This video shows the fastest and easiest way I know of for factoring quadratic trinomials. Give it a watch and see if you agree.
Now that you’ve gotten a taste for the benefits of analyzing algebraic mistakes, it’s time to explore a second common mistake. This one is so common that nearly every student commits it at least once on the road to algebra success.
As you watch the video, notice how by thinking hard about two expressions, we can think this mistake through to its very root, thus discovering the core difference between two similar-looking algebraic expressions.
And along the road, we’ll learn a general strategy for decoding the meaning of algebraic expressions. What I like about this strategy is that you can use it to understand the meaning of pretty much any algebraic expression, and you’ll see that it’s not a hard thing to do. In fact, it just involves using numbers in a nifty way.
Best of all, students usually find this approach interesting, convincing and even a bit fun. So here goes, Common Algebra Mistake #2 …
Here is the second in my series of “Fun 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 provide your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.
Please show how you worked the problem. Thanks. I will post the names of the first three people who get this right.
The Problem: Before you go out to lunch, you glance at the clock above your desk. When you come back from lunch, you glance at the clock again, and you notice something strange. The minute and the hour hand have exchanged places from the positions they had just before you went to lunch.
The question is: how long were you away?
Answer to problem about the circular and square pegs and holes.
The “fit” for each situation is the following ratio:
(Area of Inner Figure) ÷ (Area of Outer Figure)
For the square peg in a round hole —
Call the radius of the circle r.
Then the diagonal of square “peg” = 2r
Notice that by slicing the square along its diagonal,
we get a 45-45-90 triangle, with the diagonal being
the hypotenuse and the sides being the two equal legs.
Using the proportions in a 45-45-90 triangle,
side of square peg = r times the square root of 2
Multiplying this side of the square by itself gives
us the area of the square, which comes out as:
2 times the radius squared
This being the case,
Area of square is: 2 times radius squared, and
Area of circle is: Pi times radius squared, and so …
Cancelling the value of the radius squared, we get:
Ratio of (Area of square) to (Area of circle) is:
2÷Pi = 0.6366
For the round peg in a square hole —
Call radius of the circle r.
And since the diameter of the circle is the same length as
the side of the square, the side of the square = 2r
Multiplying the side of the square by itself to get the
area of the square, we find that the area of the square
is given by: 4 times radius squared.
This being the case,
Area of circle is: Pi times radius squared
Area of square is: 4 times radius squared, and so …
Ratio of (Area of circle) to (Area of square) is therefore:
Pi ÷ 4 = 0.7854
Of the two ratios, the ratio of the circular peg in a square hole
is greater than that of the square peg in a circular hole.
Therefore we can say that the circular peg in a square hole
provides a better fit than a square peg in a circular hole.
And that is the answer!