Monday, August 18, 2014

Coordinate Geometry


Coordinate Geometry: y=mx+b

Why Equals Am-ex Plus Bee?

Why not? This format for an x-y equation helps when solving coordinate geometry problems. It standardizes the look so it's easier to compare different equations. The equation itself often gives you hints to where the line the equation represents will sit on the graph.

Y and X are graph points, So what are M and B?

In the equation y=mx+b, y and x are the coordinates (as in the two ordered pairs/graph points shown here). m represents the slope of the line and b represents the y-intercept (the value of y when x is 0 (zero)).

In order to solve an equation in this format, you need a few pieces of information.
  • If the problem gives you m (the slope) and b (the y intercept or the value of y when x is zero), you can figure out the line with a few xs and matching ys (you chose values for x, plug it into the equation and figure out what the y value is)
  • If you are given an x and a matching y (an ordered pair) with either the slope or the y intercept, you can find the other
  • If you are given two ordered pairs (as in the graphic here), you can figure out m and b

Figuring out the Slope and Y-Intercept from Two Points

In order to solve a y=mx+b problem when you have two points, you first have to figure out the slope. The slope is defined as the "rise over run". This can also be figured as the change in y over the change of x.

As you can see in the graphic here, you decide which point is (x1, y1) and which is (x2, y2). It really doesn't matter which one you subtract from which, just designate the two points. Then, subtract y2 from y1 and divide it by x1-x2 (see the graphic). In this case, we have 0-6=-6 and 5-0=5. The slope, in this case, is -6/5. (Note: a positive slope moves from the lower left to the upper right and the negative slope moves from the lower right to the upper left.)

Figuring the Equation with One Point and the Slope

If you have the slope (m) and one point, you substitute the x and y in the equation and put the slope in where the m should be. So, in this case, we use 0 for x, 6 for y and -6/5 for m. The progression in the graphic shows how you would solve this algebraically.

Note: in this case, solving for b is even easier, since b is the value of y when x is 0 -- you can see from the x,y pair that, when x is zero, y is six.

A Piece of Pi


National Pi Day March 14th (3.14)

March 14th is "National Pi Day" because Pi is represented, rounded to the nearest hundredth, as 3.14.

Pi is also sometimes approximated as 22/7, or 3 and 1/7.

Pi is an irrational number that represents the ratio of the circumference of a circle to its diameter.
An irrational number is one that, no matter how many decimal places you take the number to, there will always be more decimal places you can take it to. Other examples of irrational numbers are square roots of numbers that are not perfect squares, like the square root of 2, 3 or 5. Any number that can be shown as a fraction is by definition a rational number.

The Circumference of a Circle

In order to figure out the circumference of a circle (this would be of help if you wanted to, say, put a ribbon around a cylinder), you multiply the length of the diameter (a diameter is a line from one end of the circle edge [circumference -- the thick magenta around the edge of the circle pictured here] to the other that passes through the center of the circle, like the magenta lines within the pictured circle) or twice the length of the radius (a radius is a line segment from the center of the circle to the edge of the circle) and multiply it by pi. As an estimate (pi is an irrational number, a number that has no exact value, so, to work with it, we must estimate), you can use either 3.14 or 22/7

In other words, if the diameter of a circle is 1, then the circumference is approximately 3.14 or 22/7; if the radius is 1 (then the diameter is 2), the circumference is 6.28 or 44/7 (or 6 2/7).

The Area of a Circle

To find the area of a circle, you square the radius (or 1/2 the diameter) and multiply it by pi.

For example, if the diameter is 2 (making the radius 1), 1 squared is 1, multiplied by pi is pi (or approximately 3.14 or 22/7). If the radius is 2, then 2 squared is 4, 4 times pi is about 12.56 or 88/7 or 12 4/7. If the radius is in inches, then the area is in square inches (so if the radius is 2 inches, then the area is 12.56 square inches).

Math and Me

Math has always been one of my "things". I've been tutoring math since I was in High School -- when I was a sophomore in high school (age 14), I started tutoring freshwomen (I went to an all-girl religious school) in Algebra. By my senior year, I was tutoring my own classmates (in elective senior math) along with lower classwomen. I still tutor math (and other subjects) and I still love algebra and geometry.

One of my recent tutees (who is in college now -- I tutored her from 6th to 11th grade and still work with her brother) used to ask me to create sample problems for her and then said the ones I created were harder than the ones in her book, so I said to her -- keep in mind, if you can do mine the test will be a snap -- she never again complained and always aced her tests.

Oh, BTW, one of my pet peeves is calculators. I don't allow my tutees to use them for anything that can be figured out without one. If you can do it without the calculator, doing it with the calculator should be easy, right? Well, not so much for me. When I was in grad school, I took a statistics class. During tests, I would have a pile of paper and do all the math that way. I would then check on my calculator (which I left in the corner) and if the numbers didn't match, I would redo on the calculator first (that was wrong more often than my figuring). I still managed to finish the test (and get a high A) before anyone else in the class. Never underestimate the power of your own brain.
Compugraph Designs on Printpop

See Compugraph Designs’ Printpop portfolio . Check back periodically as new designs are uploaded.


Compugraph Designs Arts Now Site

Arts Now is another “Print on Demand” site. They have a nice collection of clocks and watches, including the one pictured here (with my popular math symbols design on it). Click on the picture to see the entire site.

Difference of Squares: Answer to Quiz


If you don't know what the Difference of Squares is, look here

Here are the problems:

Here are the answers:

Algebra -- Difference of Squares


Factoring Trick

Algebra students often have to factor polynomials on their exams. Since tests are generally timed, most students would welcome anything that would save them time.

One thing that can make a test quicker is "Difference of Squares". If you can master this concept, you can save yourself a lot of time on standardized math test.

The Difference of Squares: What does it mean?

The Difference of Squares is an interesting mathematical anomaly. In most cases, when you take two binomials (that is an equation of two different bases, as in one being a constant and the other being a variable and adding them together -- for example (2x+7) or (x-5) or even (3x+4y)) that have the same bases (for example (2x+7) times (4x-5) or (y-5) times (y+3) or even (2x+y) times (x-3y)) you end up with a trinomial (that is a three base answer -- for example (8x [squared] +18x-35) or (y [squared] -2y-15) or 2x [squared] -5xy-3y [squared]) respectively.

The picture here shows this concept in its most basic form (x and y can each be any number, variable or combination of number and variable that is a perfect square, like 4, 9, 16, x [squared], 16x [to the 4th power], etc.)

An Example

In the picture here, since 49 and 36 are perfect squares (49 being 7 [squared] and 36 being 6 [squared]), this works with this equation pictured here.

Why does it work this way?

Notice what happens if you multiply (7x-6) and (7x+6) -- from right to left -- (+6) times (-6) equals (-36), (+6) times (7x) equals (+42x), (-6) times (7x) equals (-42x), (7x) times (7x) equals (49x [squared]) {see the middle section on the picture}.

If you notice, the middle products ( (+42x) and (-42x) ) will cancel each other out {see the line through them} because (+42x) + (-42x) = 0.

The ultimate product {see the bottom line in the picture} ends up being (7x) [squared] {which is 49x [squared]} minus 6 [squared] {which is 36}.

So, you can apply this to factoring -- if you see a set up that is (x+y) times (x-y) you know the answer will be (x [squared] - y [squared]) (see the picture above). And, if you see a set up of  (x [squared] - y [squared]), then you know it factors out to (x+y) times (x-y).

Note: this will NOT work with  (x [squared] + y [squared]) -- it only works with  (x [squared] - y [squared]) -- this is it only works with a difference of squares, not a sum of squares.

Difference of Squares Quiz


You can view the answers here

Fraction Problems


How Well Did You Understand Fractions?

One of my most popular lenses is an instructional lens about how to work with fractions. I thought, perhaps, presenting a quiz of problems would help, because you could then do actual problems to see if you actually understand the concepts.

If you need to go back to the the explanation lens, go to Understanding Fractions (and read or reread it).

10 Addition and Subtraction of Fractions Problems

Use your notebook or scrap paper to do your figuring. If you understood, you should be able to get at least 7 correct.

The Questions

  • 2/5 + 1/5 =
  • 1/4 + 1/8 =
  • 7/8 - 5/16 = 
  • 5/6 + 3/8 =
  • 3/7 + 4/5 +1/2 + 3/10 = 
  • 4/7 - 2/5 = 
  • 5/6 + 1/3 + 3/12 = 
  • 5/9 - 1/4 = 
  • 7/5 + 3/4 =
  • 7/12 - 1/3 =

10 Multiplication and Division of Fractions Problems

Test your understanding of multiplying and dividing fractions. Please reduce fractions.

The Questions

  • 1/3 X 1/2 = 
  • 2/5 X 3/4 = 
  • 5 / 5/6 = 
  • 6/12 X 4/12 =
  • 7/4 / 7/2 =
  • 10/3 X 9/5 =
  • 9/5 / 5/9 = 
  • 4/5 X 2/9 = 
  • 4/5 / 2/9 = 
  • 7/12 X 3/2 =


You can view the answers here

Answers to Fractions Problems


10 Addition and Subtraction Problems

The Questions and Answers

  • 2/5 + 1/5 = 3/5
  • 1/4 + 1/8 = 3/8
  • 7/8 - 5/16 =  9/16
  • 5/6 + 3/8 = 29/24
  • 3/7 + 4/5 +1/2 + 3/10 = 142/70
  • 4/7 - 2/5 = 6/35
  • 5/6 + 1/3 + 3/12 = 17/12
  • 5/9 - 1/4 = 11/36
  • 7/5 + 3/4 = 43/20
  • 7/12 - 1/3 = 1/4

10 Multiplication and Division Problems

The Questions and Answers

  • 1/3 X 1/2 = 1/6
  • 2/5 X 3/4 = 6/20
  • 5 / 5/6 = 6
  • 6/12 X 4/12 = 1/6
  • 7/4 / 7/2 = 1/2
  • 10/3 X 9/5 = 6
  • 9/5 / 5/9 = 81/25
  • 4/5 X 2/9 = 8/45
  • 4/5 / 2/9 = 18/5
  • 7/12 X 3/2 = 7/8

Understanding Fractions


Fractions can be daunting for children and adults, but with a few hints, they become less mysterious.

One of the things you need to keep in mind is that fractions are just numerical representations of parts of wholes. For example, if you cut a pie into two equal pieces, each piece is 1/2 (or one-half) of the pie.

Defining Fractions

Fractions are just another way of doing division. Think of them in terms of a pie or a pizza that needs to be cut into enough pieces to feed a group.

In the picture at the right, you see there are 8 people sitting around a pie. If you cut that pie so that each one gets a piece equal to all the other pieces, each person will get 1/8 (one-eighth) of the pie. This means that one pie is divided into 8 pieces.

If you had two pies to feed the eight people, each one would get 2/8 (two-eighths) of the pie (two pies divided into 8 pieces total). Another way of saying 2/8 is 1/4 (one-fourth) because each pie will need to be cut into 4 pieces (two pies cut into a total of eight pieces means each pie will need to be cut into four pieces).

Another way of looking at fractions is that you have groups of items that need to be split evenly between members of a group. The example at the right shows 12 marbles. Let's assume that three people are dividing a bag of 12 marbles. Each one gets 1/3 (one-third) of the total -- the total (one) divided by 3 people. If you count out the number of marbles in 1/3, you will see that each third is equal to 4 marbles (4/12 -- four-twelfths -- is equal to 1/3).

If there are two people, but one person gets two shares, then that person would get 2/3 or 8 marbles. The whole (1) would be the same as all three thirds (3/3).

Arithmetic with Fractions

How to add, subtract, multiply and divide fractions

In order to add fractions, you need to get the denominators (that is the bottom number of the fraction) of all the fractions to be added to be equal. Looking again at the pie example, we wouldn't know how to add 1/8 of the pie to 1/4 of the pie unless we remembered that 1/4 is the same as 2/8 (1 divided by 4 is the same as 2 divided by 8). Then you could see that 1/4 of the pie (or 2/8) added to 1/8 is the same as adding 1+2 (see below):

1/4 + 1/8 = 2/8 + 1/8 = 1+2 /8 = 3/8

This is pretty easy when you are talking about fractions that have denominators (bottom number on a fraction) that are related to each other (like 4 and 8, 3 and 9, for example), but it's harder when the numbers aren't related (like 8 and 9, for example).

Keep in mind that any number divided by itself equals one. 3/3, 8/8, 16/16, 298/298 -- all are equal to 1.

So what do you do if you have two unrelated numbers? Let's take 5 and 6. You need to find a number that both numbers can be divided into. In the case of 5 and 6, that number would be 30 (5X6). So, if you were trying to add 3/5 and 5/6, you would need to put both fractions in terms of x/30.

To do this, you need to multiply. Multiplying fractions is relatively easy. If you have two fractions, say 5/9 X 3/7 -- you multiply the numerators (top numbers) and put that as the numerator on the new fractions and multiply the denominators and use that answer as the new denominator -- in the example, you would multiply 5X3 for 15 on top and 9X7 -- or 63 -- on the bottom so:

5/9 X 3/7 = 5X3 / 9X7 = 15/63

Now back to our addition example.

If we want to add 3/5 and 5/6, we have to write both fractions in terms of how many 30ths they would be -- to do that, multiply 3/5 X 6/6, and 5/6 X 5/5. So you end up with:

3/5 + 5/6 = 18/30 + 25/30 (3X6/5X6 + 5X5/6X5) = 18+25 /30 or 43/30

Let's do another example

2/3 + 5/8 =

in order to add these two, we need to put both fractions in terms of 24ths (3X8=24), so....

2/3 + 5/8 = [2/3 X 8/8] + [5/8 X 3/3] = 16/24 + 15/24 = 16+15 /24 or 31/24

Try some of your own examples for practice on your own.

For more on finding the lowest common denominator, check out Finding the Lowest Common Denominator elsewhere in this blog.

Subtracting and Dividing

Subtracting is like adding; Dividing is like multiplying

One last thought about arithmetical operations on fractions: Subtraction is like addition and division is like multiplication. How? I'll explain.

With subtraction, as with addition, you need to get the fractions into a form where the denominators are the same. For example: Let's take 5/6 and 3/5 again.

5/6 - 3/5 = (5/6 X 5/5) - (3/5 X 6/6) = 25/30 - 18/30 = 25-18 /30 = 7/30

another example:

2/3 - 5/8 = [2/3 X 8/8] - [5/8 X 3/3] = 16/24 - 15/24 = 16-15 /24 or 1/24

Division is like multiplication except you need to "flip" the term after the division sign. For example:

3/4 divided by 6/5 = 3/4 5/6 = 15/24 or 5/8

To understand why this works, see the picture at the right -- this represents 5 / 5/6. This means that you are taking 5 bars (or, it means you are taking 5 cups of raisins and dividing it into 5/6 cup portions). As you can see, the bars are each divided into sixths. As you can see, you can get 6 portions of five-sixths. This is the same as if you multiply 5 X 6/5 (this is the same as 5/1 X 6/5 = 30/5 = 6/1 = 6).

I hope this has helped you understand fractions a little bit better. If you have any questions, you can either leave a comment below or you can e-mail the author at

If you want to test your understanding, I have Fractions Problems here