Fraction problem with mental math

Someone asked me,

How many times does 3/4 fit into 15 6/8?

Here are two ways to solve this:

1) These numbers look awkward, but if I changed them to easier ones, for example
"How many times does 2 fit into 834", then we'd all soon realize that we need to use DIVISION.

So the original problem is solved by fraction division:

15 6/8 ÷ 3/4

Are you ready? Remember how to divide fractions?

---

2) But wait a minute! These numbers aren't so difficult after all... because 6/8 equals 3/4. Let's use this thinking cap of ours - mental math.

3/4 goes into 1 1/2 two times. Doubling that, we find 3/4 goes into 3 four times. And so to 15... fives times that: 3/4 goes into 15 4 × 5 or 20 times!

And, of course 3/4 goes into 6/8 exactly one time.

So all total 3/4 goes into 15 6/8 exactly 21 times. No leftovers. And that was easy!

---

On another note, Denise in Illinois has made up a mnemonic poem for kids to remember better the "invert and multiply" rule.

Fraction problem with mental math

Someone asked me,

How many times does 3/4 fit into 15 6/8?

Here are two ways to solve this:

1) These numbers look awkward, but if I changed them to easier ones, for example
"How many times does 2 fit into 834", then we'd all soon realize that we need to use DIVISION.

So the original problem is solved by fraction division:

15 6/8 ÷ 3/4

Are you ready? Remember how to divide fractions?

---

2) But wait a minute! These numbers aren't so difficult after all... because 6/8 equals 3/4. Let's use this thinking cap of ours - mental math.

3/4 goes into 1 1/2 two times. Doubling that, we find 3/4 goes into 3 four times. And so to 15... fives times that: 3/4 goes into 15 4 × 5 or 20 times!

And, of course 3/4 goes into 6/8 exactly one time.

So all total 3/4 goes into 15 6/8 exactly 21 times. No leftovers. And that was easy!

---

On another note, Denise in Illinois has made up a mnemonic poem for kids to remember better the "invert and multiply" rule.

Math Mammoth Measuring worksheets ready

It seems lately that I'm bogged down by all the work with my books. One of the newest developments was I finally got online and ready two worksheets collection in the new "green" series:

Math Mammoth Measuring Worksheets


Math Mammoth Decimals Worksheets

The books in the green series are worksheets by topic, instead of by grade. The worksheets have been pulled out from the grade 3, 4, 5, and 6 collections (so contain duplicate material with those).

These per topic collections, I feel, will be very helpful for teachers who need worksheets on some particular topic, but with varying difficulty levels.

Check out the freebie sample sheets on the two pages above!

Math Mammoth Measuring worksheets ready

It seems lately that I'm bogged down by all the work with my books. One of the newest developments was I finally got online and ready two worksheets collection in the new "green" series:

Math Mammoth Measuring Worksheets


Math Mammoth Decimals Worksheets

The books in the green series are worksheets by topic, instead of by grade. The worksheets have been pulled out from the grade 3, 4, 5, and 6 collections (so contain duplicate material with those).

These per topic collections, I feel, will be very helpful for teachers who need worksheets on some particular topic, but with varying difficulty levels.

Check out the freebie sample sheets on the two pages above!

Ma and Pa Kettle math

A little fun video clip where Pa and Ma Kettle prove themselves good 'mathematicians'! He "proves" by long division that 25 ÷ 5 = 14, and she "proves" by multiplication algorithm that 5 × 14 = 25.

Ma and Pa Kettle math

A little fun video clip where Pa and Ma Kettle prove themselves good 'mathematicians'! He "proves" by long division that 25 ÷ 5 = 14, and she "proves" by multiplication algorithm that 5 × 14 = 25.

Have algebra books changed?

Recently I've gotten started with the project of writing algebra worksheets for Spidersmart tutoring company.

To guide me, I have table of contents from one book many of their students are using, plus two algebra 1 books I have at home.

The one is Algebra 1 by Houghton Mifflin, from 1989 & 1986.

The other is Merrill Algebra 1 Applications and Connections by Glencoe / McGraw Hill, 1995 & 1992.

So in essence there's 6 years in between the first publications of these books. But the books are quite different.

I've been wondering if it has been a general trend among algebra books (I don't know), or I just happened to use two very different books.

The first is black-and-white with red, the second is full color and sprinkled with photographs.

But the main difference is how much more advanced mathematically the first book is.

I've been especially looking at the second "chapters" or parts, where both books practice the four basic operations and their properties.

The Houghton Mifflin book talks about axioms and theorems, and proves a few properties of real numbers, such as (a + b) + (−b) = a, or asks the student to supply reasons for steps of proof in proving a(−1) = −a or a(b − c) = ab − ac, and even asks students to do such proofs.

It also has plenty of "easy" calculation exercises, but each lesson also has some more challenging problems, such as proofs.

The latter book has Critical Thinking problems, and easy word problems following the typical calculation problems.

I'm not saying either book is bad; it seems both are well-made, just a little different. However I would guess that axioms and proving are difficult topics for an eight- or ninth-grader.

But did all algebra 1 books used to be that way in the past? Do you know? Is algebra easier today than it was in the past?

---

Also an interesting phenomenon is that nowadays you don't have to be a prominent textbook author or mathematics professor to write an algebra book.

You've probably heard of Teaching Textbooks - especially designed for homeschoolers. These products supply FULL solutions to EVERY single problem in the book. It's a massive amount of data. So this way a homeschooling parent or student can never get 'stuck' in a problem.

It says on their site that "Teaching Textbooks was founded by two brothers, Shawn and Greg Sabouri (a Harvard graduate and former Harvard math tutor). "

I also recently ran across another Algebra 1 in workbook format written by Christy Walters - apparently a tutor as well.

So while I'm making algebra worksheets, I've kept thinking if I would ever write an algebra book... Not this year for sure. It certainly would be a challenge.

Even while making algebra problems only, I constantly face challenges. I have to constantly think about the order of topics and how they relate to each other, and how mature/advanced or how easy to make the problems to suit today's audience... I don't know. I hope to get feedback on these some day, after they're available.

Have algebra books changed?

Recently I've gotten started with the project of writing algebra worksheets for Spidersmart tutoring company.

To guide me, I have table of contents from one book many of their students are using, plus two algebra 1 books I have at home.

The one is Algebra 1 by Houghton Mifflin, from 1989 & 1986.

The other is Merrill Algebra 1 Applications and Connections by Glencoe / McGraw Hill, 1995 & 1992.

So in essence there's 6 years in between the first publications of these books. But the books are quite different.

I've been wondering if it has been a general trend among algebra books (I don't know), or I just happened to use two very different books.

The first is black-and-white with red, the second is full color and sprinkled with photographs.

But the main difference is how much more advanced mathematically the first book is.

I've been especially looking at the second "chapters" or parts, where both books practice the four basic operations and their properties.

The Houghton Mifflin book talks about axioms and theorems, and proves a few properties of real numbers, such as (a + b) + (−b) = a, or asks the student to supply reasons for steps of proof in proving a(−1) = −a or a(b − c) = ab − ac, and even asks students to do such proofs.

It also has plenty of "easy" calculation exercises, but each lesson also has some more challenging problems, such as proofs.

The latter book has Critical Thinking problems, and easy word problems following the typical calculation problems.

I'm not saying either book is bad; it seems both are well-made, just a little different. However I would guess that axioms and proving are difficult topics for an eight- or ninth-grader.

But did all algebra 1 books used to be that way in the past? Do you know? Is algebra easier today than it was in the past?

---

Also an interesting phenomenon is that nowadays you don't have to be a prominent textbook author or mathematics professor to write an algebra book.

You've probably heard of Teaching Textbooks - especially designed for homeschoolers. These products supply FULL solutions to EVERY single problem in the book. It's a massive amount of data. So this way a homeschooling parent or student can never get 'stuck' in a problem.

It says on their site that "Teaching Textbooks was founded by two brothers, Shawn and Greg Sabouri (a Harvard graduate and former Harvard math tutor). "

I also recently ran across another Algebra 1 in workbook format written by Christy Walters - apparently a tutor as well.

So while I'm making algebra worksheets, I've kept thinking if I would ever write an algebra book... Not this year for sure. It certainly would be a challenge.

Even while making algebra problems only, I constantly face challenges. I have to constantly think about the order of topics and how they relate to each other, and how mature/advanced or how easy to make the problems to suit today's audience... I don't know. I hope to get feedback on these some day, after they're available.

Quite funny: Proof by...

By way of Natural Blogarithms, we can learn some "useful" proof techniques...

For example:

"Proof by vigorous handwaving: Works well in a classroom or seminar setting.

Proof by forward reference: Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.

Proof by funding: How could three different government agencies be wrong?"

There's much more!

Quite funny: Proof by...

By way of Natural Blogarithms, we can learn some "useful" proof techniques...

For example:

"Proof by vigorous handwaving: Works well in a classroom or seminar setting.

Proof by forward reference: Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.

Proof by funding: How could three different government agencies be wrong?"

There's much more!

Interview with an astronomer

You might think this is slightly off topic, but I got the opportunity to conduct an email 'inverview' with a NASA astronomer Dr. Sten Odenwald who is heavily involved in math and science education.

Sten has been writing weekly space-related math problems for grades 9-11. You can find those here.

He has also studied space weather, which is quite interesting to learn about.



And here's the link to the interview.

Interview with an astronomer

You might think this is slightly off topic, but I got the opportunity to conduct an email 'inverview' with a NASA astronomer Dr. Sten Odenwald who is heavily involved in math and science education.

Sten has been writing weekly space-related math problems for grades 9-11. You can find those here.

He has also studied space weather, which is quite interesting to learn about.



And here's the link to the interview.

Problem on proportional reasoning

I'm just going to solve here on the blog another math problem that was sent in to me... Hopefully it helps some of you to learn how to solve problems (you can let me know!)

A man travelled 8 miles in the second hour. This is 1/7 times more than during the first hour, and 1/4 times more than he travelled during the third hour. What is the total miles he covered in three hours?

In this problem, it is easy to get "deceived" and think that you'd just go 1/7 × 8 miles or something like that.

But think first; did the man cover MORE miles during the first hour than during the second hour?

Yes; it says plainly that the 8 miles was 1/7 times MORE than what he covered during the first hour.

Is that 8 miles a LOT more, or a LITTLE BIT more than what he traveled during the first hour?

It's 1/7 times more, so it's a little bit more.

So simply mark as x the miles covered during the first hour. Then,

8 miles = 1 1/7 x

It's not 1/7 x, but 1 1/7 x, or 8/7x. If you put down 1/7 x, you're finding the seventh part, but it was seventh part MORE.

So 8 miles = 8/7x

Multiply both sides by 7, and divide by 8:

x = 7 miles.

Then the last hour. Again the miles traveled during the second hour are more than the miles traveled during the third hour. If miles traveled during the third hour are y, then we get,

8 miles = 1 1/4 y

8 miles = 5/4 y

y = 32/5 miles, or 6 2/5 miles.

Total he covered 7 mi + 8 mi + 6 2/5 mi = 21 2/5 miles or 21.4 miles.

Problem on proportional reasoning

I'm just going to solve here on the blog another math problem that was sent in to me... Hopefully it helps some of you to learn how to solve problems (you can let me know!)

A man travelled 8 miles in the second hour. This is 1/7 times more than during the first hour, and 1/4 times more than he travelled during the third hour. What is the total miles he covered in three hours?

In this problem, it is easy to get "deceived" and think that you'd just go 1/7 × 8 miles or something like that.

But think first; did the man cover MORE miles during the first hour than during the second hour?

Yes; it says plainly that the 8 miles was 1/7 times MORE than what he covered during the first hour.

Is that 8 miles a LOT more, or a LITTLE BIT more than what he traveled during the first hour?

It's 1/7 times more, so it's a little bit more.

So simply mark as x the miles covered during the first hour. Then,

8 miles = 1 1/7 x

It's not 1/7 x, but 1 1/7 x, or 8/7x. If you put down 1/7 x, you're finding the seventh part, but it was seventh part MORE.

So 8 miles = 8/7x

Multiply both sides by 7, and divide by 8:

x = 7 miles.

Then the last hour. Again the miles traveled during the second hour are more than the miles traveled during the third hour. If miles traveled during the third hour are y, then we get,

8 miles = 1 1/4 y

8 miles = 5/4 y

y = 32/5 miles, or 6 2/5 miles.

Total he covered 7 mi + 8 mi + 6 2/5 mi = 21 2/5 miles or 21.4 miles.

Trigonometry: Finding the value of sine Pi/3.

Trigonometry: Finding the value of sine Pi/3.

First we need to remember that the whole circle is 360° and in radians it is 2Pi. So then Pi is 180°, and Pi/3 is 60°.

To find sine of Pi/3, you'd want to have a right triangle with one angle 60°.

Fortunately that is easy to come by; just take an equilateral triangle and draw an altitude to it. You will have two identical 30°-60°-90° triangles.

And yes this is one of the special triangles - also used in drafting, and there are rulers in this shape.



Where on this picture is the 60° angle? Where's the 30° angle?


Now, to get sine 60° one needs side lengths. I made the sides of this equilateral triangle ABC to be 2 units. The side CD is obviously just 1 unit (easy numbers thus far!)

But what about the height h?

Well, that's where we need to dig up the goold ole' Pythagoras. Can't forget him.

You write the equation, h² + 1² = 2²

h² = 2² − 1² = 3.

So taking square roots... h = √3.

Then, to the sine.

Remember sine is a ratio of side lengths; it is the ratio of the OPPOSITE side to the hypotenuse.

....and soon you will have the answer: sine 60° is _____ (fill in the blank.)

So it was easy, just using the very basics of trigonometry.

However, I wouldn't memorize the result. Just remember the idea HOW it was derived; and you can derive it when you need it (such as in a test).

Read also:
Special Right Triangles

Trigonometry: Finding the value of sine Pi/3.

Trigonometry: Finding the value of sine Pi/3.

First we need to remember that the whole circle is 360° and in radians it is 2Pi. So then Pi is 180°, and Pi/3 is 60°.

To find sine of Pi/3, you'd want to have a right triangle with one angle 60°.

Fortunately that is easy to come by; just take an equilateral triangle and draw an altitude to it. You will have two identical 30°-60°-90° triangles.

And yes this is one of the special triangles - also used in drafting, and there are rulers in this shape.



Where on this picture is the 60° angle? Where's the 30° angle?


Now, to get sine 60° one needs side lengths. I made the sides of this equilateral triangle ABC to be 2 units. The side CD is obviously just 1 unit (easy numbers thus far!)

But what about the height h?

Well, that's where we need to dig up the goold ole' Pythagoras. Can't forget him.

You write the equation, h² + 1² = 2²

h² = 2² − 1² = 3.

So taking square roots... h = √3.

Then, to the sine.

Remember sine is a ratio of side lengths; it is the ratio of the OPPOSITE side to the hypotenuse.

....and soon you will have the answer: sine 60° is _____ (fill in the blank.)

So it was easy, just using the very basics of trigonometry.

However, I wouldn't memorize the result. Just remember the idea HOW it was derived; and you can derive it when you need it (such as in a test).

Read also:
Special Right Triangles

Curriculum development news article

Just sort of interesting:

Can Less Equal More? - Proposal to teach math students fewer concepts in greater depth has divided Md. educators.

Maryland currently has between 50 and 60 math objectives for each grade.

"The State Department of Education is now meeting with math supervisors in each jurisdiction around the state to get a consensus on whether they should follow the Focal Points."

If you remember, Curriculum Focal Points is a fairly new document released by the NCTM. It identifies three most important subject areas or 'focal points' for each grade.

Note it idenfities ONLY THREE major objectives per grade. Contrast that to typical state math curriculum objectives list...

(I don't think they mean that's the only mathematics content for each grade, just that those are the most important critical areas that teachers should put a lot of emphasis on.)

Like I've said before, the focal points document is good reading for teachers and homeschoolers alike. You can try keep those three major goals or focal points in mind as you go thru school year and plan your lessons and various activities.

See also my earlier post on this.

Curriculum development news article

Just sort of interesting:

Can Less Equal More? - Proposal to teach math students fewer concepts in greater depth has divided Md. educators.

Maryland currently has between 50 and 60 math objectives for each grade.

"The State Department of Education is now meeting with math supervisors in each jurisdiction around the state to get a consensus on whether they should follow the Focal Points."

If you remember, Curriculum Focal Points is a fairly new document released by the NCTM. It identifies three most important subject areas or 'focal points' for each grade.

Note it idenfities ONLY THREE major objectives per grade. Contrast that to typical state math curriculum objectives list...

(I don't think they mean that's the only mathematics content for each grade, just that those are the most important critical areas that teachers should put a lot of emphasis on.)

Like I've said before, the focal points document is good reading for teachers and homeschoolers alike. You can try keep those three major goals or focal points in mind as you go thru school year and plan your lessons and various activities.

See also my earlier post on this.

Microscope giveaway

No, this is not my giveaway but is happening at HsLaunch.com... they're trying to launch a new homeschooling website with a bang, and have a microscope giveaway going on, till 15th of January.

Feel free to spread the word.

This actually reminds me that I was planning to have a giveaway thingy too, in the near future... Keep tuned.

Microscope giveaway

No, this is not my giveaway but is happening at HsLaunch.com... they're trying to launch a new homeschooling website with a bang, and have a microscope giveaway going on, till 15th of January.

Feel free to spread the word.

This actually reminds me that I was planning to have a giveaway thingy too, in the near future... Keep tuned.

It's 2007!

Welcome to 2007!

Do you have any expectations for this year?
For my websites and books, I am expecting growth. I've put a lot of effort, and still am, in developing the Math Mammoth books.

But alongside those, I keep building the sites as well.

Yesterday I wrote an article explaining the square root concept for my MamutMatematicas.com site (in Spanish). It took quite some time... I hope it turned out okay.

My goal was to just write real clear and easy explanation of the square root. If you happen to know Spanish, go check it out and let me know how it reads :
¿Que es la raíz cuadrada y cómo se la calcula?

I've noticed that in Spanish, there is NOT the same wealth of mathematical information as you can find in English. We're just about flooding in English math related websites, I sometimes feel. But not in Spanish ones.

Here's some more... (of those English math websites):

The Evolution of Real Numbers
An excellent tutorial on the difference of rational and irrational numbers and how the thinking on those has "evolved" in history; covers topics such as ratio of natural numbers, continuous versus discrete, unit fractions, measurement, common measure, squares and their sides etc.

Math-Kitecture.com
Architecture and math resources. Students learn estimation, measuring skills, proportion, and ratios by hand-drafting a floor plan of their classroom to scale.

A math GAME for 2007 - Write expressions for each of the counting numbers 1 through 100, using the digits in the year 2007, standard operations, and grouping symbols. This puzzle is good for students in grades 3-12 with a general knowledge of mathematics.

Oh, yes and one more thing. If you followed the "tag along with me" conversation, I asked people to guess where I was from... not many guessed, but anyway the right answer is Finland. Here's the uni I went to.

It's 2007!

Welcome to 2007!

Do you have any expectations for this year?
For my websites and books, I am expecting growth. I've put a lot of effort, and still am, in developing the Math Mammoth books.

But alongside those, I keep building the sites as well.

Yesterday I wrote an article explaining the square root concept for my MamutMatematicas.com site (in Spanish). It took quite some time... I hope it turned out okay.

My goal was to just write real clear and easy explanation of the square root. If you happen to know Spanish, go check it out and let me know how it reads :
¿Que es la raíz cuadrada y cómo se la calcula?

I've noticed that in Spanish, there is NOT the same wealth of mathematical information as you can find in English. We're just about flooding in English math related websites, I sometimes feel. But not in Spanish ones.

Here's some more... (of those English math websites):

The Evolution of Real Numbers
An excellent tutorial on the difference of rational and irrational numbers and how the thinking on those has "evolved" in history; covers topics such as ratio of natural numbers, continuous versus discrete, unit fractions, measurement, common measure, squares and their sides etc.

Math-Kitecture.com
Architecture and math resources. Students learn estimation, measuring skills, proportion, and ratios by hand-drafting a floor plan of their classroom to scale.

A math GAME for 2007 - Write expressions for each of the counting numbers 1 through 100, using the digits in the year 2007, standard operations, and grouping symbols. This puzzle is good for students in grades 3-12 with a general knowledge of mathematics.

Oh, yes and one more thing. If you followed the "tag along with me" conversation, I asked people to guess where I was from... not many guessed, but anyway the right answer is Finland. Here's the uni I went to.