Making the fractions in a proportion

"How do you know how to make the fractions in a proportion?
When making them, how do you know where each number goes making the fraction, like which ones go on top of the fraction?"

Well, actually you can choose which quantity will go on top; the proportion WILL work either way!

But, sometimes people are used to always putting certain quantity on top and certain on the bottom. For example, if the question is about speed and the unit is "miles per hour", that tells you that miles go on top, and hours on bottom, because "per" means division (the fraction line).

However, you could still solve the proportion by putting hours in the numerator of the fractions and miles in the denominator, and the calculation will turn out alright.

Or, if the question is about "dollars per pound", then dollars go to the numerator and pounds in the denominator.

Let's look at this problem for example:

A car drives on constant speed. It can go 80 miles in 90 minutes. How long will it take for it to travel 100 miles?

You can make both fractions to be

miles
-----------
minutes

OR

minutes
------------
miles

Let's try the first way:

80 mi           100 mi
--------  =   ---------
90 min            x

To solve, cross multiply and you get 80x = 100 * 90, and then x = 900/8 = 112.5 minutes.

The other way it will be

90 min         x
--------  =  --------
80 mi         100 mi

To solve, cross multiply and you get 80x = 90 * 100

You see, the final equation ends up being the same, no matter which
quantities were on top of the fractions.

HOWEVER, one way is wrong: that is if you but "miles" on top in one
fraction, and "minutes" on top in the other... then you'll get it wrong:

90 min        100 mi
--------  =  -------
80 mi         x min

=> 90x = 100* 80 (WRONG)

Geometry problem: a tiling

UPDATED!

Today I have a geometry problem for you.

Just yesterday I showed you a proportion problem and its solution.

(I want to insert a note here, especially for parents reading this.

Don't feel that you always need to use the problems I present here with your students. Instead, as a homeschooling parent, consider them part of your "math teacher training" that you never got. Reading through problems and their solutions will help you become a better math teacher. Consider yourself an apprentice observing a master completing a task, in this case mathematics problem solving.

It just takes you a minute or two to read this, think it through, try it - and understand (I hope). Your students' or child's book will have similar problems later on when it's time for geometry studies, but if you read these that I present, you will have learned more and can better tackle those problems at that time.)

OK, here we go. Look at the tiling below. Here you can see the tiling in real life, on a floor in Germany.



The original problem asks you about the area of the small square and of the parallelogram, given that the area of the large square is 1 square meter and that the ANGLE between the large and the small square is 45°.

But, I simply ask you (or your student) to DRAW this picture. Just take out blank paper, a ruler, and a protractor, and DRAW it. Can you draw it exactly how it is in my picture?

Solution:

This tiling will work (it will tile the plane), no matter what the size of the "little square". So, it is NOT possible for you to draw it exactly like I did, unless you knew something more about the side lengths.

Here's an example where use the same information, but made the small square smaller than in the previous picture:



To solve the original problem of finding the area of the little square and of the parallelogram, one would need some more information, such as a side length or ratio of side lengths.

SO, all in all, it's a problem with incomplete information to solve it...

However, someone suggested in the comments a solution to the original problem where we assume that the parallelogram is cut into two right triangles by its diagonal. With that additional information, the area calculation would be solvable.

(BTW, For my first picture, I used golden ratio as the ratio of the side of the large square to the side of the small square.)

A certain proportion

I need help explaining how to solve:

 12      36
---- =  ----
 3w      63

Could you solve this if instead of 3w it was x?

Well you can do just that:

 12      36
---- =  ----
 x       63

Now cross multiply and go about it as usual. The answer is x = 21.

But, x is actually 3w. So 3w = 21. So w = 7.

This changing of variables is a very often used "trick" in mathematics.


Alternate solution.

Upon examining the proportion, we notice both the numbers in the first fraction are divisible by 3, and both numbers in the second fraction are divisible by 9.

 12      36
---- =  ----
 3w      63

So, we can write it as follows and simplify both fractions (the right and left sides):

3 * 4        9 * 4
------  =   -------
 3w          9 * 7


 4     4
--- = ---
 w     7

You can't find a simpler proportion than that, obviously w must equal 7.

Free e-book: Number Stories of Long ago

Catch if while you can! HomeschoolFreebie of the day offers TODAY a mathematics book called Number Stories of Long Ago . Tuesday January 20 only.

Alcumus

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

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

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

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

When will I ever need this (math)?

We all know that youngsters often have this question in mind when they go on in their math studies: where or when will I ever need all this?

It's pretty easy and simple to find applications and real-life examples for the usage of the simple, elementary math, such as multiplication tables or decimals and percents. But when we go on to algebra and beyond, this question may be harder to answer.

Here are a few answers you can give as to why study algebra:

• You need it in any occupational field that requires higher education, such as computer science, electronics, engineering, medicine (doctors), trade and commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra. You need algebra to take your SAT test or GED.
  
  
• Algebra also lets you develop logical thinking and problem solving skills. It can increase your intelligence! Actually, studying any math topic can do that (even elementary math), if the mathematics is presented and taught in such a manner as to develop a person's thinking.

Now, it's no big secret that mathematical concepts in algebra and beyond are not needed in every single occupation, especially in those of mostly manual labor. (You can check Math Careers Database for the math requirements of 277 major occupations.)

But, ask your students if they are 100% sure what they are going to do as adults? Most kids in middle school are not. If they are not 100% sure, they'd better study algebra and learn all the math they can so that when they finally have some idea, they won't be stopped from a career because of not having studied algebra, geometry, or calculus. (This has definitely happened to many folks.)

Besides, in today's world many people have to change careers even during their work life. So I would recommend young people to get a good solid basic education, just so they are well prepared and also so they.

Now, beyond these thoughts, there also exist SEVERAL find websites dedicated to these issues, so I want to highlight them here.

The website Future's Channel has short movies, lesson guides, and student worksheets that show how algebra is used in with real word applications, such as roller coasters, banking, rice production, skyscrapers, solar power, and lots more. These video clips are really well made and can motivate students to choose a career where math is used.



Another site to check is Make It Real Learning and their free worksheets. These worksheets focus on answering the question, "When am I ever going to use this?" They also sell worksheets, and as a special BONUS, visitors of this site get 30% off their regular pricing with the coupon code "save30". This applies to ANY worksheet or worksheet bundle sold on the site.

Then, Micron: Math in the Workplace contains a collection of real-world math problems and challenges contributed by a variety of businesses, demonstrating the relevance of math in today's world.

Yet more resources

Career profiles
Essays from authors from a variety of careers for which a background in the mathematical sciences is useful.

Examining How Mathematics is Used in the Workplace
Mathematics in Automobile Production; Proportional Reasoning by Nurses; Modeling the Mathematics of Banking; Mathematical Models as Seen by Biologists; How do Scientists Interpret Graphs?

Mathematical Moments - a series of one-page PDF files that promote appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture. The one-page flyers cover very diverse topics such as findign oil, solving crimes, DNA, brain, traffic, aircraft design, etc.

Where do you need exponents in everyday life? - my article.

Math in Technology
Fully written out examples of how various areas of mathematics are applied to various areas of technology.

Career profiles in mathematical sciences
If you were a math major, what kind of jobs could you end up with?

ExploreLearning open house

If you've been to ExploreLearning before, you know they offer interactive online simulations or demos for science and math, called Gizmos. These gizmos are pretty neat things!

Anyway, I'm just passing this on. They happen to have an OPEN HOUSE, in other words, a completely free 45-day trial. Use promo code OpenHouse09 to take advantage of this trial offer, which ends on January 21st.

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Math Contest at MathNotations

I was asked to pass on the word on this free TEAM-BASED math contest... geared towards middle/high school students. Teachers, take note!

Dave Marain from MathNotations has authored a Math Contest, which will take place on Tuesday February 3rd.

Students in Grades 7-12 may participate. Schools can register one team of up to 6 students to try the contest which will be different from most math competitions:

• Team-based - students will have up to 90 minutes to complete 6 questions (although many teams could finish in 45 minutes)
  
• Calculator permitted
  
• Some multi-part questions and some open-ended questions requiring explanation a la MathNotations investigations
  
• Emphasis on Algebra II although some questions are accessible to Middle Schoolers
• Registration, delivery of questions and submission of official team answers/solutions all done via email
• All students receive a Certificate of Participation and top-scoring schools/students will receive a Certificate of Merit

Here's the link to the info needed to register:
http://mathnotations.blogspot.com/2009/01/first-mathnotations-math-contest.html

It's unclear to me if homeschoolers may participate, if they form a team.

Adding It Up: Helping Children Learn Mathematics

I wanted to pass on this book link to those of you who are interested in the pedagogy of mathematics: Adding It Up: Helping Children Learn Mathematics (2001) by Center for Education (CFE)

A big plus is that you can read the whole book online for free.

This book talks a lot about how the conceptual understanding of basic arithmetic develops in children during grades 1-8. It explains common misconceptions. The book cites a lot of research about how children learn addition, subtraction, multiplication, division, fractions, and proportions. It doesn't deal with geometry or measuring, because the authors decided to concentrate on "number" concepts.

All in all I think it's good basic reading for all of us who teach math. You can read it online by chapters. Here's a direct link to page 187 which starts dealing with single-digit addition. Here's a table comparing various addition and subtraction problem types. And here's a link to the discussion about single-digit subtraction.

Follow this link to the main page of the book, and down further you'll see the links to the various chapters in order to read it online. Or, see the widget on the right.

A little trick for square roots (mental math)

Someone sent me this little mental math trick for square roots. I liked it, didn't know it before, so here goes:

I read your suggestion for calculating square root without a calculator. I teach Math for Elementary Teachers and developmental math courses (algebra) to adults. I feel that the focus should be on understanding the number rather than an exercise in following a memorized algorithm. I suggest you have the student determine the pair of perfect squares the number falls between. For example, if finding the sqrt of 645, it falls between the sqrt of 625 which equals 25 and the sqrt of 676 which equals 26. So the sqrt of 645 has to be between 25 and 26. Where does it fall between? There are 50 numbers between 676 and 625. 645 is 20 numbers beyond 625, so 20/50 = 0.4
So the sqrt of 645 is very close to 25.4
This method provides the student with a process that improves their understanding of numbers without expecting them to memorize an algorithm, and it provides an answer to the nearest tenth.

Andrea S. Levy, Ed.D.

She is referring to my article about the square root algorithm. However, I've never meant that kids would need to learn that long algorithm in school work. In the article, I'm actually advocating the method of finding the approximate square root by "guess and check".

Free algebra worksheets

Usually algebra textbooks provide lots of problems to practice the algebraic concepts and techniques, but some of you may still benefit from resources for free (or mostly so) printable algebra worksheets. Please see the list below, which I've originally compiled for my HomeschoolMath.net site.

Algebra worksheets

Worksheet Builder
Great and free worksheet maker software with nearly 7,000 built-in algebra and geometry questions.
www.jmap.org/JMAP_WORKSHEET_BUILDER_INSTALLATION_FILES.htm

Free Algebra Worksheets from KUTA Software
Free worksheets (PDF) for equations, exponents, inequalities, polynomials, radical & rational expressions and more.
www.kutasoftware.com/free.html

AlgebraHelp.com worksheets
Interactive worksheets that are checked online for most algebra 1 topics.
www.algebrahelp.com/worksheets/

Math.Com algebra worksheets generator
Generate worksheets for: linear equations, systems of equations, and quadratic equations.
www.math.com/students/worksheet/algebra_sp.htm

LessonCorner worksheets
These free worksheets include a few topics such as calculations with polynomials, factoring, and graphing lines.
www.lessoncorner.com/worksheets/

Algebra Fun Sheets
Worksheets that integrate algebra skills with fun activities including sudoku, word finds, riddles, color patterns, crosswords, games, matching cards, etc. A subscription is required.
www.algebrafunsheets.com

About.com Algebra Worksheets
An assorted collection of free algebra worksheets and answers. These pages are not very well organized, but they have lots of worksheets.
math.about.com/od/algebraworksheets/Algebra_Worksheets.htm

Algebra Worksheets from MathWorksheetCenter
Lots of worksheets for over 100 algebra topics. A few are free; most are accessible only by one-year a subscription.
www.mathworksheetscenter.com/mathskills/algebra/

A few fun algebra worksheets
These are for graphing linear equations and linear inequalities.

Online Math Work
Free multiple-choice worksheets for pre-algebra and algebra 1 topics. You can do them online, or copy to a word processor to print.
www.mathonlinework.com

Lastly... my own algebra worksheet collections, which aren't free but there are many free samples:

Math Mammoth Algebra 1 Worksheets Collection
A two-part collection (A and B) of 137 quality algebra worksheets covering all the topics in a typical algebra 1 curriculum. These worksheets are hand-crafted and contain lots of word problems and other variable problems. Free samples available. $11.50.
www.mathmammoth.com/worksheets/algebra_1.php