Where math comes alive


I received an interesting question from alert reader Ivasallay a couple of days ago … about fractions.

Responding to my post about the fraction “hack” of using the gap between fraction numbers, Ivasallay wrote: “What if the numerator is smaller than the gap?”

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Good question, and thanks for sharing it. My answer: Yes, the numerator could be smaller than the gap, and if it is, that can help us simplify fractions, too.

Now we could have a fraction like 15/6, in which the lesser of the two numbers is the denominator, so to keep our discussion general I’m going to talk, not about the numerator, but rather about the “smaller fraction number,” whether numerator or denominator.

The way this matters is as follows: like the gap, the smaller fraction number provides an upper limit, a greatest possible value, for the GCF of the fraction’s two numbers. So if the fraction is 12/90 (smaller number being 12), that means that the GCF can be no larger than 12. If the fraction is 3/1011, with lesser number 3, the GCF can be no larger than 3.

The reason should be obvious, and when I say this I really mean it. Take the fraction 6/792, for example. Could a number larger than 6 go into both 6 and 792? Well there may be a number larger than 6 that goes into 792 evenly, but nothing larger than 6 can go into 6 itself, right? A large peg can’t go through a tiny hole, right? So there you go. Nothing larger than 6 can go into both 6 AND 792. QED.

So what does this mean for you, the math student, or parent of a math student, or the teacher of math students? … I means you want to keep in mind that in actuality two different numbers will help you nail down the size of the GCF. One is the gap between the fraction numbers, and the other is our “new friend,” the smaller of the two fraction numbers.

And here’s another … hack fact. (Whenever I say that, you know we’re heading into ‘nerd-land,’ right?) For both limiting numbers, the gap and the smaller fraction number, the only numbers that can possibly go into both fraction numbers are the FACTORS of those limiting numbers. So for example, if your fraction is 6/50, with the smaller number of 6, the only numbers that can possibly go into 6 and 50 are the factors of 6: i.e., 6, 3, or 2.

A nice rule of thumb:  see which is smaller, the gap or the smaller fraction number. Then use that smaller number as your largest possible GCF. To nail this down, let’s do two example problems.

Example 1:  8/44. What’s smaller? 8 or the gap, 36. Obviously 8! So use 8. Test the factors of 8, which are 8, 4, 2. Notice that 8 doesn’t go into both 8 and 44. But 4 does, so 4 is the GCF, and using 4, the fraction simplifies down to 2/11.

Example 2:  22/36. What’s smaller? 22 or the gap, 14. Here the gap is smaller. So test the gap’s factors: 14, 7, 2. 14 doesn’t go into 22 and 36; nor does 7. But 2 does. So 2 is the GCF, and using 2, the fraction simplifies to 11/18.

Time for you all to try your hands at this fun practice, which catapults your “number sense” to new heights.

For each problem, 1) identify the fraction’s smaller number and the gap. 2) Say which of those two numbers is smaller. 3) Using that number’s factors, find the GCF. 4) Finally, using the GCF, simplify the fraction. Answers follow.

SIMPLIFY THE FRACTIONS:

a)   8/42

b)  12/20

c)  36/60

d)  18/96

e)  21/91

ANSWERS:

a)   8/42:  1)  smaller # = 8; gap = 34.  2)  8 < 34. 3)  GCF = 2. 4)  4/21

b)  12/20:  1)  smaller # = 12; gap = 8.  2)  8 < 12. 3)  GCF = 4. 4)  3/5

c)  36/60:  1)  smaller # = 36; gap = 24.  2)  24 < 36. 3)  GCF = 12. 4)  3/5

d)  18/96:  1)  smaller # = 18; gap = 78.  2)  18 < 78. 3)  GCF = 6. 4)  3/16

e)  21/91:  1)  smaller # = 21; gap = 70.  2)  21 < 70. 3)  GCF = 7. 4)  3/13

Josh Rappaport is the author of five math books, including the wildly popular Algebra Survival Guide and its trusty sidekick, the Algebra Survival Workbook. And FYI:  the 2nd Edition of the Survival Guide was just released in March, so get it while it’s hot off the press! If you’d like to get tutored by Josh, you can. Josh and his remarkably helpful wife, Kathy, use Skype to tutor students in the U.S. and Canada in a wide range of subjects. They also prep students for the “semi-evil” ACT and SAT college entrance tests. If you’d be interested in seeing your ACT or SAT scores soar, shoot an email to Josh, sending it to: josh@SingingTurtle.com  We’ll keep an eye out for your email, and in our office, our tutoring is always ON … except on Saturdays.


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.


Working with one student on functions this morning, I was reminded of how much there is that students can fail to understand.

I was trying to explain to this student that the x-value is the input, and that the f(x) value is the output. But because of the repetition of x in both of these terms, he got confused.

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I finally solved the problem by telling my student to view the parentheses around the x in (x) as like the slot that takes the coin in a vending machine. It sort of looks like a slot, too, right? So what goes inside it must be the “in”put. Now he at least understands clearly that what goes in the ( ) slot is the input.

Staying with the vending machine analogy, I told him that the f(x) is what the function machine (like a vending machine) gives you after you put the x-value in the slot.

I did need to clarify that when you’re working out the value of inputs and outputs, you must insert the x-value twice: once inside the ( ) slot, and secondly on the other side of the equation, in where the ‘x’ stands.

Obviously this student has a lot of trouble with processing the visual symbols of math. But working with him reminds me of something important. It shows how much students can get confused by math concepts and math notation. I feel that it’s important for us educators to keep this in mind as we teach. There’s so much that we take for granted in our understanding of math. But for students who struggle with notation and with the visual aspect of math, notation can be confusing.

One thing I try to do when I work with students comes from something I saw in a Great Courses class by Bruce Edwards, an excellent teacher of higher math. Mr. Edwards likes to say things like, “Now this is next part is a little bit tricky …” Just by saying this, Mr. Edwards shows that he understands that not everyone will get the concept, and that, I believe, helps students relax.

Ever since I saw Mr. Edwards use this way of talking, I’ve been using it in my tutoring work, too. And I find that it helps students. It makes them feel like no one will think badly of them for not understanding, since I, the teacher, have acknowledged that the concept is “tricky.” As a result, students relax, and that helps them be more relaxed in taking in what you’re going to tell them. A nice thing to learn from a master teacher, and another lesson in the importance of the way in which we talk to students to help them learn. There’s so much more to being a good math teacher than just being thorough and clear. The affective aspects of communicating, such as showing empathy, are very important as well.


This is a different kind of post; it deals not with math per se, but rather with the recent surge in standardized testing nationwide and with the controversy surrounding it.

As a tutor I work with a wide range of students, and lately I’ve noticed how much my high school students are talking about the standardized testing at their schools.

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In my home state of New Mexico, many students grew so concerned about the new PARCC standardized tests that they walked out of school and protested, with placards and chants, the “whole nine yards.” I drove down to meet these students at the state capitol building, where the students were voicing their concerns.

Asked about the testing, the students told me that they’ve been told that if they don’t pass the PARCC, they’ll never get their diploma. They said they were told that directly by some of their teachers and counselors. (In the local newspaper this issue is debated, and it seems that there’s no clear consensus as to the implications of not passing the PARCC.)

As a math tutor I’m aware of the math content that is in my students’ curriculum, and I’ve noticed that some of the PARCC exam’s math problems are based on concepts that are not in students’ curriculum. For example, the PARCC’s math section has many problems on statistics, and my students have not been taught statistics through their curriculum.

My concern is that this exam is testing students on topics that are beyond their curriculum, and that the consequences of their failing might be not graduating.

The students told me that if instead of the PARCC they were given the SBA, the test that they’ve taken for many years now, they would not even consider protesting. They said their concern is that the district changed the test they would take rather abruptly from the SBA to the PARCC without giving teachers adequate time to teach students the PARCS’s content, which is largely aligned with Common Core State Standards for Math. Several teachers in Santa Fe have echoed this concern in the local media.

On the other hand, other students don’t seem concerned about the tests. They take it all in stride. And of course, local district officials say that the tests help establish a basis for evaluating both the progress of individual students, and the success of individual schools and districts. They argue that there needs to be a standardized “baseline” so that people and communities can make “apples-to-apples” comparisons of students, schools and districts.

I’m writing this post in hopes of starting a dialog to find out how people around the country feel about the many standardized tests dotting the educational landscape these days. In your opinion, are the tests a good thing, a bad thing? What are your thoughts and feelings on this topic? Please share. It would be great if this blog could be a forum for discussion of this issue.


Is there a quick-and-easy way to find the LCM for three or more numbers … WITHOUT prime factorizing?

Of course! We’ll demonstrate the technique by finding the LCM for 10, 14, 20.

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To begin, use the technique for finding the GCF for 10, 14, 20 that’s shown in my post:  How to Find GCF for 3+ Numbers — FAST … no prime factorizing. If you don’t want to go to that post, no worries. I’ll re-show the technique here.

1st)   Write the numbers from left to right:

……….   10     14     20

[The periods: …… are just to indent the lines. They have no mathematical meaning.]

2nd)  If possible, rip out a factor common to all numbers. The factor 2 is common. So divide the three numbers by 2 [10 ÷ 2 = 5 and 20 ÷ 2 = 10] and show the result below:

2   |       10     14     20
……….   5      7      10

3rd)  At this point, notice there’s no number that goes into the remaining numbers: 5, 7, 10. That means you’ve found that the GCF is the number pulled out, 2. At this point we’re at a crossroads. We’re done finding the GCF, but now we’re at the start of a new process, finding the LCM.
To proceed toward getting the LCM, see if there’s any number that goes into any pair of remaining numbers. Well, 5 goes into 5 and 10. So divide both those numbers by 5 [5 ÷ 5 = 1 and 10 ÷ 5 = 2] , and show the results below:

2   |       10     14     20
5   |         5      7      10
………..  1       7       2

Notice that if there’s a number 5 doesn’t go into, you leave that number as is. So leave the 7 as 7.

4th)  Repeat. See if there’s a number that goes into two of the remaining numbers. Since nothing goes into 1, 7, and 2, we’re done. To get the LCM, multiply all of the outer numbers. That means you multiply the numbers you pulled out on the left (2 and 5), and also multiply the numbers at the bottom (1, 7 and 2). Ignoring the meaningless 1, you have:  2 x 5 x 7 x 2 = 140, and that’s the LCM.

To see the process in more depth, let’s find the LCM for … not three, not four … but five numbers:
6, 12, 18, 30, 36.

1st)   Write the numbers left to right:

………  6     12     18     30     36

2nd)  If possible, rip out a common factor.  2 is common, so divide all by 2 and show the results below:

2     |    6     12     18     30     36
………. 3      6       9     15     18

3rd)  Repeat. See if there’s a number that goes into the five remaining numbers. 3 goes into all, so divide all by 3 and show the results below:

2     |    6     12     18     30     36
3     |    3       6       9     15     18
……..   1       2       3       5       6

4th)  Repeat. See if any number goes into the last remaining numbers. Nothing goes into all of them, so now you get the GCF by multiplying the left-hand column numbers. GCF = 2 x 3 = 6.
Proceeding to find the LCM, look for any number that goes into two or more of the remaining numbers. One such number is 3, which goes into the remaining 3 and 6. Divide those numbers by 3 and leave the other numbers as they are.

2     |    6     12     18     30     36
3     |    3       6       9     15     18
3     |    1       2       3       5       6
……… 1       2       1       5       2

5th)  Interesting! Notice that 2 goes into the two remaining 2s, so pull out a 2 and show the results below:

2     |    6     12     18     30     36
3     |    3       6       9     15     18
3     |    1       2       3       5       6
2     |    1       2       1       5       2
……..   1      1        1       5       1

6th)  We’ve whittled the bottom row’s numbers so far down that finally there’s no number that goes into two or more of them (except 1, which doesn’t help). So we have all the numbers we need to find the LCM. Multiply them together. The left column gives us:  2 x 3 x 3 x 2. The bottom row gives us 1 x 1 x 1 x 5 x 1. Multiply all of those (non-1) numbers together, you get:
2 x 2 x 3 x 3 x 5 = 180, and that is the LCM! Pretty amazing, huh? And no prime factorizing, to boot.

Some people find that this process takes a bit of practice to get used to it. So here are a few problems to help you become an LCM-finding expert!

a)  12, 18, 30
b)   8, 18, 24
c)  15, 20, 30, 35
d)  16, 24, 40, 56
e)   16, 48, 64, 80, 112

And the answers. LCM for each set is:

a)   180
b)   72
c)   420
d)   1680
e)   6720


Suppose you need to find the GCF of three or more numbers, and you’d really prefer to avoid prime factorizing. Is there a way? Sure there is … here’s how.

 

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Example:  Find the GCF for  18, 42 and 96

Step 1)  Write the numbers down from left to right, like this:

………. 18     42     96

[FYI, the periods: …. are there just to indent the numbers. They have no mathematical meaning.]

Step 2)  Find any number that goes into all three numbers. You don’t need to choose the largest such number. Suppose we use the number 2. Write that number to the left of the three numbers. Then divide all three numbers by 2 and write the results below the numbers like this:

2    |  18     42     96
……..  9     21     48

Step 3)  Find another number that goes into all three remaining numbers. It could be the same number. If it is, use that. If not, use any other number that goes into the remaining numbers. In this example, 3 goes into all of them. So write down the 3 to the left and once again show the results of dividing, like this:

2    |  18     42     96
3    |    9     21     48
……… 3      7      16

Step 4)  You’ll eventually reach a stage at which there’s no other number that goes into all of the remaining numbers. Once at that stage, just multiply the numbers in the far-left column, the numbers you pulled out. In this case, those are the numbers:  2 and 3. Just multiply those numbers together, and that’s the GCF. So in this example, the GCF is 2 x 3 = 6, and that’s all there is to it.

Now try this yourself by doing these problems. Answers are below.

a)   18, 45, 108
b)   48, 80, 112
c)   32, 72, 112
d)   24, 60, 84, 132
e)   28,  42, 70, 126, 154

Answers:
a)   GCF =  9
b)   GCF =  16
c)   GCF =  8
d)   GCF =  12
e)   GCF =  14


Yesterday I named a property and claimed that it’s useful. Today I show how to put it to use.

In yesterday’s post I showed what I call the Opposite Differences Property, a property that  tells us that the answer we get when we subtract one number from another, a – b, is the numerical opposite of what we get when we switch the order of the numbers, b – a.

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Coffee, Pi and More

In other words, this property assures us that a – b  =  – (b – a)

A quick example of this with actual numbers:  7 – 4  =  – (4 – 7), since 7 – 4 = 3, and 4 – 7 = – 3.

The property tells us that the two results are mathematical opposites of each other:  3  =  – (– 3)

This is handy to know in general, but it’s especially useful when you’re simplifying complex algebraic fractions.

First, consider a simple algebraic fraction:  (x – y) / (y – x)

The Opposite Differences Property tells us that the numerator’s difference is the opposite of the denominator’s difference. That means that this fraction simplifies to – 1.

Not convinced? Try plugging in a couple of numbers: let x = 9 and let y = 2.

The fraction then gives us: (9 – 2) / (2 – 9), and that simplifies to 7 / – 7. But what is 7/– 7? It’s just – 1.

This holds true no matter what numbers we use; that’s what we mean by calling this a “property.” It works in general.

So we can confidently assert that, in general:  (x – y) / (y – x)  = – 1

[To be mathematically proper, I must add one caveat. This property holds in all situations but one. The variable x cannot equal y, for then we would have 0 / 0, which is “indeterminate.” That’s a fancy term, but it means that it’s value can change depending on the situation. Stuff we don’t need to worry about now.]

But back to the power of the property …  suppose we want to simplify the algebraic fraction:

(x + y) (x – y) / (y + x) (y – x)

Looks complicated, right? But we have the power to wrestle this to the ground.

First, using the Commutative Property, we know that the numerator’s (x + y) term  =  the denominator’s (y + x) term, so those two quantities simplify to 1/1.

And also, using our newly coined Opposite Differences Property, we see that the fraction made up of the other terms:
(x – y) / (y – x) = – 1

So (x + y)(x – y) / (y + x)(y – x)  simplifies to (1) (– 1)  = – 1

As another example, suppose you want to simplify this fraction:

abc (ab – b) /  bc (b – ab)

First, the b and c in the numerator cancel with the b and c in the denominator. So as far as these plain variables are concerned, we’re left with only an ‘a’ in the numerator.

Secondly, the numerator’s (ab – b) is the opposite difference of the denominator’s (b – ab), so that part of the fraction turns into – 1.

Putting the pieces together, we have ‘a’ times – 1, which simplifies to just:  – a

So that, – a, is the beautiful, simplified version of that big, ugly algebraic fraction.

As you do more and more algebraic fraction problems, you’ll find that the Opposite Differences Property comes in handy time and again. Learn it, use it, use it again.

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