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Pythagorean Theorem - Perigal's Visual Model

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Pythagorean Theorem - Perigal's Visual Model

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Pythagorean Theorem - Physical Model, Water Model Demo

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Pythagorean Theorem - Physical Model, Water Model Demo

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Museum of Science in Boston

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Hosted by Chris Shore of The Math Projects Journal

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1466 A Non-Simplified Yet Easy-to-Remember Unit Circle

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Look at the Patterns in this Unit Circle:

A unit circle has a radius of one and is often used to display the sines and cosines of special angles. I’ve made a unit circle that showcases the patterns that make memorizing how to complete one easier.

This unit circle is easy to memorize because many of the values have not been simplified. However, always remember to simplify the values when giving your final answer on any assignment or test!

The patterns you see in the ordered pairs representing the cosines and sines are only coincidental: they are merely an aid somebody else created to make memorizing the values easier. Over half of the square roots can be simplified and should be. However, I do love them in their non-simplified form because it is easy to remember them, and it is easy to see the sine wave and the cosine wave as the square root values increase and decrease incrementally as we go around the unit circle.

I’ve also made a similar unit circle that might help students locate the angles in the quadrants better:

Factors of 1466:

1466 is a composite number.

Prime factorization: 1466 = 2 × 733

1466 has no exponents greater than 1 in its prime factorization, so √1466 cannot be simplified.

The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1466 has exactly 4 factors.

The factors of 1466 are outlined with their factor pair partners in the graphic below.

More Facts about the number 1466:

1466 is the sum of two squares:
29² + 25² = 1466

1466 is also the hypotenuse of a Pythagorean triple:
216-1450-1466 calculated from 29² – 25², 2(29)(25), 29² + 25²

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1466 A Non-Simplified Yet Easy-to-Remember Unit Circle

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Look at the Patterns in this Unit Circle: A unit circle has a radius of one and is often used to display the sines and cosines of special angles. I’ve made a unit circle that showcases the pa…

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Image by author

Our modern society is built upon mathematics. And mathematics is built upon the achievements of thousands of outstanding individuals. Explorers, creators, problem solvers... they each pushed the boundaries of knowledge and invented new methods of thinking about the world.

Who was the greatest mathematician of all time? How would you even decide? Is it the most creative genius? The most prolific? The one who solved the most important problem?

This article presents a visualisation of the greatest mathematicians, ranked according to how much influence they have on mathematics today.

The main sections are:

The interactive data visualisation.

Comments on the overall top 10 ranked mathematicians.

A list of top-ranked mathematicians by field.

A discussion of the ranking methodology.
(InfluenceRanking™ engine by academicinfluence.com)

(Give it 10-20 seconds to load 🕵)

You can tap a mathematician's image to show their name and more information, or tap a topic button to show the top-ranked mathematicians for that topic.

1. David Hilbert

You can debate whether or not he should top the list, but Hilbert certainly played a huge role in many areas of mathematics, including geometry, number theory, logic, and mathematical physics. In 1900 he famously presented his list of "Hilbert's problems", which influenced the direction of mathematical research for years to come.

2. Albert Einstein

Ask a random person on the street to "name a mathematician" and it will probably be Einstein, whose name has become synonymous with genius. He rocked the world of space and time in 1905 with his Special Theory of Relativity, then again 10 years later with his General Theory of Relativity.

3. Leonhard Euler

Apart from founding the fields of graph theory and topology, his name permeates mathematical literature: Euler's number, Euler's identity, Euler's theorem... which one!?! Gauss said: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."

4. Carl Friedrich Gauss

From child prodigy to the "prince of mathematicians", Gauss picked up where Euler left off and influenced almost all fields of mathematics. Many developments in modern mathematics can be traced back to his work, and his name appears in terminology about as much as Euler: the Gaussian distribution, Gaussian elimination, Gaussian curvature...

5. Isaac Newton

Madman?... Perhaps. Genius?... Undoubtedly. He helped bring Europe out of the dark ages and his laws of motion allowed enormous breakthroughs in our understanding of the physical world. He gave birth to calculus (as well as a famous historic rivalry with Leibniz).

6. Bernhard Riemann

Another German genius, he devised the Riemann integral in analysis, Riemannian manifolds in geometry, and the Riemann zeta function in number theory. The Riemann hypothesis is one of the most famous unsolved problems in mathematics today.

7. Euclid

300 years before Christ, he laid down the foundations for geometry and number theory which would stand for the next 2000 years. His Elements text still forms the foundation of high school geometry.

8. Henri Poincaré

From pure mathematics to physics and engineering, Poincaré's contributions made him one of the greatest mathematicians in the late 19th century. The solution of the Poincaré conjecture, by Grigori Perelman, was one of the most famous breakthroughs in modern mathematics.

9. Aristotle

Known more for his philosophy than mathematics, his appearance on the list may be a little surprising. But he did lay the foundations of formal logic and influenced the education system of the western world for thousands of years.

10. Sophus Lie

Lie largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Lie groups and Lie algebras continue to be studied heavily in pure mathematics as well as physics, computer science, and statistics

Algebra

1. Sophus Lie
2. John von Neumann
3. David Hilbert

Combinatorics

1. Paul Erdős
2. August Ferdinand Möbius
3. Terence Tao

Differential equations

1. Leonhard Euler
2. Isaac Newton
3. Sophus Lie

Geometry

1. Euclid
2. Albert Einstein
3. Carl Friedrich Gauss

Graph theory

1. Paul Erdős
2. Leonhard Euler
3. László Lovász

Group theory

1. Évariste Galois
2. Sophus Lie
3. Felix Klein

Linear algebra

1. Alston Scott Householder
2. Alexandre Théophile Vandermonde
3. Hermann Grassmann

Number theory

1. Carl Friedrich Gauss
2. Leonhard Euler
3. G. H. Hardy

Probability

1. Pierre Simon Laplace
2. Andrey Kolmogorov
3. Siméon Denis Poisson

Set theory

1. Georg Cantor
2. Ernst Zermelo
3. Bertrand Russell

Statistics

1. Ronald Fisher
2. Karl Pearson
3. Albert Einstein

Topology

1. Henri Poincaré
2. Bernhard Riemann
3. David Hilbert

When I was planning for this article and data visualisation, I had two ideas for how to source the data for ranking:

Mathematicians mentioned on Wikipedia

Mathematicians cited in papers submitted on arxiv.org

Both Wikipedia and arXiv are free to access and provide a vast amount of data. But both present challenges for extracting data to create a meaningful ranking.

Thankfully, I discovered that the team at Academic Influence had already created a business out of ranking scholars (and universities).

Their machine learning model uses data from Wikipedia, Wikidata, Crossref, and Semantic Scholar in order to rank people, and hence institutions by affiliation. Although the model is not open source, they give a high-level overview on their website:

"In assessing influence by discipline for persons, the InfluenceRanking engine avoids making influence a popularity contest. The point is not to count the number of mentions a person receives. Rather, what is crucial to the InfluenceRanking engine is tracking the intersection of name mentions and discipline mentions, rewarding the proximity of these intersecting mentions, keeping track of relevant connecting hyperlinks and the amount of digital space devoted to the mentions. -academicinfluence.com

Something like Google's PageRank, where each website is given an authority score which is used to weight its links and content.

Of course, there will always be debate about the accuracy of the ranking. Every model has flaws, and it's hard to be objective about the concept of "influence". While creating the visualisation, I came across several rankings that seemed anomolous to me. Einstein being ranked #3 in Statistics, for example. 🤔

If you have thoughts, please don't hold back... that's what the comment button is for! 😁😁😁

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The Greatest Mathematicians - A Data Visualisation

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Our modern society is built upon mathematics. And mathematics is built upon the achievements of thousands of outstanding individuals. Explorers, creators, problem solvers... they each pushed the boundaries of knowledge and invented new methods of thinking about the world. Who was the greatest mathematician of all time? How would you even decide?

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Author:

John Golden

How are vending machines like functions? Do ideas like domain and range apply? Are both of these machines functions? If so, what's the input and output?

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A vending machine can be used as an example or nonexample of a function. This sketch has both. Use this model to introduce the idea of input and outp...

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Free offline GeoGebra apps for iOS, Android, Windows, Mac, Chromebook and Linux

Calculator Suite

Explore functions, solve equations, construct geometric shapes and 3D objects.

Graphing Calculator

Graph functions, investigate equations, and plot data with our free graphing app

3D Calculator

Graph 3D functions, plot surfaces and do 3D geometry with our free 3D Grapher

Geometry

Construct circles, angles, transformations and more with our free geometry tool

GeoGebra Classic 6

Apps bundle including free tools for geometry, spreadsheet, probability, and CAS

CAS Calculator

Solve equations, expand and factor expressions, find derivatives and integrals

GeoGebra Classic 5

Apps bundle including free tools for geometry, spreadsheet, probability, and CAS

GeoGebra (app versions included) is fully supported on Windows 11.

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Free offline GeoGebra apps for iOS, Android, Windows, Mac, Chromebook and Linux

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Interactive Class Participation - Quizziz

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Interactive Class Participation - Quizziz

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Join an activity with your class and find or create your own quizzes and flashcards.

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Compare Equations - Mixed Operations Math Game - Fruit Splat Skill Builders -Sheppard Software Educational Games

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Learn how to compare equations - using greater than, less than, equal to - with this fun arcade game.

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Teaching Math through Games

Very often, traditional methods of learning math fail to show the practical application of a mathematical concept in one's real life. This is especially true for primary school kids who are extremely curious about everything. Lately, pedagogy has shifted focus on more active forms of learning such as math games and activities, which are often an enactment of a real life situation. There are a few advantages of this shift to math games:

1. Engage the learner - Children these days have a lower attention span. Therefore, games have become a popular classroom tool, because they are a fun way to motivate and push the learner on the right path, without really making them feel like they are taking on a daunting theoretical task.

2. Encourage Creativity - While many appreciate the age old pattern of standardized curriculums and regular evaluations as the way to ensure progress of a learner, it is now believed that such a restrictive pattern of study does not allow for the creative instinct of an individual to surface. Math games allow us not only to live the real life situation but also make us more competitive and think out of the box.

Introduction to Grade 4 Math

In Grade 4, the focus is on three critical areas:

1. Gaining understanding and fluency with multi-digit multiplication, and gaining an understanding of division problems involving multi-digit dividends: The kids are introduced to place value to 1,000,000, they develop an understanding of the relative sizes of digits at each place. They compute products of multi-digit whole numbers by applying their understanding of place value, models for multiplication like arrays, area models and equal-sized groups and properties of operations, particularly the distributive property. They are able to choose appropriate methods to solve problems depending on the context and the numbers.

The kids use their understanding of place value, division models and properties of operations as they develop skills to find quotients involving multi-digit dividends. They also understand the relationship between multiplication and division and use this understanding to appropriately apply methods to estimate or mentally solve division problems.

2. Extending fractions understanding and decimal notations: The kids extend their understanding of fraction equivalence and apply it to identify and generate equivalent fractions. They develop skills to compare fractions with different numerator and different denominators. They learn addition and subtraction of mixed numbers with like denominators. They also extend their previous understanding of multiplication. They are also introduced to decimal notation for fractions.

3. Developing an understanding of analyzing and classifying geometric figures: The kids describe, analyze and classify two-dimensional shapes according to the properties of lines and angles. They compare two-dimensional shapes on the basis of their properties, such as perpendicular sides, parallel sides, symmetry, and particular angle measures. The kids also develop skills to identify line-symmetric figures.

Overview of Topics for Grade 4

Operations and Algebraic Thinking:

Multiplication & Division: The kids further develop their understanding of multiplication and interpret a multiplication equation as a comparison statement. For example, the kids interpret 24 = 6 × 4 as 24 is 6 times as many as 4 and 4 times as many as 6. They use multiplication comparison to solve word problems involving multiplication and division by using the symbol for the unknown number in the equation. They understand the difference between addition and multiplicative comparison.

Factors & Multiples: The kids extend their understanding of multiplicative comparison to understand the concepts of factors and multiples. They are able to find factor pairs of the whole number in the range 1 - 100. For example, (2, 16), (4,8) are factor pairs of 32. The kids also develop skills to identify if a given whole number is a multiple of a given one-digit number. For example, 8 is a multiple of 1, 2, and 4. They are also able to differentiate between prime and composite numbers. For examples, the numbers with exactly two factors; 1 and itself, are prime numbers. The numbers with more than two factors are composite numbers. 1 is neither prime nor composite number.

Analyze Patterns: The kids broaden their skills to generate patterns of shapes or numbers that follow a given rule. They are also able to recognize apparent features of the patterns that are not obvious in the pattern. For example, the starting number being 1, given the rule add 5, will generate a sequence of numbers that alternate between even and odd numbers. The kids are able to informally explain the alternating pattern of even and odd numbers.

Number & Operations in Base Ten:

Generalize place value understanding: The kids recognize that a digit in one place represents ten times of what it represents in the place to its right. For example, the value of the digit 8 in the number 880 is 80 whereas the value of the digit 8 is 800. The kids are also able to read and write multi-digit whole numbers using number names and base-ten numerals. They compare numbers based on place values, using >, =, and < symbols. The kids also use place value understanding to round whole numbers to any place.

Apply arithmetic properties of operations: The kids are able to fluently add and subtract multi-digit whole numbers. They use properties of operations and place value strategies to multiply up to four digits by one digit whole number or two two-digit numbers. The kids also learn to divide up to four digit numbers by one digit using properties of operations and place value strategies.

Number & Operations - Fractions:

Fraction Equivalence and ordering: The kids extend their understanding of fraction equivalence and use this understanding to identify and generate equivalent fractions. They apply their understanding of fraction equivalence to compare two fractions with different numerators and different denominator. This is done by making either the denominators or the numerators common for both the fractions. The kids comprehend that comparisons are possible only when the fractions refer to the same whole.

Build fractions from unit fractions: The kids understand that a fraction a/b where a > 1 can be written as a sum of fractions 1/b. This enables them to perform addition and subtraction of fractions. They apply the skill of decomposition of whole numbers learned in the previous grades to fractions and decompose a fraction in more than one way. For example, 4/5 = 1/5 + 1/5 + 1/5 + 1/5 or 2 3/6 = 1 + 1 + 1/6 + 1/6 + 1/6.

Understand decimal representation for fractions: The kids' knowledge of fraction equivalence enables them to understand decimal representation for fractions with denominator 10 or 100. For example, 3/10 = 0.3 and 32/100 = 0.32. They also develop skills to locate decimal fractions on number line diagram. They also learn to compare two decimals to hundredths.

They apply their understanding of multiplication and multiply a fraction by a whole number. They relate decomposition of addition with the decomposition of multiplication and understand that a fraction a/b is a multiple of fraction 1/b. For example, 3/8 = 3 × (1/8). The kids are able to solve word problems involving addition, subtraction, and multiplication of fractions.

Measurement & Data:

Conversion of measurements: The kids understand the relative sizes of measurement units within a single system of measurement. For example, 1 km > 1 m > 1 cm etc. They solve word problems involving lengths, volumes, masses of objects, money, and intervals of time using the four operations. They also develop skills to represent measurement quantities using number line diagrams. They also learn the area and perimeter formulas for rectangles.

Represent and Interpret Data: The kids learn to plot line plots with measurement units in fractions. They perform addition and subtraction of line fractions to solve problems.

Geometric Measurement: The kids recognize angles as geometric shapes. They learn to measure angles in degrees using a protractor. They solve addition and subtraction problems to find unknown angles in a diagram.

Line and Angles: The kids recognize rays, lines, line segments, angles and parallel and perpendicular lines. They classify two-dimensional figures on the basis of angle measurement and presence or absence of parallel or perpendicular lines. They are also able to identify line symmetric figures and draw lines of symmetry.

Geometry:

SplashLearn: Making Teaching and Learning Math Fun

Technology has invaded all spheres of life, including how education is imparted to young learners. Using the interactive activities and fun games on phones, desktops and tablet devices, SplashLearn guarantees a unique step-by-step Math learning for kids.

Step 1: The child begins by taking a Placement Test to determine the concepts they already know.

Step 2: Follow a curriculum developed by experts at SplashLearn that helps the learner smoothly transition from topic to topic.

Step 3: As and when the child performs well, SplashLearn increases the level of difficulty such that even the brightest feel challenged and develop a capacity to solve high order thinking questions and therefore excel both in school evaluations and at Math competitions.

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Math Games for 4th Grade Kids Online - SplashLearn

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Very often, traditional methods of learning math fail to show the practical application of a mathematical concept in one's real life. This is especially true for primary school kids who are extremely curious about everything. Lately, pedagogy has shifted focus on more active forms of learning such as math games and activities, which are often an enactment of a real life situation.

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Number Properties - identify prime factors

Common Core Connection for Grades 4, 5, 6

Determine whether a given whole number in the range 1-100 is prime or composite.
Find common factors and multiples.
Find the greatest common factor of two whole numbers less than or equal to 100.
Find the least common multiple of two whole numbers less than or equal to 12.

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Factor Trees and Prime Factorization | Math Playground

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Number Properties - identify prime factors Common Core Connection for Grades 4, 5, 6 Determine whether a given whole number in the range 1-100 is prime or composite. Find common factors and multiples. Find the greatest common factor of two whole numbers less than or equal to 100.

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The Abacus - The Hand as the Visible Part of the Brain

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The Abacus - The Hand as the Visible Part of the Brain

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Could the hands-on approach of an abacus be the secret to metacognition? Watch this video and see what you think.

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Japan's ancient secret to better cognitive memory - BBC REEL

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Hard to find an allergy-free cake that is square - but if you can - this is a wonderful project for math friends to work on in groups.

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This is not our video - but the inspiration for our Math Club to transform the walls of the second floor school into binary (the 4-6th grade) and Roman (3rd grade) zones.

In Math Club, we looked at number systems other than ours, including binary (base 2), octal (base 8), and Roman Numerals.  Just like the world has many languages (e.g. English, Spanish, Latin and more), we also have many number systems that we can choose to use when working with numbers.  Problems are sometimes easier to solve when we work in binary or octal and Number Sense contests always have Roman Numeral questions. But more importantly, the value we gain in letting our math friends play with number systems beyond decimal, is that they have the opportunity to compare and contrast our decimal number system with others to understand its strengths and weaknesses.

The Third Grade Club created a clock in Roman Numerals -- it was the cutest thing to see that group work fervently on converting each number on the clock face and then look at the end result and go -- "Oh I have seen one of these clocks before!"  Another group converted the telephone in the room into a Roman Numeral phone and were befuddled with what to put in for the "0" as the Roman Numeral System does not have a concept of 0.  Another group converted Number 2 pencils to Number II pencils.  The last group converted the classroom door.

The Elementary Club collectively created a computer that converts a number to binary.  The math friends upgraded a real clock to run in binary, translated the telephone interface, created posters that explain what binary is and how we convert to decimal and back, converted a number 2 pencil to number 10 and converted all of the fifth locker numbers (whew!)

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Math Video Challenge - The Bi-Light Zone (2013-2014 Finalist)

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"Mathematical PI" (The PI song with lyrics!)

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"Mathematical PI" (The PI song with lyrics!)

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Learners that struggle with recalling mathematics vocabulary, facts and notation are limited in their ability to be successful in mathematics. Creating vocabulary puzzles can be an engaging approach, including interactive games like Pictionary, and tools like Quizlet, to building retention. We want learners to be able to retrieve what they learn in their mathematics education, to have it all at their fingertips. Providing retention and retrieval methods is a significant part of what a mathematics education should be about.

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Online Puzzlemaker | Create Your Own Puzzle | Discovery Education Puzzlemaker

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Discovery Education’s Puzzlemaker provides teachers, students, and parents, the tools necessary to create crossword, puzzles, word search puzzles, mazes and more online!

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A square will always maximize the area.
The vertex will be the maximum area.
Calculus can define the maximum area.

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Solve a Problem, Develop a Technique.

The perimeter of a rectangle is 24 cm. Find the maximum area of the rectangle.

The solution is an area of 36 square units achieved using a rectangle which is a square, in this case, a 6 unit by 6 unit rectangle.

For a given perimeter, the square ALWAYS yields the MAXIMUM AREA.

We will examine the reason from 3 points of view -- arithmetic, algebra II, calc.

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MATH SPOKEN HERE! Classes Quadratic Equations, 320

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Solve a Problem, Develop a Technique. The perimeter of a rectangle is 24 cm. Find the maximum area of the rectangle. The solution is an area of 36 square units achieved using a rectangle which is a square, in this case, a 6 unit by 6 unit rectangle.

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Set the probability of heads (between 0 and 1.0) and the number of tosses, then click "Toss". The outcomes of each toss will be reflected on the graph. Check the box to show a line with the true probability on the graph. Click "Reset" at any time to reset the graph.

Click the "Quiz Me" button to complete the activity.

When you toss a coin, there are only two possible outcomes, heads or tails. On any one toss, you will observe one outcome or another-heads or tails. Over a large number of tosses, though, the percentage of heads and tails will come to approximate the true probability of each outcome.

In this applet, you can set the true probability of heads for your virtual coin, then toss it any number of times. Notice how the proportion of tosses that produce heads can be quite variable at first, but will eventually settle down to the true probability.

Question 1.1

If you set the true probability of heads to $ph:

What should be the true probability of tails? voljcClHuMI=

How many heads would you expect to observe after 10 tosses? PPnexCYE7RiF+h29

Question 1.2

Perform this "experiment" yourself: set the probability of heads to $ph, then observe how many heads you've gotten after 10 tosses. Enter this number here: VI8Rxynw+GI=

Great.

Question 1.3

If the true probability of heads is $ph, how many heads would you expect to observe after 50 tosses? B2FCg8YCcoDmLNQd

After 500 tosses? DELel3IDPr5OUqfr

Correct.

Incorrect.

Question 1.4

8S1R8+R8GAkoUXG1Twuhn4Tbft/0xdfjrzgjo/AiV1AhGdt4R5jYRHbpKdbzsqd5ERkbSsegTASvrS8IVxMg51xo/bQn5kVnu6KhrJWOsLul40bp7yJrOQzPrzcQYniMocl2Z0vjUYeKZD7NIYK6r8KHmGAIASSrjAr4pshNrM4jF175YmzZ73EovEj5x0nxJXc62r8bTaj40P7j9gpkMFaQyZ3SI+iC88tJkq8YoWbalw2SFzZmGZjkpTcGQZhfzQYX1A/QPQYuu/aNhoIFEEAvV6IQjc8tAOiJb6nDQu+kTvL50b07xpoudcbLs4FMlajyZf8IWJIy0buoMlztTPPRP0Z6/azAsrk9kvYZKX7TX1zXMUND34P7KNWhQl/Pt/CUbho3J7/ZO+LvAXSBtA==

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Set the probability of heads (between 0 and 1.0) and the number of tosses, then click “Toss”. The outcomes of each toss will be reflected on the graph. Check the box to show a line with the true probability on the graph. Click “Reset” at any time to reset the graph.

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https://berkeleyeverett.com/images/number-talk-images/

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A Song About Pi

In 1999, at the age of 82, Irving Kaplansky (1917-2006) remained actively engaged in mathematical research.

Then Director Emeritus of the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., Kaplansky spent much of his time in the library, poking into various nooks and crannies of mathematical history. Tidying up loose ends and filling in unaccountable gaps in the mathematical literature, he patiently worked through mathematical arguments, proved theorems, and prepared papers for publication. His remarkably wide-ranging efforts belied the oft-repeated notion that mathematicians are most productive when they are young.

A distinguished mathematician who made major contributions to algebra and other fields, Kaplansky was born in Toronto, Ontario, several years after his parents had emigrated from Poland. In the beginning, his parents thought he was going to become a concert pianist. By the time he was five years old, he was taking piano lessons. That lasted for about 11 years, until he finally realized that he was never going to be a pianist of distinction.

Nonetheless, Kaplansky loved playing the piano, and music remained one of his passions. "I sometimes say that God intended me to be the perfect accompanist-the perfect rehearsal pianist might be a better way of saying it," he said. "I play loud, I play in time, but I don't play very well."

While in high school (Harbord Collegiate in Toronto), Kaplansky started to play in dance bands. During his graduate studies at Harvard University, he was a member of a small combo that performed in local night clubs. For a while, he hosted a regular radio program, where he played imitations of popular artists of the day and commented on their music. A little later, when Kaplansky became a math instructor at Harvard, one of his students was Tom Lehrer, later to become famous for his witty ditties about science and math.

In 1945, Kaplansky moved to the University of Chicago, where he remained until 1984, when he retired, then became MSRI director.

Songs had always interested him, particularly those of the period from 1920 to 1950. These songs tended to have a particular structure: the form AABA, where the A theme is repeated, followed by a contrasting B theme, then a return to the original A theme.

Early on, Kaplansky noticed that certain songs have a more subtle, complex structure. This alternative form can be described as AA'BAA'(B/2)A", where A is a four-bar phrase, A' and A" are variants, and B is a contrasting eight-bar phrase. "I don't think anyone had noticed that before," he remarked. Kaplansky's discovery is noted in a book about the American musical by University of Chicago film scholar Gerald Mast (1940-1988).

Kaplansky argues that the second structure is really a superior form for songs. To demonstrate his point, he once used it to turn an unpromising source of thematic material-the first 14 decimal digits of pi-into a passable tune. In essence, each note of the song's chorus corresponds to a particular decimal digit. When Chicago colleague Enid Rieser heard the melody at Kaplansky's debut lecture on the subject in 1971, she was inspired to write lyrics for the chorus.

A SONG ABOUT PI

Through all the bygone ages,
Philosophers and sages
Have meditated on the circle's mysteries.
From Euclid to Pythagoras,
From Gauss to Anaxag'ras,
Their thoughts have filled the libr'ies bulging histories.
And yet there was elation
Throughout the whole Greek nation
When Archimedes did his mighty computation!
He said:

Chorus

3 1 4 1 Oh (5) my (9), here's (2) a (6) song (5) to (3) sing (5) about (8,9) pi (7).
Not a sigma or mu but a well-known Greek letter too.
You can have your alphas and your great phi-bates, and omegas for a friend,
But that's just what a circle doesn't have-a beginning or an end.
3 1 4 1 5 9 is a ratio we don't define;
Two pi times radii gives circumf'rence you can rely;
If you square the radius times the pi, you will get the circle's space.
Here's a song about pi, fit for a mathematician's embrace.

The chorus is in the key of C major, and the musical note C corresponds to 1, D to 2, and so on, in the decimal digits of pi.

The music and lyrics are unpublished. However, singer-songwriter Lucy Kaplansky (Irving Kaplansky's daughter) sometimes includes a rendition of "A Song About Pi" in her programs. Although she has her own distinctive style, she doesn't mind occasionally showcasing her father's old-fashioned tunesmanship. Video.

In 1993, Irving Kaplansky wrote new lyrics for the venerable song " That's Entertainment" ( video) to celebrate his enthusiasm for mathematics. He dedicated the verses to Tom Lehrer.

THAT'S MATHEMATICS

The fun when two parallels meet
Or a group with an action discrete
Or the thrill when some decimals repeat,
That's mathematics.
A nova, incredibly bright,
Or the speed of a photon of light,
Andrew Wiles, proving Fermat was right,
That's mathematics.
The odds of a bet when you're rolling two dice,
The marvelous fact that four colors suffice,
Slick software setting a price,
And the square on the hypotenuse
Will bring us a lot o' news.
In genes a double helix we see
And we cheer when an algebra's free
And in fact life's a big PDE.
We'll be on the go
When we learn to grow with mathematics.
With Lagrange everyone of us swears
That all things are the sums of four squares,
Like as not, three will do but who cares.
That's mathematics.
Sporadic groups are the ultimate bricks,
Finding them took some devilish tricks.
Now we know-there are just 26.
That's mathematics.
The function of Riemann is looking just fine,
It may have its zeros on one special line.
This thought is yours and it's mine.
We may soon learn about it
But somehow I doubt it.
Don't waste time asking whether or why
A good theorem is worth a real try,
Go ahead-prove transcendence of pi;
Of science the queen
We're all of us keen on mathematics.

Originally posted July 12, 1999.

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A Song About Pi In 1999, at the age of 82, Irving Kaplansky (1917-2006) remained actively engaged in mathematical research. Then Director Emeritus of the Mathematical Sciences Research Institute (MSRI) in Berkeley, Calif., Kaplansky spent much of his time in the library, poking into various nooks and crannies of mathematical history.

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Graphing Utility - How to use the SOLVER function to solve an equation

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Graphing Utility - How to use the SOLVER function to solve an equation

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0:00 / 4:47•Watch full video

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How to use the SOLVER function on a TI-84

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To limit domain put  /(x>0) immediately after function.  The logic conditions are found in test (i.e. 2nd math).

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Graphing Utility - How to Restrict Domain and Range on a Function

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Graphing Utility - How to Restrict Domain and Range on a Function

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https://www.youtube.com/watch?v=4axxjSm05ic&t=180s

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Learning Prime Numbers By Decomposing Space Rocks

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Learning Prime Numbers By Decomposing Space Rocks

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My Prime Rocks game is an initial try at developing models for how we can think about number theory. Gamers shoot down rocks that are composite, breaking them apart until they are prime numbers. They beam up the prime rocks to their space ship, winning when all primes up to 100 are collected.

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Back in 1987, NCTM published a paper, "The Calculator is a Problem-Solving Concept Developer", (Comstock, The Arithmetic Teacher, 34(6), 48–51. ). The authors could never imagined the power that Desmos would bring to helping learners develop an understand functions. Learners can visually learn how a function and myriads of transformations can work through real-time manipulation, while learning the rules of the equation.

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Playing with Big Chalk Ladders (i.e. Number Lines) to Introduce GCF and LCM

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Playing with Big Chalk Ladders (i.e. Number Lines) to Introduce GCF and LCM

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Once a year, the Mathematics Club goes outside with a set of problems in hand and big chalk rakes to draw number lines. A number Llne is a great way to solve problems that involve counting and factors. Words like “up” or “down”, “forward” or “back” and “more” or “gives away” provide a clue that a Number Line might be a good way to solve the problem.  Often the starting point is not given in the problem, but we learn that we can climb up and down it like a ladder to figure the answer out from the clues provided.  The learners love to act out animals skip-hopping across the number line at various speeds -- to see when they will meet up again.

No monkey math, subsequent division, factorization or knowledge of GCF and LCM is required. This is the physical stage of learning about the factors and multiples to come.

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Understanding Parabolas - Will it Hit the Hoop?

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Understanding Parabolas - Will it Hit the Hoop?

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https://teacher.desmos.com/activitybuilder/custom/56e0b6af0133822106a0bed1?collections=featured-collections%2C5e73b204d560367270838c4b

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Learning How We Solve Problems - The Big No Left Turn Maze

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Learning How We Solve Problems - The Big No Left Turn Maze

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We build out a life-size No Left Turn Maze, using chalk, paint or border paper.  A walk through of the No Left Turn Maze provides for the opportunity for learners to think about "How Do You Solve a Problem?"  We learn about:

1.  Problem solving approaches - Just Do Something, Eliminate the Impossibilities and Working Backwards.  Learners always start with Just Do Something, racing through the maze looking for an exit, until they realize this puzzle is going to require a recipe-based approach.

2.  The Problem Solving Process (Understand, Plan, Try It, Look Back)  - We employ deliberate practice and think about what worked and what didn't work. We consider what we understood at the start and end, what plans we made to conquer the puzzle, how each try worked or didn't , and we look back on what led us to our success. Learners construct for themselves how they will solve problems moving forward.

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Annenburg - Against All Odds - Lesson 9 - Checking Assumptions of Normality

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Real World Statistics - How Pete & Gerry's Eggs Uses Statistics to Size their Eggs

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Real World Statistics - How Pete & Gerry's Eggs Uses Statistics to Size their Eggs

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Concrete representations can be difficult to achieve in the classroom without building up a complicated field trip.  Annenburg brings the physical manipulation aspect of mathematics to the classroom through their real-world problems with video and data sets into the classroom and engaging manipulatives and activities.

I have had the opportunity to work with Annenburg in the classroom in Probability and Statistics and in Online Personal Finance.

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Checking Assumption of Normality - Annenberg Learner

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Classroom Resources > Against All Odds: Inside Statistics > 9. Checking Assumption of Normality

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My Desmos Activity - What Transformation is That? for Probability & Statistics

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Probability & Statistics - What Transformation is That?

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Compare two distributions to uncover what, if any, transformation occurred.

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My Desmos Activity Created for Probability and Statistics

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Correlating Manatee Deaths to the Number of Powerboats Registered in Florida

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Correlating Manatee Deaths to the Number of Powerboats Registered in Florida

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Understanding R (The Correlation Coefficient)

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Understanding R (The Correlation Coefficient)

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"Welcome to a journey. It is a mathematical journey based on a story from me, James, that isn’t true. When I was a child I invented a machine – not true - and this machine is nothing more than a row of boxes that extends as far to the left as I could ever desire. I gave this machine of mine a name. I called it a “two-one machine” both written and read in a funny backward way. (I knew no different as a child.) ..." -- James Tanton.

In Mathematics Club, we developed and employed James Binary Machine to translate the entire floor into binary numbers - including lockers, number two pencils, clocks, phones, room numbers and more.  The younger students converted their classroom into Roman Numerals and were stumped on what to put on the telephone for the number 0!

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Understanding Pi by Calculating it with Pies

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Understanding Pi by Calculating it with Pies

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This is an activity that I have longed to run in elementary Mathematics Club. I fully believe that we need to spend more time, at an earlier age, deeply understanding the meaning and value of pi. From how we write its symbol, to how its estimated value has evolved over the millennium, to its various representations (e.g. fractional, decimal), and all of the wonderful things we can do with this important ratio.

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Calculating Pi with Real Pies - Numberphile

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Reinforcing Lessons with Movement - Vertical Angles

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Reinforcing Lessons with Movement - Vertical Angles

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All sessions from the workshop can be found on this YouTube Playlist!!

They can also be found at

https://mathcommunities.org/teachers-online-workshop/

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Stallings-Roberts, V. (1991). An absolute-ly value-able manipulative. The Mathematics Teacher, 84(4), 303–307. doi.org/10.5951/mt.84.4.0303 

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"Learners can make the AVS from a sheet of ruled notebook paper. The paper should be creased lengthwise so that students can tear two one-inch-by-eleven-inch strips. One strip will represent the number line. The other strip will represent a distance scale that corresponds to the number line.  Using the ruled portion of one strip to maintain equal distances between numbers, students can write the integers lengthwise along the edge of the strip with 0 at the middle. The negative integers should be clearly written with a negative sign or be written in red. The label "Number line"
should be written on the lower portion of the manipulative.

The second strip should have the natural numbers written lengthwise along the bottom. Zero is placed in the middle with the natural numbers increasing to the right and left of 0." The distance scale is similar to that of the number line except that natural numbers should replace the negative integers (to indicate a positive distance from zero in both directions). This strip, which is the upper portion of the manipulative, should be labeled "Distance from." When the base of the upper strip is placed a long the top of the lower strip, the scales should match. The center marker on the distance scale will be moved horizontally along the number line.

The AVS can be used initially to illustrate Ia - bl = lb - a!. The result of l-4 - 31 and 13 - ( - 4)1. To demonstrate lx - 21 = 3 as "all values of x whose distance from 2 is equal to 3
units," the "Distance from" strip is placed above the "Number line" strip and moved horizontally until the 0 on the upper strip lines up with the positive 2 on the lower strip."

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Understanding Roman Numeral in Tees with Math Friends

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Understanding Roman Numeral in Tees with Math Friends

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We learn so much more about our decimal number system when we compare and contrast it with other number systems, such as Roman Numerals.

Easy to make - fun to learn.  Make a bunch of t-shirts with painted or ironed-on Roman Letters.  Hand them out to the learners in the class and create groups.  Call out a number (e.g. 25) and have the teams compete to create the number by arranging themselves in the correct ordering (e.g. the two learners with an X tee stand next to a learner with a V tee).

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Using marshmallows as vertices and toothpicks as edges, learners construct polyhedrons from a polygon base into a pyramid or prism. Recipe cards can help students get started. A shape building contest is a wonderful exit-ticket for this learning event.

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Wonderful resource to explore the conservation of matter (e.g. area). From this square (a^2) of tangrams, we can see that a^2=(a-b)^2+b^2 by building two smaller squares (i.e. (a-b) and (a)). Perfect as students enter Piaget'sconcrete operational stage (7-11) from the preoperational stage.

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Learn to count in the number system of Minnie Mouse, Base 4. A wonderful tool for learners to acclimatize themselves to a new number system and expound upon their understanding of the hindu-arabic decimal number system.

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These plastic blocks mirror the base ten set with cubic centimeter units and rods, flats and cubes, except only four units form a rod. The flat, therefore, is only four rods in width (16 units), and the cube is four flats tall (64 units). Of course using multi-base blocks is a terrific extension for students needing to solidify their understanding of place value. Surprisingly, this manipulative also great for younger students who are just learning to count and trade. Base ten blocks might involve too many pieces to work with so teach the trading process using smaller bases. Other base block sets available are base 6, base 5, base 3 and base 2. Be aware, the smaller the base the more frequent the trades. Extra practice, but lots more tricky too. Try base 2 yourself for a fun challenge!

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These make for possibly the best way to finally understand adding, subtracting, multiplying and dividing fractions! Each set comes with 16ths, 12ths, 8ths, 6ths, 4ths, 3rds, halfs, and a whole. Of course, it's more challenging and fun to create projects that change the value of the "whole," thus changing the value of the other pieces! Young children use these transparent tiles to begin seeing pieces that are larger and smaller than the others, greater/less than comparisons, then move to equivalency and into basic number operations. This is an outstanding manipulative to use in your classroom.

Online versions also exist (https://nrich.maths.org/4519) or can be printed out (see Unit Fractions)

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Cuisenaire rods and counting tracks can be used for an abstract level of counting (without attaching a value), addition and subtraction (number bonds), multiplication (multiples and factors), division, prime and composite relationships, area, perimeter, volume, geometry and algebra. These rods have a high level of abstraction; Unifix cubes level of abstraction for is lower, and students will be able to work more easily using them.

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Explore our Cuisenaire environment, or scroll down for some instructions.

Click to reveal a short video which explains how to use the Cuisenaire environment.

Alternatively, here are some instructions.

Click on the coloured circles to choose a Cuisenaire rod. Drag it wherever you like on the background.
You can rotate a rod by clicking to select it, then clicking on the 'Rotate' button.
You can move any rod by dragging or you can select a rod and use the arrows.

To delete a rod, drag it to the bin. You can clear all the rods by pressing the 'Clear rods' button.
You can turn the grid and labels on and off using the purple buttons.
Use the + and - buttons to zoom in and out.
You can save your table of rods and load one that you have saved using the 'Save' and 'Load' buttons.

'Prefabs' gives you a selection of ready-made arrangements.
'Activities' offers you a selection of tasks to explore.
'Collaborate' allows you to set up or join a shared table.
'Hide' moves the menu out of the way to give you more space.

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An environment which simulates working with Cuisenaire rods.

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BrainPop brings engaging real world applications of mathematics into the elementary school classroom.

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Comparing Fractions - One Unit Fraction from a Whole

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Comparing Fractions - One Unit Fraction from a Whole

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Students should have flexible strategies for comparing fractions, and the strategies should be based on reasoning and fractions sense. In the first article of this series, we looked at comparing fractions with like numerators or denominators. You'll also find the vertical alignment of fraction comparison standards from 2nd through 4th grade in that post.

The next comparison strategy involves fractions that are one unit fraction from a whole. A unit fraction is a fraction with a numerator of 1, for example 1/8 or 1/12. Examples of fractions that are one unit fraction from a whole are 7/8 or 11/12. The key to this type of comparison is to think about, and visualize, the distance each fraction is from one whole. For example, 7/8 is 1/8 from a whole and 11/12 is 1/12 from a whole. Since 1/12 is smaller than 1/8, 11/12 is closer to one whole, so it is the greater fraction.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

To really understand this strategy, students need lots of hands-on practice and they need to understand the meaning of the denominator-the larger the denominator, the smaller the piece. Take a look at this graphic:

By using the visual, you can see that 11/12 is closer to the whole, because 1/12 is smaller than 1/8.

After we had practiced comparing fractions one unit fraction from a whole using fraction tiles, we played a good, old-fashioned game of War. I made two copies of this sheet for each pair of students on cardstock, laminated them, and cut them out. Each player turned over one card, and the player with the greater fraction had to explain why theirs was greater. Students had the fraction tiles handy to refer to, if they needed them. Check out this video of my students in action!

Click here for the next strategy! 🙂

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Comparing Fractions: One Unit Fraction from a Whole - Math Coach's Corner

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Students should have flexible strategies for comparing fractions, and the strategies should be based on reasoning and fractions sense. In the first article of this series, we looked at comparing fractions with like numerators or denominators. You'll also find the vertical alignment of fraction comparison standards from 2nd through 4th grade in that post.

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"A prime number is a number measured by a unit alone." -- Euclid, third century BC.  The concept of prime and composite numbers has fascinated mathematicians since the time of the ancient Greeks.    Primes are the atoms from which all counting numbers are comprised.

There are no patterns known that can help us to determine where the next prime number will come up in a sequence of numbers.  There are, however, strategies that can help us more quickly figure out if a number is or isn’t prime.

There are only 25 numbers to memorize under 100.  And three simple rules to help if you forget.  A number is prime if it meets these 3 conditions:

● Its unit digit is a 1, 3, 7 or 9.
● It is not divisible by 3.
(i.e. The sum of its digits is not divisible by
● It is not 49, 77 or 91.

We have been playing with primes since at least 300 B.C. simply because we are curious.  An applied use for prime numbers only came up in the 1970’s (i.e. after the invention of cryptographic algorithms like RSA).

But isn’t that how Mathematics works?  Mathematics is a constant pull and tug between pure mathematics and applied mathematics; pure means that we play with math because it is intriguing and we are curious, and applied means that we use that play to solve everyday problems.

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Math Concepts That Stick