In the first “Slaying Proportions” post, you learned how to save time by canceling horizontally as well as vertically. In this post you’ll learn how to save even more time with another shortcut. Let’s look at an example to refresh our memory.
Given a proportion such as this:
15 = 5
a 3
most people would do the traditional “cross-multiplying” step, to get:
5 x a = 15 x 3 (the x here is a times sign)
Using the usual steps, the next step is to ÷ both sides by 5, to get:
a = (15 x 3) ÷ 5
But let’s look more closely at this answer expression: (15 x 3) ÷ 5
We can view this expression more clearly if we see he original proportion:
15 = 5
a 3
as containing two DIAGONALS.
One diagonal holds the 15 and the 3; the other holds the ‘a’ and the 5.
Let’s call the diagonal with the ‘a’ the ‘first diagonal.’ And since ‘5’ accompanies ‘a’ in that diagonal, we’ll call 5 the “variable’s partner.”
We’ll call the other diagonal just that, the “other diagonal.”
Now I know you’re getting ‘antsy’ for the shortcut, so just know it’s right around “the bend.”
Using our new terms, we can better understand the expression we got up above:
a = (15 x 3) ÷ 5
The (15 x 3) is the product (result of multiplication) of the “other diagonal,”
and ‘5’ is the “variable’s partner.
So the answer,
(15 x 3) ÷ 5
is simply (and here’s the shortcut):
(product of other diagonal) ÷ by (“variable’s partner.”)
We’ll call this the Diagonal Trick.
The Diagonal Trick saves a BIG STEP. Using it, we no longer need to write out the cross-multiplication product the usual way, as:
5 x a = 15 x 3
Instead, using the Diagonal Trick, we can go straight from the proportion to an expression for ‘a‘:
a = (15 x 3) ÷ 5
Let’s see how the Diagonal Trick works in another proportion, such as:
9 = 45
13 a
What’s the “variable’s partner”? 9.
What’s in the “other diagonal”? 13 and 45.
So using the Diagonal Trick, the answer is this:
a = (13 x 45) ÷ 9
This simplifies to 65, of course. Isn’t it nice not to have to “cross-multiply” any more?
Another nice thing: the Diagonal Trick works no matter where the variable is located in the original proportion. All you need to do is identify the “variable’s partner,” and the “other diagonal,” and then you’re all good go with the Diagonal Trick.
Try a few of these to see how easy and convenient the Diagonal Trick makes it to solve proportions.
PROBLEMS:
1) a = 15
12 36
2) 18 = a
24 4
3) 21 = 75
14 a
ANSWERS (using the Diagonal Trick first):
1) a = (12 x 15) ÷ 36
a = 5
2) a = (18 x 4) ÷ 24
a = 3
3) a = (14 x 75) ÷ 21
a = 50
So c’mon … everything that can be said about simplifying fractions has been said … right?
Not quite! Here’s something that might just be original … a hack to smack those fractions down to size.
Suppose you’re staring at an annoying-looking fraction: 96/104, and it’s annoying the heck out of you, particularly because it’s smirking at you!
But it won’t smirk for long. For you open up your bag of hacks (obtained @ mathchat.me) and …
1st) Subtract to get the difference between numerator and denominator. I also like to call this the gap between the numbers. Difference (aka, gap) = 104 – 96 = 8.
NOTE: Turns out that this gap, 8, is the upper limit for any numbers that can possibly go into BOTH 96 and 104. No number larger than 8 can go into both. And this is a … HACK FACT: The gap represents the largest number that could possibly go into BOTH numerator and denominator. In other words, the gap is the largest possible greatest common factor (GCF).
2nd) Try 8. Does 8 go into both 96 and 104? Turns out it does, so smack the numerator and denominator down to size: 96 ÷ 8 = 12, and 104 ÷ 8 = 13.
3rd) State the answer: 96/104 = 12/13.
Is it still smirking? I think … NOT!
Try another. Say you’re now puzzling over: 74/80.
1st) Subtract to get the gap. 80 – 74 = 6. So 6 is the largest number that can possibly go into BOTH 74 and 80.
2nd) So try 6. Does it go into both 74 and 80? No, in fact it goes into neither number.
NOTE: Turns out that even though 6 does NOT go into 74 OR 80, the fact that the gap is 6 still says something. It tells us that the only numbers that can possibly go into both 74 and 80 are the factors of 6: 6, 3 and 2. This, it turns out, is another … HACK FACT: Once you know the gap, the only numbers that can possibly go into the two numbers that make the gap are either the factors of the gap, or the gap number itself.
3rd) So now, try the next largest factor of 6, which just happens to be 3. Does 3 go into both 74 and 80? No. Like 6, 3 goes into neither 74 nor 80. But that’s actually a good thing because now there’s only one last factor to test, 2. Does 2 go into both 74 and 80? Yes! At last you’ve found a number that goes into both numerator and denominator.
4th) Hack the numbers down to size: 74 ÷ 2 = 37, and 80 ÷ 2 = 40.
5th) State the answer. 74/80 gets hacked down to 37/40, and that fraction, my dear friends, is the answer. 37/40 the final, simplified form of 74/80.
O.K., are you ready to smack some of those fractions down to size? I believe you are. So here are some problems that will let you test out your new hack.
As you slash these numbers down, remember this rule. In some of these problems the gap number itself is the number that divides into numerator and denominator. But in other problems, it’s not the gap number itself, but rather a factor of the gap number that slashes both numbers down to size. So if the gap number itself doesn’t work, don’t forget to check out its factors.
Ready then? Here you go … For each problem, state the gap and find the largest number that goes into both numerator and denominator. Then write the simplified version of the fraction.
a) 46/54
b) 42/51
c) 48/60
d) 45/51
e) 63/77
Answers:
a) 46/54: gap = 8. Largest common factor (GCF) = 2. Simplified form = 23/27
b) 42/51: gap = 9. Largest common factor (GCF) = 3. Simplified form = 14/17
c) 48/60: gap = 12. Largest common factor (GCF) = 12. Simplified form = 4/5
d) 45/51: gap = 6. Largest common factor (GCF) = 3. Simplified form = 15/17
e) 63/77: gap = 14. Largest common factor (GCF) = 7. Simplified form = 9/11
Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. Josh has been tutoring math for more years than he can count — even though he’s pretty good at counting after all that tutoring — and he now tutors students in math, nationwide, by Skype. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada, preparing them for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores rise dramatically, shoot an email to Josh, addressing it to: josh@SingingTurtle.com We’ll keep an eye out for your email, and our tutoring light will always be ON.
Algebra Survival Guide — Second Edition
by Josh Rappaport
illustrated by Sally Blakemore
The Algebra Survival Guide — Second Edition is here, but not yet released to the general public. Now’s your chance to order it at a 25% discount through April 10th. Just go to SingingTurtle.com.
But first, let me tell you about what’s new … a massive, 62-page chapter on advanced story problems.
It’s no secret that algebra gives students the ‘jitters,’ and word problems give them the ‘shakes.’ As a dastardly duo, the word problems of algebra are just about as nerve-wracking as anything in the teenage years.
The Algebra Survival Guide — Second Edition takes a hard look at algebra’s word problems and offers time-tested advice for cracking them. With a new 62-page chapter devoted to these word problems, the new edition tackles the ultimate math nightmares of the puberty years: problems involving rate, time and distance, work performed, and mixture formulas, among others. Added to the pre-existing 20-page introduction-to-story-problems chapter (in the Algebra Survival Guide — 1st Edition), it’s like having a book within a book.
The Algebra Survival Guide — Second Edition also includes:
The many cartoons not only provide well-deserved comic relief for math learners, they also offer a visual way to grasp algebra’s challenging abstractions. Example: The above cartoon illustrates a real-world mixture problem by showing different percent concentrations of paint.
With all of these features, the Second Edition Algebra Survival Guide is ideal for homeschoolers, tutors and students striving for algebra excellence.
The Second Edition also aligns with the current Common Core State Standards for Math, so it’s ideal for today’s teachers, as well. Its content chapters tackle the trickiest topics of algebra:
Properties, Sets of Numbers, Order of Operations, Absolute Value, Exponents, Radicals, Factoring, Cancelling, Solving Equations, and the Coordinate Plane.
So, have some fun learning algebra!
My book, the Algebra Survival Guide, has just won an award from a student tutoring organization in Detroit, Michigan.
The “Front Porch,” of Detroit, Michigan, has given the Algebra Survival Guide its distinguished “Golden Porch Award” for helping “kids in Detroit through their first year of algebra successfully.”
Here is the notification about the award:
GOLDEN PORCH AWARD for Algebra Survival Guide
The eVersion of the Algebra Survival Guide is HERE!
Amazon.com just listed the eBook TODAY, and so you can now get this book for just $9.95, and read it on ANY of these devices:
Kindle / iPhone / iPad / Android / Blackberry / PC / MAC
The standard price of the paperback book is $19.95, but now you can get the eVersion of this book, and have it with you electronically, for HALF the price $9.95!
Algebra Survival Guide, now in convenient eVersion!
The Algebra Survival Guide, which debuted in 2000, has sold more than a quarter of a million copies. It has also garnered a Parents Choice award, and it has been used in school districts all across the country as the cornerstone curriculum for Algebra Professional Development Workshops.
The book is read and used by struggling students, teachers, tutors, homeschoolers, and parents. It is an easy book to read, as it is written in a friendly Q&A conversational style. The companion workbook, soon to be available in an eVersion as well, provides thousands of additional practice problems.
Check this book out on Amazon.com at this site!
My recent posts about “Dear Aunt Sally” have, I hope, shown how dangerous it is to teach the memory trick of Please Excuse My Dear Aunt Sally — at least without some additional explanation.
Today I propose a way to save Dear Aunt Sally, for those of you who still like her.
As you know, the memory sentence is often abbreviated PEMDAS, which stands for: PARENTHESES, the EXPONENTS, then MULTIPLICATION, then DIVISION, then ADDITION, then SUBTRACTION.
The problem with PEMDAS is that it makes kids think they always multiply before dividing, and that they always add before subtracting.
For students attached to PEMDAS, I let them use it, but I have them write it a novel way, so they realize they must pay attention to the left-right orientation of the operation symbols.
In the new way of writing PEMDAS, I put M and D in the same place, separate by the word “or.” Then I do the same for the A and S. So the whole memory device looks like this:
Alternative for PEMDAS
My suggestion is that teachers who like PEMDAS try this and see if your students start making fewer mistakes with the order of operations.
For those of you who never liked PEMDAS on the first place, I recommend that you check out the order of operations as presently in a clearer way, as it is in my Algebra Survival Guide. To get a feel for that, you can download the chapter of the book on Positive and Negative numbers here. Just click where it says: See Sample Chapters of the Algebra Survival Guide and Workbook.
Then you’ll want to get the Survival Guide for the chapter on Order of Operations, which you can do through Amazon.com, here. Sorry, I can’t make everything available for free … I do have a business to run, with new products I’d like to create and make available.
Includes new 62-page chapter on Advanced Word Problems
Comic-style math book teaches how integers work.
Math Tutors, Santa Fe, NM
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