The Flashy Maths website has a good interactive game called Fraction Matcher, where 2 teams pit their wits against each other and match equivalent fractions. You can also download this and play it offline.

The story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.

In a previous article, I recommended books that you may find at your local library or be able to order through inter-library loan. Now, let me introduce you to the wealth of math history resources on the Internet.

[These will also be added to my math resources page.]

Math history on the Internet

Quicklinks for easy browsing:

• Most valuable sites
• General resources
• Math topics & significant individuals
• Specific cultures or time periods
• Math history for elementary/middle-school students

Math history > Most valuable sites

The MacTutor History of Mathematics Archive
My favorite place to begin any foray into math history. Highlights include:

• An Overview of the History of Mathematics
• Biographies Index
• History Topics Index
• Famous curves index
• Mathematicians of the day
• A Time Line of Mathematicians

La Habra High School's Math History Timeline
Math discoveries, publications, and other tidbits --- from paleolithic number bones to the present.

• Pre-historic and Ancient Times 1,000,000 B.C. - 500 A.D.
• Middle Ages 500 - 1400 A.D.
• Renaissance 1400 - 1550 A.D.
• Reformation 1517-1598 A.D.
• Baroque Era 1600-1700 A.D.
• Enlightenment 1700-1789 A.D.
• Age of Revolutions 1789-1848 A.D.
• Age of Liberalism 1848-1914 A.D.
• 20th Century ... 1914-present A.D.

[Back to Math History quicklinks.]

Math history > General resources

Biographies at Wolfram MathWorld
Long, long list, and each biography is linked to explanations of the mathematician's major discoveries.

Biographies of Women Mathematicians
Indexed alphabetically, chronologically, and by country of birth. Includes modern news tidbits, too.

A Completely Inadequate Bibliography of the History of Mathematics
"Most of the following books are aimed at the professional non-mathematician (i.e., someone to whom the land of mathematics is an interesting place to visit, but you wouldn't want to live there)."

Convergence
An online magazine from the MAA: "Where mathematics, history, and teaching interact."

Fred Rickey's History of Mathematics Page
Includes Teaching a Course in the History of Mathematics and An Annotated Bibliography.

Galileo and Einstein: Overview and Lecture Index
Lecture notes on the history of math and physics.

Mathematicians of the 17th and 18th Centuries
Adapted from A Short Account of the History of Mathematics, by W. W. Rouse Ball.

Mathematicians of the African Diaspora
Black men and women of mathematics, in history and in the present.

Mathematical Quotation Server
I love quotations! No matter what I want to say, somebody else has probably already said it better.

Math Forum History Listings
"651 items found." No, I have not checked them all. Go browse for yourself!

Math History and Mathematicians Pages
Julie Brennan at Living Math is building an index of links to biographical information, famous quotes, activities and book suggestions to accompany a homeschool math history course. [Sample lessons.] The last time I visited, she had almost finished the first year's listings.

Philosophy of Science
Many assorted links to readings for a college class, including several chapters from String, Straightedge & Shadow.

[Back to Math History quicklinks.]

Math history > By topic

Abacus: The Art of Calculating with Beads
The abacus through history, how to make and use an abacus, and classroom ideas.

Archimedes
"This site is a collection of Archimedean miscellanea under continual development." See also: Archimedes' Approximation of Pi, and The Archimedes Palimpsest.
Edited to add: Prehistoric Calculus: Discovering Pi.

Carl Friedrich Gauss
Detailed biography, quotations, and more.

Earliest Known Uses Of Common Mathematical Symbols and Words
Research is ongoing (I found a page that had been modified last week), so don't assume that a citation is the earliest use unless indicated as such.

Euclid’s Elements
David E. Joyce brings the text of Euclid's 13 Books to life with Java applets. See also: An Introduction to the Works of Euclid.

Famous Problems in the History of Mathematics
This site includes problems, paradoxes, and proofs that have inspired mathematicians through the ages, plus links for further exploration.

A Golden Sales Pitch
"There is little evidence to suggest that the golden ratio has any special aesthetic appeal... When a myth is repeated over and over, it begins to sound like truth."

The History of Measurement
"There were unbelievably many different measurement systems developed in early times, most of them only being used in a small locality."

Hypatia of Alexandria
Lots of links, including The Primary Sources for the Life and Work of Hypatia of Alexandria. See also: Hypatia, the First Known Woman Mathematician.

MacTutor Topical Indexes

• Algebra
• Analysis
• Geometry and Topology
• Numbers and Number Theory
• Mathematical Astronomy
• Mathematical Physics

Mathematical games and recreations
"The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics."

The Mathematical Problems of David Hilbert
With a link to Hilbert's 1900 address to the International Congress of Mathematicians in Paris, surely the most influential speech ever given about mathematics. Wolfram MathWorld has an annotated list of all 23 problems.

MathPages History Topics
A wide assortment of tidbits for advanced students.

Slide Rule History
"The slide rule has a long and distinguished ancestry … from William Oughtred in 1622 to the Apollo missions to the moon."

Who was Fibonacci?
"A brief biographical sketch of Fibonacci, his life, times and mathematical achievements."

[Back to Math History quicklinks.]

Math history > Cultures or time periods

Ancient Africa
Part of the Mathematicians of the African Diaspora website.

History of Egyptian and Mesopotamian Mathematics Page
An excellent resource for my Alexandria Jones stories.

History of Mathematical Education
What topics of mathematics have been taught in different cultures and time periods? Why have these changed?

MacTutor Mathematics in Various Cultures

• Ancient Babylonian mathematics
• Ancient Egyptian mathematics
• Ancient Greek mathematics
• Arabic mathematics
• Chinese mathematics
• Indian mathematics
• Mayan mathematics
• American mathematics
• Mathematics in Scotland

Mathematics in Specific Cultures, Periods or Places
A short collection of links. This site also contains: Websites relevant to the History of Mathematics.

Mesopotamian Mathematics
"From the earliest tokens, through the development of Sumerian mathematics to the grand flowering in the Old Babylonian period, and on..."

[Back to Math History quicklinks.]

Math history > For elementary/middle-school students

Adding with the Abacus
"What did people do to save time working out more difficult problems before the calculator existed?"

Ancient Greek Mathematics
Selections from String, Straightedge & Shadow:

• Chapters 8, 9: Thales
• Chapters 11, 12: Pythagoras and his Theorem
• Chapter 13: Platonic Solids
• Chapter 14: The Irrationals
• Chapter 15: The Golden Mean
• Chapter 16: Archimedes
• Chapter 17: Eratosthenes

Archimedes & Large Numbers
A brief look at Archimedes, Avogadro, and Cantor. See Approximating Pi for an interactive demonstration.

Calendars
"Calendars were one of the earliest calculating devices developed by civilizations."

Egyptian Math
Could you survive in the world of Egyptian numerals and mathematics? [Note to teachers: The Egyptian Math Worksheet Creator looks like fun!]

Eratosthenes' sieve
Click on any number, and all its multiples (except the number itself) will disappear from the chart. See also: Murderous Maths Prime Numbers Page.

Eureka! The Achievements of Archimedes
Click "next" to read the pages one by one, or browse through the Index.

Famous Problems in the History of Mathematics
This site includes problems, paradoxes, and proofs that have inspired mathematicians through the ages, plus links for further exploration.

Fibonacci Activities
For explanations and more fun, see: Fibonacci Numbers and Nature.

Solid Gold Gnarly Math: The Gnarly Gnews
Free bi-monthly newsletter of math history with a twist of humor.

History of Fractions
"Did you know that fractions as we use them today didn't exist in Europe until the 17th century?"

History of Measurement
To work effectively and share goods fairly, people had to find ways to measure their stuff. See also: Measure for Measure.

Leonardo da Vinci Activity
"Is the ratio of our arm span to our height really equal to 1?" See also: Teacher Lesson Plan and Leonardo's Vitruvian Man.

Negative Numbers
"Among the earliest people to use negative numbers in calculations were the ancient Chinese." See also: The History of Negative Numbers.

Pascal's Triangle
Lessons and links for all grade levels. See also: All You Ever Wanted to Know About Pascal's Triangle.

Pi, a Very Special Number
Over the centuries, mathematicians kept looking for better values for pi.

Platonic Solids
With printable nets, so you can make your own models. Part of the wonderful Maths is Fun site --- take some time to explore!

Pythagoras
"Pythagoras believed that everything in the world could be explained by numbers." See also: All Is Number.

ThinkQuest History of Mathematics
Brief overview of math history, with biographies of influential mathematicians and short online quizzes.

Women in Maths
"Ever wondered why stories about mathematicians always seem to be about men? ...There were a few women who dared to go against the flow."

[Back to Math History quicklinks.]

:: :: :: :: :: :: :: :: :: :: ::

• A Very Short History of Mathematics
• Hooray for (Math) History
• An Ancient Mathematical Crisis
• The Puzzling Pythagorean Pebbles
• The secret of Egyptian fractions

Using my title as an example, a rational number is basically a fraction like 1/2. And of course fractions are basically division problems. Sticking with 1/2, if you divide one pie by 2 you get half a pie. That is a rational line of thought and thus 1/2 is a rational number. On the other hand if you think about it, each half of a pie is a distinct thing unto itself, you could almost think of each piece of a pie as a pie or just as a piece rather than say 1/8 of a pie (that's a stingy sized piece for most pies I say). And if I understand correctly that's the way that other animals would look at it. They don't have the ability to rationalize that one piece of pie sitting there is the same thing as the piece of pie sitting over here, all they know is that it's food and they probably want to eat it. Actually they probably know other things about it that we don't like subtler scents that escape the human nostrils.

OK so this has turned into something that doesn't make any sense. But my point is that humans have come up with all kinds of things that aren't necessarily real and that has allowed them to come up with things that are. Other animals appear to lack this ability to imagine, to rationalize, to make expansive plans. If they had it, then they would be able to fully communicate with us, they would be able to write, they would be able to do math.

I suppose there are a lot of people who would just as soon not do math, and there is a case to be made for the way of nature being ideal compared to the perversions of humanity. You don't get necrophiliac serial-killer lions after all. But we humans are what we are and there possibly has never been anything similar that came before us, and when we die out there might not be anything else like us that will follow.

Of course in writing this I'm saying nothing of god but that's because god says nothing about it to me. Perhaps the christians and jews are right and moses was given the 10 commandments by god whenever that was and to god that was supposed to be be good enough for all of us plus or minus the gospels and so on (I don't really know how all that stuff works and couldn't care less). Same with allah and muhammad, to an omnipotent being perhaps 2-4,000 years is the blink of an eye and to one who is omnipresent all thoughts and actions are carefully timed and orchestrated. Or *gasp perhaps god fed people a bunch of bs or the people who heard god didn't understand and made things up. Not to mention other people who just lied to serve their own agendas and perversions. I am reminded of a thought I once had that perhaps humans are made in god's own image in the sense that god is an aberration and so are humans. We're god's mistake and god is the universe's or something like that. It would explain a lot.

Well whatever kids. I guess my point is to appreciate things like math and books because they're what we have that maybe nothing else does. I met someone the other day who didn't think they'd ever read an entire book. People like that are out there, it's just not important to them, they don't make anything of it, they don't have the ability to grasp it. I honestly don't know whether that's a shame or not, whether this person had missed out on anything or not. After all I've read thousands of books but perhaps I missed out on thousands of hours of other activities as a result. Certainly it's hard to imagine that I gained any unique knowledge by reading the likes of Mo Hayder's "Birdman" that either benefits or affords me a special advantage over anyone else. If anything I think it just sparked all kinds of depressive thoughts. So whatever. I guess I don't have a point. It gets hard after you reach a certain... let's call it a "state" to separate things out and be concise. Possibly everything is related after all. Or is that probably? Whichever one works for you.

INTRODUCTION

Why this proposal?

The United States is facing a critical shortage of graduates who major in mathematics, science, engineering, and technology. As children progress through the educational system, far too many shy away from science and mathematics. Other nations, particularly in Asia, have far more success in producing successful students with strong backgrounds in science and mathematics.

This proposal presents The Mathematics Room Project, a three-week enrichment program that stresses mastering mathematical skills that underpin success in higher mathematics and ultimately, in high tech and mathematics centered jobs. But to no lesser degree, it addresses social responsibilities and the City’s rich cultural heritage.

THE PROJECT DESCRIPTION

What is The Mathematics Room Project?

The Mathematics Room Project (TMRP) is a three-week summer enrichment program that stresses

· fluency with whole numbers,

· mastery of fractions,

· relationships to others and to the XXXXXX neighborhood,

· the City’s rich musical heritage.

Why are these emphases over others?

In a recent Washington Post article, a presidential panel of mathematicians, education experts and psychologists reported that, students need a deeper understanding of fluency with whole numbers and fractions, if they are to be success in completing Algebra II. Further, the panel stressed, that many students are befuddled by fractions. A fundamental focus of TMRP is to develop fluency of whole numbers and mastery of fractions, thereby responding, to correct the problems identified by the presidential panel.

“Market emphasis” is a fact of life. We approach education, the arts and even our spiritual institutions with a business plan. This approach has its advantages but it reduces all things to a bottom line mentality. Often in human endeavors, the process leading to the bottom line is as important as the bottom line, for example, in building relationships, in working in team, in caring for others, in caring for things over which we have charge, in developing plans for the future, etc. In all a spirit is built that is necessary to relate to others as well as our community. TMRP will teach participants to solve mathematical problems within a group structure, thus highlighting the importance of relationships and highlighting our responsibilities in human affairs.

What is the teaching philosophy of TMRP?

The teaching philosophy of TMRP is rooted in the famed Singapore Method of teaching mathematics. Like the Singapore approach, TMRP focuses on teaching mathematical thinking to produce mathematical thinkers. It does this, by walking the learner through all the components of a problem and then presenting her with the whole problem to solve; all done, with the learner’s preparation in mind while poised to elevate her to the next level.

Is this method better than the traditional?

The Mathematics Room Project believes and data shows this method is optimal for learning “how to” frame a problem in a concrete stage, and then, “ how to” usher the learner into the abstract stage, thereby enabling him to learn mathematics in a meaningful way. It encourages an active thinking process, effective problem solving skills and clear communication of mathematical ideas. All of the attributes are necessary to develop a foundation that students need to successfully study more advanced mathematics.

TMRP supports this method with outstanding, experienced, caring teachers and with excellent textbooks and workbooks that stresses the

Concrete-> Pictorial-> Abstract approach.

Is the Singapore Method proven?

Since 1995, Trends in International Mathematics and Science Study tests (TIMSS) have been administered to students of thirty-six developed nations. These nations used the TIMSS test to measure the success of its student preparations in mathematics and the sciences. Singapore, a small nation of 45 million, has never scored less than first position in the TIMSS test, while the United States has never placed higher than 16th position.

Follow-up results of the TIMSS attest that students taught by the Singapore method have a higher success rate in completing algebra and higher mathematics.

Who will lead the Mathematics Room Project?

The Director and Coordinator of The Mathematics Room Project, XXXXX, have acclaimed success in directing mathematics enrichment programs.

The support staff includes XXXXXX - a retired grade teacher of 33 years experience, having taught 25 years at the XXXXXX, one of California’s Distinguished Schools.

Where will the program be held?

The XXXXXX Church has committed to host The Mathematics Room Project. This Christian institution has ministered to the XXXXXXXX Community for XXX years.

How will the participants be recruited?

Thirty students will be recruited by invitations sent to:

· Area churches

· Area recreations facilities

· Advertisements in a weekly newspaper

· The CITY NEWSPAPER

PROJECT NEEDS

The Mathematics Room Project needs:

• meeting space to accommodate 30 children
• eight tables: seating four per table
• Singapore Method mathematics workbooks: grades 4 through 6
• Singapore Method teacher manuals: grades 4 through 6
• cost of admittance for 30 participants to:
• one of the City's Philharmonic performance and
  
• one of the City's Choral performance

BENEFICARIES OF THE PROJECT

The Students: The immediate beneficiaries of the project are its participants. Mastery of problem solving skills enhances student self-esteem and cementing student–school bonds. Studies have demonstrated that girls having strong mathematics skills tend to resist predatory male advances and amoral peer pressure. In like manner, boys having strong mathematics skills and strong parental guidance tend to successfully shun gang advances. Thus, TMRP will be another effort to engage students in academic life and to build a student more resistant to social distractions.

Over the years, students trained under the Singapore Method boast high rates of success in higher mathematics. Algebra II, the gateway to higher mathematics, is a proven predictor of success in college. The recent Washington Post article reported that The National Mathematics Advisory Panel - the Presidential Panel composed of mathematicians, educators and psychologists - declared that success in Algebra II is predicated upon mastering fractions and fluency in whole number operations; primary focus topics of The Mathematics Room Project. Thus, TMRP is another local effort to increase the number of students who earn successful completion in Algebra II, and thus earn success in college.

The Society is a long-range beneficiary of TMRP. As problem solving is a necessary prelude to develop a thoughtful citizenry, and as thoughtful citizens build strong and safe communities, The Mathematics Room Project will take part in providing the citizens for better Sacramento communities.

The Country: The science and medical communities of the United States have experienced a severe shortage of prepared graduates in science and mathematics, due in large part to the poor performances in the sciences and mathematics. This is visibly reflected in the medical, computer science and engineering communities. The Mathematics Room Project is a Sacramento effort to increase the number of U.S. students who are successful in Algebra II, and hence increasing the number of candidates in the medical, mathematics and science pipeline.

FITS TO FUNDING AGENT MISSION.

As The Mathematics Room Project is a program with an academic,

cultural and social impact that targets bettering students in a particular Sacramento neighborhood, it is consonant with the vision of FUNDING AGENT. to develop awareness and care for neighborhoods.

It does this:

· By teaching its participants to master mathematics identified as a building block to success in college.

· Through its planned ice cream social in the local park.

· With scheduled attendances to a performance of the Sacramento Philharmonic and the Sacramento Chorale Society.

Familiarity breeds acceptance. So during its instruction time, TMRP will play as background music those selections designated by the City Philharmonic and City Chorale Society to be performed during the scheduled attendance of TMRP. This predisposition to the selected music provides for greater student enjoyment as they witness familiar music dramatized in a cultural setting.

Our activities will be held in the XXXXXXXX Church facilities. This structure is a recognized historic landmark to the neighborhood. Gaining life changing academic experiences, while experiencing positive events and joyful times in a place of historic value in one’s neighborhood promotes positive and manifold connections to the neighborhood, thus this program fits the vision of FUNDING AGENT.

AFTER TMRP, WHAT THEN?

After the program, our participants will have:

· learned “how to” learn mathematics.

· mastered mathematics crucial to learn higher mathematics.

· a better appreciation of mathematics, and stronger self-esteem.

· experienced great music performed by local musicians.

· recognized the historic value and importance of the building that housed project.

In summary, our participants will have exposures that communities desire of its youth.

REQUESTED FUNDS

The Mathematics Room Project requests $ xxxxxx to cover operational expenses, student textbooks, student workbooks, teacher instruction manuals, 12 nutritious snack for 30 students, transportation cost to the music events, t-shirts that stress TMRP common focus and fosters a sense of belonging; and certificates of completion.

FIELD TRIPS

The Project Director will coordinate all field trips during the project year.

PROPERTIES OF THE MATHEMATICS ROOM PROJECT

All properties of The Mathematics Room Project, except student workbooks, revert to FUNDING AGENT. upon cessation of the project.

TOTAL PROJECT BUDGET $XXXXXX

PROJECT START/COMPLETION DATE

ORIENTATION: June 7, 2008

Accept applications, distribute rules of operation, administer pretests to measure student preparedness and measure for t-shirts.

PROJECT TIME FRAME

MESUREMENTS AND EVALUATIONS

Upon entering the program, all students will be administered a pretest to determine whole number fluency, mastery of fractions, and problem solving skills.

TMRP will permit its learners to advance to a new topic only upon 85% or better mastery of his latest workbook assignment.

During the last three days of the enrichment program, the Director will assign to each student a problem he will present at a closing ceremony. Parents and the public will be encouraged to attend the closing ceremony. The presentations will be graded on clarity, correctness and problem insights.

On the final day of TMRP, student will take a post-test. Its results will be compared with the pretest results to measure the impact of the enrichment intervention.

At the completion of the first year of operation, the Director will submit an annual report to the Board of FUNDING AGENT. The report will summarize student performances and the project’s impact on student educational and cultural growth.

CONTACT INFORMATION

• Shorter is Larger: with this students often treat the decimal as a denominator. So 0.64 is larger than 0.798 because when you divide a pie 64 times the pieces are larger.
• Longer is Larger: Students are basing this on whole numbers. So 0.698 is larger than 0.79 because 698 is larger than 79.

Why?

It comes down to the way students are commonly taught decimals. In addition and subtraction they are first taught with using the same amount of decimal places. Usually this starts with tenths, then hundredths and so on.

What this means for me is similar to what I have commented on in a journal article from Maths 1. Mix it up a little. Why do students need to be introduced to decimals in incrementing lengths? Also give students activities such as

• patterns
  • Continue this sequence 0.07, 0.08, 0.09……………………..
• Sort
  • 1.300, 00.13, 100.3, 1300, 3.001, 3100
• Predictions with Calculators
  • List a series of decimals and have students predict what will happen when you multiply or divide that number by 10, 100 etc. and then check those answers with a calculator.

There are many other activities that can be done but the major implication for me in my classroom is to never rest on the traditional steps and procedures when teaching mathematics. Find new and interesting methods that address any problems students may be having when learning basic math concepts.

Condon, C. & Hilton, S. (1999). Decimal dilemmas. Australian Primary Mathematics Classroom, 4(3), 26-31

We have went through a lot of different curriculums whilst homeschooling, the first being Abeka and the last used being Christian Light Education with a mixture of Rod and Staff. We've also done Paces and Switched-On-Schoolhouse (SOS). SOS was a computer based one.
According to grade level, I am in the 9th grade. I know the basics of math (subtraction, division, addition, multiplication, fractions, etc) English, Reading, Writing, and a bit of Science. I also have computer knowledge, and I know how to operate 'The GIMP' which is a image program.

In the schoolbooks, you basically kept repeating and doing over most of the same things you learned last year. Every morning we'd be hoping.. "Maybe Mom will forget about school for today!" As you can well imagine, that didn't happen. :P The books called it "Review" I believe, but you did the review year after year of some of the stuff you learned years ago. How often do you diagram a sentence a day? Other than in school, how often do you use Algebra?

But now, our 'school' will be different. Mom had finally found something that will work for us, and it's called Unschooling (which you may have heard me reference to in my posts). Which is basically unstructured, freeform, daily experience, interest based learning. There will be no more waking up in the morning, dreading the upcoming 'school hours', or begging Mom to let us have the day 'off'. We will learn from our daily life experiences. Math, from playing Monopoly to counting money for allowance (or babysitting jobs!), and, well, just about everything can be put into a category.. LOL.

Or if there is some specific subject we're curious about, we'll look it up, and actually learn from it because we wanted to learn it! Somehow Levi and I have just never really learned anything from school books. If we need to learn something to complete a project or just to do something in life, we will learn it in our own way, and probably remember it a lot better than learning it from books.

So, I guess my highest 'grade' education will stop at 9th grade!!! :) That doesn't bother me though, as I never could figure out that when you told someone you were in 9th grade or so, they automatically figured you had learned this, this, and this. Or, it was haven't you ever done this yet? No? And you're in 9th grade!? Or, "You're in 9th grade? Neat, you only have a few more years of school left!"

Now people will ask what we do, since we don't use textbooks or workbooks anymore. What do I do? Every day is full of different things and surprises. Mainly I am being trained by my mother on how to take care of the house, being a keeper at home, obedient, sewing, cooking, cleaning. I can learn these by watching her, helping her, and asking her. When I'm not inside helping, I can be outside managing the barnyard, making sure all the animals have food and water, letting some animals out, playing with my new roosters, taking the dog out for a walk. And if I'm not doing that, I can be riding my bike with my brother, reading books, or practicing my psaltery! The days are too different to be able to say what we do every day!

Shouldn't I be more worried about studying the Word of God and hiding it in my heart, then trying to remember the formulas of algebra, nouns, adverbs, verbs, diagrams, and history dates?

For the wisdom of this world is foolishness with God. For it is written, He taketh the wise in their own craftiness.
1 Corinthians 3:19

In all thy ways acknowledge him, and he shall direct thy paths.
Proverbs 3:6

Be not wise in thine own eyes: fear the LORD, and depart from evil.
Proverbs 3:7

decimals, especially when related to things like litres and millimetres,

fractions and percentages

mean, median, mode and range

angles and measuring them

perimeter and area

I asked my classroom assistant (the amazing one I mentioned last week) to supervise them in the open area and I pretty much left them to it apart from giving them a couple of different levels of textbooks and the internet so they could research help tips.  We had spent a lot of time discussing how they would do this before so I was not just leaving them with no background help whatsover and Alyson was great at helping them.

When I went out to the open area after about 30 mins I was absolutely amazed at how they had sorted themselves, how well they were working together and how much they had achieved.  Between all of the groups they had made up sheets, lesson plan notes, explanations of how to do things, powerpoint presentations and had saved websites of different games etc to be used on the ebeam to help with learning.

I am thrilled at how  well this went on the first day and how intersested the children were.  I think one of the tricks of making this work will be not to let the planning stage for them go on too long as that is when they could start getting bored with it.  It has occurred to me while writing this that maybe this will help the children realise how important it is to listen as they will be experiencing the amount of prep that goes into a lesson.  That was by no means the reason for doing this though and I felt really excited today to be facilitating the children in leading the learning.  I really hope it goes well in the end as I will be really disappointed if it doesn't, and I think the children will be too!

My middle group then also worked much better today as well although it was hard going as it was equal fractions and we all made rather heavy weather of it!  I think a lot of this might be to do with parents nights being last night and I could not over-emphasise enough there how much work that group had to do.

We had a parent council meeting tonight and it was really interesting to see how the new headteacher worked with the parents; it will be a great learning experience for me to see that in action.

I loved my job today as I felt I was doing something new and interesting for me and the children; we are all on a learning curve with this!  It is something I really hope will make a difference to their learning and will probably (hopefully) give me some new ideas on how to explain concepts in maths.  A teacher learning from the children!

I have one group of great gals that all have kids the same age as the Big One. The kiddos will all be turning three in the next month or so. Our situations are all different. Some of us are experiencing the “terrible twos” for the first time, while others are celebrating it being the last time they have to deal with them.

One of the girls sent an e-mail the other day lamenting how wonderful this age is. She wrote: {please don’t get offended Renee that I am sharing this}

Are all of you just loving the stage that our May babies are at? I am sooooo enjoying Kaden so much. I don’t know if it is because he is my last baby or what but I just “inhale” all of his cuteness everyday. I just can’t get enough sometimes. I am sooo thankful for him in our life. I just could not imagine my life without him.

Based on the last few days in my house, I have to say the she is on crack! Okay, just kidding. But I do have to say I think it is Renee’s perspective that makes her gush with joy over toddler-ville. Since he is her last baby, this wonderful time is a joy to behold.

Now for me, since the Big One is my first I have to say I could really live without this stage. At least once a day, she frustrates me almost to tears. She is sassy, whiny, demanding and generally rotten to be around. Yes, I love her to pieces, I just wish I could eliminate certain pieces of her– especially her mouth!

I do love that she is learning to be independent, voice her opinions and develop her little personality. However I just wish she could do it without yelling, doing the whiney dance, and crying.

I know she is only two and that while she is a great talker, she doesn’t know how to express her frustrations and feelings appropriately. I also know that it is my job to help her learn how to do this effectively (note: the screaming and whiney dance are not effective). But how many times a day to have to say, “STOP! Don’t whine, talk to me in a big girl voice and tell me what you want.”

Based on the number of times a day I have to say it, I am beginning to fear that my child is a slow learner. Thank God the Husband is a math geek. I know I don’t have the patience to teach fractions to this one!

Another Interesting Fact:

Or rather, dear reader, the first "Another Interesting Fact" of what may turn into an ongoing part of this Blog, dependent on two factors, one being the extent to which it is popular with you, dear reader. The second factor, however, is significantly more random, in a holistic sense at least. This factor is based on the number of "Interesting Facts" (or IFs) that I experience in any twenty-four hour period. Thus, there is an infinite potential for IFs to occur in the vicinity of this blog, yet if you wish to calculate the exact proportion or probability of IFs per calendar day, it would involve dividing the average rate of IFs by some external indicator. This, I would suggest, would best be understood as a quantitative scale measuring the proportions of interesting to uninteresting phenomenology which I encounter in any one twenty four hour period. Thus, this PI2uIP scale could objectively discern the potential for IFs (and uIFs, for that matter) in my life, for potentially any specified time period. Of course, we then come to the difficult stage of cross-referencing the real world PI2uIP scale onto some measure of other datasets which could conceivably form a part of this blog. We will call this, for ease and clarity, the EBD set (Extraenous Blog Data set). Furthermore, for our formula to produce accurate and quantifiable results it would be necessary to map on certain other real life factors. For instance, the probability of a blog actually being created. This could be a simple binary indication where 0 = no blog created and 1 = blog created (the BC indicator, for want of a better term). Thus, if

PI2uIP = 1:32.56 and IFs = 2, but BC = 0, then the number of IFs recorded in the BS (Blogspace) would = 0 since it is impossible to record an IF in the BS if BC = 0.

In contrast, if

BC = 0, but PI2uIP = 1:1, thus making the IF = 0.5, then the potential for the IF being present in the BS = 0.75, and thus it is entirely probably on a level graded scale of 0 to 1 that the IF will happen to be recorded in the BS.

Such a system, however, lacks a certain theoretical rigour and mathematical finesse. Thus, I would humbly propose the system is altered to include some means of recording the level of procrastination, distraction, cooking or other distractive phenomenon which form the causal nexus behind the BC score. For clarity, I label this theoretical construct n. Thus, a proportionally higher n value would reduce the BC score proportionally, thus meaning that if:

n = BC / IF

then IF = BC * n.

And there we have it:

BC = IF / n, where IF = 1 / P2IuIP * 100.

Byeee!

Class, we have been studying different countries. Look at the flags that we have hanging at the front of the room. Which flag is the flag of Italy? About how much of the flag of Italy is green? Which flag is Poland's? How much of the Poland's flag is red? How about the U.S. flag? About how much of our flag is blue?

In order to receive full credit for your post, you will need to provide at least one web link to an image.

View "Understanding Fractions" and "Just a Fraction" to see how British teachers are using plastic cups to model (addition and subtraction of) the benchmark fractions 1/2 and 1/4 to their students.

Deshalb - jetzt mal zur Verdeutlichung - auch Bruchteile. Denn wenn man die addiert, bekommt man ein ganzes, vollkommenes. Im Idealfall, versteht sich. Man sollte sich nicht immer nur auf eine Sache konzentrieren, sondern immer auch auf andere. Auf alles, was einem wichtig ist. Wenn man sich zur Hälfte nur auf eine Sache konzentriert und zur Hälfte auf den Rest des Lebens - das kann und ist nicht gut. Denn wenn diese Hälfte weg ist, ist eine Hälfte vollkommen durcheinander. Das ist furchtbar, wirklich.

Natürlich kann man auch mit den Differenzen der Konzentration variieren - je nach dem, wie wichtig einem verschiedene Sachen sind. Doch es sollte immer ausgeglichen bleiben.

Ich wollte an dieser Stelle mein Lieblingsbuch zitieren (bessergesagt: Meine Lieblingsbücher; die Bis(s)-Reihe), aber ich glaube, das lasse ich besser. Das ist nicht ganz so aussagekräftig wie ich es in Erinnerung hatte.

Es passiert mir öfter, dass ich denke, dass diese Philosophie ziemlich richtig ist. Ich sehe, höre, oder denke etwas und daraufhin denke ich dann ein einziges Wort - Fractions.

Eine witzige Geschichte ist übrigens, dass ich durch das Wort Fractions vor gar nicht mal so langer Zeit herausgefunden habe, warum die Fraktionen im Budestag (oder so) Fraktionen heißen. Peinlich.

Punkt ist: Die Balance halten. Oder, wie ich es formulieren würde: Fractions definieren.

Ich glaube, das hätte ich vor einen halben Jahr machen müssen. Dann wäre ich jetzt nicht so am Ende, weil etwas aus meinem Leben verschwunden ist...

Freak.

The cold came back and knocked me flat, but there are compensations. The downtime gave me a chance to browse my overflowing bookmarks folder, and I found something to add to my resource page. Princess Kitten and I enjoyed exploring these games and quizzes from Ambleweb.

Math games by elementary students

Each game was designed by (or at least with the help of) 4th-6th grade students at Ambleside CE Primary School.

Callum's Addition Pyramid
"Can you open the Mummy's Tomb?" With three difficulty levels, this will challenge any elementary student's mental addition skills --- and many adults' skills, too!

The Table Trees
Practice your times tables, with a twist --- it's not always the answer that's missing.

Button Beach Challenge
"A taster of our forthcoming Oddsocks --- The Land of the Lost interactive web based mental maths adventure." This addition puzzle is harder than it looks. I can't wait to try the rest of the adventure...

Ambleweb Function Machine
Choose the type of problem you want to guess, or go random for more challenge. My math club kids LOVE function machines.

The Fractotron
You will have to work fast with this game. The questions are not super-difficult, but it doesn't give you much time to think.

Test your problem-solving skills

Challenging mental arithmetic quizzes by the kids at Ambleside Primary.

Brainwave
For early elementary students: "What is half of 10?..."

Braintwister
For middle elementary students: "There are 35 children in a class. 16 are boys. How many are girls?..."

The Braindrainer
For upper elementary or middle school students: "Is 4 a factor of 106?..."

The Totally Mental Machine
Random numbers make a new set of problems every time: "What is 15.7 and 4.7 together?..."

Problems? No Problem!
Includes some geometry: "What is the name of an angle greater than 90 degrees but less than 180 degrees?..."

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• In Between Sneezes...
• Story Problem Challenge Revisited
• Are You Smarter than a 3rd-6th Grader?
• Pre-Algebra Picture Puzzles
• Math Quotes III: Five out of Four People

Well I might have an answer
His name:  Mr. Duey
His Occupation: Teacher
His Passion: Teaching by any means neccessary.

What has he done that is so different?  Well he decided to take his lessons and put them to hip hop music to try and reach his students. 

I had a friend email me this link www.mrduey.com to have me take a look at this new approach to reaching students.  He has recorded 17 songs, I mean raps, to help his students learn subject matter.  He also has a Fractions video on Teacher Tube,  giving you a visual of what he is doing to reach his students.  http://www.teachertube.com/view_video.php?viewkey=ebd7c1e1b7118af88edc

My hat is off to you Mr. Duey for doing what ever it takes to reach your students.

We had a math emergency at my house the other day. I teach math to fifth graders, and since my son has taken trigonometry and pre-calculus, I assumed that he and my husband could both handle everyday household math.  I was wrong.

Chaos ensued.

We have very few vices in my household, but one of them happens to be Monkey Bread. If you've never tried Monkey Bread, you've been missing out on something akin to the nectar of the gods. It's not healthy for you, which is why I call it a vice, but it's definitely worth the sin.

My son made Monkey Bread for a school potluck. The next day, we all decided we wanted some as well, but we only had one can of refrigerated biscuits. The recipe calls for three. I didn't think that was a problem, because we could just cut the recipe into thirds.

The biscuits weren't the problem. The trouble began when we got to the butter and sugar. The recipe calls for ½ cup.

"Mom, what's 1/3 of ½ cup of butter?"

"You should be able to figure that out."

Blank stare.

My husband intervened. "Here, give me the stick of butter. It has lines on it. I'll just cut it on the fourth line."

I calmly said, "But that won't be the correct amount. That's four tablespoons. Besides, what are you going to do when you have to reduce the sugar by 2/3?"

Two blank stares.

"Just put a bunch in a bowl and I'll stir in some cinnamon," my son suggested.

They turned their backs to me and proceeded to make their modified version of Monkey Bread.

It still turned out delicious, even without the recipe. But the frustration of the math part was almost more than my two men could bear.

And in case you're wondering, 1/3 of ½ cup is 1/6 cup, which works out to 2 tablespoons + 2 teaspoons.

Monkey Bread

Ingredients:

3 cans refrigerated biscuit dough

½ cup of sweet cream butter

½ cup of granulated sugar

½ cup of packed brown sugar

2 tablespoons cinnamon

2 teaspoons of nutmeg

Oven: 375˚

Tear the biscuits into fourths and spread one can in a layer on the bottom of a greased bundt pan. Mix the granulated sugar and cinnamon together and sprinkle a third of the mixture over the biscuits. Repeat two more times. On medium heat, melt the butter and brown sugar together, stirring constantly until smooth. Pour over the top of the biscuits and sugar mixture in the pan. Sprinkle the top with nutmeg. Bake for 30 minutes or until the top is golden brown. Immediately turn the pan upside down on a plate and let the Monkey Bread slip out. Cool for ten minutes before eating.

Paraphrased from a homeschooling math discussion forum:

Help me teach fractions! My son can do long subtraction problems that involve borrowing, and he can handle basic fraction math, but problems like $latex 9 - 5 \frac{2}{5} $ give him a brain freeze. To me, this is an easy problem, but he can't grasp the concept of borrowing from the whole number. It is even worse when the math book moves on to $latex 10 \frac{1}{4} - 2 \frac{3}{7} $ .

Several people replied to this question, offering advice about various fraction manipulatives that might be used to demonstrate the concept. I am not sure that manipulatives are needed or helpful in this case. The boy seems to have the basic concept of subtraction down, but he gets flustered and is unsure of what to do in the more complicated mixed-number problems.

Let's take a quick look at what is involved in a calculation like this...

7 steps to subtract mixed numbers

$latex 10 \frac{1}{4} - 2 \frac{3}{7} = ? &s=3 $

1. Ignore the whole numbers for a moment, and focus on the fraction parts.
   $latex \frac{1}{4} - \frac{3}{7} &s=3 $
2. Convert both fractions to a common denominator. That in itself involves several steps:
   • Look at the two denominators. Can you see a super-easy common denominator? For example, is one of the numbers a multiple of the other, or are they both divisors of some common number like 12?
   • If not, then realize that the Least Common Denominator is an irrelevant sidetrack and don't worry about it.
   • Use the Easiest Common Denominator: multiply the denominators of your fractions. It always works.
   • Multiply the numerators of your fractions by the same amount to create equivalent fractions.
   • Cross out the original fractions and write your new fractions next to them.

   $latex \frac{1}{4} - \frac{3}{7} = \frac{1 \times 7}{4\times 7} - \frac{3\times 4}{7\times 4} = \frac{7}{28} - \frac{12}{28} &s=3 $

3. Compare the fractions. Notice that the one being subtracted (which is called the subtrahend) is larger than the one you are taking it away from (the minuend).
4. Borrow/rename from the whole number part of the minuend to make a bigger fraction. This involves several more steps:
   • Subtract 1 from the whole number, crossing it out and writing your new number above.
   • Mentally convert the "borrowed" 1 to an improper fraction with the common denominator.
   • Mentally add that converted fraction to the fraction part of the minuend.
   • Cross out the fraction part of your minuend (yes, again!) and write your newer value beside it.

   $latex 10 \frac{7}{28} - 2 \frac{12}{28} = 9 \frac{7 + 28}{28} - 2 \frac{12}{28} = 9 \frac{35}{28} - 2 \frac{12}{28} &s=3 $

5. Subtract the fraction part of the subtrahend from the fraction part of the minuend. Write this answer down below.
   $latex \frac{35}{28} - \frac{12}{28} = \frac{35 - 12}{28} = \frac{23}{28} &s=3 $
6. Check to make sure the fraction part of your answer is in lowest terms. Whew! This time it is.
7. Now look at the whole number part of the calculation and follow the standard rules for whole number subtraction...

I can see how a student might get confused, can't you? And transferring the problem to manipulatives will only add steps, making it even more complicated!

Photo by gotplaid?

Make it simpler, step 1

I would like to try a different approach. Sometimes, taking the long way around a problem actually leads to a shorter solution. As an added bonus, the round-about method may help a student make sense out of the calculation, rather than just trying to memorize and follow a complex recipe of steps.

Here is the basic principle:

Whenever you get stuck on a problem, think about anything you can do to make it simpler.

With that in mind, let's look again at our mixed-number calculation:

$latex 10 \frac{1}{4} - 2 \frac{3}{7} = ? &s=3 $

How can we make it simpler? Ignore the fractions for now. If it was 10 - 2, that would be easy:

$latex 10 - 2 = 8 &s=3 $

Now put the fractions back where they were:

$latex 8 \frac{1}{4} - \frac{3}{7} = ? &s=3 $

Can you see that this calculation will have exactly the same answer as the problem before?

This is a key principle of working with arithmetic --- do whatever you can to make your calculation simpler, without changing the basic problem. Most of the first year of algebra is spent learning how to simplify equations in this way.

Make it simpler, step 2

Okay, now what else can we do? If we had a really short problem like $latex 1 - \frac{3}{7} $, that would be easy, right? I am sure you could do it in your head. So let's "steal" 1 from the 8 and ignore everything else:

$latex 1 - \frac{3}{7} = \frac{4}{7} &s=3 $

Now we have to put that left-over $latex \frac{4}{7} $ back with the other part that we were ignoring. This is what our calculation really looked like:

$latex 8 \frac{1}{4} - \frac{3}{7} = 7 \frac{1}{4} + [ 1 - \frac{3}{7} ] = 7 \frac{1}{4} + \frac{4}{7} = ? &s=3 $

So far, we have made the problem simpler (only one mixed number instead of two), and we have taken away everything we were supposed to subtract. Our last step is a relatively straight-forward addition problem, putting the two fraction pieces together.

At this point, we are forced to find a common denominator --- but at least we got the hard subtraction work out of the way first!

$latex 7 \frac{1}{4} + \frac{4}{7} = 7 \frac{1 \times 7}{4 \times 7} + \frac{4 \times 4}{7 \times 4} = 7 \frac{7}{28} + \frac{16}{28} = 7 \frac{23}{28} &s=3 $

Show your work

How would this work in the "real life" of math homework? Start by crossing out the 10 - 2 and writing in the 8. It is easy to figure 8 - 1 = 7 mentally. I think most students can do $latex 1 - \frac{3}{7} = \frac{4}{7} $ in their heads, but teachers will probably want them to show that step --- especially at first.

The main thing students would need to write down would be that last line of addition:

$latex 7 \frac{1}{4} + \frac{4}{7} = 7 \frac{7}{28} + \frac{16}{28} = 7 \frac{23}{28} &s=3 $

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• Quiz: Those Frustrating Fractions
• Fraction Division — A Poem
• Order of Operations
• Percents: Key Concepts and Connections
• Putting Bill Gates in Proportion

Here’s a word problem that involves the multiplication/division of fractions.

_________________________

Problem – One third of the athletes in the local youth racquetball club are girls.  The rest are boys.  There are 30 girls in the club.  How many members are there in the club?

_________________________

Methodology -

_________________________ 

What is the question? - “How many members are there in the club?”

_________________________

Step One – Set up the equation.  Kids learn fractions before they take Pre-Algebra, so they’re not going to know what an “equation” is.  But every word problem at any level, elementary school on up, involves setting up some kind of equation or fomula to help solve the problems.  Elementary school kids set up equations all the time; they just don’t realize that they’re doing it.

The basic wording for a typical fraction problem is:

“Fraction” "OF" “Total” "IS/ARE" “Result”

Typically, the "OF" translates to "x" (multiplication sign), and

the "ARE" (or "IS") tranlates to "=" (equal sign).

Therefore the “equation" would be:

“Fraction” x “Total” = “Result”

In this case, it’s…"Fraction" "OF" "Total Membership" "ARE" "No. of Girls"

Again, look for key words or phrases in the word problem; in this case, the problem states … “One third of the athletes … are girls”.  The wording in this problem basically tells you exactly how to set up the equation. 

1/3 "of" __ "are" 30

1/3 x  __  =  30

Given the elementary school audience of this blog entry, we’ll dispense with the “x” and “y” variables of the prototypical equation, and use a blank or “__” instead.  But really a blank or “__” is a variable, just like an “x” or a “y”.

(By the way, as your child ages into middle school math, they’ll see a lot of P of B = A problems.  P of B = A means Percent of Base = Result, or P x B = A.  To solve P of B = A problems, the student has to translate the percent to a decimal or fraction, and then plug it into P.  Parents know that percents, decimals, ratios and fractions are different ways of expressing the same thing, so P of B = A is the middle school version of the formula I describe above.)

 __________________________ 

Step 2 – Solve the equation.

1/3 x  __ = 30

There’s no need to introduce the concept of “isolating” the variable to solve for it.  Kids know that if 2 x 3 = 6, then 6 divided by 2 = 3.  So most kids will know that … __ = 30 divided by 1/3

If they don’t already know that dividing by a fraction is the same as multiplying by the reciprocol, now might be a good time to introduce that concept.

__ = 30 x 3 = 90 athletes all together

______________________________

Step 3 – Answer the question – There are 90 athletes in the club. 

Meet Tom -- Peeping Tom. Tom is a raven who visits my classroom everyday. He walks on a roof and looks in the window to the classroom. He kronks up there loudly. Walking back and forth in front of my window, kroking. The window is up high in my classroom which is why it's hard for us to see Tom well.

Tom is looking at me here with particular intent because he knows that I totally beat him this time. I have been trying for weeks to get a picture of Tom to share with you all. Even though I'm in the classroom on the other side of the room, he is gone the minute I pull out my camera. Rotten bird! :) But I got him this time and he knows it.

Can't you just hear him thinking, "Drat that woman and her camera!" Tom shows up during Math almost everyday. The kids totally lose focus and point and talk about him (They call him "Peanut Butter" which is just silly -- his name is clearly Tom! Sigh!). Their easy slip into distraction is completely my fault -- I shared my enthusiasm for Ravens with them and they found that more compelling than my enthusiasm for adding and subtracting like and unlike fractions. I had to tell them that if Tom is too big a distraction that the blinds will have to close during math. It is pretty cool to watch them see him up there, want to point and shout out about it, struggle to stay calm and at least partially focused on their work while still sneaking peeks up to Tom. They are such great kids!

These ravens at school are wonderful. I do worry a little bit though -- they are so smart and Tom is watching through the window during Math -- what if he's picking up how to add and subtract like and unlike fractions? Are we really ready for a population of educated Ravens? The mind boggles.

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And, while you are recovering from boggled mind, head on over to I and the Bird #71 - The Quotable I and the Bird. Clare of The House and other Arctic Musings did a fantastic job. Lots of good weekend reading -- enjoy!

Anyways... We have a horse of a problem here. Looks like Cp is releasing something at the very last minute. Hmmmm... I'm certain it will be western. OMG! It might be a new shop or catalog! :shock:

Dana 7977 general of MACP

This is part of an actual conversation during an "AIS" math class. If you are unfamiliar with "AIS," it stands for Academic Intervention Services. The AIS math class is designed to help students who are struggling with various math concepts (we have ELA AIS as well). Most students end up in AIS because of a below proficient score on the previous years state exam. I am not necessarily opposed to testing per se, as long as it is used appropriately. If we are teaching using best practice and developing students at their level, the tests are something we will just deal with as the time comes. One would think that just by listening to the conversation, that these particular students do not need remediation for this particular topic.

I believe that the teaching of concept and skill are linked and cannot stand alone in isolation. One without the other would be like peanut-butter without the jelly. Each decent on their own, but together a classic. As we pilot, we must look at what will be the best fit for our students. We need to take what we know is best practice and incorporate the new program with that.

For example, when developing the idea of division of fractions. Some teachers will just teach "invert and multiply." Others will just teach it through a picture representation. Well what happens when students move on to more difficult fractions and algebra? Why aren't we discussing the generalized formula and the inverse operation of division, multiplication. What do I mean?

The focus needs to be on the students and teaching them what they need, not a test, a program, or a workbook.