Sep 052014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as teachers transition to Common Core standards. Here’s the direct link to quotes on Mr. L’s site. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Here are some quotes that address these 8 Practices. I hope you find some that “speak to you,” and more importantly, to your students.

MP7 Look for and make use of structure. The following quotes are about structure.

Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be “seen” with the “eyes of the mind”. It is as if we had no sense of hearing, so that only someone able to sight-read music would be able to appreciate its patterns and harmonies.
- Keith Devlin, from Mathematics: the Science of Patterns
Out of clutter, find simplicity. From discord, find harmony. In the middle of difficulty lies opportunity.
- Albert Einstein
The rules of logic are to mathematics what those of structure are to architects.
- Bertrand Russell
Mathematics as the art of creating and exploring mathematical structures.
- Michael Enciso
Studying math is like building a castle in your head. When building a castle, you first must learn to build a brick, and once that is mastered, you can use it to build a wall. Once you can build a wall, you can build a tower. Stronger bricks allow for higher walls and bigger towers. Additionally, powerful tools allow you to build faster and more efficiently.
- Michael Enciso
Some people argue that math is just a tool invented by scientists to explain the natural world. But (Max) Tegmark contends the mathematical structure found in the natural world shows that math exists in reality, not just in the human mind.
- Tanya Lewis
The higher your structure is to be, the deeper must be its foundation.
- Saint Augustine
Even the structure of the atom has been found by the mind.
- Ramana Maharshi

Sep 052014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as teachers transition to Common Core standards. Here’s the direct link to quotes on Mr. L’s site. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Here are some quotes that address these 8 Practices. I hope you find some that “speak to you,” and more importantly, to your students.

MP8 Look for and express regularity in repeated reasoning. The following quotes are about regularity and patterns.

What humans do with the language of mathematics is to describe patterns … To grow mathematically, children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.
– Lynn Arthur Steen, from On the Shoulder of Giants
One of the most amazing things about mathematics is the people who do math aren’t usually interested in application, because mathematics itself is truly a beautiful art form. It’s structures and patterns, and that’s what we love, and that’s what we get off on.
– Danica McKellar
My big thesis is that although the world looks messy and chaotic, if you translate it into the world of numbers and shapes, patterns emerge and you start to understand why things are the way they are.
– Marcus du Sautoy
Patterns repeat themselves in history
– Rick Riordan
Math is sometimes called the science of patterns.
– Ronald Graham
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
– G. H. Hardy
A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?”
– Martin Gardner
The most distinct and beautiful statements of any truth must take at last the mathematical form.
– Henry David Thoreau
The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.
– Eugene Wigner

Sep 052014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as teachers transition to Common Core standards. Here’s the direct link to quotes on Mr. L’s site. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Here are some quotes that address these 8 Practices. I hope you find some that “speak to you,” and more importantly, to your students.

MP6 Attend to precision. The following quotes are about precision.

Mathematics is a linguistic activity; its ultimate area is preciseness of communication.
– William L. Schaff
Writing is thinking. To write well is to think clearly. That’s why it’s so hard.”
– Bruce Cole
Be precise. A lack of precision is dangerous when the margin of error is small.
– Donald Rumsfeld
There’s no sense in being precise when you don’t even know what you’re talking about.
- John von Neumann
There is no science in this world like physics. Nothing comes close to the precision with which physics enables you to understand the world around you. It’s the laws of physics that allow us to say exactly what time the sun is going to rise. What time the eclipse is going to begin. What time the eclipse is going to end.
- Neil deGrasse Tyson
In football as in watchmaking, talent and elegance mean nothing without rigour and precision.
- Lionel Messi
Clarify your vision . . . plan with precision!
- www.themindstretch.com
Good writing is where precision meets passion.
- Nancy Rue
Golf is a game of precision, not strength.
- Jack Nicklaus

Sep 052014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as school resumes across the U.S. Here’s the direct link to quotes on Mr. L’s Math. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Math Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Using teaching quotations in the Common Core math classroom allows language to deepen, broaden, and clarify the math content. Here’s the statement of the 5th Common Core Math Practice along with some selected quotes.

MP5 Use appropriate tools strategically. The following quotes are about strategy.

However beautiful the strategy, you should occasionally look at the results.
– Winston Churchill
Sound strategy starts with having the right goal.
– Michael Porter
When you’re prepared, you’re more confident. When you have a strategy, you’re more comfortable.
– anonymous
There is always a better strategy than the one you have; you just haven’t thought of it yet.
– Sir Brian Pitman
What’s the use of running if you are not on the right road.
– German proverb
Strategy without process is little more than a wish list.
– Robert Filek
It’s not the plan that’s important, it’s the planning.
– Graeme Edwards
Organizing is what you do before you do something, so that when you do it, it’s not all mixed up.
– A.A. Milne
Greatness is not where we stand, but in what direction we are moving . . .
– Oliver Wendell Holmes
Hope is not a strategy.
– USAF Special Ops pilot
We have a “strategic” plan. It’s called doing things.
– Herb Kelleher

Sep 052014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as school resumes across the U.S. Here’s the direct link to quotes on Mr. L’s Math. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Math Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Using teaching quotations in the Common Core math classroom allows language to deepen, broaden, and clarify the math content. Here’s the statement of the 4th Common Core Math Practice along with some selected quotes.

MP4 Model with mathematics. The following quotes are about modeling.
What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of “model,” is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors.
– John Casti, from Reality Rules
I know this world is ruled by infinite intelligence. Everything that surrounds us- everything that exists – proves that there are infinite laws behind it. There can be no denying this fact. It is mathematical in its precision.
– Thomas A. Edison
Music rhythms are mathematical patterns. When you hear a song and your body starts moving with it, your body is doing math. The kids in their parents’ garage practicing to be a band may not realize it, but they’re also practicing math.
– Kareem Abdul-Jabbar
Mathematics is the door and key to the sciences.
– Roger Bacon
The calculus is the greatest aid we have to the application of physical truth in the broadest sense of the word.”
– W. F. Osgood
Geometry is the foundation of all painting.
– Albrecht Durer
Nature’s great book is written in mathematics.
– Galileo
A mathematical model is a simplified version of the real world that employs the tools of mathematics – algebraic equations, probability, statistics, graph theory, etc.
– Joseph Malkevitch
When a mathematical structure such as a graph is used to describe and study a real world problem we call such a structure a mathematical model for the original problem.
– Peter Tannenbaum and Robert Arnold
Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.
– Bertrand Russell

Sep 012014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as school resumes across the U.S. Here’s the direct link to quotes on Mr. L’s Math. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Math Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Using teaching quotations in the Common Core math classroom allows language to deepen, broaden, and clarify the math content. Here’s the statement of the 3rd Common Core Math Practice along with some selected quotes.

MP3 – Construct viable arguments and critique the reasoning of others.
The following quotes are about reasoning.

It is a capital mistake to theorize before you have all the evidence. It biases the judgment.
- Sherlock Holmes
There is nothing like first-hand evidence.
- Sherlock Holmes
It has long been an axiom of mine that the little things are infinitely the most important.
-Sherlock Holmes
It is of the highest importance in the art of detection to be able to recognize, out of a number of facts, which are incidental and which vital. Otherwise your energy and attention must be dissipated instead of being concentrated.
- Sherlock Holmes
How often have I said to you that when you have eliminated the impossible, whatever remains,however improbable, must be the truth?
- Sherlock Holmes
Nothing clears up a case so much as stating it to another person.
- Sherlock Holmes
I never guess. It is a shocking habit — destructive to the logical faculty.
- Sherlock Holmes
In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.
- Galileo Galilei
Children must be taught how to think, not what to think.
- Margaret Mead
Everything should be made as simple as possible, but not simpler.
- Albert Einstein
Reading doesn’t mean accepting everything you read, it means reasoning everything you read.
- Amit Kalantri
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems—general and specific statements—can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
- Sir Erik Christopher Zeeman

Aug 302014
 

A trapezoid can be transformed into a rectangle. The formula for the area of a trapezoid can look confusing, so this app shows how to simplify the formula to look like the area of a rectangle: (base)x(height). The Common Core curriculum promotes the use of transformations (translations, rotations, and reflections). A simple rotation is used here to show the area of a trapezoid.
The area formula for a trapezoid can be written as ¹/₂(base₁+base₂)*h.
The average of the bases, ¹/₂(base1+base2), is called the median.
So the area can be seen as the median*h, a much simpler rectangular formula.
Use the rotation slider to transform by rotation the trapezoid into a rectangle.
Adjust the width of the trapezoid by dragging on the corners; this app works when the base2 angles are acute.
Adjust the height of the trapezoid by using the slider.

The downloadable GeoGebra file can be found here.  
My other GeoGebraTube apps can be found here.

Aug 302014
 

Math & Teaching Quotations has continued to be one of the most popular pages on my site, and it’s appropriate to revisit this as school resumes across the U.S. Here’s the direct link to quotes on Mr. L’s Math. A good quote can go a long way towards brightening my day as well as give students some brain food as they work through their daily tasks. Since the 8 Common Core Math Practices talk about “big ideas” and overarching goals, quotes fill this niche by allowing the discussion of wider goals – kind of like looking at the forest as opposed to looking at individual trees. Using teaching quotations in the Common Core math classroom allows language to deepen, broaden, and clarify the math content. Here’s the statement of the second Common Core Math Practice along with some selected quotes.

MP2: Reason abstractly and quantitatively. The following quotes are about abstraction.
Abstraction is one of the greatest visionary tools ever invented by human beings to imagine, decipher, and depict the world.
– Jerry Saltz
It is an excellent habit to look at things as so many symbols.
― Gustave Flaubert
Remember that every science is based upon an abstraction. An abstraction is taking a point of view or looking at things under a certain aspect or from a particular angle.

― Fulton J. Sheen
Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates.
― Melvin Schwartz
Abstraction forces you to reach the highest level of the basics.
– Alan Soffer
Abstraction is a mental process we use when trying to discern what is essential or relevant to a problem; it does not require a belief in abstract entities.
– Tom G. Palmer
To abstract is to draw out the essence of a matter.
– Ben Shahn
The longer you look at an object, the more abstract it becomes, and, ironically, the more real.
– Lucian Freud
Pictures, abstract symbols, materials, and colors are among the ingredients with which a designer or engineer works. To design is to discover relationships and to make arrangements and rearrangements among these ingredients.
– Paul Rand
Abstraction is the prelude of reason.
– Vinícius De Marco Medina

Aug 302014
 

A parallelogram can be transformed into a rectangle, and the formulas for the area of both of these figures are the same: (base)x(height). The Common Core curriculum promotes the use of transformations (translations, rotations, and reflections). A simple translation is used here to show the area of a parallelogram.
Move slider n to show the translation. Move point C to create different parallelograms.

The downloadable GeoGebra file can be found here.  
My other GeoGebraTube apps can be found here.

Aug 292014
 

A triangle can be transformed easily into a rectangle, and the area of the triangle can be found to be half its (base)x(height). One of the strengths of the Common Core curriculum in the area of Geometry is in transformations (translations, rotations, and reflections). A simple rotation is used here to show the area of a triangle.

The downloadable GeoGebra file can be found here.  
My other GeoGebraTube apps can be found here.