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Invisible Misunderstandings: Square roots of 2 and 3


Would you say that the square root of two is an important number in math? Hmmm … and would you agree that the square root of three, while perhaps not quite so important, is still a quantity whose value students should be able to estimate?

Why not, right? After all, these numbers play key roles in the 30-60-90 and 45-45-90 “special triangles.” And therefore they both appear a lot in geometry, and a great deal in trig. And on top of that, root two, widely believed to be the first irrational number discovered, shows up in a wide range of other math contexts as well.

First letter of a text about the square root o...

square root of 2 w/ "parent" triangle

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How to tell if a Number is Divisible by 8


I’ve explained a number of divisibility rules lately, offering tricks to tell if numbers are divisible by 2, 3, 4, 5, 6 and 7.

There is also a trick for divisibility by 8, and that’s what I’d like to explain in this post.

Essentially the trick for 8 is a lot like the trick for 4. If you’d like to refresh your memory on how that trick works, just go here. (more…)

(Divisibility) Practice Makes Perfect


As the saying goes, practice makes perfect.

And boy is that true in math! Of the standard school subjects, math requires the most practice, if you want to excel at it.

That being the case, this strikes me as a great time to practice the divisibility tricks we’ve just learned.

There are many skill areas where divisibility tricks are useful — solving proportions, factoring polynomials, multiplying fractions — but one of the most obvious is the critical skill of reducing fractions.

So now I’m offering you a chance to practice your divisibility skills for 2, 3, 4, 5 and 6. We will save the trick for 7 till we have a few more tricks “up our sleeves.”

For the following problems, answer these four questions:

1)  Which of these numbers — 2, 3, 4, 5 or 6 — divides evenly into the numerator (NM)?

2)  Do the same for the denominator (DNM).

3)  Then choose the largest number that divides into both NM and DNM. For these problems, this number will be the GCF.

4)  Finally, reduce the fraction by dividing both NM and DNM by this number.

Here’s an example that shows what you’d write:

ex)  24/42

1)  NM:  2, 3, 4, 6
2)  DNM:  2, 3, 6
3)  GCF = 6
4)  Answer:   4/7

NOW TRY THESE PROBLEMS:

a)  20/24
b)  25/40
c)   18/48
d)  26/60
e)  21/72
f)  30/85
g)  36/66
h)  56/92
i)  84/102
j)  99/141

ANSWERS:

a)  20/24
1)   NM:  2, 4, 5
2)  DNM:  2, 3, 4, 6
3)  GCF =  4
4)  Answer:   5/6

b)  25/40
1)   NM:  5
2)  DNM:  2, 4, 5
3)  GCF =   5
4)  Answer:  5/8

c)   18/48
1)   NM:  2, 3, 6
2)  DNM:  2, 3, 4, 6
3)  GCF =  6
4)  Answer:  3/8

d)  26/60
1)   NM:  2
2)  DNM:  2, 3, 4, 5, 6
3)  GCF =  2
4)  Answer:  13/30

e)  21/72
1)   NM:  3
2)  DNM:   2, 3, 4, 6
3)  GCF =   3
4)  Answer:   7/24

f)  30/85
1)   NM:  2, 3, 5, 6
2)  DNM:  5
3)  GCF =  5
4)  Answer:  6/17

g)  36/66
1)   NM:  2, 3, 4, 6
2)  DNM:  2, 3, 6
3)  GCF =  6
4)  Answer:  6/11

h)  56/92
1)   NM:  2, 4
2)  DNM:  2, 4
3)  GCF =  4
4)  Answer:  14/23

i)  84/102
1)   NM:  2, 3, 4, 6
2)  DNM:  2, 3, 6
3)  GCF =   6
4)  Answer:   14/17

j)  99/141
1)   NM:   3
2)  DNM:  3
3)  GCF =  3
4)  Answer:   33/47

Multiplication Trick: x 25


This is a simple trick that anyone can easily learn. It is just a trick for
multiplying a number by 25.

If someone asked you what 25 times 36 equals, you’d probably be tempted
to reach for a calculator and start punching buttons. But remarkably, you’d
probably be able to work it out even faster in your head.

Since 25 is one-fourth of 100, multiplying by 25 is the same thing as
multiplying by 100 and dividing by 4. Or, even more simply:
first divide by 4,
then add two zeros.

Here’s the example:

Problem: 36 x 25
First divide 36 by 4 to get 9.
Then add two zeros to get: 900.
That, amazingly enough, is the answer.

Another example: 88 x 25
First divide 88 by 4 to get 22.
Then add two zeros to get: 2,200.

Now try these problems in your head:

a) 25 x 12
b) 25 x 28
c) 25 x 48
d) 25 x 60
e) 25 x 84
f) 25 x 96

Here are the answers:
a) 300
b) 700
c) 1,200
d) 1,500
e) 2,100
f) 2,400

But, you say, what if the number you start with is not divisible by 4.
No problem. Just use this fact:
if the remainder is 1, that is the same as 1/4 or .25
if the remainder is 2, that is the same as 2/4 or .50
if the remainder is 3, that is the same as 3/4 or .75

So take a problem like this: 25 x 17
dividing 17 by 4, you get 4 remainder 1.
But that is the same as 4.25
Now just move the decimal right two places (same as multiplying by 100)
Answer is: 425

Another example: 25 x 18
dividing 18 by 4, you get 4 remainder 2.
But that is the same as 4.50
Now move the decimal right two places.
Answer: 450

Another example: 25 x 19
dividing 19 by 4, you get 4 remainder 3.
But that is the same as 4.75
Now move the decimal two places to the right.
Answer is: 475

Now try these in your head:
A) 25 x 21
B) 25 x 26
C) 25 x 35
D) 25 x 42
E) 25 x 63
F) 25 x 81

And here are the answers:

A) 525
B) 650
C) 875
D) 1,050
E) 1,575
F) 2,025

How to Survive College Algebra


Why is algebra hard for college students? Let me count the reasons …

First, the college students who struggle with with algebra are usually the same folks who struggled with algebra in high school, only older now. They hated it then, and they dread it now. It didn’t make sense then, and it still doesn’t. So they are already predisposed to struggle with this class from painful past experiences.

Another problem stems from the vocabulary of algebra. The words that are used to describe algebra are — let’s face it — intimidating! Words like: polynomial, quadratic, radicals. This is a specialized language, written in annoyingly polysyllabic Latin. And when you start to dislike a subject it is natural that you start to dislike the vocabulary of that subject. And the vocabulary of algebra is somewhat remote and cold, easy to dislike.

Another problem is the tone of the textbooks that teach algebra. I mean, if you want to make a fire, you’d probably do well to burn an algebra book, for the simple reason that the text on the pages is so DRY. I mean, take a sentence from a typical algebra book, and it will sound like this (actual quote):
“The difference between two integers is defined as the absolute value of of the difference of the absolute value of the integers.” I mean, all you’d have to do is pull out a match. It will virtually light itself when placed next to this kind of prose.

The final reason has to do with the teachers who taught algebra back in high school. Now I am not picking on all teachers, but the ones I hear about over and over in my tutoring are those that droned on and on, “Then you subtract 17 from both sides, and finally you divide both sides by 3 …” just like the textbooks, never trying to make the ideas come to life. If you have a great algebra teacher, that can make the textbook bearable. But if the teacher is as dry as the textbook, it can be impossible for some people to make the critical connections.

So what is the solution? Good teaching in high school, and great teaching in college can make a huge difference.

For students who don’t have access to great teaching, however, there is my book, the Algebra Survival Guide. As a tutor who has worked with hundreds of high schoolers and scores of college students, I know how hard these students try, and how little progress they sometimes make. So I wrote a book in plain English, a book everyone can read as easily as you’d read a good novel. My goal was to take the edge off of the intimidating quality of algebra, and from the emailed responses I’ve received, it has worked.

I also threw some humor into the book. O.K., I’m not Jack Benny, but I do make a few jokes, here and there. I mean, we all need to have a little bit of fun, even doing math.

And I worked hard to connect the ideas of algebra to real life, to make them make sense.

For example, take the problem of – 8 + 3. I liken this situation to a tug of war. There are two teams in the tug of war, a Positive Team and a Negative Team. The – 8 means that the Negative Team has 8 people pulling. The + 3 means the Positive Team has 3 people pulling. All you have to do to solve the problem is answer two questions: first, which team will win, if everyone is equally strong? The Negatives, since they have more people pulling. And then, by how many does the larger team outnumber the other smaller team? By 5, since 8 is 5 more than 3. Put your answers together, and you’ll see that the Negatives win by 5, so the answer is – 5. See how easy it can be when you relate the concepts to real life?

The Algebra Survival Guide explains many ideas this way, and it gives algebra a friendly human face.

Here’s one example of a quote from a college student who found the book helpful:

I’m a returning college adult now in the 4th week of my College Algebra course. Your book has FINALLY filled in the gaps in my earlier education … thank you (to the third power)! Thanks for the great book … 30 years of math phobia gone in 3 hours of reading … really, thank you very much!
College Student, Mary Ellen Kirian, Lake Oswego, OR

If you’d like to check out this book, just go to this Amazon page,where you can read lots of reviews.

While you’re @ Amazon, don’t forget to check out the companion Algebra Survival Workbook, with thousands of additional practice problems, which take you from understanding to mastery. You’ll find that page here.

As they used to say on TV, “Try it, you’ll like it!”

— Josh

Multiplication Trick #5 — How to Multiply Two-Digit Numbers by 11


This is the fifth in my series on multiplication tricks. I suggest that you make mental math “tricks” a steady part of your math instruction. Benefits students will reap include:

—  delight with the tricks themselves

—  enhanced confidence in working with numbers

—  students who otherwise don’t like math — or don’t like it much — often find the tricks irresistibly fun and interesting

TRICK #5:

WHAT THE TRICK LETS YOU DO: Multiply two-digit numbers by 11.

HOW YOU DO IT:  To multiply a two-digit number by 11, first realize that the answer will have three digits. The first (left-most) digit of the answer is the first digit of the number; the last (right-most) digit of the answer is the last digit of the number; and the middle digit is the sum of the first and last digits.

But those are just words … here’s a living, breathing example …

Example:  11 x 25

 

Look at 25. The first digit is 2; the last digit is 5.

First digit of answer is 2, so thus far we know the answer looks like:  2 _ _

Last digit of answer is 5, so now we know the answer looks like:  2 _ 5

Middle digit is 7, since 2 + 5 = 7.

The answer is the three-digit number:  2 7 5, more casually known as 275.

It’s that easy!

ANOTHER EXAMPLE:  11 x  63

First digit of answer is 6, so thus far we know the answer looks like:  6 _ _

Last digit of answer is 3, so now we know the answer looks like:  6 _ 3

Middle digit is 9, since 6 + 3 = 9.

The answer is the three-digit number: 6 9 3, or just 693.

Try these for practice:

11 x 24

11 x 31

11 x 52

11 x 27

11 x 34

11 x 26

11 x 62

 Answers:

11 x 24 = 264

11 x 31 = 341

11 x 52 = 572

11 x 27 = 297

11 x 34 = 374

11 x 26 = 286

11 x 62 = 682

NOTE:  If you’re clever (and we’re sure that you are), you have probably realized that this trick, as described, works only when the digits add up to 9 or less. So what do you do when the digits add up to 10 or more? Some of you may figure this out on your own. For those who need a little help, the answer to this will be included in an upcoming blog post.

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 

Multiplication Trick #3 — How to Multiply by 25 FAST!


Here’s the third in my series of multiplication tricks. The first was a trick for multiplying by 5. The second a trick for multiplying by 15, and now this one, a trick for multiplying by 25. Anyone see a pattern?

TRICK #3:

WHAT THE TRICK LETS YOU DO: Quickly multiply numbers by 25.

HOW YOU DO IT:  The key to multiplying by 25 is to think about quarters, as in “nickels, dimes, and quarters.”

Since four quarters make a dollar, and a dollar is worth 100 cents, the concept of quarters helps children see that 4 x 25 = 100.

Since four quarters make one dollar, children can see that twice that many quarters, 8, must make two dollars (200 cents). And from that fact children can see that 8 x 25 = 200.

Following this pattern, children can see that twelve quarters make three dollars (300 cents). So 12 x 25 = 300. And so on.

Fine. But how does all of this lead to a multiplication trick?

The trick is this. To multiply a number by 25, divide the number by 4 and then tack two 0s at the end, which is the same as multiplying by 100.

A few more examples:

16 x 25. Divide 16 by 4 to get 4, so the answer is 400. [In money terms, 16 quarters make $4 = 400 cents.]

24 x 25. Divide 24 by 4 to get 6, so the answer is 600. [In money terms, 24 quarters make $6 = 600 cents.]

48 x 25. Divide 48 by 4  to get 12, so the answer is 1200. [In money terms, 48 quarters make $12 = 1200 cents.]

Try these for practice:

20 x 25

32 x 25

36 x 25

16 x 25

24 x 25

44 x 25

52 x 25

76 x 25

Answers:

 20 x 25  =  500

32 x 25  =  800

36 x 25  =  900

16 x 25  =  400

24 x 25  =  600

44 x 25  =  1100

52 x 25  =  1300

76 x 25  =  1900

But wait, you protest … what about all of the numbers that are not divisible by 4? Good question! But it turns out that there’s a workaround. You still divide by 4, but now you pay attention to the remainder.

If the remainder is 1, that’s like having 1 extra quarter, an additional 25 cents, so you add 25 to the answer.

Example:  17 x 25. Since 17 ÷ 4 = 4 remainder 1, the answer is 400 + 25 = 425.

If the remainder is 2, that’s like having 2 extra quarters, an additional 50 cents, so you add 50 to the answer.

Example: 26 x 25. Since 26 ÷ 4 = 6 remainder 2, the answer is 600 + 50 = 650.

If the remainder is 3, that’s like having 3 extra quarters, an additional 75 cents, so you add 75 to the answer.

Example:  51 x 25. Since 51 ÷ 4 = 12 remainder 3, the answer is 1200 + 75 = 1275.

Now try these for practice:

9 x 25

11 x 25

14 x 25

19 x 25

22 x 25

25 x 25

34 x 25

49 x 25

Answers:

9 x 25  =  225

11 x 25  =  275

14 x 25  =  350

19 x 25  =  475

22 x 25  =  550

25 x 25  =  625

34 x 25  =  850

49 x 25  =  1225

Happy teaching!

—  Josh

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information! 


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