Crazy 4 Math contest

You might feel that your kid/students will never be able to take part in any math contest, but this one is different. It is for everybody, and you don't have to solve any math problems at all.

Crazy 4 Math contest simply asks you to...

"Describe how you use math in any activity you love to do - a sport, game, craft, hobby or anything else. Send in a description of the activity and how you use math. You can also include a drawing or diagram. Class entries are also accepted. All participants will receive an MP3 of Googol Power's new song "Crazy 4 Math" when you enter."

So this might be a nice way to motivate a student, plus underline the fact, like I've mentioned before, that math is part of our life - not some obscure 'no-one needs it' type of thing.

I encourage you to check some of the last year's entries at Crazy4Math.com.

Teachers: you can submit a class entry too.

Crazy 4 Math contest

You might feel that your kid/students will never be able to take part in any math contest, but this one is different. It is for everybody, and you don't have to solve any math problems at all.

Crazy 4 Math contest simply asks you to...

"Describe how you use math in any activity you love to do - a sport, game, craft, hobby or anything else. Send in a description of the activity and how you use math. You can also include a drawing or diagram. Class entries are also accepted. All participants will receive an MP3 of Googol Power's new song "Crazy 4 Math" when you enter."

So this might be a nice way to motivate a student, plus underline the fact, like I've mentioned before, that math is part of our life - not some obscure 'no-one needs it' type of thing.

I encourage you to check some of the last year's entries at Crazy4Math.com.

Teachers: you can submit a class entry too.

Mitsubishi Endeavor Origami Commercial

I just saw a unique thing on TV today, it's the commercial of the 2006 Mitsubishi Endeavor that features origami, with trees, a dragon, and deers. It's similar to the United Airlines "Dragon" commercial during Super Bowl 2006 which also had paper as the star. This video is hosted and brought to you by the Mitsubishi Motors company, it is available thru their website:

Origami Commercial - http://www.mitsubishicars.com/endeavor/tvcommercials.html

Refreshing algebra books from Dr. Math®

I've just finished writing a review on Dr. Math® algebra books by The Math Forum. There are two: one for pre-algebra and one for algebra 1 students.

They are NOT algebra textbooks because they don't have any exercises. Rather, they're supplemental.

BUT they're very nice AND fairly inexpensive (around $9-10 new, and maybe $6-7 used). Also they weren't awfully long; both were under 200 pages with clear layout. I really think you should take a look at these if you or your student has trouble with algebra!

Dr. Math's books are written in a very easy-reading and friendly tone. They are compiled from question-answers that real students have asked the Dr. Math® service at The Math Forum over the years. It's like reading letters, in a sense, but they deal with math questions. And these letters or explanations are not reading like a textbook but more like a real teacher talking.

One thing I liked was that the books included some 'philosophical' questions, too, such as why 0.999999..... = 1 or what is infinity or why we need variables in algebra.

The main editor, Suzanne Alejandre, wrote to me in an email that she thinks they would be great books for pre-service (particularly elementary and middle school level) teachers to use to get a fuller understanding of algebra.

And, parents could really use them to get a fuller understanding of algebra but also to read with their children.

Anyway, you can read more in my complete review.

Or check out Dr. Math Gets You Ready For Algebra and Dr. Math Explains Algebra by The Math Forum, at Amazon bookstore.

Tags: math, algebra

Refreshing algebra books from Dr. Math®

I've just finished writing a review on Dr. Math® algebra books by The Math Forum. There are two: one for pre-algebra and one for algebra 1 students.

They are NOT algebra textbooks because they don't have any exercises. Rather, they're supplemental.

BUT they're very nice AND fairly inexpensive (around $9-10 new, and maybe $6-7 used). Also they weren't awfully long; both were under 200 pages with clear layout. I really think you should take a look at these if you or your student has trouble with algebra!

Dr. Math's books are written in a very easy-reading and friendly tone. They are compiled from question-answers that real students have asked the Dr. Math® service at The Math Forum over the years. It's like reading letters, in a sense, but they deal with math questions. And these letters or explanations are not reading like a textbook but more like a real teacher talking.

One thing I liked was that the books included some 'philosophical' questions, too, such as why 0.999999..... = 1 or what is infinity or why we need variables in algebra.

The main editor, Suzanne Alejandre, wrote to me in an email that she thinks they would be great books for pre-service (particularly elementary and middle school level) teachers to use to get a fuller understanding of algebra.

And, parents could really use them to get a fuller understanding of algebra but also to read with their children.

Anyway, you can read more in my complete review.

Or check out Dr. Math Gets You Ready For Algebra and Dr. Math Explains Algebra by The Math Forum, at Amazon bookstore.

Tags: math, algebra

Challenging / open-ended math problems

I just read a very inspirational article Teenager or Tyke, Students Learn Best by Tackling Challenging Math (PDF) (html).

It tells about two teachers who frequently employ open-ended problem-solving sessions in their teaching - and the students (almost all) like it well and are very motivated.

In math education, OPEN-ENDED problem usually means it doesn't have a specific step-by-step solution. You can solve it in many different ways. Or, it may have more than one solution.

The problems these teachers use are often from real life, and not quick to solve. Instead it can take some time and struggling to get anywhere. (Hey, that's how problems in real life often are, too!)

But, struggling can be valuable. One of the teachers featured in the article, Heidi Ewer, says:

"Struggling helps them see this as an investment of their own time and energy. It makes them more willing to learn," Ewer says. "Struggling to solve problems requires students to use their intuitive skills to investigate concepts, she explains,
and, in this way, they gain a deeper and more lasting understanding of the mathematics."

There's even some value in trying impossible problems! Douglas Twitchell says in The Value of 'Impossible' Problems, "...if you have students who are interested in trying, think of the mathematics they might learn in the process of attempting these! [some very difficult problems]"

I realize dealing with open-ended problems is not easy to do if you're not an experienced math teacher - and not even then. Like the other teacher from the article, Judith Carter, says, a problem-solving activity is not something she can fit into every day, or even every week.

But, parents organize field trips in other subjects. Maybe an occasional afternoon dedicated to a challenging open-ended math problem can serve as a 'mathematical field trip'!

Find out more:
Teenager or Tyke, Students Learn Best by Tackling Challenging Math (PDF)

Teenager or Tyke, Students Learn Best by Tackling Challenging Math (html)

The article is from a publication "Northwest Teacher". Their volumes seem to have interesting titles.

If you're ready to give some challenging problems for your student(s), check out these:
Open-Ended Math Problems from The Franklin Institute Online (middle school level). Check also the links in my previous post.

And, by the way, there's nothing wrong in working together. If you as the homeschooling parent don't yet know the answer, you can work the problems together with your child.


Tags: math, mathematics, teaching

Challenging / open-ended math problems

I just read a very inspirational article Teenager or Tyke, Students Learn Best by Tackling Challenging Math (PDF) (html).

It tells about two teachers who frequently employ open-ended problem-solving sessions in their teaching - and the students (almost all) like it well and are very motivated.

In math education, OPEN-ENDED problem usually means it doesn't have a specific step-by-step solution. You can solve it in many different ways. Or, it may have more than one solution.

The problems these teachers use are often from real life, and not quick to solve. Instead it can take some time and struggling to get anywhere. (Hey, that's how problems in real life often are, too!)

But, struggling can be valuable. One of the teachers featured in the article, Heidi Ewer, says:

"Struggling helps them see this as an investment of their own time and energy. It makes them more willing to learn," Ewer says. "Struggling to solve problems requires students to use their intuitive skills to investigate concepts, she explains,
and, in this way, they gain a deeper and more lasting understanding of the mathematics."

There's even some value in trying impossible problems! Douglas Twitchell says in The Value of 'Impossible' Problems, "...if you have students who are interested in trying, think of the mathematics they might learn in the process of attempting these! [some very difficult problems]"

I realize dealing with open-ended problems is not easy to do if you're not an experienced math teacher - and not even then. Like the other teacher from the article, Judith Carter, says, a problem-solving activity is not something she can fit into every day, or even every week.

But, parents organize field trips in other subjects. Maybe an occasional afternoon dedicated to a challenging open-ended math problem can serve as a 'mathematical field trip'!

Find out more:
Teenager or Tyke, Students Learn Best by Tackling Challenging Math (PDF)

Teenager or Tyke, Students Learn Best by Tackling Challenging Math (html)

The article is from a publication "Northwest Teacher". Their volumes seem to have interesting titles.

If you're ready to give some challenging problems for your student(s), check out these:
Open-Ended Math Problems from The Franklin Institute Online (middle school level). Check also the links in my previous post.

And, by the way, there's nothing wrong in working together. If you as the homeschooling parent don't yet know the answer, you can work the problems together with your child.


Tags: math, mathematics, teaching

Two Choices - story

I got this in an email... so I'm "forwarding" it to all of you. A touching story about choices in all of our lives.

---

What would you do? You make the choice! Don't look for a punch line; there isn't one! Read it anyway. My question to all of you is: Would you have made the same choice?


At a fundraising dinner for a school that serves learning disabled children, the father of one of the students delivered a speech that would never be forgotten by all who attended. After extolling the school and its dedicated staff, he offered a question:

"When not interfered with by outside influences, everything nature does is done with perfection. Yet my son, Shay, cannot learn things as other children do. He cannot understand things as other children do. Where is the natural order of things in my son?"

The audience was stilled by the query.

The father continued. "I believe, that when a child like Shay, physically and mentally handicapped comes into the world, an opportunity to realize true human nature presents itself, and it comes, in the way other people treat that child" Then he told the following story:

Shay and his father had walked past a park where some boys Shay knew were playing baseball. Shay asked, "Do you think they'll let me play?"

Shay's father knew that most of the boys would not want someone like Shay on their team, but the father also understood that if his son were allowed to play, it would give him a much-needed sense of belonging and some confidence to be accepted by others in spite of his handicaps.

Shay's father approached one of the boys on the field and asked if Shay could play, not expecting much. The boy looked around for guidance and said, "We're losing by six runs and the game is in the eighth inning. I guess he can be on our team and we'll try to put him in to bat in the ninth inning."

Shay struggled over to the team's bench, put on a team shirt with a broad smile and his Father had a small tear in his eye and warmth in his heart the boys saw the father's joy at his son being accepted. In the bottom of the eighth inning, Shay's team scored a few runs but was still behind by three.

In the top of the ninth inning, Shay put on a glove and played in the right field. Even though no hits came his way, he was obviously ecstatic just to be in the game and on the field, grinning from ear to ear as his father waved to him from the stands. In the bottom of the ninth inning, Shay's team scored again. Now, with two outs and the bases loaded, the potential winning run was on base and Shay was scheduled to be next at bat.

At this juncture, do they let Shay bat and give away their chance to win the game? Surprisingly, Shay was given the bat. Everyone knew that a hit was all but impossible 'cause Shay didn't even know how to hold the bat properly, much less connect with the ball.

However, as Shay stepped up to the plate, the pitcher, recognizing the other team putting winning aside for this moment in Shay's life, moved in a few steps to lob the ball in softly so Shay could at least be able to make contact. The first pitch came and Shay swung clumsily and missed. The pitcher again took a few steps forward to toss the ball softly towards Shay. As the pitch came in, Shay swung at the ball and hit a slow ground ball right back to the pitcher.

The game would now be over, but the pitcher picked up the soft grounder and could have easily thrown the ball to the first baseman. Shay would have been out and that would have been the end of the game.

Instead, the pitcher threw the ball right over the head of the first baseman, out of reach of all team mates. Everyone from the stands and both teams started yelling, "Shay, run to first! Run to first!" Never in his life had Shay ever ran that far but made it to first base. He scampered down the baseline, wide-eyed and startled.

Everyone yelled, "Run to second, run to second!"

Catching his breath, Shay awkwardly ran towards second, gleaming and struggling to make it to second base. By the time Shay rounded towards second base, the right fielder had the ball, the smallest guy on their team, who had a chance to be the hero for his team for the first time. He could have thrown the ball to the second-baseman for the tag, but he understood the pitcher's intentions and he too intentionally threw the ball high and far over the third-baseman's head. Shay ran toward third base deliriously as the runners ahead of him circled the bases toward home. As Shay neared third base, the opposing shortstop ran to help him and turned him in the direction of third base, and shouted, "Run to third! Shay, run to third! "

All were screaming, "Shay, Shay, Shay, all the Way Shay"

As Shay rounded third, the boys from both teams and those watching were on their feet were screaming, "Shay, run home! " Shay ran to home, stepped on the plate, and was cheered as the hero who hit the "grand slam" and won the game for his team.

That day, said the father softly with tears now rolling down his face, the boys from both teams helped bring a piece of true love and humanity into this world.

Shay didn't make it to another summer and died that winter, having never forgotten being the hero and making his Father so happy and coming home and seeing his Mother tearfully embrace her little hero of the day!

---

AND, NOW A LITTLE FOOTNOTE TO THIS STORY: We all send thousands of jokes through the e-mail without a second thought, but when it comes to sending messages about life choices, people think twice about sharing. The crude, vulgar, and often obscene pass freely through cyberspace, but public discussion about decency is too often suppressed in our schools and workplaces.

If you're thinking about forwarding this message, chances are that you're probably sorting out the people on your address list that aren't the "appropriate" ones to receive this type of message.

Well, the person who sent you this believes that we all can make a difference. We all have thousands of opportunities every single day to help realize the "natural order of things." So many seemingly trivial interactions between two people present us with a choice: Do we pass along a little spark of love and humanity or do we pass up that opportunity to brighten the day of those with us the least able, and leave the world a little bit colder in the process?

A wise man once said every society is judged by how it treats it's least fortunate amongst them.

You now have two choices:
1. Delete
2. Forward

May your day, be a Shay Day, sunny today tomorrow & always!

Two Choices - story

I got this in an email... so I'm "forwarding" it to all of you. A touching story about choices in all of our lives.

---

What would you do? You make the choice! Don't look for a punch line; there isn't one! Read it anyway. My question to all of you is: Would you have made the same choice?


At a fundraising dinner for a school that serves learning disabled children, the father of one of the students delivered a speech that would never be forgotten by all who attended. After extolling the school and its dedicated staff, he offered a question:

"When not interfered with by outside influences, everything nature does is done with perfection. Yet my son, Shay, cannot learn things as other children do. He cannot understand things as other children do. Where is the natural order of things in my son?"

The audience was stilled by the query.

The father continued. "I believe, that when a child like Shay, physically and mentally handicapped comes into the world, an opportunity to realize true human nature presents itself, and it comes, in the way other people treat that child" Then he told the following story:

Shay and his father had walked past a park where some boys Shay knew were playing baseball. Shay asked, "Do you think they'll let me play?"

Shay's father knew that most of the boys would not want someone like Shay on their team, but the father also understood that if his son were allowed to play, it would give him a much-needed sense of belonging and some confidence to be accepted by others in spite of his handicaps.

Shay's father approached one of the boys on the field and asked if Shay could play, not expecting much. The boy looked around for guidance and said, "We're losing by six runs and the game is in the eighth inning. I guess he can be on our team and we'll try to put him in to bat in the ninth inning."

Shay struggled over to the team's bench, put on a team shirt with a broad smile and his Father had a small tear in his eye and warmth in his heart the boys saw the father's joy at his son being accepted. In the bottom of the eighth inning, Shay's team scored a few runs but was still behind by three.

In the top of the ninth inning, Shay put on a glove and played in the right field. Even though no hits came his way, he was obviously ecstatic just to be in the game and on the field, grinning from ear to ear as his father waved to him from the stands. In the bottom of the ninth inning, Shay's team scored again. Now, with two outs and the bases loaded, the potential winning run was on base and Shay was scheduled to be next at bat.

At this juncture, do they let Shay bat and give away their chance to win the game? Surprisingly, Shay was given the bat. Everyone knew that a hit was all but impossible 'cause Shay didn't even know how to hold the bat properly, much less connect with the ball.

However, as Shay stepped up to the plate, the pitcher, recognizing the other team putting winning aside for this moment in Shay's life, moved in a few steps to lob the ball in softly so Shay could at least be able to make contact. The first pitch came and Shay swung clumsily and missed. The pitcher again took a few steps forward to toss the ball softly towards Shay. As the pitch came in, Shay swung at the ball and hit a slow ground ball right back to the pitcher.

The game would now be over, but the pitcher picked up the soft grounder and could have easily thrown the ball to the first baseman. Shay would have been out and that would have been the end of the game.

Instead, the pitcher threw the ball right over the head of the first baseman, out of reach of all team mates. Everyone from the stands and both teams started yelling, "Shay, run to first! Run to first!" Never in his life had Shay ever ran that far but made it to first base. He scampered down the baseline, wide-eyed and startled.

Everyone yelled, "Run to second, run to second!"

Catching his breath, Shay awkwardly ran towards second, gleaming and struggling to make it to second base. By the time Shay rounded towards second base, the right fielder had the ball, the smallest guy on their team, who had a chance to be the hero for his team for the first time. He could have thrown the ball to the second-baseman for the tag, but he understood the pitcher's intentions and he too intentionally threw the ball high and far over the third-baseman's head. Shay ran toward third base deliriously as the runners ahead of him circled the bases toward home. As Shay neared third base, the opposing shortstop ran to help him and turned him in the direction of third base, and shouted, "Run to third! Shay, run to third! "

All were screaming, "Shay, Shay, Shay, all the Way Shay"

As Shay rounded third, the boys from both teams and those watching were on their feet were screaming, "Shay, run home! " Shay ran to home, stepped on the plate, and was cheered as the hero who hit the "grand slam" and won the game for his team.

That day, said the father softly with tears now rolling down his face, the boys from both teams helped bring a piece of true love and humanity into this world.

Shay didn't make it to another summer and died that winter, having never forgotten being the hero and making his Father so happy and coming home and seeing his Mother tearfully embrace her little hero of the day!

---

AND, NOW A LITTLE FOOTNOTE TO THIS STORY: We all send thousands of jokes through the e-mail without a second thought, but when it comes to sending messages about life choices, people think twice about sharing. The crude, vulgar, and often obscene pass freely through cyberspace, but public discussion about decency is too often suppressed in our schools and workplaces.

If you're thinking about forwarding this message, chances are that you're probably sorting out the people on your address list that aren't the "appropriate" ones to receive this type of message.

Well, the person who sent you this believes that we all can make a difference. We all have thousands of opportunities every single day to help realize the "natural order of things." So many seemingly trivial interactions between two people present us with a choice: Do we pass along a little spark of love and humanity or do we pass up that opportunity to brighten the day of those with us the least able, and leave the world a little bit colder in the process?

A wise man once said every society is judged by how it treats it's least fortunate amongst them.

You now have two choices:
1. Delete
2. Forward

May your day, be a Shay Day, sunny today tomorrow & always!

Prove that irrational*non-zero rational number is equal to an irrational number

How can you prove that irrational*non-zero rational number is equal to an irrational number?

I suspect this is a university student asking me this.

One important step in how to make proofs is to understand clearly what you're asked to prove. It often involves statements that are true for ALL numbers in a certain set (such as all real numbers, all rational numbers etc.).

This one does not include the word "all" nor the word "any", but it still is of that type. I can reword it like this:

Prove that any irrational number multiplied by any non-zero rational number is equal to some irrational number.

Or, this way:

Prove that for all irrational numbers x and for all rational numbers y excluding 0 the following is true: xy is an irrational number.

This question basically ASKS for indirect proof.

In indirect proof, we assume the OPPOSITE is true, and show that would lead to a contradiction.

We are supposed to prove that if you take ANY irrational number times ANY non-zero rational number, then the result is an irrational number.

So assume the opposite: assume there is SOME irrational number X and SOME non-zero rational number R so that when you multiply those two, you get a RATIONAL one as an answer:

XR = S and S is rational

Well, from this equation X = S/R (remember R is not zero so this is legal). But then X would be a rational number (S) divided by another rational number (R).

You probably already know that if you divide a rational number by a rational number, the result is going to be a rational number - never an irrational number, so this is the contradiction needed for the proof. ALL DONE!

But just for a reminder: a rational number is like fraction: a/b where a and b are integers, b nonzero. So if you take a/b ÷ c/d, you know how that is done: multiply by d/c and it produces another fraction (rational number).

Tags: math, mathematics, college

Prove that irrational*non-zero rational number is equal to an irrational number

How can you prove that irrational*non-zero rational number is equal to an irrational number?

I suspect this is a university student asking me this.

One important step in how to make proofs is to understand clearly what you're asked to prove. It often involves statements that are true for ALL numbers in a certain set (such as all real numbers, all rational numbers etc.).

This one does not include the word "all" nor the word "any", but it still is of that type. I can reword it like this:

Prove that any irrational number multiplied by any non-zero rational number is equal to some irrational number.

Or, this way:

Prove that for all irrational numbers x and for all rational numbers y excluding 0 the following is true: xy is an irrational number.

This question basically ASKS for indirect proof.

In indirect proof, we assume the OPPOSITE is true, and show that would lead to a contradiction.

We are supposed to prove that if you take ANY irrational number times ANY non-zero rational number, then the result is an irrational number.

So assume the opposite: assume there is SOME irrational number X and SOME non-zero rational number R so that when you multiply those two, you get a RATIONAL one as an answer:

XR = S and S is rational

Well, from this equation X = S/R (remember R is not zero so this is legal). But then X would be a rational number (S) divided by another rational number (R).

You probably already know that if you divide a rational number by a rational number, the result is going to be a rational number - never an irrational number, so this is the contradiction needed for the proof. ALL DONE!

But just for a reminder: a rational number is like fraction: a/b where a and b are integers, b nonzero. So if you take a/b ÷ c/d, you know how that is done: multiply by d/c and it produces another fraction (rational number).

Tags: math, mathematics, college

Carnival of Homeschooling

This week's Carnival of Homeschooling is up at PHAT Mommy. Again, lots of interesting stuff to read about homeschooling.

Carnival of Homeschooling

This week's Carnival of Homeschooling is up at PHAT Mommy. Again, lots of interesting stuff to read about homeschooling.

Harvest Moon Papercraft - Dog

This is the second part from bokumono.com, it is a basic paper model of a farm dog. Again this is from the Harvest Moon Series of video games and was created by Keicraft. You can look here for the first part of this paper model series (which was a cow). The paper model layout is in PDF format and comes with instructions.




Harvest Moon Papercraft - Dog [via Mediafire]

Harvest Moon Papercraft - Cow [Related Post]

Harvest Moon Papercraft - Cow

We've got a new cute paper model of an "ushi" (Japanese for cow) from bokumono.com, this is the site of a very popular "farming" video game in Japan known here in the US as Harvest Moon. For those not familiar with this game, it is a simulation of how to run a farm, from breeding cows to planting crops and everything in between. For more information about the game check out the links below. This paper model is in PDF format and is pretty basic so beginners would have an easy time building it, once again this was created by the very skilled people at Keicraft.

Harvest Moon Papercraft - Cow [via Mediafire]

Harvest Moon Papercaft - Dog [Related Post]

Review of Time4Learning

Time4Learning.com is an online curriculum covering not only math but language arts, science, and social studies.

I've done a review on their mathematics curriculum (their content is provided by Compasslearning Odyssey).

It's quite motivating, because kids get to do these animated lessons and on lower grades they can aftewards go to "playground". My child sure likes it a lot!

Read my review and see some pics at Homeschoolmath.net/reviews/time4learning.php

Tags: math, lessons, curriculum

Review of Time4Learning

Time4Learning.com is an online curriculum covering not only math but language arts, science, and social studies.

I've done a review on their mathematics curriculum (their content is provided by Compasslearning Odyssey).

It's quite motivating, because kids get to do these animated lessons and on lower grades they can aftewards go to "playground". My child sure likes it a lot!

Read my review and see some pics at Homeschoolmath.net/reviews/time4learning.php

Tags: math, lessons, curriculum

Strategies to solve simple math equations

What are the strategies for solving simple equations?

I got this question in the mail just today.

I assume the person means LINEAR equations - those where you only have one variable (usually x), and that x is not raised to second or third or any other power, nor is it in the denominator or under square root sign or anything. Just x's multiplied by numbers and numbers by themselves, such as:

2x - 14 = 9x + 5

OR

1/3x - 3 = 2 - 1/2x

OR

2(5x - 4) = 3 + 5(-x + 1)

Here are the strategies for solving these:
* You get rid of paretheses using distributive property
* You may multiply both sides of the equation by the same number
* You may divide both sides of the equation by the same number
* You may add the same number to both sides of the equation
* You may subtract the same number from both sides of the equation

You might think, "Which one of those will I use, and in which order?"

That depends. There is no clear cut-n-dried answer.

Whatever you do, you try to transform your equation towards the ultimate goal: where you have x on one side alone. Also whatever you do, your goal is to transform the equation to one that you already know how to solve. It might take several steps.

For example, your first step with these equations, could be to...

1/3x - 3 = 2 - 1/2x	... multiply both sides by 6 to get rid of the fractions
2(5x - 4) = 3 + 5(-x + 1)	...multiply out the parentheses
2x - 14 = 9x + 5	...add 14 to both sides (or subtract 9x)
1/4(2x - 27 + 0.5x) = 2/5(8x + 3)	...multiply by 20 to get rid of the fractions

As with most things, practice makes perfect. Check also the websites below:

Tutorial on linear equations has a 4-step strategy for solving linear equations which summarizes it real well.

Algebra 1 Review - Solving Simple Equations - a step-by-step slideshow.

Ask Dr. Math ® - Solving simple linear equations - lots of examples to read here.


Tags: math, mathematics, teaching, algebra

Strategies to solve simple math equations

What are the strategies for solving simple equations?

I got this question in the mail just today.

I assume the person means LINEAR equations - those where you only have one variable (usually x), and that x is not raised to second or third or any other power, nor is it in the denominator or under square root sign or anything. Just x's multiplied by numbers and numbers by themselves, such as:

2x - 14 = 9x + 5

OR

1/3x - 3 = 2 - 1/2x

OR

2(5x - 4) = 3 + 5(-x + 1)

Here are the strategies for solving these:
* You get rid of paretheses using distributive property
* You may multiply both sides of the equation by the same number
* You may divide both sides of the equation by the same number
* You may add the same number to both sides of the equation
* You may subtract the same number from both sides of the equation

You might think, "Which one of those will I use, and in which order?"

That depends. There is no clear cut-n-dried answer.

Whatever you do, you try to transform your equation towards the ultimate goal: where you have x on one side alone. Also whatever you do, your goal is to transform the equation to one that you already know how to solve. It might take several steps.

For example, your first step with these equations, could be to...

1/3x - 3 = 2 - 1/2x	... multiply both sides by 6 to get rid of the fractions
2(5x - 4) = 3 + 5(-x + 1)	...multiply out the parentheses
2x - 14 = 9x + 5	...add 14 to both sides (or subtract 9x)
1/4(2x - 27 + 0.5x) = 2/5(8x + 3)	...multiply by 20 to get rid of the fractions

As with most things, practice makes perfect. Check also the websites below:

Tutorial on linear equations has a 4-step strategy for solving linear equations which summarizes it real well.

Algebra 1 Review - Solving Simple Equations - a step-by-step slideshow.

Ask Dr. Math ® - Solving simple linear equations - lots of examples to read here.


Tags: math, mathematics, teaching, algebra

Addition facts with 6 and 8

Sums with 8

Well today's post will not contain any deep mathematical insights... Just two pics of the things we're doing at our house. We're studying simple addition facts. I just keep writing problems in her notebook. Have a happy weekend!

Sums with 6

Addition facts with 6 and 8

Sums with 8

Well today's post will not contain any deep mathematical insights... Just two pics of the things we're doing at our house. We're studying simple addition facts. I just keep writing problems in her notebook. Have a happy weekend!

Sums with 6

Word problem situations in elementary grades

I still want to continue on the topic of word problems - not because it's my 'soap box' subject but because I feel parents and students can use lots of help with them.

If you didn't catch the earlier post on how your typical math books subtily teach kids NOT to think carefully with word problems, read it here.

---

The easy 'routine' word problems in early grades usually require just one operation to solve. I would recommend studying a bunch of such word problems WITHOUT calculating the answers but only thinking and finding which operation is needed to solve each problem.

After you do that enough times, the student should start associating the types of situations with the appropriate operations:

• Total is divided into so many parts/containers, each part having same amount.
  
  This is the multiplication/division situation:
  (number of parts) × (amount in each) = total
  
  
  • If you know how many parts and how much in each, MULTIPLY.
  • If you know the total and the number of parts, DIVIDE.
  • If you know the total and the amount in each, DIVIDE.



• Total is divided into unequal groups.
  
  This is the addition/subtraction situation:
  (amount in group 1) + (amount in group 2) + (amount in group 3) + etc. = total.
  
  
  • If you know the amounts in groups but not the total, ADD.
  • If you know the total and the amounts in all but one group, SUBTRACT. This is the opposite of addition.
    
    Of 187 pictures, 45 were black-and-white. How many were color pictures?
    There were 57 pumpkins and 15 of them were ripe. How many were not ripe?
    
    Notice that NOTHING is 'going away' or being 'taken away'. They are typical "addition situations":
    
    color pictures + black-white pictures = total pictures
    ripe pumpkins + non-ripe pumpkins = all pumpkins
    They are solved by subtracting because the total is already known and in essence we're trying to find the missing addend.

Then there are some other subtraction situations:

• You know what the total used to be, but part of it went away or got used - the easy, classic 'take away situation'.
  Jenny had $14.56 and she bought a doll for $2.55. How much money is left?


• How many more (= difference)
  Joe has 24 stamps and Bill has 13. How many more does Joe have?
  Note nothing is 'taken away'.

I would also devote extra attention to problems involving time.

I'm not claiming this is a complete coverage of all the possible situations that are solved by single operation (let me know of others), but I wonder how much it would help the kids if these ideas were explicitly taught to them.

Tags: math, elementary, teaching

Word problem situations in elementary grades

I still want to continue on the topic of word problems - not because it's my 'soap box' subject but because I feel parents and students can use lots of help with them.

If you didn't catch the earlier post on how your typical math books subtily teach kids NOT to think carefully with word problems, read it here.

---

The easy 'routine' word problems in early grades usually require just one operation to solve. I would recommend studying a bunch of such word problems WITHOUT calculating the answers but only thinking and finding which operation is needed to solve each problem.

After you do that enough times, the student should start associating the types of situations with the appropriate operations:

• Total is divided into so many parts/containers, each part having same amount.
  
  This is the multiplication/division situation:
  (number of parts) × (amount in each) = total
  
  
  • If you know how many parts and how much in each, MULTIPLY.
  • If you know the total and the number of parts, DIVIDE.
  • If you know the total and the amount in each, DIVIDE.



• Total is divided into unequal groups.
  
  This is the addition/subtraction situation:
  (amount in group 1) + (amount in group 2) + (amount in group 3) + etc. = total.
  
  
  • If you know the amounts in groups but not the total, ADD.
  • If you know the total and the amounts in all but one group, SUBTRACT. This is the opposite of addition.
    
    Of 187 pictures, 45 were black-and-white. How many were color pictures?
    There were 57 pumpkins and 15 of them were ripe. How many were not ripe?
    
    Notice that NOTHING is 'going away' or being 'taken away'. They are typical "addition situations":
    
    color pictures + black-white pictures = total pictures
    ripe pumpkins + non-ripe pumpkins = all pumpkins
    They are solved by subtracting because the total is already known and in essence we're trying to find the missing addend.

Then there are some other subtraction situations:

• You know what the total used to be, but part of it went away or got used - the easy, classic 'take away situation'.
  Jenny had $14.56 and she bought a doll for $2.55. How much money is left?


• How many more (= difference)
  Joe has 24 stamps and Bill has 13. How many more does Joe have?
  Note nothing is 'taken away'.

I would also devote extra attention to problems involving time.

I'm not claiming this is a complete coverage of all the possible situations that are solved by single operation (let me know of others), but I wonder how much it would help the kids if these ideas were explicitly taught to them.

Tags: math, elementary, teaching

Pi Day

I almost forgot...! Today is Pi Day! See, it's 3-14 (March 14) and pi ≈ 3.14.

To celebrate, check some pi day activities.

Pi Day

I almost forgot...! Today is Pi Day! See, it's 3-14 (March 14) and pi ≈ 3.14.

To celebrate, check some pi day activities.

The problem with word problems 2

For the purpose of this post, we could divide word problems to three different categories:
1) routine word problems
2) non-routine word problems
3) algebra word problems

Actually you could divide algebra word problems to routine and non-routine as well, but I want to now talk about word problems kids encounter in school before algebra - in grades 1-8 usually.

J.D. Fisher suggested in the comments section of my previous post on word problems that kids are encouraged to think linearly, step-by-step. Then, when the word problems they encounter don't anymore follow any step-by-step recipe, they are lost. You might want to go back and read that.

Don't typical math book lessons kind of follow this recipe:

LESSON X
---------------------
Explanation and examples.
Numerical exercises.
A few word problems.

In other words, the word problems are usually in the end of the lesson. (That might make solving them a rush.)

Then, have you ever noticed... If the lesson is about topic X, then the word problems are about the topic X too!

For example, if the topic in the lesson is long division, then the word problems found in the lesson are extremely likely to be solved by long division.

And, typically the word problems only have two numbers in them. So, even if you didn't understand a word in the word problem, you might be able to do it. Just try: let's say that the following made-up problem is found within a long division lesson. Can you solve it?

La tienda tiene 870 sabanas en 9 colores diferentes. Hay la misma cantidad en cada color. Cuantos sabanas de cada color tiene la tienda?

My thought is that over the years, when kids are exposed to such lessons over and over again, they kind of figure it out that it's mentally less demanding just not even read the problem too carefully. Why bother? Just take the two numbers and divide (or multiply, or add, or subtract) them and that's it.

I'm not saying that such word problems are not needed in the end of division lessons. I'm sure they have their place. But too much of those simple 'routine' problems can be a problem... I feel kids then "learn" a rule: "Word problems found in math books are solved by some routine or rule that you find in the beginning of the corresponding lesson."

It might teach their minds to be lazy and not willing to tackle non-routine problems.

Maybe it would help to give students a bunch of short routine word problems, and NOT ask them to find answer. Instead, ask them to tell what operation(s) are needed to find the answer.

Maybe it would help to have separate lessons with mixed word problems, including some non-routine, and devote some time to them.

I'm curious to hear your thoughts on this.

And, lastly some (most are free) word problem resources if you need more than what's in your math book:

Word problems for kids
A great selection of word problems for grades 5-12. A hint and a complete solution available for each problem.

Aunty Math
Math challenges in a form of short stories for K-5 learners posted bi-weekly.

Problem of the Week home page
Links to 'problem of the week' websites organized by grade levels. These are excellent for finding more challenging problems and to motivate.

Primary Mathematics Challenging Word Problems
For grades 1-6 from Singapore Math. The books include answer key, worked examples, practice problems, and challenging problems. About $8 per book.

I have some more resources listed at my own site.

Tags: math, elementary, curriculum

The problem with word problems 2

For the purpose of this post, we could divide word problems to three different categories:
1) routine word problems
2) non-routine word problems
3) algebra word problems

Actually you could divide algebra word problems to routine and non-routine as well, but I want to now talk about word problems kids encounter in school before algebra - in grades 1-8 usually.

J.D. Fisher suggested in the comments section of my previous post on word problems that kids are encouraged to think linearly, step-by-step. Then, when the word problems they encounter don't anymore follow any step-by-step recipe, they are lost. You might want to go back and read that.

Don't typical math book lessons kind of follow this recipe:

LESSON X
---------------------
Explanation and examples.
Numerical exercises.
A few word problems.

In other words, the word problems are usually in the end of the lesson. (That might make solving them a rush.)

Then, have you ever noticed... If the lesson is about topic X, then the word problems are about the topic X too!

For example, if the topic in the lesson is long division, then the word problems found in the lesson are extremely likely to be solved by long division.

And, typically the word problems only have two numbers in them. So, even if you didn't understand a word in the word problem, you might be able to do it. Just try: let's say that the following made-up problem is found within a long division lesson. Can you solve it?

La tienda tiene 870 sabanas en 9 colores diferentes. Hay la misma cantidad en cada color. Cuantos sabanas de cada color tiene la tienda?

My thought is that over the years, when kids are exposed to such lessons over and over again, they kind of figure it out that it's mentally less demanding just not even read the problem too carefully. Why bother? Just take the two numbers and divide (or multiply, or add, or subtract) them and that's it.

I'm not saying that such word problems are not needed in the end of division lessons. I'm sure they have their place. But too much of those simple 'routine' problems can be a problem... I feel kids then "learn" a rule: "Word problems found in math books are solved by some routine or rule that you find in the beginning of the corresponding lesson."

It might teach their minds to be lazy and not willing to tackle non-routine problems.

Maybe it would help to give students a bunch of short routine word problems, and NOT ask them to find answer. Instead, ask them to tell what operation(s) are needed to find the answer.

Maybe it would help to have separate lessons with mixed word problems, including some non-routine, and devote some time to them.

I'm curious to hear your thoughts on this.

And, lastly some (most are free) word problem resources if you need more than what's in your math book:

Word problems for kids
A great selection of word problems for grades 5-12. A hint and a complete solution available for each problem.

Aunty Math
Math challenges in a form of short stories for K-5 learners posted bi-weekly.

Problem of the Week home page
Links to 'problem of the week' websites organized by grade levels. These are excellent for finding more challenging problems and to motivate.

Primary Mathematics Challenging Word Problems
For grades 1-6 from Singapore Math. The books include answer key, worked examples, practice problems, and challenging problems. About $8 per book.

I have some more resources listed at my own site.

Tags: math, elementary, curriculum

Logarithms in a nutshell

Someone asked me recently to make a post about logarithms. So here goes. I already answered the person in an email but I thought I could include some interesting history tidbits here, too.

Logarithms are simply the opposite operation of exponentation.

For example, from 2³ = 8 we get log₂8 = 3, and we read it "base 2 logarithm of 8 equals 3".

So it's not difficult: if you understand how exponents work, logarithms have the same numbers, just in a little different places.

Just as in exponentiation, a logarithm has a base (2 in the above example). Remember that in 5³, 5 is called the base and 3 is the exponent.

Other examples:

5³ = 125 and log₅125 = 3.
10⁴ = 10,000 and log₁₀10,000 = 4.
2^x = 345 and log₂345 = x.

As you can see from the last example above, you can use logarithms to solve equations where the x is the exponent:

4^x = 1001
x = log₄1001.

Then you'd get the value of x from a calculator.

However, the base of the logarithm can be anything and most calculators only have two buttons: one for base 10 logarithm and another for base e logarithm (also called natural logarithm).

So you might have to convert the answer log₄1001 to one of those first.

And aren't we happy it CAN be done: log₄1001 = log₁₀1001/log₁₀4. So now you have two base 10 logs that you can punch into your calculator. And the answer is: x ≈ 4.983613129. Check: 4^4.983613129 = 1000.99999942, yeah, reasonably close I guess.

History

Initially logarithms were used as calculation aids - before the age of calculators, of course. The way they work, people were able to avoid multiplying by using logarithms.

They would have a BIG book called the table of logarithms. And, if you had two really BIG numbers to multiply, you can imagine how tedious a task it is by hand. But with logarithms, let's say you had a and b to multiply as your numbers. You'd find their logarithms in the book, and simply ADD those logarithms - which is way easier to do by hand than multiplying: you'd find what log a + log b was.

Then, after you had the sum, you'd again check in the book "backwards" and find which number's logarithm was the sum you'd just calculated.

How does that work? If you're ready for more math and symbols, see below:

As you may remember, exponents have a property that:

b^t b^s = b^s+t

Logarithms are the opposite of exponentiation... If you call A = b^t and B = b^s, then t = log_bA and s = log_bB.
Rewrite the original exponent property now using A and B:

AB = b^s+t

And writing that in logarithm form you get:

log_b AB = s + t = log_bB + log_bA.

So base b logarithm of a product (AB) is the same as the sum of the logarithms of the numbers. And this property allowed people to find products of large numbers by just adding the logarithms and checking 'backwards' which number had as its logarithm the obtained sum.

See more about the history of logarithms, or read more about them in general.

Tags: math, mathematics

Logarithms in a nutshell

Someone asked me recently to make a post about logarithms. So here goes. I already answered the person in an email but I thought I could include some interesting history tidbits here, too.

Logarithms are simply the opposite operation of exponentation.

For example, from 2³ = 8 we get log₂8 = 3, and we read it "base 2 logarithm of 8 equals 3".

So it's not difficult: if you understand how exponents work, logarithms have the same numbers, just in a little different places.

Just as in exponentiation, a logarithm has a base (2 in the above example). Remember that in 5³, 5 is called the base and 3 is the exponent.

Other examples:

5³ = 125 and log₅125 = 3.
10⁴ = 10,000 and log₁₀10,000 = 4.
2^x = 345 and log₂345 = x.

As you can see from the last example above, you can use logarithms to solve equations where the x is the exponent:

4^x = 1001
x = log₄1001.

Then you'd get the value of x from a calculator.

However, the base of the logarithm can be anything and most calculators only have two buttons: one for base 10 logarithm and another for base e logarithm (also called natural logarithm).

So you might have to convert the answer log₄1001 to one of those first.

And aren't we happy it CAN be done: log₄1001 = log₁₀1001/log₁₀4. So now you have two base 10 logs that you can punch into your calculator. And the answer is: x ≈ 4.983613129. Check: 4^4.983613129 = 1000.99999942, yeah, reasonably close I guess.

History

Initially logarithms were used as calculation aids - before the age of calculators, of course. The way they work, people were able to avoid multiplying by using logarithms.

They would have a BIG book called the table of logarithms. And, if you had two really BIG numbers to multiply, you can imagine how tedious a task it is by hand. But with logarithms, let's say you had a and b to multiply as your numbers. You'd find their logarithms in the book, and simply ADD those logarithms - which is way easier to do by hand than multiplying: you'd find what log a + log b was.

Then, after you had the sum, you'd again check in the book "backwards" and find which number's logarithm was the sum you'd just calculated.

How does that work? If you're ready for more math and symbols, see below:

As you may remember, exponents have a property that:

b^t b^s = b^s+t

Logarithms are the opposite of exponentiation... If you call A = b^t and B = b^s, then t = log_bA and s = log_bB.
Rewrite the original exponent property now using A and B:

AB = b^s+t

And writing that in logarithm form you get:

log_b AB = s + t = log_bB + log_bA.

So base b logarithm of a product (AB) is the same as the sum of the logarithms of the numbers. And this property allowed people to find products of large numbers by just adding the logarithms and checking 'backwards' which number had as its logarithm the obtained sum.

See more about the history of logarithms, or read more about them in general.

Tags: math, mathematics

Example of failed problem solving

My intention for today's post was to solve a problem, and write down my thinking process, so it would be an example of problem solving process.

But I failed to solve the problem!

I went to this site first but the problems there looked routine, plus I didn't want to choose a calculus problem but keep it somewhat simple.

Then I went to MathsChallenge.net and looked at their one-star problems... they seemed kind of easy, I was able to "see" the solution immediately. I wanted a problem that I personally could not solve just by reading it.

I finally settled on a two-star geometry problem.

It has a right triangle and lines drawn in it, and you're supposed to prove that a certain angle is equal to another angle.

The details of this are not important so I'm not going to post here all that I typed... (I tried to type in my thinking process while solving it). I quickly saw some right triangles there, and that instead of proving the angles equal, I could try prove a certain line segment to have the same lenght as another line segment.

Then I tried sines, a bit of algebra, similar triangles, all sorts of things... I tried something, then would think "this doesn't work", and try something else. The problem seemd to be so simple so I was looking for a simple solution and didn't want to try
"heavy tools" such as cosines and sines.

After a long while (well it seemed like it, maybe 1/2 hour) I thought it was getting too lenghty, because after all, I cannot take hours to make a single blogpost. So I clicked on the solution.

It had a CIRCLE! Seeing the circle drawn in the picture, I saw the solution immediately.

It was kind of a "think-out-of-the-box" problem. There was no circle draw in the problem, yet the solution had it.

So from now on, I will not forget to think about circles when solving geometry problems, even if the problem only shows me triangles or whatever.

---

The morale of this story? Choose one:

*Do not ask Mrs. Miller any geometry problems - she can't solve them.
*Do not use MathsChallenge.net two-star geometry problems for your word problems - you might get stumped and get a low self-esteem for the rest of your life. (just joking! :^)
*Do not use geometry problems as examples of word problems since they are not word problems!

---

Well, actually the "take home messages" I want you to remember are these:

• Do NOT give up at non-routine problems too soon.


• Monitor your progress: expert problem solvers keep 'mental track' of whether they're getting closer of the solution. If on wrong path, they abandon it and start with something different.


• After struggling for a while, even if you do 'cheat' and look at the solution, the time was not lost - you often learn something valuable that you don't forget easily.

---

And here's part of the "failed attempt" thinking process... if you're interested:

"...I remember from recent stuff that triangles ACC' and BCC' have same area - if that happens to help any.

don't know if the angles aplha etc are related to the areas of the three subtraingles.

Or if, since C is right angle and ABC is a right triangle, woudl it happen to be similar to one ofthe subtriangles?
Well they share side AC (that won't matter) but they SHARE an angle (angle A). So yes they rae similar.

So angle alpha = angle B.
So if alpha = gamma (supposed to rpove that) then triangle CC'B is isosceles.

But converse also true; if I can prove triangle CC'B isosceles, from which follwos that gamma = B.

Thinking some more while keenly looking at the pic. How could i prove those two sides are equal lenght? Could I prove CC' = C'A ? That would be isosceles triangle there too (ACC')

Thinking again of the areas of triangles... since two of them are equal./.. using other kind of height..?

I might be on wrong path. Look again at the angles alpha gamma beta. Since alpha = angle B, then beta + gamma = 90- alpha = angle A. There is another pair of similar trianlges ther e(CDB and the big one). Hee should have found that earlier.

Would simple algebra help?

beta + gamma = A = 90 -alpha.
alpha = angle B

B+A =90 so B =
Substitue

beta + gamma = A = 90 -alpha.
alpha = 90-A

need to prove alpha = gamma so get rid of beta please.

No this is not getting anywhere.

The righgt triangles might still lead to the answer.
How about sine theorem:

sine alpha/AD = sine 90/AC

sine gamme /C'B =sine B/CC'

Sine B/AC =sine 90/AB = sine 90/2C'B

sine 90 is 1.

substitute

sine alpha/AD = 1/AC => sine alpha = AD/AC. Heh knew this.

sine gamme /C'B =sine B/CC' = >Sine gamma = sine B/CC' * C'B.

Sine B/AC =1/AB = 1/2C'B
Sin B = AC/AB and is same as sine alpha


Sine alpha =
Not getting anywhere! I feel embarrassed.

Somehow get alpha/beta from the areas or from the side length ratios?
Sine alpha = AD/AC
AI also have to use the fact C' is middle point.
Areas? COsine law?
...."

Tags: math, mathematics

Example of failed problem solving

My intention for today's post was to solve a problem, and write down my thinking process, so it would be an example of problem solving process.

But I failed to solve the problem!

I went to this site first but the problems there looked routine, plus I didn't want to choose a calculus problem but keep it somewhat simple.

Then I went to MathsChallenge.net and looked at their one-star problems... they seemed kind of easy, I was able to "see" the solution immediately. I wanted a problem that I personally could not solve just by reading it.

I finally settled on a two-star geometry problem.

It has a right triangle and lines drawn in it, and you're supposed to prove that a certain angle is equal to another angle.

The details of this are not important so I'm not going to post here all that I typed... (I tried to type in my thinking process while solving it). I quickly saw some right triangles there, and that instead of proving the angles equal, I could try prove a certain line segment to have the same lenght as another line segment.

Then I tried sines, a bit of algebra, similar triangles, all sorts of things... I tried something, then would think "this doesn't work", and try something else. The problem seemd to be so simple so I was looking for a simple solution and didn't want to try
"heavy tools" such as cosines and sines.

After a long while (well it seemed like it, maybe 1/2 hour) I thought it was getting too lenghty, because after all, I cannot take hours to make a single blogpost. So I clicked on the solution.

It had a CIRCLE! Seeing the circle drawn in the picture, I saw the solution immediately.

It was kind of a "think-out-of-the-box" problem. There was no circle draw in the problem, yet the solution had it.

So from now on, I will not forget to think about circles when solving geometry problems, even if the problem only shows me triangles or whatever.

---

The morale of this story? Choose one:

*Do not ask Mrs. Miller any geometry problems - she can't solve them.
*Do not use MathsChallenge.net two-star geometry problems for your word problems - you might get stumped and get a low self-esteem for the rest of your life. (just joking! :^)
*Do not use geometry problems as examples of word problems since they are not word problems!

---

Well, actually the "take home messages" I want you to remember are these:

• Do NOT give up at non-routine problems too soon.


• Monitor your progress: expert problem solvers keep 'mental track' of whether they're getting closer of the solution. If on wrong path, they abandon it and start with something different.


• After struggling for a while, even if you do 'cheat' and look at the solution, the time was not lost - you often learn something valuable that you don't forget easily.

---

And here's part of the "failed attempt" thinking process... if you're interested:

"...I remember from recent stuff that triangles ACC' and BCC' have same area - if that happens to help any.

don't know if the angles aplha etc are related to the areas of the three subtraingles.

Or if, since C is right angle and ABC is a right triangle, woudl it happen to be similar to one ofthe subtriangles?
Well they share side AC (that won't matter) but they SHARE an angle (angle A). So yes they rae similar.

So angle alpha = angle B.
So if alpha = gamma (supposed to rpove that) then triangle CC'B is isosceles.

But converse also true; if I can prove triangle CC'B isosceles, from which follwos that gamma = B.

Thinking some more while keenly looking at the pic. How could i prove those two sides are equal lenght? Could I prove CC' = C'A ? That would be isosceles triangle there too (ACC')

Thinking again of the areas of triangles... since two of them are equal./.. using other kind of height..?

I might be on wrong path. Look again at the angles alpha gamma beta. Since alpha = angle B, then beta + gamma = 90- alpha = angle A. There is another pair of similar trianlges ther e(CDB and the big one). Hee should have found that earlier.

Would simple algebra help?

beta + gamma = A = 90 -alpha.
alpha = angle B

B+A =90 so B =
Substitue

beta + gamma = A = 90 -alpha.
alpha = 90-A

need to prove alpha = gamma so get rid of beta please.

No this is not getting anywhere.

The righgt triangles might still lead to the answer.
How about sine theorem:

sine alpha/AD = sine 90/AC

sine gamme /C'B =sine B/CC'

Sine B/AC =sine 90/AB = sine 90/2C'B

sine 90 is 1.

substitute

sine alpha/AD = 1/AC => sine alpha = AD/AC. Heh knew this.

sine gamme /C'B =sine B/CC' = >Sine gamma = sine B/CC' * C'B.

Sine B/AC =1/AB = 1/2C'B
Sin B = AC/AB and is same as sine alpha


Sine alpha =
Not getting anywhere! I feel embarrassed.

Somehow get alpha/beta from the areas or from the side length ratios?
Sine alpha = AD/AC
AI also have to use the fact C' is middle point.
Areas? COsine law?
...."

Tags: math, mathematics

The problem with story problems

I was trying to think today why it is that so many kids feel that word problems or story problems are difficult.

It didn't make sense to me initially. See,

• Most kids just love stories.
• Usually kis love words, too, based on the fact they use them a lot.
• And problems - I can't imagine that kids don't like word problems just because they need find an answer to something. Most of us even adults get fascinated by puzzles, for example.

So what is the problem with word problems?

It surely can't start on 1st grade. You know, someone tells you a story problem such as: There are five ducks on the lake and three on the shore. How many ducks are there total? And often the math book has a nice picture there to accompany it. Surely kids don't think that as being difficult.

My child has gotten to like "subtraction stories" pretty well - just simple situations where someones or some things go away. She has even made up some herself.

Could it be that they don't understand the language? Or, that they are hurried to solve them too quickly?

Please send in your thoughts on this. Also, what is the advice you most often hear on word problems?

And lastly, go check Snowmen decorations challenge problem from Aunty Math's collection. Reading through it, do you think kids would be able to feel anything else but motivated/excited to try solve it? (It's for grades K-5)

Tags: math, philosophy, elementary

The problem with story problems

I was trying to think today why it is that so many kids feel that word problems or story problems are difficult.

It didn't make sense to me initially. See,

• Most kids just love stories.
• Usually kis love words, too, based on the fact they use them a lot.
• And problems - I can't imagine that kids don't like word problems just because they need find an answer to something. Most of us even adults get fascinated by puzzles, for example.

So what is the problem with word problems?

It surely can't start on 1st grade. You know, someone tells you a story problem such as: There are five ducks on the lake and three on the shore. How many ducks are there total? And often the math book has a nice picture there to accompany it. Surely kids don't think that as being difficult.

My child has gotten to like "subtraction stories" pretty well - just simple situations where someones or some things go away. She has even made up some herself.

Could it be that they don't understand the language? Or, that they are hurried to solve them too quickly?

Please send in your thoughts on this. Also, what is the advice you most often hear on word problems?

And lastly, go check Snowmen decorations challenge problem from Aunty Math's collection. Reading through it, do you think kids would be able to feel anything else but motivated/excited to try solve it? (It's for grades K-5)

Tags: math, philosophy, elementary

Howl's Moving Castle Papercraft

One of my favorite movies from Hayao Miyazaki is finally on DVD, Howl's Moving Castle ( Hauru no ugoku shiro) was an Oscar nominee for Best Animated Feature Film for 2005. This is probably one of the best free papercraft models in my collection and it took me four days to complete it. The model is highly detailed and has moving parts, particularly the wings, this is considered a paper craft model and at the same time a paper automata because it's got the mechanisms for movement. If you haven't seen the movie, I highly recommend you see it, buy or rent it on DVD. This Howl's Moving Castle papercraft is hosted by Epson Japan and is available for both Windows and Mac users. The files come in two parts, a guide and the pattern itself and is really huge, the pattern is 54MB and the guide is 15MB for Windows users, it's even bigger for the MacOS crowd.
This Howl's Moving Castle papercraft was printed on A4 paper by a Canon Pixma iP3000, If your having a hard time understanding the instruction guide(Japanese), you can download an English translation for it below, courtesy of rose-rote. Check out some detailed photos I took of my model at my Flickr page.

Howl's Moving Castle Papercraft [via Heywink]

Howl's Moving Castle Papercraft Photos [via Flickr]
Howl's Moving Castle Papercraft - English Instructions [via Mediafire]

Addition fact flashcards

This morning, my baby found a set of addition flashcards I had bought at some yardsale, and scattered them on the floor.

While picking them up, I was lamenting in my mind the fact how they are not usable with my other daughter. The set has 32 cards so obviously it doesn't cover all basic addition facts. It seems to have just random ones, such as 7 + 6 = __ or 8 + 4 = __ or 2 + __ = 2 or 1 + __ = 5 all mixed together.

But what I'd want for right now is those cards only where the sum is five:
0 + __ = 5 and 5 + __ = 5
1 + __ = 5 and 4 + __ = 5
2 + __ = 5 and 3 + __ = 5

That's the approach I have used in my ebook Addition 1: to practice the facts in small groups, grouping them by the sum.

First you could study those facts where the sum is 4, then those with sum 5, and so on.

That would make more of a logical approach to all this fact practicing. You know, everybody understands that you're supposed to memorize multiplication tables by the tables and not by 'all at once' or in random order.

So obviously some sort of organization is good when practicing addition facts, too.

Did your math book have that?

Addition fact flashcards

This morning, my baby found a set of addition flashcards I had bought at some yardsale, and scattered them on the floor.

While picking them up, I was lamenting in my mind the fact how they are not usable with my other daughter. The set has 32 cards so obviously it doesn't cover all basic addition facts. It seems to have just random ones, such as 7 + 6 = __ or 8 + 4 = __ or 2 + __ = 2 or 1 + __ = 5 all mixed together.

But what I'd want for right now is those cards only where the sum is five:
0 + __ = 5 and 5 + __ = 5
1 + __ = 5 and 4 + __ = 5
2 + __ = 5 and 3 + __ = 5

That's the approach I have used in my ebook Addition 1: to practice the facts in small groups, grouping them by the sum.

First you could study those facts where the sum is 4, then those with sum 5, and so on.

That would make more of a logical approach to all this fact practicing. You know, everybody understands that you're supposed to memorize multiplication tables by the tables and not by 'all at once' or in random order.

So obviously some sort of organization is good when practicing addition facts, too.

Did your math book have that?

10th Carnival of Homeschooling

10th Carnival of Homeschooling is online at PalmTree Bundit. Anne is celebrating spring!

10th Carnival of Homeschooling

10th Carnival of Homeschooling is online at PalmTree Bundit. Anne is celebrating spring!

What to do when math gets dry

Have you ever felt like your child needed something different for math than whatever you were currently doing? Like your math program didn't work anymore: your child didn't learn, or was bored? What to do?

Here a little while back I received a question along those lines. We exchanged a few emails and I got permission to post her situation here on this blog. I think it is a good example of what many times happens in home school.

"We have 4 children; boy-12, girl-8, girl-5 and boy-3. With oldest we have used the gamut of curricula for math and eventually settled out with Abeka for the past 3 years.

However, the material seemed to not sufficiently explain new material, but run old material in the ground until the grinding sensation was numbing. That in itself seems to be contradictory, but my son was really not getting much better at the basics, mainly I think because he was bored with the repetition.

I narrowed his assignments down to every other one and if I found he missed a problem, he had to do the complete set of that type of problem. He was terribly slow in doing his assignments and spacing out everytime I looked over toward him.

I found your e-books somehow during a search for explaining some concept in decimals. He needed better
explanation than what his text was offering and I could only put it so many
ways.

The samples seemed promising and the price was worthwhile so we ordered the Decimal book. His interest, comprehension and speed picked up noticeably. I ordered the Multiplication one for my daughter then.

Now we are nearly finished with the Geometry one we ordered when he finished the decimals book. He really likes the Geometry book and I wish there were more to go on to such as Pre-Algebra and further.

His ability to do basic math seems to be fine, it was just he wasn't focusing or interested and making silly mistakes all the time. The break away and into the two books we did from you was what he needed!
Sonya

I then suggested to her a change in curriculum, perhaps to Singapore or Math-U-See. Or, giving him more responsibility in completing his lessons. I also suggested RightStart Geometry since he seemed to like geometry, and Developmental Math workbooks, or Key To... workbooks.

She responded back:

We've pretty much done the gamut of approaches. He is responsible for his lessons, but math is the one thing that zones him out. We've done Singapore and he did like it, but I didn't think it was complete and took too long to get to certain concepts that are typically achieved at a certain level here.

The first math curriculum we tried was Math-U-See and we use it to introduce math and reinforce certain concepts like place value. I still had the same complaint with it as I did the Singapore.

I look forward to looking into the curriculum you sent a link to [meaning RightStart Geometry]. I am not really looking for a new curriculum though, but rather something more like your individual books covering, and covering well, the individual concept levels. I was given some math software awhile ago and looked into its approach this morning. It may be helpful for now. Thanks for answering my questions and trying to help, its appreciated! Sonya

Then, as of last week, she said, "...we've had a bit of a break and been experimenting with some software and online resources. We have not settled on any one thing yet."

THE MORALE... Sometimes a change in approach or material is beneficial. Or even just a break from math. I've written along these lines before, in my article How to choose a homeschool math curriculum.

You don't always need to hunt for a new curriculum, though. Internet is full of all sorts of math tutorials, as you can imagine. And learning games.

Remember the principle of "variable learning": use various materials or resources to teach the same thing. When the child encounters the same topic several times with slightly different wordings, it helps to solidify his knowledge.

How about using some software or interactive online math curriculum to supplement the existing curriculum? The child could get instruction from those lessons, then read the corresponding material in the existing textbook and do SOME practice problems found in there.

Parents don't always realize this, but math textbooks include lots of practice problems to accommodate all kinds of learners. You certainly don't have to have your child do all of them.