Math Contest: Fall 2011

This week we've started an online challenge for Swift Middle School.  8th grade students select games to play from the website, Sumdog.com and earn points for each time they score.  The games not only involve gaming ability, but have some problem solving and math knowledge as well. 
Students can play as much as they like and earn extra credit homework points for their scores or comments on our class blog.  

Please comment below to share some of your favorite games, and tips for this math challenge. 

Stretch your skills at fractions and probability

Check out the online versions of Connect Four and Racing Game!  Click a link below to begin.
Connect Four 
Racing Game

Below are 3 easy steps for finding the sum or difference of fractions include:
1) Find a common denominator-  list the multiples of both denominators and find the lowest one in common. 
example  2/5  +  1/2     5 and 2 both have  a common multiple of 10
2) Make equivalent fractions- Divide the original denominators by 10 (common denominator).  Use the quotient to multiply by the numerator. 
example  10/5=2   2(2)=4  equivalent fraction 2/5 = 4/10;   10/2=5  5(1)=5  equivalent fraction 5/10=1/2
3) Add/Subtract numerators- Keep the denominator the same, and take the sum or difference of the numerator
example   4/10 + 5/10 =     4 + 5    =  9
                                          ------      ---
                                            10         10

Real life use of Fractions involves beats in music, cooking recipes, clothing design, city buildings, and sports statistics.  What are some ways you see fractions used in real life?  How does your science fair experiment involve the use of fractions, data, or math displays?       

Equations and Surface Area of a Figure

Equations are a way we represent a problem by using numbers and symbols.  After we develop an equation it becomes easy to apply it to many similar problems. 
Pythagoras, a famous mathematician developed a famous equation to find the side lengths of any right triangle.  His equation is called the Pythagorean Theorem. A theorem is a math rule that is developed from tests over time.  It is kind of like a science experiment in that it has to be proven through repeated tests. 
The equation shows that the square of the sides of right triangles forms a pattern.  It says that a right triangle has a side across from the right angle which is equal to the sum of the other two sides squared.  In equation form:  a^2 + b^2 = c^2  This equation is shown in picture form at this web link- Pythagorean theorem
We can use the Pythagorean theorem to solve real life problems that involve finding the sides of triangles.  I think it's interesting how the web link above has problems about finding the distance on a baseball diamond, and finding the length of a ladder needed to reach a window.  Careers in medicine, construction, engineering, and architecture use equations to solve problems.
One example is how 3D figures like square pyramids use the Pythagorean theorem. For example, how do I find the surface area of a square pyramid?  Surface area is found when we want to know the amount of material needed to cover a 3D shape.

A square pyramid has four triangles and one square as shown in the net of the 3D shape above. The 3D shape becomes folded out in a "net" or "net drawing". The website Interactives 3-D Shapes shows a video clip of how to make a net.
The Pythagorean theorem can help us find the side lengths of the triangles if we know the side lengths but need to find the height.  The base of a yellow triangle needs to be bisected, or divided in half with a perpendicular line, to form a right angle.  If the base is 6 cm and we bisect it, then the side of the right triangle formed will be 3 cm.  If the hypotenuse, or side across from the right angle is 5 cm then we can find the height with the Pythagorean theorem.
We use the equation 3^2 + b^2 = 5^2 to find the height of the triangle.  When the equation is used to solve for the missing side we can find the exact length quickly!
Can you find the missing side using the equation above?  Which city buildings or designs use the square pyramid shape? 

Attack Integer Computation with Success

Integers are made up of natural numbers 0, 1, 2, 3... and negative whole numbers  -1, -2, -3...
The number line is a familiar way to recall the rules of computing integers.   In the example pictured above we see the numeral two as our starting point which is symbolized with a dot.  This is followed by a subtraction sign, which leads us to move to the left three spaces as shown with curved line and arrow.
Helpful hints are another way that we can remember how to add and subtract integers.
We learned the first hint: Two like signs become a positive sign.
Example  2 - (-3) = ?   This can be rewritten as 2 + 3 = ? since there are 2 like signs in the original problem. 
Another example  -3 - (-4) = ?   Rewritten as - 3 + 4 =? Again like signs - and - turn into a positive sign.

And conversely the second hint: Two unlike signs become a negative sign.
Example 2 + (-2) = ?  This is rewritten as 2 - 2 = ? because there are unlike + and - signs in the problem. 
Another example  -4 + (-2) = ?  Rewritten as -4 - 2 ?  We start at -4 on a number line, and move to the left (away from zero) 2 spaces.  This leads us to -6 as the difference. 

Now, we find the product or quotient of two integers to be:
Positive only when both integers are positive, or both are negative.
Examples:  -4(-4) =  -4 * -4 = 16     -24/-2 =  12
Negative when one integer is positive and the other is negative.
Examples  30/(-5)= -6  or   -5(7) =  -5 * 7 = -35

Which type of integer calculation do you find easiest to solve?   
Try your skill with the interactive version of Integer Battleship found at the link below.  After you make a locate a Battleship you'll be challenged with an integer problem to see if you can hit the target.
Integer Battleship   


What are some other tips that you have for ways to calculate integers?  Share a website, strategy you've learned from a teacher, or a way that helps you keep integer computation clear.   What other games or activities have you done with Integers?  Work at it and Integer computation will become automatic.