Introducing Make It Real Learning workbooks

I have recently had the pleasure to add Make It Real Learning workbooks to my site. These books contain real-life math activities with real-life data, companies, and situations. They are written by Frank Wilson.

Arithmetic I



Fractions, Percents, and Decimals I



Linear Functions I


Calculus I



Periodic and Piecewise Functions I

Some examples of the topics included in these activities are: cell phone plans, autism, population growth, cooking, borrowing money, credit cards, life spans, population growth, and music downloads. But there are many more, more than I can list here.

As students work through the problems, they can use the math skills and concepts they have learned in their math curriculum (such as the concept of average or graphing), and apply those to a situation from real life.

Each activity-lesson in the book contains several questions about the situation, starting with basics and going into more in-depth evaluations, and should be adequate for one-two complete class periods.

Why does that benefit you, the teacher? It will motivate the students far more than dull, boring word problems from the textbook. It will show students how mathematics is truly USEFUL. Also:

• The problems are written by an experienced math teacher (Frank Wilson)


• The problems are matched to the learning objectives of the National Council of Teachers of Mathematics (NCTM). This means that the concepts and skills required to complete the problems ARE found in typical middle and high school mathematics curriculum. You can simply replace some of the problems in your textbook with these real-life scenarios.
  
  
• These activities are excellent to be used in a small-group setting.
  
  
• Typically, the activities contain challenging parts and therefore allow students to practice real problem solving - not just apply knowledge from textbook examples to other almost identical problems.
  
  
• Gifted students can enjoy the challenge of solving all the questions on their own.

All books include complete solutions to all activities and problems. Please note that Make It Real Learning workbooks do not contain the instruction or explanations of the concepts.

List of available books:

Arithmetic I - for grades 3-6
Fractions, Percents, and Decimals I - for grades 4-8
Fractions, Percents, and Decimals II - for grades 6-11
Sets, Probability, and Statistics I - for grades 6-10

Linear Functions I - for algebra 1 and algebra 2
Linear Functions II - for algebra 1 and algebra 2
Quadratic Functions I - for algebra 1 and algebra 2

Exponential and Logarithmic Functions I
- for algebra 2/precalculus
Periodic and Piecewise Functions I
- for algebra 2/precalculus
Polynomial, Power, Logistic, and Rational Functions I
- for algebra 2/precalculus
Calculus I - for grade 12

Introducing Make It Real Learning workbooks

I have recently had the pleasure to add Make It Real Learning workbooks to my site. These books contain real-life math activities with real-life data, companies, and situations. They are written by Frank Wilson.

Arithmetic I



Fractions, Percents, and Decimals I



Linear Functions I


Calculus I



Periodic and Piecewise Functions I

Some examples of the topics included in these activities are: cell phone plans, autism, population growth, cooking, borrowing money, credit cards, life spans, population growth, and music downloads. But there are many more, more than I can list here.

As students work through the problems, they can use the math skills and concepts they have learned in their math curriculum (such as the concept of average or graphing), and apply those to a situation from real life.

Each activity-lesson in the book contains several questions about the situation, starting with basics and going into more in-depth evaluations, and should be adequate for one-two complete class periods.

Why does that benefit you, the teacher? It will motivate the students far more than dull, boring word problems from the textbook. It will show students how mathematics is truly USEFUL. Also:

• The problems are written by an experienced math teacher (Frank Wilson)


• The problems are matched to the learning objectives of the National Council of Teachers of Mathematics (NCTM). This means that the concepts and skills required to complete the problems ARE found in typical middle and high school mathematics curriculum. You can simply replace some of the problems in your textbook with these real-life scenarios.
  
  
• These activities are excellent to be used in a small-group setting.
  
  
• Typically, the activities contain challenging parts and therefore allow students to practice real problem solving - not just apply knowledge from textbook examples to other almost identical problems.
  
  
• Gifted students can enjoy the challenge of solving all the questions on their own.

All books include complete solutions to all activities and problems. Please note that Make It Real Learning workbooks do not contain the instruction or explanations of the concepts.

List of available books:

Arithmetic I - for grades 3-6
Fractions, Percents, and Decimals I - for grades 4-8
Fractions, Percents, and Decimals II - for grades 6-11
Sets, Probability, and Statistics I - for grades 6-10

Linear Functions I - for algebra 1 and algebra 2
Linear Functions II - for algebra 1 and algebra 2
Quadratic Functions I - for algebra 1 and algebra 2

Exponential and Logarithmic Functions I
- for algebra 2/precalculus
Periodic and Piecewise Functions I
- for algebra 2/precalculus
Polynomial, Power, Logistic, and Rational Functions I
- for algebra 2/precalculus
Calculus I - for grade 12

Alcumus

Alcumus is a brand new online math program - currently free - by the Art of Problem Solving folks. This program is specifically designed to provide a challenging enrichment program for gifted math students.

Alcumus interacts with the student and adapts to his or her level. Currently, the program contains over 1100 problems with fully worked-out solutions and over 60 video lessons.

Anyone can join - whether you are just interested in exploring math a little further, or you have student who is training for some math contest, etc.

However, currently one can only join the program as part of a registered class. If you are a single person, this can be a problem. Fortunately, Denise at Let's Play Math has set up a class which is open to all interested homeschoolers and self-directed learners. Please click here to find out more about joining Alcumus as part of Denise's class. Otherwise, if you have a class, just go straight to Alcumus and join there.

Alcumus

Alcumus is a brand new online math program - currently free - by the Art of Problem Solving folks. This program is specifically designed to provide a challenging enrichment program for gifted math students.

Alcumus interacts with the student and adapts to his or her level. Currently, the program contains over 1100 problems with fully worked-out solutions and over 60 video lessons.

Anyone can join - whether you are just interested in exploring math a little further, or you have student who is training for some math contest, etc.

However, currently one can only join the program as part of a registered class. If you are a single person, this can be a problem. Fortunately, Denise at Let's Play Math has set up a class which is open to all interested homeschoolers and self-directed learners. Please click here to find out more about joining Alcumus as part of Denise's class. Otherwise, if you have a class, just go straight to Alcumus and join there.

Math enrichment problems at MathNotations

Today I want to highlight a blog called Math Notations by Dave Marain that specializes in math enrichment problems such as various type challenges and investigations. And they are quite good!


For example, recently Dave posted a geometry problem, and even gave the answer in his post, but he asked the readers to find DIFFERENT ways to solve the problem.

I encourage you to go see the problem and see how many different ways you can find to solve it!

That is exactly what we can ask students too - especially if you have many students and some of them solve the original problem in no time. You can use the same problem as a basis for further investigations for them.

His blog has "labels" in the sidebar, which makes it easy to find problems on a particular topic or concept.

Math enrichment problems at MathNotations

Today I want to highlight a blog called Math Notations by Dave Marain that specializes in math enrichment problems such as various type challenges and investigations. And they are quite good!


For example, recently Dave posted a geometry problem, and even gave the answer in his post, but he asked the readers to find DIFFERENT ways to solve the problem.

I encourage you to go see the problem and see how many different ways you can find to solve it!

That is exactly what we can ask students too - especially if you have many students and some of them solve the original problem in no time. You can use the same problem as a basis for further investigations for them.

His blog has "labels" in the sidebar, which makes it easy to find problems on a particular topic or concept.

Performing well below grade level

I am leading a training next week on how to introduce grade level concepts/standards when the students are performing well below grade level, and I am sure that math is going to be an issue. Any suggestions?

These are my 2 cents on teaching under-performing students.

Let's imagine we have an 8th grader performing on 3rd grade perhaps.

I would dismiss for starters geometry and measuring topics, and concentrate on this train of topics, in THIS ORDER:

addition
subtraction
multiplication
division
fractions
decimals.

... the goal being to cover the basic arithmetic up to pre-algebra.

Think of mathematics as a building. You need to have the foundation building blocks before you can go forward.

Maybe the child stopped understanding the math on 2nd grade or 3rd. We need to find the exact point after which he has not understood everything.

You can gauge this by the way by asking the child simple questions such as,

• I give you an addition 8 + 2 = 10, you give me a subtraction sentence (1st grade knowledge).


• How much do you need to add to 600 to make a thousand? (2nd grade place value)


• Draw a picture of 4 x 3 (3rd grade knowledge)


• There are 293 flowers and 145 are red. How many are not red - how do you find that? (using subtraction in finding parts)


• Draw a picture of 20 : 4. (division concept, 3rd grade)

etc.

So for such imaginary 8th grader, I'd make up a separate curriculum that goes through addition, subtraction, multiplication, division, fractions and decimals, in this order.

A teenager can go through those topics in 1 year, and understand it all, if he wants to. Obviously you'd want to show the student the train of topics too so they know what's coming next. You'd want to motivate them that hey, in one year I can learn all this basic math. You'd want to show them how the basic topics lead to the next ones.

Addition: you'd start at adding single-digit numbers, then memorizing addition facts, then 2-digit addition, then multi-digit addition.

If the student does NOT know addition facts, then spend a week practicing them! But not in random order. Like I do in my books (Addition 1 and Subtraction 1) you need to put those facts into contexts, study them in systematic, logical fashion such as sums of 5, sums of 6, etc. (using fact families). Or, adding to 9, adding to 8, etc.

Not knowing the addition facts makes the student SLOOOOOWWWW in doing simple addition problems, and makes it hard to get into subtraction or multiplication. So it is important, even if he's on 8th grade, and it is 1st grade stuff.

On to subtraction. Cover the three situations where subtraction is used, even if it is 1st grade stuff. Cover multi-digit subtraction. Shouldn't take many weeks.

Then multiplication concept. Times tables. Multi-digit. Absolutely remember to show what principles 23 x 38 is based on (multiplying in parts). Spend several weeks here.

And so on.

In a school year's time, it absolutely is possible to cover the basic arithmetic if the student is a teenager. I just would cover it all, including 1st grade topics, to make sure to catch those points where the student "dropped" off.

Any comments anyone?

Maria Miller

Performing well below grade level

I am leading a training next week on how to introduce grade level concepts/standards when the students are performing well below grade level, and I am sure that math is going to be an issue. Any suggestions?

These are my 2 cents on teaching under-performing students.

Let's imagine we have an 8th grader performing on 3rd grade perhaps.

I would dismiss for starters geometry and measuring topics, and concentrate on this train of topics, in THIS ORDER:

addition
subtraction
multiplication
division
fractions
decimals.

... the goal being to cover the basic arithmetic up to pre-algebra.

Think of mathematics as a building. You need to have the foundation building blocks before you can go forward.

Maybe the child stopped understanding the math on 2nd grade or 3rd. We need to find the exact point after which he has not understood everything.

You can gauge this by the way by asking the child simple questions such as,

• I give you an addition 8 + 2 = 10, you give me a subtraction sentence (1st grade knowledge).


• How much do you need to add to 600 to make a thousand? (2nd grade place value)


• Draw a picture of 4 x 3 (3rd grade knowledge)


• There are 293 flowers and 145 are red. How many are not red - how do you find that? (using subtraction in finding parts)


• Draw a picture of 20 : 4. (division concept, 3rd grade)

etc.

So for such imaginary 8th grader, I'd make up a separate curriculum that goes through addition, subtraction, multiplication, division, fractions and decimals, in this order.

A teenager can go through those topics in 1 year, and understand it all, if he wants to. Obviously you'd want to show the student the train of topics too so they know what's coming next. You'd want to motivate them that hey, in one year I can learn all this basic math. You'd want to show them how the basic topics lead to the next ones.

Addition: you'd start at adding single-digit numbers, then memorizing addition facts, then 2-digit addition, then multi-digit addition.

If the student does NOT know addition facts, then spend a week practicing them! But not in random order. Like I do in my books (Addition 1 and Subtraction 1) you need to put those facts into contexts, study them in systematic, logical fashion such as sums of 5, sums of 6, etc. (using fact families). Or, adding to 9, adding to 8, etc.

Not knowing the addition facts makes the student SLOOOOOWWWW in doing simple addition problems, and makes it hard to get into subtraction or multiplication. So it is important, even if he's on 8th grade, and it is 1st grade stuff.

On to subtraction. Cover the three situations where subtraction is used, even if it is 1st grade stuff. Cover multi-digit subtraction. Shouldn't take many weeks.

Then multiplication concept. Times tables. Multi-digit. Absolutely remember to show what principles 23 x 38 is based on (multiplying in parts). Spend several weeks here.

And so on.

In a school year's time, it absolutely is possible to cover the basic arithmetic if the student is a teenager. I just would cover it all, including 1st grade topics, to make sure to catch those points where the student "dropped" off.

Any comments anyone?

Maria Miller

Developing positive attitude

What are the incentives needed in order to develop a positive attitude in children and other students towards mathematics?

I am not sure if (special) incentives is the main factor in developing a positive attitude towards mathematics.

I feel it is probably sufficient to get a few of the basics right, and then that alone will take care of most of it, and students can like math just fine.

Disliking math is not something that is inherent in us or in our kids. Little kids don't dislike math or numbers. They're just fine with math. As we know, this "I hate math" or "I don't like math" attitude seems to develop during school years.

Now, I also don't think that children are disliking reasoning, because they're happy to do puzzles and go through games where you have to think.

And, students' negative attitude towards math also is NOT due to (school) math being difficult. The math we learn in school is not difficult. You don't have to be a math whiz to understand it.

If you can learn to read and to use computer software, surely you can learn basic math. It's not that complex.

So... here are the two main factors that I feel contribute most to what attitude children develop towards math:

1) The teacher's attitude.

If you love math and are enthusiastic about it, it is seen in your teaching, and your attitude will be somewhat contagious.

It's true the other way around as well: if you don't like math and are teaching it, students will sense it. (I've written about the teacher's attitude in this article.)


2) How the math is being taught.

Children can end up not liking math when it is taught in such a way that they don't end up understanding it.

When they don't understand it, then they don't like studying more of it.

The teacher is obviously hugely influental in how the math is taught, but curriculum or the math book also plays a role.

So if we as teachers can get these two things straightened out and thus have the basics covered, then some special incentives and other extra "math goodies" won't hurt either.

---

See also:
Four habits of highly effective math teaching

Is your math curriculum coherent?

How to motivate and prevent math anxiety

Developing positive attitude

What are the incentives needed in order to develop a positive attitude in children and other students towards mathematics?

I am not sure if (special) incentives is the main factor in developing a positive attitude towards mathematics.

I feel it is probably sufficient to get a few of the basics right, and then that alone will take care of most of it, and students can like math just fine.

Disliking math is not something that is inherent in us or in our kids. Little kids don't dislike math or numbers. They're just fine with math. As we know, this "I hate math" or "I don't like math" attitude seems to develop during school years.

Now, I also don't think that children are disliking reasoning, because they're happy to do puzzles and go through games where you have to think.

And, students' negative attitude towards math also is NOT due to (school) math being difficult. The math we learn in school is not difficult. You don't have to be a math whiz to understand it.

If you can learn to read and to use computer software, surely you can learn basic math. It's not that complex.

So... here are the two main factors that I feel contribute most to what attitude children develop towards math:

1) The teacher's attitude.

If you love math and are enthusiastic about it, it is seen in your teaching, and your attitude will be somewhat contagious.

It's true the other way around as well: if you don't like math and are teaching it, students will sense it. (I've written about the teacher's attitude in this article.)


2) How the math is being taught.

Children can end up not liking math when it is taught in such a way that they don't end up understanding it.

When they don't understand it, then they don't like studying more of it.

The teacher is obviously hugely influental in how the math is taught, but curriculum or the math book also plays a role.

So if we as teachers can get these two things straightened out and thus have the basics covered, then some special incentives and other extra "math goodies" won't hurt either.

---

See also:
Four habits of highly effective math teaching

Is your math curriculum coherent?

How to motivate and prevent math anxiety

Online math resources

Resources

These are some of the links I've added to my site recently. Maybe there's some that interest you.

Mathopenref.com
Free online textbook for high school geometry; not finished.

GapMinder
Visualizing human development trends (such as poverty, health, gaps, income on a global scale) via stunning, interactive statistical graphs. This is an interactive, dynamic tool and not just static graphs. Download the software or the reports for free.

How to write proofs
A 12-part tutorial on proof writing. Includes direct proof, proof by contradiction, proof by contrapositive, mathematical induction, if and only if, and proof strategies.

Money Math
Crystal clear tutorial on interest.

Graph Mole
A fun game about plotting points in coordinate plane. Plot points before the mole eats the vegetables.

---

All sorts of sites to explore! But if those didn't fit your bill, if you're in need of a game or tutorial about specific math topic, check my link lists of online math resources; they're organized by level and topic:

Math Help & Tutoring

Math help, Homework help, tutoring

General Math Resources

Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep

Elementary Math 1

Basic operations, Times tables, Place value

Elementary Math 2

Time, Money, Measuring, Geometry

Middle School Math 1

Fractions, Decimals, Percent, Integers

Middle School Math 2

Measuring, Coordinate Plane, Geometry

High School 1

Algebra, Graphing, Calculus

High School 2

Geometry,  Trigonometry, Statistics, Logic

Fun & Games Math Sites

Math games, quizzes, or interactive tutorials websites

Online math resources

Resources

These are some of the links I've added to my site recently. Maybe there's some that interest you.

Mathopenref.com
Free online textbook for high school geometry; not finished.

GapMinder
Visualizing human development trends (such as poverty, health, gaps, income on a global scale) via stunning, interactive statistical graphs. This is an interactive, dynamic tool and not just static graphs. Download the software or the reports for free.

How to write proofs
A 12-part tutorial on proof writing. Includes direct proof, proof by contradiction, proof by contrapositive, mathematical induction, if and only if, and proof strategies.

Money Math
Crystal clear tutorial on interest.

Graph Mole
A fun game about plotting points in coordinate plane. Plot points before the mole eats the vegetables.

---

All sorts of sites to explore! But if those didn't fit your bill, if you're in need of a game or tutorial about specific math topic, check my link lists of online math resources; they're organized by level and topic:

Math Help & Tutoring

Math help, Homework help, tutoring

General Math Resources

Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep

Elementary Math 1

Basic operations, Times tables, Place value

Elementary Math 2

Time, Money, Measuring, Geometry

Middle School Math 1

Fractions, Decimals, Percent, Integers

Middle School Math 2

Measuring, Coordinate Plane, Geometry

High School 1

Algebra, Graphing, Calculus

High School 2

Geometry,  Trigonometry, Statistics, Logic

Fun & Games Math Sites

Math games, quizzes, or interactive tutorials websites

Logic course for the gifted

Some of you might be interested:

Recently I've had the pleasure to review a logic course by IMACS - Institute of Mathematics and Computer Science. This course is meant for the mathematically precocious middle/high schoolers. It truly is not for everyone - there is even an aptitude test before entering the course.

The logic course I reviewed is the first course in the Elements of Mathematics series for gifted secondary school students. Top universities are keen on recruiting students who have studied this math curriculum.

Read the review here: Introduction to Logic course (Propositional Logic) by IMACS.

Tags: math, logic, gifted

Logic course for the gifted

Some of you might be interested:

Recently I've had the pleasure to review a logic course by IMACS - Institute of Mathematics and Computer Science. This course is meant for the mathematically precocious middle/high schoolers. It truly is not for everyone - there is even an aptitude test before entering the course.

The logic course I reviewed is the first course in the Elements of Mathematics series for gifted secondary school students. Top universities are keen on recruiting students who have studied this math curriculum.

Read the review here: Introduction to Logic course (Propositional Logic) by IMACS.

Tags: math, logic, gifted

Planets

This blogpost is inspired and especially written for Homeschooling Carnival, as they are having a galaxy theme.

---

Astronomy and mathematics have always been closely related. Astronomers have always been using the latest mathematical knowledge and theories. Many mathematicians in the past have also done research in astronomy.

All sciences strive to help us understand the world we live in, but astronomy does it on the largest possible scale.

I have always found astronomy fascinating, and so have multitudes of other people, too. But I just wonder if mathematicians might have even a little bit stronger fascination or interest in that direction; somehow those two just seem to fit together very well.

It is part of well-rounded education, I feel, to know some basics of history of astronomy. And it sure is very interesting too! What I've written below is just some thoughts on the subject of planetary orbits.

---

Greeks believed that planets go around the earth, in circular orbits. This view was held true all thru many centuries and Dark Ages till the 1500s, though the idea of sun-centered solar system had been proposed by various people.

Nicolaus Copernicus was one of the astronomers who after the Dark Ages was firmly of the opinion that the Earth goes around the sun. But, it was the German astronomer Johannes Kepler who really worked on the idea with Tycho's observational data.

After years of calculations, Kepler realized - and found it hard to believe himself - that the planets do NOT orbit in circular orbits, but in elliptical ones. This is called Kepler's first law.

The planetary orbits are quite close to circles though. In other words, the ellipses are not very 'eccentric', and that explains why astronomers before him were able to believe them to be circles.

Please see this web page, Kepler's Laws of Planetary Motion, for an excellent explanation and diagrams about ellipses and planetary orbits. This topic is worth understanding, I feel.

Kepler also found that the planets don't travel around the sun at constant speed. Instead, they speed up when they are closest to the sun, and slow down when they are farthest from the sun. Kepler found a simple mathematical relationship that is called Kepler's second law: that a line between the planet and the Sun sweeps out equal areas in equal times.

I found a neat applet illustrating this speeding and slowing on an elliptical orbit. Go check it out! You can change the eccentricity of the ellipse; the traditional planets in our solar system all have eccentricity less than 0.3.

And, Kepler also settled the question whether all planets travel at the same speed. No, not at all. The planets that are furthest from the sun travel slower than the planets closer to sun. This is expressed in Kepler's third law of planetary motion.

The story of planetary motion does not end with Kepler, though. Astronomers kept finding irregularities in the orbits of planets - and this led to the discoveries of new planets.

For example, Neptune was found in 1840s because the orbit of Uranus didn't fit the predictions. In late 1800s, scientists started speculating that Neptune's orbit, too, was perturbated by some other then unknown planet. Pluto was finally discovered in 1930.

And the discoveries have not stopped there. In recent years there have been thousands of new "things" discovered that orbit our sun - they are called "trans-Neptunian objects" or "Kuiper-belt objects" based on their location.

One of them is larger than Pluto and has been called the tenth planet. Currently it only has the official number 2003 UB₃₁₃.

The International Astronomical Union (IAU) is scheduled to publish the definition of the term "planet" in early September 2006 and after that we will know whether 2003 UB₃₁₃ is going to be called the tenth planet or just a Kuiper-belt object.

And after that, the appropriate committee can go on deciding about its name.

At any rate, astronomical knowledge is currently expanding at an ever-increasing rate, it seems. It is one of the oldest sciences and still going strong!


See also:

Orbits and gravitation - a good article explaining the history of those topics.

Planets

This blogpost is inspired and especially written for Homeschooling Carnival, as they are having a galaxy theme.

---

Astronomy and mathematics have always been closely related. Astronomers have always been using the latest mathematical knowledge and theories. Many mathematicians in the past have also done research in astronomy.

All sciences strive to help us understand the world we live in, but astronomy does it on the largest possible scale.

I have always found astronomy fascinating, and so have multitudes of other people, too. But I just wonder if mathematicians might have even a little bit stronger fascination or interest in that direction; somehow those two just seem to fit together very well.

It is part of well-rounded education, I feel, to know some basics of history of astronomy. And it sure is very interesting too! What I've written below is just some thoughts on the subject of planetary orbits.

---

Greeks believed that planets go around the earth, in circular orbits. This view was held true all thru many centuries and Dark Ages till the 1500s, though the idea of sun-centered solar system had been proposed by various people.

Nicolaus Copernicus was one of the astronomers who after the Dark Ages was firmly of the opinion that the Earth goes around the sun. But, it was the German astronomer Johannes Kepler who really worked on the idea with Tycho's observational data.

After years of calculations, Kepler realized - and found it hard to believe himself - that the planets do NOT orbit in circular orbits, but in elliptical ones. This is called Kepler's first law.

The planetary orbits are quite close to circles though. In other words, the ellipses are not very 'eccentric', and that explains why astronomers before him were able to believe them to be circles.

Please see this web page, Kepler's Laws of Planetary Motion, for an excellent explanation and diagrams about ellipses and planetary orbits. This topic is worth understanding, I feel.

Kepler also found that the planets don't travel around the sun at constant speed. Instead, they speed up when they are closest to the sun, and slow down when they are farthest from the sun. Kepler found a simple mathematical relationship that is called Kepler's second law: that a line between the planet and the Sun sweeps out equal areas in equal times.

I found a neat applet illustrating this speeding and slowing on an elliptical orbit. Go check it out! You can change the eccentricity of the ellipse; the traditional planets in our solar system all have eccentricity less than 0.3.

And, Kepler also settled the question whether all planets travel at the same speed. No, not at all. The planets that are furthest from the sun travel slower than the planets closer to sun. This is expressed in Kepler's third law of planetary motion.

The story of planetary motion does not end with Kepler, though. Astronomers kept finding irregularities in the orbits of planets - and this led to the discoveries of new planets.

For example, Neptune was found in 1840s because the orbit of Uranus didn't fit the predictions. In late 1800s, scientists started speculating that Neptune's orbit, too, was perturbated by some other then unknown planet. Pluto was finally discovered in 1930.

And the discoveries have not stopped there. In recent years there have been thousands of new "things" discovered that orbit our sun - they are called "trans-Neptunian objects" or "Kuiper-belt objects" based on their location.

One of them is larger than Pluto and has been called the tenth planet. Currently it only has the official number 2003 UB₃₁₃.

The International Astronomical Union (IAU) is scheduled to publish the definition of the term "planet" in early September 2006 and after that we will know whether 2003 UB₃₁₃ is going to be called the tenth planet or just a Kuiper-belt object.

And after that, the appropriate committee can go on deciding about its name.

At any rate, astronomical knowledge is currently expanding at an ever-increasing rate, it seems. It is one of the oldest sciences and still going strong!


See also:

Orbits and gravitation - a good article explaining the history of those topics.

Russian geometry book

Recently I received the following note,

I would like to bring to your attention the following new
geometry textbook:

"Kiselev's Geometry / Book I. Planimetry" by A.P. Kiselev,
ISBN 0977985202 Publisher: Sumizdat

It is an English translation and adaptation of a classical Russian textbook in plane geometry, which has served well as to several generations of students of age 13 and up, and their teachers in Russia.

The English edition is intended for those students, homeschooled or not, who want to achieve a good command of elementary geometry, and learn to appreciate for its intellectual depth and beauty.

More information about the book and its author is available through the publisher's webpage: www.sumizdat.org.

The book is currently available at: www.sumizdat.org and Singaporemath.com.

I posted this note here because some of you might be interested - a classical Russian geometry book translated into English. You can browse quite many sample pages to get an idea of the book.
It was first published in 1892 has been revised and published more than forty times altogether.

The original author Kiselev wrote several math textbooks. QUOTING from the preface:

"...and a few years prior to Kiselev's death in 1940, his books were officially given the status of stable, i.e. main and only textbooks to be used in all schools to teach all teenagers in Soviet Union.

The books held this status until 1955 when they got replaced in this capacity by less successful clones written by more Soviet authors. Yet "Planimetry" remained the favorite under-the-desk choice of many teachers and a must for honors geometry students. In the last decade, Kiselev's "Geometry," which has long become a rarity, was reprinted by several major publishing houses in Moscow and St.- Petersburg in both versions: for teachers as an authentic pedagogical heritage, and for students as a textbook tailored to fit the currently active school curricula. In the post-Soviet educational market, Kiselev's "Geometry" continues to compete successfully with its own grandchildren."

Quite an accomplishment for a single book.

If you look at the type of exercises found in the book, I think you will easily see that there is a difference when comparing to modern American books (this book is meant for 7-9th graders).

For example (these are from the sample pages provided on the website)

79. Suppose that an angle, its bisector, and one side of this angle in one triangle are respectively congruent to an angle, its bisector, and one side of this angle in another triangle. Prove that such triangles are congruent.

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

Prove theorems:

400. If a diagonal divides a trapezoid into two similar triangles, then this diagonal is the geometric mean between the bases.

401. If two disks are tangent externally, then the segment of an external common tangent between the tangency points is the geometric mean between the diameters of the disks.

402. If a square is inscribed into a right triangle in such a way that one side of the square lies on the hypotenuse, then this side is the geometric mean between the two remaining segments of the hypotenuse.

576. The altitude dropped to the hypotenuse divides a given right triangle into smaller triangles whose radii of the inscribed circles are 6 and 8 cm. Compute the radius of the inscribed circle of the given triangle.

577. Compute the sides of a right triangle given the radii of its circumscribed and inscribed circle.

578. Compute the area of a right triangle if the foot of the altitude dropped to the hypotenuse of length c divides it in the extreme and mean ratio.

Personally, I feel they are interesting sounding problems! (I will probably solve some in future blogposts, as examples.)

But how many US high school students would be willing and able to do them? Feel free to comment.

Now, this book could serve for a high school geometry course for sure. It does have one big disadvantage though if you're a homeschooler: there is no answer key. But the book appears to possess intellectual depth and beauty, just like its subject matter!


Tags: math, geometry, curriculum

Russian geometry book

Recently I received the following note,

I would like to bring to your attention the following new
geometry textbook:

"Kiselev's Geometry / Book I. Planimetry" by A.P. Kiselev,
ISBN 0977985202 Publisher: Sumizdat

It is an English translation and adaptation of a classical Russian textbook in plane geometry, which has served well as to several generations of students of age 13 and up, and their teachers in Russia.

The English edition is intended for those students, homeschooled or not, who want to achieve a good command of elementary geometry, and learn to appreciate for its intellectual depth and beauty.

More information about the book and its author is available through the publisher's webpage: www.sumizdat.org.

The book is currently available at: www.sumizdat.org and Singaporemath.com.

I posted this note here because some of you might be interested - a classical Russian geometry book translated into English. You can browse quite many sample pages to get an idea of the book.
It was first published in 1892 has been revised and published more than forty times altogether.

The original author Kiselev wrote several math textbooks. QUOTING from the preface:

"...and a few years prior to Kiselev's death in 1940, his books were officially given the status of stable, i.e. main and only textbooks to be used in all schools to teach all teenagers in Soviet Union.

The books held this status until 1955 when they got replaced in this capacity by less successful clones written by more Soviet authors. Yet "Planimetry" remained the favorite under-the-desk choice of many teachers and a must for honors geometry students. In the last decade, Kiselev's "Geometry," which has long become a rarity, was reprinted by several major publishing houses in Moscow and St.- Petersburg in both versions: for teachers as an authentic pedagogical heritage, and for students as a textbook tailored to fit the currently active school curricula. In the post-Soviet educational market, Kiselev's "Geometry" continues to compete successfully with its own grandchildren."

Quite an accomplishment for a single book.

If you look at the type of exercises found in the book, I think you will easily see that there is a difference when comparing to modern American books (this book is meant for 7-9th graders).

For example (these are from the sample pages provided on the website)

79. Suppose that an angle, its bisector, and one side of this angle in one triangle are respectively congruent to an angle, its bisector, and one side of this angle in another triangle. Prove that such triangles are congruent.

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

Prove theorems:

400. If a diagonal divides a trapezoid into two similar triangles, then this diagonal is the geometric mean between the bases.

401. If two disks are tangent externally, then the segment of an external common tangent between the tangency points is the geometric mean between the diameters of the disks.

402. If a square is inscribed into a right triangle in such a way that one side of the square lies on the hypotenuse, then this side is the geometric mean between the two remaining segments of the hypotenuse.

576. The altitude dropped to the hypotenuse divides a given right triangle into smaller triangles whose radii of the inscribed circles are 6 and 8 cm. Compute the radius of the inscribed circle of the given triangle.

577. Compute the sides of a right triangle given the radii of its circumscribed and inscribed circle.

578. Compute the area of a right triangle if the foot of the altitude dropped to the hypotenuse of length c divides it in the extreme and mean ratio.

Personally, I feel they are interesting sounding problems! (I will probably solve some in future blogposts, as examples.)

But how many US high school students would be willing and able to do them? Feel free to comment.

Now, this book could serve for a high school geometry course for sure. It does have one big disadvantage though if you're a homeschooler: there is no answer key. But the book appears to possess intellectual depth and beauty, just like its subject matter!


Tags: math, geometry, curriculum

Math resources for gifted students

I added a few weblinks and books for gifted or talented students in math on the online resources list at my website.

Math resources for gifted students

I added a few weblinks and books for gifted or talented students in math on the online resources list at my website.