PI() ) * 60 * 1.1515 * 1.609344 ); END$$ DELIMITER ;
The format of the arguments are as follows:
|@oLat||The latitude from which the distance will be calculated.|
|@oLon||The longitude from which the distance will be calculated.|
|@dLat||The latitude to which the distance will be calculated.|
|@dLon||The longitude to which the distance will be calculated.|
So, how do I use it?
It’s really easy. Simply execute the SQL provided above. If your database user doesn’t have the necessary privileges to be able to create functions, then you won’t be able to use this. After having run the SQL, you can find the distance between two arbitrary points like that shown below:
SELECT CALCULATE_DISTANCE(-33.91429, 18.42389, -34.079120, 18.446882);
Thanks to zcentric.com for the SQL calculation.


Sarah Schwartz graduated from Indiana University Southeast in 2014 with a bachelor’s degree in education and a concentration in math. There weren’t many positions available for math teachers near her home in southern Indiana, so she continued to work part-time in the IU Southeast math lab. She also got a second job at a Goodwill retail store to earn extra income.
Shortly after, she was diagnosed with muscular dystrophy (MD), a disease that causes progressive weakness and degeneration of the skeletal muscles, making it difficult to do things like walking, climbing stairs and sitting and standing. However, Sarah didn’t let her disability stop her from pursuing her passion.
“When I learned that Goodwill was opening an Excel Center in Clarksville, I told one of my co-workers that I was going to be one of the math teachers,” Sarah said. “Not only am I now one of the math teachers, but I also ended up being the first teacher hired at the school.”
The Excel Center is a tuition-free high school for adults that helps students remove barriers to completing their diploma. Ms. Sarah, as her students call her, has had a huge impact on those around her.
“I struggled with math growing up, but Ms. Sarah helped me understand things in a clear way, which helped me build confidence and the belief that I can learn,” said Ashley Neal, one of Sarah’s students at The Excel Center.
Sarah’s position at The Excel Center has provided her with a means to serve by doing what she loves and also allows her to live independently. In addition, she is currently working to complete her master’s degree in math education to help with The Excel Center’s dual credit program.
“Muscular dystrophy doesn’t define me and won’t keep me from doing the things I want to do,” Sarah said.
To learn more about The Excel Center or to enroll, visit: excelcenter.org


From June 20 to 22, 2020 the Departments of Information Technology (IT) and Computer Science (CS) jointly organised a RUSA-sponsoredthree-day webinar on “Data Science and Machine Learning”, which was attended by a total of 118 participants.
Mr. Shishir Dubey, Data Scientist at Spocto Solution, was the resource person. Over three days, Mr. Dubey covered various topics of data science and machine learning with Python by discussing Jupiter Notebook along with Panda. He also gave a brief introduction about the various implementation tools available in Python, such as Scikits, Panda, Keras, TenserFlow and Numpy. In addition, he discussed concepts of central tendencies, measures of dispersion, probability distribution, linear regression and K-Nearest Neighbour (KNN) algorithm. He also explained the importance of mathematics in data science by discussing probability and linear regression. Finally, he shared some basic ideas about random forest.
The participants benefited greatly from the Webinar.
All Copyrights Reserved © 2017 PATKAR VARDE COLLEGE | Design by PRAJAKTA SOFTWARE


Numerical integration of light paths in a Schwarzschild metric
Differential equations of orbit
The Schwarzschild metric is one of the most famous solutions to the Einstein field equations, and the line element in this metric (in natural units ) is given by:
We are interested in the trajectory of a light ray in such a metric. Since the metric is spherically symmetric, any light ray that starts with a certain must stay in the same plane, hence we can arbitrarily set and do away with all the terms.
Light follows a null (lightlike) trajectory given by . In the absence of external forces, it should also travel along a geodesic. These are governed by the geodesic equations, which can be derived using Euler-Lagrange equations. Due to the symmetry of the metric, applying the Euler-Lagrange equations to the metric gives us two conserved quantities:
where refers to derivative with respect to an affine parameter.
Using the null condition, we have
This can be expressed in terms of , and differentiating again gives the second-order differential equation for :
This can be easily converted into a first-order differential equation to be solved numerically by setting a variable . So we have these 3 differential equations to compute numerically:
(This can of course be solved analytically in the weak gravity limit, which gives the light bending equation .)
In principle, we need the initial values of , , and to start the numerical simulation. However, if we fix the incoming velocity to be horizontal, then we would only need to specify the initial and coordinates.
The initial conditions then can be given as follows:
Then, the only free parameters to specify and , in addition to mass.
For a mass of (corresponding to Schwarzschild black hole radius of 2), this is a plot of the trajectories with different impact parameters :
And they do fit quite well with the theoretical deflection angle, for large impact parameters:


The Firing Squad Problem (FSP) is about designing a 1D cellular automaton, with n possible states for each cell, such that all cells will end up in the same state, having never been in that state before. Each cell follows the same set of rules (the "automaton"), which take the previous state of itself and its two neighbors, and outputs the next state for that cell. For cells at the edge, their neighbor is a special "null" state, which does not count to n.
A 6-state solution exists for all possible widths. It has been proven that there is no single 4-state solution for all possible widths, although 4-state solutions exist for powers of 2. It is unknown whether there is a single 5-state solution that works for all widths.


Thanks to a grant from the All Points North Foundation, Breakthrough Collaborative has updated its academic curriculum in all four subject areas – literature, writing, math, and science. The new curriculum includes updated materials and resources, assessment tools for all subject areas and grades so teachers can monitor students’ learning, differentiated practice opportunities in math so students at all levels can continue to grow, and a new literature text, The Other Wes Moore, so teachers have more options in tailoring the curriculum to the community’s needs. Last summer, 593 teaching fellows at 19 affiliates used these new resources as they taught a collective 2,950 students. The new materials received positive feedback from from leaders across the Collaborative who appreciated the opportunity to track local growth and differentiate instruction. We are grateful to the All Points North Foundation for making this work possible, and we’re excited to continue to refine our curriculum in the years to come.


Hi! I'm Paul, an undergraduate majoring in computer science, mathematics, and statistics at the University of Pennsylvania. I am fortunate to be doing research with Nadia Heninger. I'm applying to computer science Ph. D. programs this fall. Generally, I'm interested in cryptography, computer security, and related systems building. My research experiences include implementing and designing bignum parallelized algorithms related to cryptography, and theoretical exploration of lattice reduction algorithms.
My other hobbies include running, yoga, listening to podcasts, drinking hot chocolate, and making a variety of oatmeal. I am also fond of animals.


Graph Theory I: Intro And Data Structure Examples
Why use Graph Theory?
Here’s an example from the firehose project:
Say you want to build a data representation of a tweet. How would you represent retweets and favorites? This problem sounds easy in theory, but naturally starts moving towards advanced graph theory searching algorithms. For example, for a particular tweet you may need to write a depth-first-search algorithm to determine if someone with the twitter handle “barackobama” retweeted the tweet that is supplied.
So what exactly is graph theory? Well first of all, computer science graphs have nothing to do with your garden variety real-life graphs. These graphs are effectively ways to represent a bunch of dots connected by lines. Think of a map of cities connected by roads. Got that? Now replace “city” with “node”, and “road” with “edge”. Now you’re talking graph language!
The first thing you need to do to understand graph theory is use a tool to visually represent graphs, otherwise you’ll go insane with confusion. My favorite tool is VisuAlgo (see what they did there?). The runner-up prize goes to Graph Online.
VisuAlgo is great because you can create your own graphs and see how they are represented in the three standard varieties:
- adjacency list
- adjacency matrix
- edge list
Adjacency lists are most useful when we mostly want to enumerate outgoing edges of each node. This is common in search tasks, where we want to find a path from one node to another or compute the distances between pairs of nodes.
So with adjacency lists, you list the edges (connections) for each node in a separate array, starting with node 0:
var adjList = [ , // Node 0 is only connected to Node 1 [0, 2, 3], // Node 1 is connected to everyone // (except itself!) , // Node 2 is connected to Node 1 , // You get the idea ];
With an adjacency matrix, every single possible connect between every node is listed. If there is no edge (connection), a
0 is used. If there is an edge (connection), a
1 is used.
var adjMatrix = [ [0, 1, 0, 0], // Node 0 is only connected to Node 1 [1, 0, 1, 1], // Node 1 is connected to everyone // (except itself!) [0, 1, 0, 0], // Node 2 is connected to Node 1 [0, 1, 0, 0], // You get the idea ];
Edge lists, as the name implies, just lists the edges (connections) between each node.
var edgeList = [ [0, 1] // The first edge is from 0 to 1 [1, 2] // The second edge is from 1 to 2 [1, 3] // The third list is from 1 to 3 ]; // And that's it! Show's over, folks.
You can see that the edge lists are often the most efficient way to represent a graph.
So now we know how to represent graphs. What do we do with them? We can use them to repesent all kinds of network relationships.
The two most popular algorithms to use with graphs are depth-first search and breadth-first search. I’m going to look at implementing a breath-first search in the next part. If you can’t wait for that, you can look at Khan Academy’s excellent introduction to depth-first search.


Today we’re going to look at what it means to have the Macd above zero.
Can we use it as a reliable indicator for long or short trades?
Moving average convergence divergence is a widely used indicator for analyzing market dynamics.
Although it doubles as an oscillator, it is not typically used to identify overbought and oversold conditions.
Instead, it is used to identify trends.
The indicator uses two moving averages and tries to predict the formation of a new trend.
The indicator appears in charts as two lines; the MACD line and the signal line.
There is also a zero line above and below which underlying trends emerge.
When the two moving average lines cross, they create crossover patterns that traders want to profit from.
The two lines swing without limits.
The shorter line is usually a 12-period exponential moving average that moves faster.
The longer one is usually a 26 exponential moving average that moves more slowly.
In addition, the MACD indicator has a histogram that shows the number of bars used to calculate the moving average and the difference between the faster and slower moving averages.
The MACD line is usually the difference between the two exponential moving averages, 12 and 26.
It also represents the difference between the two lines.
In the MACD indicator, the MACD line is usually the faster moving average.
On the other hand, the signal line is usually the slower moving average.
The signal line represents the average of the previous MACD line.
In most cases, it is usually the 9-period exponential moving average.
The signal line is used to smooth the sensitivity of the MACD line.
On the other hand, the histogram shows the difference between the MACD line and the signal line, which is shown in bars.
Depending on how the bars form, they can signal that a crossover is imminent.
Whenever the MACD line is above the signal line, the histogram is above the zero line.
Consequently, whenever the MACD line is below the signal line, the histogram is below the zero line.
The histogram basically shows the market dynamics. When the momentum is high, the histogram becomes much larger.
As the dynamics decrease, the histogram also decreases in size.
As the distance between the MACD line and the signal line increases, the histogram becomes larger and leads to Deviations when the MACD line moves away from the signal line,
Similarly, as the moving averages approach, the histogram becomes smaller, resulting in convergence.
This implies that the faster moving average is converging and approaching the slower moving average
Crossing the zero line of the MACD from below is often viewed as a bullish signal.
In most cases, this indicates that upward momentum is building and the price of the underlying security is likely to increase significantly.
The higher the MACD line is from zero, the stronger the signal and the greater the likelihood of price movement.
In this case, traders will try to enter Long positions in anticipation of a price increase.
Likewise, every time the MACD line crosses the zero line from below from above, it is interpreted as a bearish signal, signaling that the price is likely to fall.
In this case, the further the MACD line is from the zero line, the stronger the bearish signal.
Using two moving averages for the same indicator creates a crossover phenomenon.
The faster moving average, in this case the MACD Line, will always react faster to price movements than the slower one, the signal line.
Similarly, every time a new trend occurs, the MACD line, which is the quickest to react to price changes, reacts faster and crosses the slower line, the signal line.
If this crossover occurs in both directions or the MACD line begins to break away from the signal line, it indicates that a new, much stronger trend has formed.
A bullish MACD crossover is manifesting in the NVDA chart above, with the MACD line crossing the signal line right near the zero line.
In this case, it signals that an uptrend is starting on the fast moving average and reacting faster to price changes.
The bullish trend is only confirmed as soon as the MACD line rises above the zero line after crossing the signal line from below.
The price rose significantly from that point on when traders took long positions.
In times of increased market volatility, will lash the MACD line and cross the signal line back and forth.
MACD users often avoid trading during such periods as the unreliable signals come into play.
After the market has completely digested the development that caused the wild fluctuations, the MACD signal line will cross the signal line and signal in which direction the price is likely to move.
The MACD line also crosses the signal line from below and moves above the zero level increased volatility would essentially signal the formation of an uptrend.
The MACD line farther away from the signal line would confirm a stronger signal.
Divergence is a powerful indicator of trend reversals.
Fidelity’s chart below shows that the stock is making a significantly lower low.
At the same time, the MACD indicator is reaching a higher low.
This suggests that the downtrend is losing strength and could soon be reversed.
MACD is a reliable trend following momentum indicator that is widely used in technical analysis.
The indicator is often used to show new trends when the fast moving average rises above or below the slow moving average.
The indicator that crosses the zero line is considered bullish while it crosses below the zero line bearish.
Disclaimer: The information above is for For educational purposes only and should not be treated as investment advice. The strategy presented would not be suitable for investors who are unfamiliar with exchange-traded options. All readers interested in this strategy should do their own research and seek advice from a licensed financial advisor.


Pascal’s Triangle is a treasure hunt for sequences and patterns.
Our study of Pascal’s Triangle was inspired by my son’s Folding the Circle class. A father in our co-op hosted this class (it made a great zoom homeschooling activity), and the kids really dug into some amazing geometric art.
During this class my son folded and built giant tetrahedrons out of smaller tetrahedrons, and we searched for a pattern. How many smaller tetrahedrons are added as the giant shape grows a level? How many total, are there? It turns out these patterns — the triangle numbers and the tetrahedral numbers — can be found in Pascal’s Triangle.
So we dropped everything for a couple days to investigate Pascal’s triangle.
As a jumping off point, we used the short-story biography of Pascal, from Mathematicians Are People, Too. The biographies in this book are only a few pages each, and my son really enjoys them. (It’s also great as inspiration for finding special math topics.)
After reading, my son filled in the numbers of Pascal’s Triangle, and searched for patterns.
He discovered the symmetry of the triangle, the counting numbers, the triangle numbers, the tetrahedral numbers, and the doubling of sums when you go down rows. As he learns more math concepts, we’ll be sure to revisit this.
It turns out, Pascal’s Triangle also includes probability values. But when I got out a penny to do some coin tosses, my son shouted “Noooooo! Not THAT kind of math” and ran away. Apparently he’s developed an aversion to statistics… Oh well! Maybe another day.
We also practiced some binomial notation, where he found values based on their position in the triangle. Older kids could learn the binomial theorem and use this to predict tetrahedron and triangle numbers.
For a nice overview of the math concepts in Pascal’s Triangle and a definition of the binomial theorem, check out Math Is Fun.
Below are worksheet downloads for filling out Pascal’s triangle if you want to try this activity with your kids:


Operations Research For BMS 6th Semester(Mumbai Univ.) Full Course By Madhusudan Sohani
₹2,650.00 – ₹4,875.00
- This Course covers full course of BMS Mumbai University for 6th Semester As per latest course
- 1 printed handout of 32 pages
- Over 85 numerical problems have been solved with full explanation
- One video is devoted to solving April 2017 QP fully
- Numerical questions are kept approximately at the level as contained in standard text-books on the subject. Some more difficult problems have been included for brighter students
- About 85 to 90 numerical questions have been solved.


I graduated from Warren High School is 1991. I attended college at University of Arkansas in Monticello from 1991 to 1994. I started completing on-line classes at Arkansas State University in the Fall of 2016.
I completed my BA in Elementary Education in the Spring 1994. I’ve taught math at Brunson for 23 years. I am currently working on my MSE in Curriculum and Instruction at Arkansas State University.
I began teaching fourth grade math in 1994 at what used to be Westside Elementary. During my time there, the name was changed to honor Thomas C. Brunson. Three years ago we became a charter school, and we renamed the school Brunson New Vision Charter School. I have taught fourth grade math for 23 years. I am also teaching third and fifth grade math skills depending upon the class. School has changed quite dramatically since I first began teaching, but the changes are good. They’ve made me a better instructor.
I was married in 1991 and August 10th will be my 26th wedding anniversary. I have two daughters, Cassidy who’s 19 and Carrigan who’s 12. Carrigan is in the 7th grade at Warren Middle School. Cassidy is married and lives in Tennessee.
I love to read. I absolutely love scary movies.I love to run. I competed on the race circuit last summer and loved it. I have a total of four first place medals, one second place master’s medal, one first place age group medal, and one third place age group medal.


Creativebug offers online video arts and crafts workshops and techniques. Learn how to paint, knit, crochet, sew, screen print, and more.
EdX offers free interactive online classes from the world’s best universities like MITx, HarvardX, BerkeleyX, and UTx. Topics include biology, business, chemistry, computer science, economics, finance, electronics, engineering, food and nutrition, history, law, literature, math, medicine, music, philosophy, physics, science, statistics and more.
Khan Academy is a not-for-profit with the goal of changing education for the better by providing a free world-class education for anyone anywhere. All of the site’s resources are available to anyone. It doesn’t matter if you are a student, teacher, home-schooler, principal, or an adult returning to the classroom after 20 years. Khan Academy’s materials and resources are available to you completely free of charge.
With over 100 instructional videos each, plus interactive drills and quizzes, ProCitizen provides the civics, vocabulary, plus reading and writing practice that new citizens need to succeed on the citizenship test.


iMAGINE STEAM Festival
The iMAGINE Upstate festival, which has occurred in Greenville over the past 4 years, is coming to Greenwood. The festival will feature interactive exhibits and shows emphasizing Science, Technology, Engineering, Arts and Math (STEAM) to Greenwood Uptown and Uptown Market on Saturday, March 16, 2019.
The purpose of iMAGINE STEAM Festival, fueled by the Greenwood Arts Center, is to ignite the interest of Pre-K through 12th-grade students (and their families) in STEAM subjects by providing up-close engagement with the latest technologies and giving them exposure to high-skill industries.
iMAGINE partners with corporations, schools, nonprofits, and volunteers to make STEAM come to life through hands-on exhibits and interactive stage shows.
Date & Time
March 16, 2019
10:00AM - 2:00PM
The Arts Center of Greenwood 120 Main St Greenwood 29646 SC US http://www.greenvillebusinessmag.com/businesses/sc-greenwood-the-arts-center-of-greenwood


Cauchy-Schwarz for outer products as a matrix inequality
If you read the Wikipedia page on the Cramér-Rao bound in statistics, there is an elegant and concise proof given of the scalar version of the bound. However, no proof of the full multivariate case is given there.
Indeed, it seems at first like the same approach will not work, because multivariate Cramér-Rao is a matrix inequality, while the scalar proof relies on the Cauchy-Schwarz inequality, which is a statement about inner products. Since an inner product is just a real-valued number, surely a different approach is required for proofs about matrices?
But after reading this 1980 paper of Bultheel I think the same short proof goes through, if we generalise the definition of “inner product” slightly. In fact, this form of Cauchy-Schwarz holds for the familiar outer product and the inner product version is just a special case!
Below we’ll confine ourselves to the reals for simplicity, unlike Bultheel who works more abstractly.
We’ll review the scalar case, then extend it to matrices.
Inner product definition
An inner product, on a vector space over the reals takes two vectors and returns a real number. The prototypical example is the dot product on , , but we can allow others if they satisfy these requirements:
- Positive definiteness: , with equality if and only if .
Cauchy-Schwarz from inner products
The Cauchy-Schwarz inequality is the following statement about products of inner products:
We can show this using the definition of the inner product above. Take a vector where is a real scalar.
Positive definiteness says:
We can use bilinearity to expand this:
and symmetry to obtain
Now if we make the choice and simplify:
We obtain the desired inequality:
Matrix-valued inner product axioms
Now we’d like something a little more powerful. We can get this if we are willing to generalise the notion of an inner product to something that returns a matrix instead of a number. I’ll denote this new “inner product” by .
We also need to generalise our axioms slightly for this wider definition. I will follow Bultheel’s definition, simplifying by considering only real-valued matrices. So:
- Symmetry holds up to a transpose. Now is a matrix, we need to add matrix transposition if we swap the arguments, but we still have:
- Bilinearity still applies, not only with scalar coefficients but also with matrices. We have to be careful about whether we are multiplying on the left or right, because matrix multiplication is not commutative. So we have:
- Positive definiteness: we will demand that is itself positive definite, i.e. as a matrix inequality. We can also insist that implies .
Now a multivariate Cauchy-Schwarz follows from these axioms just as it did in the scalar case, though again we must take care of the transpositions.
We substitute :
Using transpose symmetry to tidy up:
we obtain the matrix form of Cauchy-Schwarz:
In particular, in the scalar case, this reduces to the usual scalar form of the inequality.
I think this is cute! For one thing, we’ve just defined the outer product to be an inner product!
(The outer product between two dimensional vectors is the matrix , while the Euclidean dot product is the scalar .)
Yet since the outer product is transpose symmetric, bilinear, and results in a positive definite matrix for a single vector, it’s a perfectly good inner product for these purposes. I wonder why Cauchy-Schwarz is more commonly known in the less general inner product form?
I’m also intrigued by the geometric connotations of “matrix-valued inner products”. The inner product is an algebraic construction which is geometrically motivated, and so bridges these two aspects of mathematics. The inner product is at the core of geometry and defines:
- length of vectors (from the induced norm )
- angles between vectors (from ), and in particular orthogonality when
- projections onto sets (by minimizing the norm, or by orthogonality)
So – what would it mean geometrically for a length or an angle to be matrix valued?
I don’t know! But it does occur to me that if you have two ordinary, independent scalar metrics and , you can always compose these into a new “matrix-valued metric” . (This is still positive definite in the sense above). That declares two vectors to be orthogonal when both of the component metrics are: this means that the trace of our matrix, which is the sum of its eigenvalues, will be zero. (If we had used the determinant instead, it would declare orthogonality whenever any of the constituents did.)
Furthermore, the trace of the outer product recovers the inner product. In fact, the trace already gives a proper inner product between square matrices, thought of as a vector space: . So we can squash our matrix-valued inner product back to an ordinary scalar inner product by taking the trace. And if we do this for our diagonal matrix of independent metrics, we recover the usual metric on ! It was there all along, but the matrix-valued metric additionally preserves more information about along which basis directions the vectors agree and disagree.


Year 6 2020 - 2021
Welcome to our Year 6 Page
Mrs Rogers is the teacher in Class 6R and Mr Paterson is the teacher in Class 6P.
|Mrs Rogers||Mr Paterson|
We follow the objectives set in the National Curriculum. Below are links to these documents for the core subjects (mathematics, English and science).
- Mathematics: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf
- English: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/335186/PRIMARY_national_curriculum_-_English_220714.pdf
- Science: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/425618/PRIMARY_national_curriculum_-_Science.pdf
Click here to download the topic letter for Autumn 1.
Here are some guides to help you access some of the educational websites listed on this page.
- Mathletics guide: https://secure.schoolspider.co.uk/uploads/518/grade/444899_grade_file.doc
- Oxford Owl guide: https://secure.schoolspider.co.uk/uploads/518/grade/444906_grade_file.pptx
- Espresso guide: https://secure.schoolspider.co.uk/uploads/518/grade/444894_grade_file.doc
Support for Home Learning during Coronavirus Pandemic:
On this page, you will find home learning resources, links to websites and a suggested timetable that have been put together to support families having to work with children at home due to school closure. We have also included links to National Curriculum documents for core subjects and half-termly overviews so you have an idea of what is covered in school. Please make sure you scroll through the whole page to see what is included. During this time of school closure, we ask that you check the school website regularly for any important updates or information. We will continue to add to the home learning resources for each year group over this time.
Below are some links to websites that have various resources and interactive games to help with your child’s learning. These are a great way to consolidate what we have learnt in school. Children have been given usernames and passwords for Oxford Owl, Mathletics and Times Tables Rock Stars.
Oxford Owl: https://www.oxfordowl.co.uk
Times Tables Rock Stars: https://ttrockstars.com
Topmarks Education: https://www.topmarks.co.uk
Twinkl (free subscription, enter the code UKTWINKLHELPS): www.twinkl.co.uk/offer
BBC Bitesize Revision: https://www.bbc.co.uk/bitesize/levels/zbr9wmn
Maths Frame: https://mathsframe.co.uk/
- Master the Curriculum: https://masterthecurriculum.co.uk/categories/?year=year-6
- Espresso (username student 22144 password mauldeth): https://www.discoveryeducation.co.uk/
- Hamilton (free English and maths home learning packs updated weekly): https://www.hamilton-trust.org.uk/blog/learning-home-packs/
- Scratch App (free download): https://scratch.mit.edu/
- Number Fun Portal: https://parent.numberfunportal.com
- White Rose Maths (free home learning resources with videos and worksheets): https://whiterosemaths.com/homelearning/
- Literacy Company: http://www.theliteracycompany.co.uk/free-resources/
- NRICH (free online mathematics resources for ages 3 to 18 set up by the Faculties of Mathematics and Education at the University of Cambridge): https://nrich.maths.org/primary
- Robin Hood Academy (weekly home learning packs): https://www.robinhoodmat.co.uk/learning-projects/
- The World of David Walliams (free audio story everyday at 11am): https://www.worldofdavidwalliams.com/elevenses/?fbclid=IwAR1RqmHrK6WyzTDnvyvgkdg6_6Q76wfr3xW6O8E5Zs9qG60K6OLmce4dKg8
- The Maths Factor (free maths website created by Carol Vorderman): http://www.themathsfactor.com
Files to Download
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Date(s) - 01/26/2019
9:30 am - 11:30 am
Concordia Parish Library
When: Saturday, January 26th @ 9:30am
Where: Ferriday Branch
Details: Tips and tricks for scoring higher on the ACT, understanding you ACT score, and finding scholarship opportunities. Featuring an ACT Math Practice Test, guided math practice and strategies for maximizing your time. Instructor Stephen Collins is Louisiana & Mississippi certified in Mathematics, Business Education, Computer Science and Computer Literacy. Call (318) 757-3550 to register. Free and open to the public!


Suppose that we managed to assign some signs to quadrisecants of a knot (i.e., lines cutting it 4 points) so that their total number, counted with signs, does not change under knot isotopy. What kind of invariant is it, and how to assign such signs? While the answer in this case is known, it motivates more general attempts to count (with signs) various geometric objects inscribed into a knot or a plane curve. In all cases invariants are easily seen to be of finite type. I will explain a general setting to produce such invariants (using evaluation maps of configuration spaces and homology intersections) and will formulate some results and conjectures. No prior knowledge of these themes will be assumed.


The ninth-largest lottery jackpot in U.S. history, $1.04 billion, will be on the line in tonight’s Powerball drawing after no tickets with all six numbers were sold for the 30th consecutive drawing.
There hasn’t been a drawing with a grand prize winner since July 19 when a ticket worth $1.08 billion was sold at a downtown Los Angeles mini-market, the seventh-largest jackpot in U.S. history.
Ticket sales end at 7 p.m., and the drawing will be held at 7:59 p.m. Monday, Oct. 2.
The odds of matching all five numbers and the Powerball number is 1 in 292.2 million, according to the Multi-State Lottery Association which conducts the game. The overall chance of winning a prize is 1 in 24.9.
Buying tickets at a store where tickets with large jackpots have been sold in the past will not increase a purchaser’s chance of winning a jackpot, according to USC mathematics professor Ken Alexander.
“The chance that a given place will sell a winning lottery ticket is just related to how many tickets they sell,” Alexander told City News Service.
However, players wanting a better chance of avoiding sharing the jackpot should choose numbers that aren’t selected as often, Alexander said. Lottery players frequently choose the date of their birthdays as one of their numbers, so numbers higher than 31 would be played less, Alexander said.
Monday’s jackpot is the fourth-largest in the history of the Powerball game, which began in 1992. There have been five Mega Millions drawings with larger jackpots.
The Powerball game is played in 45 states, the District of Columbia, Puerto Rico and U.S. Virgin Islands.
Source: Orange County Register


Trinajstić Nenad, F.C.A.
- October, 26, 1936, in Zagreb
- August 27, 2021, in Zagreb
Trinajstić Nenad, F.C.A.
- Fellow of the Croatian Academy of Sciences and Arts
- Doctor of Science
- scientific adviser, retired – Ruđer Bošković Institute
- full professor, retired – Faculty of Science, University of Zagreb
Functions in Academy:
- member of the Presidency – Croatian Academy of Sciences and Arts (12/17/2015 – 12/31/2018)
- deputy secretary – Department of Mathematical, Physical and Chemical Sciences (01/01/2011 – 12/31/2014)
- member of the Presidency – Croatian Academy of Sciences and Arts (01/01/2004 – 12/31/2010)
Membership in Academy:
- full member – Department of Mathematical, Physical and Chemical Sciences (06/18/1992 -8/27/2021)
Professor Nenad Trinajstić was born on October 26, 1936 in Zagreb, where he finished elementary school (1951) and high school (1956). He graduated from the Department of Chemical Technology of the Technological Faculty in Zagreb in 1960. He won his master’s degree in 1966; his master-thesis supervisor was Professor Milan Randic. He got his doctor’s degree from the Faculty of Science and Mathematics of the University of Zagreb in 1967. His doctoral thesis, entitled “Electronic Structure of Some Polyatomic Molecules”, was the first Croatian dissertation in quantum chemistry.
After receiving his BSc. Chem. Techn. degree, he was employed by the Pliva Research Institute (1960–1961). He moved to the Rugjer Boskovic Institute (RBI) in early 1962, where he spent all his working life. For many years he was the head of the Theoretical Chemistry Group in the Department of Physical Chemistry. He retired at the end of 2001. In 1992 he was elected for a full membership of the Croatian Academy of Sciences and Arts. He was also a member of the International Academy of Mathematical Chemistry from its foundation in 2005, the same year he became emeritus scientist at the Rudjer Boskovic Institute.
He published more than 500 scientific papers, few hundreds of professional articles, and 12 books, among which is the most famous Chemical Graph Theory (CRC Press, 2nd ed. 1992). He is one of the most cited Croatian chemists, with h-factor 59.
His professional interest was in the fields of quantum-chemical and mathematical methods in chemistry. In mathematical chemistry, he worked on the development and application of graph theory and topology to chemistry of conjugated systems. He was also involved in the development of molecular descriptors known as topological indices (Zagreb indices, Harary index, various distance indices, detour index, etc.) and in the development of quantitative relations between the structure, properties and activities of organic molecules and biomolecules. His contributions
in the fields of chemical graph theory and the graph-based computer chemistry made him well known as one of the pioneers in these branches of mathematical chemistry.
Professor Trinajstić will be fondly remembered by all his friends and colleagues.


Geometric Constraint Solving
Product design for manufacture is an activity driven by descriptive information. Increasingly, design includes the use of design constraints that impose conditions on the shape of the product. That is, the designer states specific constraints without telling the system in detail how to satisfy them. It is the task of the underlying constraint solver to derive a plan by which to solve the constraint system.
Constraint solving will be presented in two parts. In the first part, the architecture of a simple planar constraint solver is described that anyone can implement. We then discuss ways in which to extend the solver, both regards geometric coverage as well as more complex constraints patterns.
The second part addresses spatial constraint solving focusing in particular on the algebraic side and some of the techniques that are known for solving the equations.
Before joining the Purdue faculty, Professor Hoffmann taught at the University of Waterloo, Canada. He has also been visiting professor at the Christian-Albrechts University in Kiel, West Germany (1980), and at Cornell University (1984-1986). His research focuses on geometric and solid modeling, its applications to manufacturing and science, and the simulation of physical systems. The research includes, in particular, research on geometric constraint solving and the semantics of generative, feature-based design. Professor Hoffmann is the author of Group-Theoretic Algorithms and Graph Isomorphism, Lecture Notes in Computer Science, 136, Springer-Verlag and of Geometric and Solid Modeling: An Introduction, published by Morgan Kaufmann, Inc. Professor Hoffmann has recieved national media attention for his work simulating the 9/11 Pentagon attack.
He is on the editorial boards of


The square root symbol (√) is an important mathematical function used to denote the principal square root of a number. For example, √9 = 3, since 3 squared equals 9. Being able to quickly and easily type the square root symbol on an iPhone can be invaluable for students, engineers, mathematicians, and anyone else who frequently needs to express square roots in their work.
In this comprehensive guide, we’ll cover multiple methods for typing square root on an iPhone, including using the default keyboard, the character map, and third-party apps. We’ll also provide tips, tricks and examples for using the square root symbol in various iPhone applications like Notes, Calculator, and Pages. Read on to become an expert at typing square root on your iPhone!
- The square root symbol can be typed on an iPhone using the default keyboard, character map, or third-party apps
- On the default keyboard, quickly type square root by pressing and holding the “0” key
- The Character Map provides quick access to symbols like square root
- Third-party apps like Symbol Keyboard allow full customization of the keyboard with math symbols
- Use square root in Notes for math homework assignments and equations
- The Calculator app can evaluate square roots when the symbol is typed
- Square roots can be used in Pages documents and reports to express mathematical concepts
- Customizing your keyboard or using text shortcuts are handy tricks for quick square root typing
Using the Default Keyboard
The default iOS keyboard on iPhone contains many hidden symbols and characters that can be accessed by pressing and holding certain keys. This includes the square root symbol, which is readily available but not visible on the normal keyboard layout. Here are some methods for typing square root using only the default iPhone keyboard:
Press and Hold the “0” Key
The fastest way to type square root on the default iPhone keyboard is to press and hold on the “0” key. After a second or two, a popup will appear showing various symbol options, including the square root symbol. With your finger still pressed on the “0”, slide to select the √ symbol and then release to type it.
This is by far the quickest and most straightforward way to type square root on your iPhone’s default keyboard. The symbol pops up instantly, and you can commit the “press-and-hold 0” movement to memory quite easily with a bit of practice.
Use the Numbers and Symbols Keyboard
Alternatively, you can access the square root symbol from the numbers and symbols keyboard:
- Tap the “123” key on the bottom left of the keyboard to switch from the regular layout to the numbers and symbols keyboard.
- Press and hold the “=” key until a popup appears.
- Slide your finger to the square root symbol and release to type it.
This method adds an extra step compared to pressing and holding “0”, but can be useful if you ever need quick access to other math symbols like fractions or pi.
Switch to the Emoji Keyboard
Here’s one more way to access square root from the default keyboard:
- Tap the emoji icon on the bottom left of the keyboard to switch to the emoji layout.
- Press and hold the “0” key to bring up the symbol popup.
- Slide to and select the square root symbol.
This has the same effect as pressing and holding “0” from the regular keyboard, with the benefit that you can quickly get back to emoji if needed in your message.
As you can see, accessing the square root symbol is quite simple on the default keyboard, especially with the press-and-hold “0” trick. With minimal practice, you’ll be typing √ like a pro on your iPhone in no time!
Using the Character Map
In addition to the default keyboard, your iPhone also includes a character map containing symbols, letters, and characters from a wide variety of languages. Here’s how to use it to type square root:
- Go to Settings > General > Keyboard > Keyboards > Add New Keyboard.
- Tap “Character Map” in the list of available keyboards.
- The character map will now be accessible from your list of keyboards. To access it, tap the globe icon on your keyboard to switch between languages/keyboards.
- On the character map, tap the search icon and type “square root” to find it quickly.
- Tap the square root symbol to type it.
While a bit more complex than the default keyboard method, the character map provides a helpful way to lookup just about any symbol and know you’ll find it. For quick square root typing the keyboard is still faster, but the map can be great for occasional use or finding less common symbols.
Third-Party Keyboard Apps
In addition to the system keyboards, third-party iPhone apps can provide fully customizable keyboard layouts with special math and science symbols included. Some top options include:
Symbol Keyboard provides a highly customizable keyboard experience focused specifically on symbols and characters for math, science and other technical topics.
You can easily add the square root symbol (and many other math operators) to the main keyboard layout for quick access anytime. Advanced options even allow you to create shortcut keys to type the square root symbol in just a tap or two.
If you need to type sqrt and other math symbols frequently, a specialized keyboard like Symbol Keyboard is extremely helpful.
Unicode Map offers another customizable keyboard with an emphasis on symbols from a wide range of languages and writing systems.
It includes a Math Operators section with quick access to the square root symbol, as well as many other math and technical symbols. Easy copy/paste and favorites features help you type special characters fast.
General custom keyboard apps like Fonts Keyboard also provide options to add math symbols like square root to your keyboard layout.
While less specialized for math compared to Symbol Keyboard and Unicode Map, robust custom keyboard apps remain a flexible option for quick sqrt access.
If you need to type square roots and math symbols regularly on iPhone, investing a couple dollars in a specialized keyboard app can improve your typing speed and efficiency significantly.
Typing Square Root in Notes App
The versatile Notes app on iPhone allows you to type notes, homework assignments, reminders, equations, and more. Here are some ways the square root symbol can be particularly helpful in Notes:
Math Homework and Class Notes
For students doing algebra, geometry, trigonometry, or calculus homework, the Notes app already provides ideal math support with features like handwritten notes, shape recognition, and equation editing.
Being able to quickly type square root symbols using the methods in this guide makes completing math assignments and taking class notes even easier. No more confusion over unclear handwritten sqrt symbols – just type it clearly and correctly on your iPhone keyboard.
Reviewing Math Concepts
Beyond completing specific assignments, Notes is also helpful for generally practicing and learning math concepts that involve square roots, like:
- Simplifying radicals
- Finding square roots of perfect squares
- Rationalizing denominators
- Solving quadratic equations
With the flexibility to type and edit equations on the fly, the Notes app on iPhone is a great way for students to learn and review mathematical square root concepts.
Inserting Square Root Examples
Notes allows you to create rich text documents mixing formatted text, math symbols, images, and other media.
For example, you could create a study guide on algebraic concepts and insert square root symbol examples to demonstrate ideas visually:
The square root of 169 is 13, because 13 squared equals 169:
√169 = 13
Being able to accurately type √ into your notes makes incorporating symbolic examples a breeze.
Sometimes examples from textbooks or class slides can further aid learning. In Notes, you can take a screenshot of a relevant math concept and annotate it with your own typed notes, like inserting a square root to illustrate part of an example:
To solve this, we first simplify the sqrt term:
√36 = 6
Notes makes it straightforward to import images and easily add your own explanatory notes including proper sqrt symbols typed from the keyboard.
Whether you’re completing assignments, taking class notes, making study guides, or annotating screenshots, being able to type square root symbols into the Notes app can be a big help for learning math on your iPhone.
Using Square Root in the Calculator App
The Calculator app on iPhone allows you to perform complex math operations with just a few taps. With square root readily accessible on the keyboard, you can evaluate sqrt functions in the Calculator app:
- Open the Calculator app and tap the standard calculator layout (not scientific).
- Tap the number keys to enter a number like 9.
- Press and hold the 0 key to type the square root symbol.
- The app will evaluate √9 to equal 3!
You can follow similar steps to evaluate square roots of other perfect squares. For non-perfect squares and decimal roots, you’ll need to switch to the scientific calculator layout:
- Open the Calculator app and swipe left on the calculator models to select the scientific layout.
- Use the number keys and sqrt symbol to enter an expression like √12
- Tap the “=” sign to evaluate the decimal result (√12 = 3.464)
The Calculator app makes it a breeze to evaluate any square root, perfect square or not. Combined with quick sqrt typing from the keyboard, it becomes easy to experiment with all kinds of square root calculations on iPhone.
As you can see, the Calculator app recognizes and evaluates square root functions when you type the symbol from the keyboard. This makes it super simple to find square roots and experiment with radical expressions.
Using Square Root in Pages for iPhone
Pages is Apple’s word processing app, similar to Microsoft Word or Google Docs. On iPhone, Pages provides robust tools for writing documents, reports, letters, resumes, and more. Here are some ways the square root symbol can be helpful when using Pages on your iPhone:
Science and Math Reports
For science or math-related assignments, being able to type square root into Pages is essential for presenting concepts cleanly and accurately.
Rather than relying on unclear handwritten sqrt symbols, you can insert crystal clear square roots using the methods in this guide – ideal for science lab reports. The equation editor even allows you to lay out square root functions properly.
Beyond school reports, being able to insert square roots is also useful when creating technical documents for work.
For example, a computer engineer may need to discuss clock speeds increasing at the square root of the number of transistors. By typing √ cleanly in Pages, technical concepts are communicated precisely.
Research Papers and Theses
Square roots can also appear in academic papers in mathematics, physics, engineering, and more.
Rather than compromising clarity by relying on handwritten symbols or workarounds, just type √ directly on your iPhone keyboard to properly convey complex concepts in research papers and theses.
Brochures, Flyers and Posters
For promoting STEM programs, math tutoring services, technical products, and similar offerings, Pages can be used to design informational flyers, brochures, posters and more.
Inserting square root symbols typed from the iPhone keyboard is an easy way to add technical flair and convey expertise in marketing materials.
Whether you’re a student creating reports, an engineer drafting technical documents, or a teacher designing classroom materials, the ability to easily insert square roots is extremely beneficial when using Pages on your iPhone.
Handy Shortcuts and Tips
Beyond the basic methods for typing square root on iPhone, there are some handy shortcuts and tips worth learning:
Create a Shortcut in Settings
For extremely frequent usage, you can create a custom text shortcut so typing “sqrt” autocorrects to the √ symbol:
- Go to Settings > General > Keyboard > Text Replacement
- Tap the “+” to create a new shortcut
- Type “sqrt” as the phrase and “√” as the shortcut
Now just typing “sqrt” will automatically produce the square root symbol, saving multiple taps and presses.
Add Square Root to the Shortcuts Bar
In some third-party keyboard apps, you can add dedicated sqrt shortcut buttons to the top shortcuts bar for single tap insertion. Very handy for heavy math usage.
Memorize the Press-and-Hold Movement
On the default keyboard, commit the “press-and-hold 0” motion to muscle memory so it becomes second nature when you need to quickly type square root symbols.
Use Square Root Templates
Apps like Pages often include ready-made equation templates. Use these to quickly insert square root functions rather than building them from scratch.
By mastering shortcuts, tweaks, and tips like these, you can optimize your iPhone to type √ symbols faster than ever.
Being able to cleanly type the square root symbol is essential for students, mathematicians, scientists, engineers, and other professionals who work with math concepts and notation. On the iPhone’s default keyboard, square root can be quickly accessed by pressing and holding “0”. The character map provides an alternate way to lookup and insert sqrt and other symbols. Third party keyboard apps can fully customize your typing experience for easy sqrt insertion.
Within the iPhone’s everyday apps, square root comes in handy when completing math homework assignments in Notes, evaluating radical expressions in Calculator, creating scientific reports in Pages, and more. With the typing tricks and tips covered in this guide, you can optimize your iPhone to help insert square roots seamlessly anytime they are needed in your work and studies. So tap into the power of square root – your new math superpower!
Frequently Asked Questions
Q: How do I type square root on iPhone?
A: You can use the keyboard, character map, or third-party apps to type square root on iPhone.
Q: How do I use the square root symbol in iPhone notes?
A: You can copy and paste the symbol from another application or use a third-party keyboard that includes the symbol.
Q: Can I customize the iPhone keyboard to include the square root symbol?
A: Yes, you can add a custom shortcut to the keyboard to type the square root symbol.
Q: How do I type square root on iPhone calculator?
A: You can use the character map to copy and paste the symbol into the calculator or use a third-party calculator app that includes the symbol.
Q: How do I type square root on iPhone Pages?
A: You can use the keyboard, character map, or third-party apps to type square root on iPhone Pages.
Q: What is the shortcut to type square root on iPhone?
A: You can create a custom shortcut to type the square root symbol on iPhone.
Q: How do I use the square root symbol in other iPhone applications?
A: You can copy and paste the symbol from another application or use a third-party keyboard that includes the symbol.
Q: What is the square root character on iPhone?
A: The square root character on iPhone is the symbol used to represent the square root of a number.
Q: How do I type square root on iPhone third-party apps?
A: You can use the keyboard, character map, or third-party apps to type square root on iPhone third-party apps.
Q: Can I use the square root symbol on iPhone without a third-party app?
A: Yes, you can use the character map or create a custom shortcut to type the square root symbol on iPhone without a third-party app.


About the test
CSPN uses a proprietary examination which covers reading comprehension and basic math. The reading comprehension portion is 30 minutes in length, and requires a score of 70% or higher to pass.
The math portion consists of 30 basic math problems including:
- Multiplying and dividing whole numbers
- Addition, subtraction, multiplication, and division of fractions
- Division of decimals
- Converting from percentage to decimal to fraction to ratio
Click here to see some sample math problems.
This section of the test is 45 minutes in length, and requires a score of 70% or higher to pass. CSPN will provide a calculator for you to use.
You may take the test twice per enrollment period.
How to register for the entrance exam
The entrance exam fee ($40) is non-refundable.
Please see this page for the most up-to-date available days and times. Registering and paying online assures you of a seat on the date and time of your choice. Alternatively, you may register and pay in cash (or via money order) in person. Limited walk-in seats may or may not be available on the day of the test.
We regret that we cannot accept personal checks the entrance exam fee.
Reviewing for the test
Any book that has reading comprehension practice questions will be helpful. A GED review book would be one example.
For the math portion of the test, remember to download the sample math questions above. Math workbooks and/or web sites are plentiful.
CSPN will consider transcripts of HESI or WorkKeys scores of 70% or higher in place of the CSPN entrance exam. There is no charge for this review.


Hey, this is Presh Talwalkar. This problem was part of a written interview test for Amazon in India. A drawer contains twelve identical black socks and twelve identical white socks. If you pick two socks at random: what is the probability of getting a matching pair? Can you figure it out? Give this problem a try and when you’re ready keep watching the video for the solution. Many people think when you pick two socks they will either match or they will not match. Since the number of black and white socks are equal, the probability of matching should be 50%. But this tempting answer is wrong! The probability of making a pair is actually slightly less than 50%. Let’s calculate and figure out why? There are 12 white socks and 12 black socks. By symmetry the probability of picking a black pair of socks equals the probability of picking a white pair of socks. So let’s calculate the probability of picking a black pair of socks. There is a 12 over 24 chance the first sock is black. This is because there are 12 black socks out of 24 socks total. 12 over 24 simplifies to be 1/2. Now what’s the chance of picking a black sock on the second draw? For the second sock the chance will be 11 out of 23, because there are 11 black socks remaining out of a total of 23 socks remaining. The asymmetry is because we are sampling without replacement. So what does this all mean? It means, the probability of picking a black pair is 1/2 times 11 over 23. This will also be the probability of picking a white pair by symmetry. So, we can put this all together to get the odds of picking a pair of matching socks. The probability of picking a matching pair will equal the probability of picking a black pair plus the probability of picking a white pair. Each of these is equal to 1/2 times 11 over 23. And therefore the probability of picking a pair of matching socks will be 11 over 23, which is approximately 47.8%. This is slightly less than 1/2. Did you figure it out? Thanks for watching this video. Please subscribe to my channel. I make videos on math. You can catch me on my blog mindyourdecisions. If you like this video you can support me on patreon… …and you can check out my books which are listed in the video description. You can also follow me on social media either at mindyourdecisions or at preshtalwalkar depending on the site.


[Notation: The obvious notational choice for proportion or probability is p. The standard convention is to use Roman letters for sample quantities and the corresponding Greek letter for population quantities. Some books do just that. However, the Greek letter has its own special place in mathematics. Therefore, instead of using pfor sample proportion and for population proportion, many authors use p for population proportion and p with a hat (caret) on it, (called p-hat), as the sample proportion. The use of "hat" notation for differentiating between sample and population quantities is quite common.]
There's really nothing new to learn to compare two proportions because we know how to compare means. Proportions are just means! The proportion having a particular characteristic is the number of individuals with the characteristic divided by total number of individuals. Suppose we create a variable that equals 1 if the subject has the characteristic and 0 if not. The proportion of individuals with the characteristic is the mean of this variable because the sum of these 0s and 1s is the number of individuals with the characteristic.
While it's never done this way (I don't know why not*), two proportions could be compared by using Student's t test for independent samples with the new 0/1 variable as the response.
An approximate 95-% confidence interval for the difference between two population proportions (p1-p2) based on two independent samples of size n1 and n2 with sample proportions and is given by
Even though this looks different from other formulas we've seen, it's nearly identical to the formula for the "equal variances not assumed" version of Student's t test for independent samples. The only difference is that the SDs are calculated with n in the denominator instead of n-1.
An approximate 95-% confidence interval for a single population proportion based on a sample of size n with sample proportion is
Comparing Two Proportions
There is a choice of test statistics for testing the null hypothesis H0: p1=p2 (the population proportions are equal) against H1: p1p2 (the population proportions are not equal). The test is performed by calculating one of these statistics and comparing its value to the percentiles of the standard normal distribution to obtain the observed significance level. If this P value is sufficiently small, the null hypothesis is rejected.
Which statistic should be used? Many statisticians have offered arguments for preferring one statistic over the others but, in practice, most researchers use the one that is provided by their statistical software or that is easiest to calculate by hand.
All of the statistics can be justified by large sample statistical theory. They all reject H0 100(1-)% of the time when H0is true. (However, they don't always agree on the same set of data.) Since they all reject H0 with the same frequency when it is true, you might think of using the test that is more likely to reject H0 when it is false, but none has been shown to be more likely than the others to reject H0 when it is false for all alternatives to H0.
The first statistic is
The second is
where is the proportion of individuals having the characteristic when the two samples are lumped together.
A third statistic is
The test statistic z1 is consistent with the corresponding confidence interval, that is, z1 rejects H0 at level if and only if the 100(1-)% confidence interval does not contain 0.
The test statistic z2 is equivalent to the chi- square goodness-of-fit test, also called (correctly) a test of homogeneity of proportions and (incorrectly, for this application) a test of independence.
The test statistic z3 is equivalent to the chi- square test with Yates's continuity correction. It was developed to approximate another test statistic (Fisher's exact test) that was difficult to compute by hand. Computers easily perform this calculation, so this statistic is now obsolete. Nevertheless, most statistical program packages continue to report it as part of their analysis of proportions.
Common sense suggests using z1 because it avoids conflicts with the corresponding confidence interval. However, in practice, the chi-square test for homogeneity of proportions (equivalent to z2) is used because that's what statistical software packages report. I don't know any that report z1. However, z2 (in the form of the chi-square test) has the advantage of generalizing to tests of the equality of more than two proportions.
When testing the null hypothesis H0: the population proportion equals some specified value p0 against H1: the population proportion does not equal p0, there is, once again, a choice of test statistics.
all of which are compared to the percentiles of the standard normal distribution.
Again, z1 gives tests that are consistent with the corresponding confidence intervals, z2 is equivalent to the chi-square goodness-of-fit test, and z3 gives one-sided P- values that usually have better agreement with exact P-values obtained, in this case, by using the binomial distribution.
These techniques are based on large sample theory. Rough rules of thumb say they may be applied when there are at least five occurrences of each outcome in each sample and, in the case of a single sample, provided confidence intervals lie entirely in the range (0,1).