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. Here’s the statement of the second Common Core Math Practice along with some selected quotes. Using teaching quotations in the Common Core math classroom allows language to deepen, broaden, and clarify the math content.

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.

Aug 292014
 

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 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 the first of these 8 Practices. I hope you find some that “speak to you,” and more importantly, to your students. Here’s the statement of the first Common Core Practice.

MP1: Make sense of problems and persevere in solving them. The following quotes are about perseverance.

Perseverance is not a long race; it is many short races one after the other.
- Walter Elliot

Patience and perseverance have a magical effect before which difficulties disappear and obstacles vanish.
- John Quincy Adams

No one succeeds without effort… Those who succeed owe their success to perseverance.
- Ramana Maharshi

Great works are performed not by strength but by perseverance.
- Samuel Johnson

Just remember, you can do anything you set your mind to, but it takes action, perseverance, and facing your fears.
- Gillian Anderson

Perseverance is failing 19 times and succeeding the 20th.”
- Julie Andrews

You may encounter many defeats, but you must not be defeated. In fact, it may be necessary to encounter the defeats, so you can know who you are, what you can rise from, how you can still come out of it.
- Maya Angelou

Champions keep playing until they get it right.
- Billie Jean King

When you have a great and difficult task, something perhaps almost impossible, if you only work a little at a time, every day a little, suddenly the work will finish itself.
- Isak Dinesen

It does not matter how slowly you go as long as you do not stop.
- Confucius

Aug 292014
 

Here’s a piece of math art based on Jerzy Mil’s work. I’m still deciphering his code that makes this art. One of the nice features of geogebratube is that you can download GeoGebra apps and look at code that others have written to gain new knowledge (math tricks) to create designs and math curricula.

For more of Jerzy’s artistic mathematics work, here’s his geogebratube site

Here’s the link for the app below made from Jerzy’s fine work.

Aug 282014
 

kenken logo and puzzles 300x185 KenKen   Common Core Number Sense Activity

Jackie Murawska, Ed.D., of North Central College - Naperville, Illinois gave a presentation at the Illinois Council of Teachers of Mathematics Annual Meeting October 18, 2013 and I’ve shared some of her slide content below. The link to her entire presentation is below; take a look – she has some nice stuff!

Why Is KenKen Good?
Fun! I mean – engaging; Builds confidence; Encourages positive attitudes; Aligns with Common Core Standards for Mathematical Practice K-12 Content

Here are the Standards for Mathematical Practice for reference:

Mathematically proficient students:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make sure of structure
8. Look for express regularity in repeated reasoning

KenKen Promotes…

Problem solving and perseverance
MP. 1 Make sense of problems and persevere in solving them

Reasoning
MP. 2 Reason abstractly and quantitatively

Communication and teamwork
MP. 3 Construct viable arguments and critique the reasoning of others

Argumentation and justification
MP. 3 Construct viable arguments and critique the reasoning of others

KenKen Helps Build Skills Required for Common Core Mathematics K-12…

Basic math facts
2.OA.2 Fluently add and subtract within 20 using mental strategies.
3.0A.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division

Number theory topics: divisibility, factors, primes
4.OA.4 Find all factor pairs for a whole number in the range 1–100…Determine whether a given whole number in the range 1–100 is prime or composite.

Combinatorics
K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way
HS.F-IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
HS.S-CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems

When and How Could I Use KenKen? 

Warm-up
End of class activity for skills practice
Focus for team problem solving
Just-in-case box of problems
Math club
Competition
Extra credit

Teacher education
Content course: factors and divisibility activity
Methods course: games to facilitate engagement and understanding

Here is the link to her downloadable file.

Jackie’s email address: jmurawska@noctrl.edu

Aug 272014
 

kenken logo 2386ad2c11ade12e9a394e2e12570d72 Free Teaching Tool   KenKen Classroom

One of the best things I ever did for my students was to teach them how to do KenKen puzzles. They learned problem solving, focus, persistence, number sense, and achieved that “feel good” satisfaction when completing a puzzle. I subscribed to a weekly email from the makers of KenKen that gave our class a set of puzzles to work on; my students always looked forward to the challenge! Here’s the info for this year’s puzzles:

On Friday, August 29, a whole new year of the KenKen Classroom Program will begin, bringing you exciting new KenKen puzzles and brainteasers you can use with your students every week. Over 30,000 teachers like you have signed up for our FREE weekly program and use KenKen in their classrooms.

KenKen was born in a math classroom—and we want it to stay there! That’s why we introduced our KenKen Classroom Program for teachers. Every week, we’ll email you a set of free KenKen puzzles that is perfect for student use! As the challenge level subtly shifts week to week, students develop their math and logical thinking skills without even knowing it. And it’s all absolutely FREE!

Students love KenKen for the same reasons you do: It’s fun, it’s challenging, no two puzzles are alike, and you get to use different parts of your brain. As a teacher, you can be confident that your students are mastering basic computation skills while employing advanced problem-solving strategies.

Click here to sign up for a real treat for you and your students.

 

 

Aug 252014
 

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping.
The inverted cycloid (a cycloid rotated through 180°) is the curve of fastest descent under gravity.

One arch of a cycloid generated by a circle of radius r has area = 3πr²
The arc length S of one arch of the cycloid = 8r.

- credit to Guillermo Bautista and his website mathandmultimedia.com
The downloadable GeoGebra file can be found here.  
My other GeoGebraTube apps can be found here.

Aug 242014
 

Archimedes determined that the true value of pi lies between 3.1408 and 3.1428, a remarkable achievement for a man living over 2,200 years ago.
His method used polygons inscribed in a circle. Here you can envision some of what he did by seeing the areas of inscribed polygons approach the area of the circle as the number of sides increases.
– credit to Guillermo Bautista and his website mathandmultimedia.com
The downloadable GeoGebra file can be found here.  
My other GeoGebraTube apps can be found here.