Graphing Calculators and Problem Solving

A highlight of student work this week was using the Slope-Intercept make equations, find patterns in data, and create scatter plot graphs.  It was also neat to see how students began to think of solving a system of equations using graphs of the equations:
y=.49x +10
y=.99x
Students found that the graphing calculator tool, meta-calculator.com, produced quick and accurate results. 

The solution to the 2 equations above is the point where the lines intersect.  This point is at approximately (20,20).   So, how does this graph help answer last week's Music Download word problem? 
Our definition for the variables... x= the number of songs, and the variable y= the cost of the songs.
We discovered that ...When 20 songs are purchased the price of the 2 plans happens to be the same. (around $20).
Therefore...When more than 20 songs are purchased the first plan is a better deal, when less than 20 songs are purchased the 2nd plan is cheaper.

Use the graphing calculator application to solve this problem: What solution do you find for the system of equations?
 y= -3x+2
 y= 2x-3
How would you check to make sure this solution is correct?
Find more interactive practice problems with answers at http://regentsprep.org/REgents/math/ALGEBRA/AE3/PracGr.htm

Slope Fits into Equations

How do we solve problems that involve changing costs, or other things that are in flux in our world?  In Math equations are used to figure out all the possible scenarios that may occur.  The equation y=mx+b shows a linear pattern that can be drawn on a graph or put into a table. 
This week in Middle School we wanted to purchase several items that cost the same amount.  The cost of each item was then multiplied by the # of items that we wanted to buy.  This cost is called the slope, or the rate of each item.  If we had a delivery fee, or other fixed cost that was included we had to add this to the equation too.  When making a decision it is helpful to use the slope-intercept equation to explore the cost.

Here's a problem to get started thinking about how slope fits into real life problem solving...
Musicmatch, an online music store, charges a $10 dollar membership fee plus $.50 cents per song. i Tunes sells songs for $.99 cents each, but doesn't
charge a membership fee. 
Which company offers a better deal for the music lover?
Set up an equation in slope intercept form
that can be used to find the cost for any number of songs.
 
The following website offers consumer reviews of different online music companies that may be an even better deal. Check them out at Music Download Reviews.

First we begin by comparing each music plan.  In looking at the problem, What things could change as a person begins using each plan?  We should be able to find two variables that can be represented by x and y in the problem.

After we have found the variables we look at the cost of each song.   This unit cost, or slope is represented by the letter m in the slope-intercept equation. (y= mx+b)  Slope is also called the unit cost, or rate of change.  When setting up the equation we use slope to show the rate of increase or decrease.  Slope is also how fast the rate increases or decreases.  An interesting observation that students made while graphing lines on a coordinate plane was the steeper the line the greater the number of the slope.  Can you identify what the slope is for the different music plans?
If not, or for more about the slope click on this hyperlink--- The slope.

The Y-Intercept on a graph shows where the line crosses the y-axis.  In problem solving this can be also expressed as a fixed cost or starting point for how much a person would pay up front.  For example, if a club has a membership fee that must be paid in addition to buying songs. For example, if i tunes had charged a $5.00 membership fee this would be added on to the equation.  y=.99x + 5  Can you tell what the y-intercept for Musicmatch would be?  Does i Tunes have a Y-intercept, why or why not?

The slope and y-intercept are written in Slope Intercept Form. (y=mx+b)  As we try different values and put them in for the x=#of songs, or y=total cost, we can further explore the advantages of each music plan.
Which plan did you find to be the better deal for the music lover?  What other strategies can use to solve this problem?

I found it interesting to explore music plans that are available online.  It would be great to hear about other plans that are advantageous, or other links that would give data to support this inquiry.

Aladdin Jr and Algebra Word Problems

Swift School is putting on a musical performance entitled, "Aladdin Jr" next week, and we are expecting many children and adults to attend.  The ticket counters at the play will surely be wondering how many adults and children attended Aladdin Jr.  I found a similar question about "Theater tickets" that a student posted on http://www.algebra.com/

Try reading the question and solving it on your own before reading through the solution.  Check your answer with the one shown below. 
What other strategies do you know to solve this problem?  

Sumdog Stats! Math Challenge Update

Statistics is an area of Math that centers on data collection, presentation, and analysis.  For example, the data from Sumdog.com Math Competitions show how Swift Middle School Homerooms' compare.  Room 312 has set a goal of 55,000 points by 4-30, do you think that they'll reach their goal given the data shown below?  The Math competition also shows room 313 in 1st place for the Fall Competition, will they be able to regain the lead in the Spring?   Room 315 has set a goal of 30,000 points, is this goal attainable?

One popular game on Sumdog.com is "Junk Pile"- the game pictured on the logo above:)  If you haven't tried out the games on this free web site, why not give it a try?! 

Spring 2012 Spring Math Competition            Overall Totals for Fall and Spring
Homeroom     Total Points 4-30-12      Rooms   Total Points   Percent
312                   43,221                                       312         55,937            57%
313                     7,632                                        313        30,160            32%
314                     2,218                                        315         7,370               8%
315                     5,242                                        314         3,052               3%
Total                  55,122                                 Total       96,519            100%


Fall 2011 Sumdog Math Competition
Homeroom        Final Points 
313                     22,528
312                     12,716
315                      2,128
314                         834
Total                   38,206

Compare the Surface Area

The area that covers the outside of a figure is its surface area.   Surface area can be compared to wrapping a present or covering the outside of the object with paper. 

The interactive tool at Annenbeg's web page shows how to find the surface area of cylinders and prisms.  The surface area is the sum of 2 ends of the cylinder and the middle section.
Middle= Circumference * height  +  Ends (2 * Area)

Which container below has the greater surface area, and how much more area?
How would you verify and make sure that both containers have a volume of 750ml?

How Tall is That? Using Similar Triangles

 

The figure above shows a problem of: How tall is the flagpole?  Our 8th grade Math investigation showed a way to find out the height of a cliff using similar triangles like the ones pictured above.  It is important to remember that similar figures share corresponding sides and corresponding angles.  Corresponding angles have equal measures, while corresponding sides have the same ratio.  In the picture above, the corresponding sides on the base of the similar triangles show a ratio of 15.5/4 which equals 3.875

Several eighth graders discovered a strategy to figure out the missing side by using the ratio multiplied by the corresponding side.  For example if we wanted to find the height of the flag pole, this strategy says we take 3.875 times 3, which gives the height of 11.625.  

If you want to find out more about how to use this method to find the height of familiar objects, such as skyscrapers, trees, or towers check out the website Connecting Geometry   

Another fun pair of links to sharpen your math skills are Math Videos at Khan Academy.  Check out the video links below and more great videos on Khan Academy.

 All about Circles  and Testing out Similar Triangles

Interactive Integer Flash Cards and Slope Practice

Try your skill at integer operations.  Solve the integer operations flash cards.  Check your answer by clicking the card after you solve the problem.  Then move the flash card to the correct or incorrect pile.  Try again and see how many you can get correct. 
Afterwards, add a comment below and some strategies you like to use for solving integer operations. 



More flashcards and educational activitites at StudyStack.com

Click on the Link Below to practice finding the slope of a line
Interactive slope of a line from Mathwarehouse.com

Popcorn Friday's: 8th Grade Fundraising

Eighth grade graduation fundraising has involved the business minded skills of the students selling taffy apples, buttons, and popcorn.   The sales will off set some of the costs for eighth grade graduation activities.  Students show teamwork by working together towards their goal of reducing graduation costs.  Friday popcorn sales involve the students measuring ingredients, collecting money, keeping records of sales, and problem solving for optimal sales each week.

What are some ways that math is used as students help manage a fundraiser?  Problems and questions often surface when we are popping away, or looking back at the figures.  For example, this week sales of popcorn were a total of $146 dollars combined between the early morning and afternoon shifts.  The first shift sold ten dollars more than the second shift.  How much money did each of the shifts make during their sales? 

The overall sales of shift 1 (x) and shift (2) y, equalled 146 dollars.   The equation is expressed as x+y=146
Another part of this problem involves the money earned by the second shift. y= x+10.
Solving a pair of equations can be done by solving for x algebraically, using guess and check, or a host of other strategies.  The equation way will substitute the second equation into the first equation.  Namely, x+(x+10)=146.   2x=136, x=68.   Then if x=68, y+68=146.  So y=$78.  Check involves 78+68= 146. 

The math involved in figuring out this equation likely relates to other situations or questions that have come up in managing events like a fundraising.  It'd be great to hear from you.  Share some ways that you have seen math being used.

Contest Winners and Graphing Links

Congratulations to all the participants of the Sumdog.com Winter Challenge! 
The top 10 winners were...
10  Devon
9  Alvaro
8  Elma
7  Medina
6  Victor
5  Edin
4  Priscila
The top 3 winners were...
3  Neerida
2  Luis
1 Amir

Here are some math links to check out, enjoy!
Compare quadratic function graphs: -5x^2 + -5x + 5
How do I simplify an expression like 3x + 2 + 4x - 2y - 1?
Free Graphing Calculator

Advertisements and Percents of Change

Is this the best deal?  What discounts are being offered?   The advertisements online and in print are full of discounts that can be beneficial to savy shoppers.  
For example, Target's website shows discounted prices on movies.   http://www.target.com/sb/entertainment-movies/-/N-5xsx0Z5zktyZ5zja2.  One movie that is adverstised is Harry Potter and the Deathly Hallows.  The original price is shown as $28.98, and the sale price is $12.99.  This can be estimated to be a 3/5 or 60% decrease in price.  The prices listed on the webpage make it is easy to find the percent decrease.  What other percent decreases are there on DVD's at Target?  Is there a better deal (more than 60% off DVD's at Target?  One way to find out is to compare the percent change.
If you'd like to learn more about percent change and spend some time playing a computer game, here's a weblink from The Percent Shopping Game.
http://www.mathplayground.com/percent_shopping.html

The Power of Patterns

Art ignites the imagination, and one of the fascinating aspects of art is the use of patterns.
Pascal's Triangle shows many patterns that are useful in mathematics.  The third diagonal from the left shows the triangular numbers (3, 6, 10, 15...)  These numbers show the number of dots that are needed in order to create a triangular shape.
Another pattern that can be seen with triangular numbers is when adding the counting numbers together.
(1+2= 3,  1+2+3= 6,   1+2+3+4= 10, and 1+2+3+4+5=15,...) 
When adding together a the numbers 1-10 in the same way a pattern can be found.
For example the sum of the numbers 1-10 is the tenth triangular number on the pyramid above. What patterns do you notice in the numbers leading up to the triangular number 10? 
Here's a hint:  if you take the 2nd triangular number 6, and compare it to the ones in front of it and behind it, what do you notice?
Another clue is if you list the addends and add the first and last numbers, what do you notice?
Mathematicians notice patterns and experiment with writing equations to show a general rule.  These patterns can create pyramid's of numbers that continue to grow.  

Happy Holidays!

The Swift School International Assembly began our week before Christmas holiday cheer.  Congratulations to all the performers who shined on stage.  The folk tale renditions from different regions of the world introduced each of the talented groups.  The mix of visual arts with cultural clothing and decorations brought a lively environment for all of us to enjoy.  Swift School musical talents included performing on musical instruments, singing, and dancing.  This celebration of culture showed our pride in teamwork, artistic expression, and diverse backgrounds.

Holiday crafts are a relaxing way to enjoy the season.   Create your own unique snowflake on this website and then share it along with a holiday greeting for a friend.   You can find this at Snowdays

Congratulations to the Math Contest Participants!

Congratulations go out to all the Swift Middle School students who participated in the Sumdog.com Math Contest.  The contestants earned a combined total of approximately 38,671 points!  Top classroom contestants for the math competition include:
312:  Neerida
         Devon
313:  Linda
         Amari
314:  Ayana
         Elma
315:  Jennifer B.
         Sandra
The overall winners for the contest earned a combined total of 26,484 points! 
1st place-  Linda,   2nd place- Neerida,  3rd place- Amari
Congratulations to all the 7th and 8th graders who have played the Math games.  Games are a great way to relax and have fun doing the math.  Illinois is having a state wide Sumdog.com contest coming up soon, and there will be many other chances to use math games this year.  

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.

Student Math Projects- Amazing Facts and Feats: Data Displays

Have you ever wondered, Is that even possible? Or, was that statement really true? Amazing facts can sound too incredible to believe.
Our projects will show some of these amazing facts and feats. Take for instance the fact that "Humans breathe about 23,000 times a day!" If you measured a person breathing for one minute you may find that they breathe about 20 times. That same person would breathe about 1200 times in an hour, and 28,800 times in a day. The Math and Data Displays behind the fact helps us to judge whether this sounds believable or not.
We will be able to find information to support the amazing facts and feats from books, computers sources, or teachers/librarians/adults that we know. For example, a website called, Interesting Facts lists some Science related facts:

• Each person sheds 40lbs of skin in his or her lifetime.
• Koalas sleep an average of 22 hours a day, two hours more than the sloth.
• Light would take .13 seconds to travel around the Earth.
• The largest desert in the world, the Sahara, is 3,500,000 square miles.

21facts.com also has some interesting facts about the Civil War that you may or may not believe?

• The total deaths in the United States Civil War was over 1,000,000 people
• During the Battle of Cold Harbor in the Civil War there were 7,000 people that died within 20 min.
• There were over 550 generals who served during the Civil War.
• A Confederate prison camp called Andersonville had 12,000 people die.

Books like the World Almanac, Atlases, as well as Encyclopedias could be used to find out more about these amazing facts. There is often a lot of background information and facts that can help to make the amazing facts more believable. On the other had the research often helps us point out errors or misconceptions. Either way our projects will have fun with the investigations.

Extra Support from Loyola Teacher Candidates

Swift 7th and 8th graders advance their math skills with the help of Loyola's tutoring support and team teaching.
Our Loyola teacher candidates include:
8th- 313 Jessica & Elizabeth
8th- 312 Jessica
8th- 315 Hana
7th- 314 Haley & Christina
Emphasis for these lessons is on building mathematical vocabulary, developing math concepts, strengthening math fluency, and supporting math problem solving.
Loyola University has a library of math videos. Click on this link for- Countdown Videos
These videos include review of math content and support students'
development in mathematical thinking.
What are some of your favorite activities that you've done in math?
Do you find videos helpful in reviewing and learning math? Why or Why not?

Best Fit Lines on a Scatter Plot

Best Fit Lines show the general direction of points on a data display. The best fit line doesn't connect the points, but instead is shown to be close to as many points as possible within the constraints of a straight line.

This week students measured their heights and jumping heights using meter sticks. These two variables were compared to see if one of them had any effect on the other. For example, does the height of a person affect how high the can jump? The Scatter Plot display of our data should help us see whether the answer to this question is true, or false.

Here is a website that has information and questions about scatter plots. I like the fun graphics and the way the questions give feedback about your answer. Click here to find an interactive Scatter Plots Website

Another website called The Challenge Club shows the steps to create a scatter plot.
Besides comparing the example above about whether tall people have larger hands or not, you can make scatter plots about things you're into. For example whether your favorite band's new album sales relate with the amount of attention that they're getting on the radio or Internet.
Does one thing affect another? When we make a scatter plot and draw a best fit line, we can see the relationship more clearly.