Kiss those Math Headaches GOODBYE!

Archive for the ‘Proportions’ Category

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

(more…)

Advertisements

FUN MATH PROBLEM — Circling the Square & Vice-Versa


From time to time I will post interesting math problems.

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit!

I will post the answer to the problems two days later, after people have had time to respond.

To post your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.

The problem: Which provides the fuller fit? Putting a circular peg in a square hole, or putting a square peg in a circular hole? To get credit, show all work, and justify your answer by expressing each “fit” as a percent.

A few term-clarifications, to help you do this correctly:

a) By “fit,” I mean the ratio of the smaller shape to the larger shape, expressed as a percent. For
example, if a ratio is 4 to 5, that would represent a “fit” of 80 percent.

b) For the circular peg in the square hole, assume that the diameter of the circle equals the side of the
square. For the square peg in a circular hole, assume that the diameter of the circle equals the diagonal of the square.

c) By “fuller fit,” I mean the larger of the two ratios.

Have fun!

“Hacks” for Slaying Proportions, Part 1: the Amazing Horizontal Canceling Trick


Proportions can seem intimidating, but they’re actually one of the easiest types of word problems to master. In this series I’ll offer a number of tips that help you conquer algebraic proportion problems. 

But first, a cool shortcut you can use whenever you’re facing down an algebraic proportion …

High-Octane Boost for MathIn working with proportions, I’m amazed that few students know how a canceling process that would help them find the solution more quickly and efficiently.

So I want to share the trick, for all who’ve never seen it.

Of course, given a problem like:  6/x  =  24/32,

most of us know that we can cancel vertically with the two numbers in the fraction on the right, to get:

6/x  =  3/4

Then we just cross-multiply to get:

3x  =  24, and see that x  =  8.

In other words, we know we can cancel vertically given a proportion, just as we can cancel vertically with any fraction.

What many people don’t know though, is that there’s another way we can cancel when solving proportions — horizontally!

— What? you say.

Horizontally, I say. And no, I’m not joshing.

For example,  given the proportion:  7/4  =  21/x

you can cancel horizontally with the two numbers in the numerator:  the 7 and the 21. These reduce to 1 and 3.

The proportion then becomes:

1/4  =  3/x  [I’m really not kidding.]

Cross-multiplying, you get the answer in one quick step:   x = 12.

What’s really convenient is that you can also cancel both vertically and horizontally in the same problem. For example, in

6/x  =  42/28,

you could first cancel horizontally, to get:

1/x  = 7/28

Then you can cancel vertically, to get:

1/x  =  1/4

Cross-multiplying, you get the answer in just a step:  x = 4

I find that when students cancel before cross-multiplying, they’re more apt to get the right answer, and to get less frustrated, for the numbers they deal with remain small.

For example, in the last problem, if the student had not canceled at all, he would have a cross-multiplication mess of:

6 x 28 = 42x

That sort of problem just opens up the door to arithmetic mistakes. But canceling before cross-multiplying shuts that door since it makes the numbers smaller and easier to manage.

So now you get a chance to practice horizontal cancelling!

First use horizontal cancelling to get the answer to these
proportions. Those who’d like an added challenge might like to try them in their head:

a)   x/12  =  3/4

b)  3/7  =  x/35

c)   z/48  =  7/12

d)  y/56  =  7/8

Now go really wild! Use both horizontal and vertical canceling to make quick work of these proportions:

e)  x/9  =  16/36

f)   x/22  =  30/66

g)  32/56  =  y/14

h)  13/q  =  65/35

And here are the answers to all of these problems:

 a)  x  =  9

b)  x  =  15

c)   z  =  28

d)   y  =  49

e)   x  =  4

f)   x  =  10

g)   y  =  8

h)  q  =  7