Summer Training Plans

How do you get ready for an upcoming bike ride, walk, or run?  The Training Plan below is a way to help people prepare for a big event.  This plan also be used for finishing a book, writing a story, or a big project.

The training plan uses something called The 10 Percent Rule.  This rules says you should increase your current activity level by 10 percent per week so that you don't over train (during running)

For example, if you were wanting to do a 20 mile run by the end of the summer, and currently you are running around 10 miles per week, would this be possible? Let's say that the end of the summer was 7 weeks away.  First, you might prepare by making a Table of how many miles you'll be running each week.

Table

Weeks    Equation     Miles Ran  
1            10*1.1              11    
2            11*1.1            12.1
3           12.1*1.1          13.3
4           13.3*1.1          14.63
5           14.63*1.1      16.093
6          16.093*1.1       17.7
7          17.7*1.1           19.47
x           10*1.1^x      

In the table you can see that each week you increase the Miles Ran by 10%.  The equation uses 10% written as a decimal 0.10.  Beginning with week 1 you find what is 10% more than 10 miles?  A shortcut in using percents, when multiplying by 1 you get the same number so instead of multiplying times 0.10 we can switch it to 1 + 0.10 or 1.10.
The equation column shows 10*1.10= 11, So in week 1 you plan to run 11 miles.  Then, in week 2 you start with 11 and find out what 10% more is by doing 11*1.10=12.1 miles. When doing the math you probably noticed that you multiply the total for the week by 1.10. You could keep going with this pattern through week 7 to find the total number of miles by week 7.

Another strategy you may have already thought of is Using an Equation involving exponents. The exponents can help you find out if it is possible to reach the goal by a certain number of weeks. Let's say 6 weeks?
Since you are using an equation you use the starting distance which was 10 miles and the percent increase 1.10,  the number of weeks is the exponent function in the equation. Starting distance times  1.1 ^ x weeks
10 miles*1.1^6 weeks =  17.712 miles. (Notice this is close to the value in the table, but not exactly)
10 miles*1.1^7 weeks= 19.47 miles
After 7 weeks you are still slightly less than the 20 mile goal at 19.47 miles. However, if you round up to the nearest tenth of a mile, you're at 19.5 miles.  You could Check your Answer by filling in the other values in the table above.

This plan of increasing by a percentage each week can be used for other projects. Right now you're writing 3.5 pages a week, but want to finish a young author's project that is going to be 25 pages long in 4 weeks.  If you increase your rate by 10% each week, will you finish by the deadline?
 Week              Pages per week
Starting                    3.5
1          3.5*1.1      3.85
2         3.85*1.1     4.235
3         4.235*1.1   4.6585
4        4.6585*1.1  5.124
Total Pages written  21.4 pages written

Want to learn more about How Percents Work check out this video and other fun activities
on Brain Pop.com Interest % Movie

End of School Year Projects and Weblinks

Display Boards 

Middle School Hero's Projects

Hero Day Celebration

Class of 2013 Career Day Trip

The month of June has been full of celebrations!  Students used technical drawing & coordinate geometry to create a portrait of their Heros.  We used the coordinate grid to draw an identical image of the hero.  Students shared their projects on our Hero's Day celebration.  

Another celebration was a Career Day Field Trip.  This trip was made possible by Gear Up Chicago.  Students won prizes for team building activities and learned about careers.

Weblinks:

Mr. Martini's Classroom-  Online practice with long division and equations.  Allows you to keep track of your score.  You can also learn step by step long division without using paper and pencil.  

Extreme Math- A 3-D game with Rock Music.  Watch the "Extreme" character perform ski tricks when you get the answers right.  

Mathopolis-  Math Contests with kids from all over the Globe. Earn merits for games won, and correct answers.  

How do you find the side length of a square when you know the area?   Area of a square is side times side.  For example if the area is 25 cm² and 5 x 5 = 25, the side length = 5 cm. 

Square roots are found using the perfect squares.

1² = 1            √1 = 1           Square root of 1 = 1

2² = 4            √4 = 2           Square root of 4 = 2
3² = 9            √9 = 3           Square root of 9 = 3
4² = 16         √16 =4           Square root of 16 = 4

Can you find the square roots for 25, 36, 49, 81, 100, 121, and 144?  What patterns do you notice?

Try the Square Root Game on SoftSchools.com

Interest Earned by Saving and Investing

    

Exponential Growth-  change that happens when a beginning amount grows by a consistent rate over time.

Develop a Savings or Investment Plan

      1. Make a goal for saving or investing money.  

2.   Decide on a beginning amount to put toward your savings or investment goal.

3. Choose an interest rate.  Savings account (.005-.01),  Investment Account (.01-.09)

Website Links: Savings interest rates

                        Investing interest rates

4.  Construct a table to record your data.  Include beginning amount, interest earned, and ending amount

5.   Create a line graph to show changes over time. 

Example Savings Goal

Goal- $2,000 down payment on a used Toyota Prius in 5 years

Beginning Amount $800.00

Interest Rate: 6% (.06)  Mid-Term Bond Mutual Fund

Year       Beginning       Yearly        Interest       Ending 

                 Amount        Savings     Earned         Amount

1              800                 100                 54                      954

2              954                 100                 63                     1117

3              1117               100                 73                      1290

4              1290               100                 83                      1473

5              1473               100                 94                      1667

6    6              1667               100                106                     1882

      7              1882               100                119                     2101

Student Created Review Questions

Work out these math questions.  Check in the comments section for the answers.
1.  A right triangle has legs of 15 cm and 8 cm. Find the longest side. Formula a² + b² = c²
2. Solve for x:  3x + 5 = 20, and Solve for y:  9y - 15 = 52
3. Find the surface area and volume of a rectangular prism; L= 10mm, W= 8mm, H= 7mm
4. Find the supplement of each angle. a) 65 degrees, b) 90 degrees, c) 100 degrees, d) 55 degrees
5. What is the surface area and volume for a pyramid; L= 13 in., W= 13 inches, and H= 9 inches.
Formulas for square pyramid: V=⅓ x L x W x H,  SA= (L x W) + 4 x (½ x L x slant height); 
Slant height, a² + b² = c², (½ x length)² + (height)² = c²
6. Find the volume of a cone.  Height 26 feet, Radius 15 feet, Formula for cone: ⅓ x π x r² x height
7. Find volume and surface area of cylinder. Radius 3 cm, Height 15 cm,  
Formula for cylinder V=  π x r² x height, SA = 2 x π x r² + 2 x π x r x h
8. Find the area of the similar figures.  The ratio of similar figures is 3 m = 12 cm.  The larger triangle has a base of 3 m, height 4 m. 
Formula: Area of triangle ½ x b x height.  Ratio of Similar figures is squared for area
9. Find the perimeter of the similar triangles. The ratio of similar figures is 15 inches = 3 feet.  The smaller triangle has sides of 15 in, 24 in, and 9 in.  
10. Find the missing angles from the picture below. Angle 1 = 75 degrees. Angle 2=___,        
 Angle 3=___, Angle 4= ___, Angle 5 = ___, Angle 6= ___, Angle 7= ____, Angle 8 =____

Which Angle Is Which?

When you hear the word angle, you probably think of right angles, acute angles, and obtuse angles, but what you may not know is that there are a lot more angles that you can find easily! 

Vertical Angle: Two angles that have are opposite each other when their two lines cross, they share the same vertex, and their sides are opposite  rays.  Vertical angles make the shape of an X. If you look at the picture above "C" and "B" are examples of vertical angles. 

Corresponding angles: Two angles that are matching each other, an example of corresponding angles are angle "G" and angle "C"

Alternate Interior Angle: When two lines are cut by a transversal, these angles are between the two lines and are on opposite sides of the transversal. Angle "C" and angle "F" are Alternate Interior angles.

Alternate Exterior Angle:  When two lines are cut by a transversal, these angles are outside of the two lines and are on opposite sides of the transversal. Angle "G" and angle "B" are alternate exterior angles.

Complementary Angles: Two angles whose measures have a sum of 90 Degrees.  The angle labeled 45° in the diagram, combined with Angle "E" which is also 45° would be complementary since their sum is 90°

Supplementary Angles: Two angles whose measures have a sum of 180 Degrees. Angle "E" and angle "F" are supplementary angles. 

You can practice finding Supplementary and Complementary Angles at AAAMath.com 

What's the best deal for you? Systems of Equations

Buying an expensive gadget, computer, or phone?  Doing the math can make you feel better about getting the best deal.  What

An Apple I-Phone 4 costs $400 plus a $20 monthly fee.  A Samsung Galaxy Phone costs $350 plus a $25 monthly fee.  Which phone would be the best deal if both phones required a 12 month contract?

Some students sold popcorn for a fundraising project.  On the first week they sold 6 small bags, and 20 large bags of popcorn, and they took in $31.00 the first week. The second week they sold 12 small bags and 28 large bags of popcorn. Sales totaled $47.00 the second week.  What was the price for a small bag, and what was the price for the large bag of popcorn? 

Step 1: Define the variables

s = small bag

L= large bag

Step 2: Put word problem into equations

6s + 20L = 31

12s + 28L = 47

Step 3: Use algebra to solve for one variable 

2 (6s + 20L = 31)

12s + 28L = 47

Subtract the two equations (simplify expressions) First, Eliminate the variable S

12s + 40L = 62

-12s + 28L = 47 

___________________________  Then, Solve for L

12L = 15

L = 1.25

A large bag costs $1.25

Step 3:  Substitute L = 1.25 and solve for S

6s + 20*(1.25)= 31

6s + 25 = 31

6s = 6

s= 1

Small bags costs $1.00 each

Try your hand at some equation problems at www.khanacademy.org

Level 2: Linear Equations

Level 3: Linear Inequalities

  

Visiting a College Campus? Campus Buildings and Designs

College campuses have many attractions.  As we visit a college campus its fun to check out interesting sites.

Reading Room at Cathey Learning Center 

 Cathey Library as seen from Midway Plaissance

We'll find out more about these college buildings and designs on the tour.  Here's a blog post that shows some amazing buildings and designs. A couple of the featured buildings are at the University of Chicago. I wonder what it would be like to read a favorite book inside the immense reading room at the U of Chicago's Cathey Learning Center which stands 39 feet tall?  This building has 2 towers that have an interesting design.  Another library on the campus has robotic arms that take books off shelves for students and guests.  This library is inside a picture perfect glass dome at the U of Chicago's Mansueto Library.  It's neat how historical buildings stand next to modern designs.  What are some buildings and designs that you like?  What Math is involved in creating buildings and designs?  

Functions: Inputs, Outputs, and Beyond!

Try to find patterns below.  How do the numbers compare?  What do you notice from the input to output?
Chicago Parking                 Secure Parking
Input       Output                          Input     Output              

1                7                                      1             5
2                9                                      2             8
3                11                                    3             11
4                13                                  4             14
10              25                                 10            32
20              45                                 20            62
 n              2*x + 5                        n            3*x + 2

Question:  When trying to park in downtown Chicago we are trying to decide between 2 options. Which company offers the better deal if you want to park for a short trip, and which one for a longer trip?

Chicago Parking charges a 5 dollar entry fee and then 2 dollars per hour

Secure Parking offers a 2 dollar entry fee and then a 3 dollars per hour rate.

Equations and Graphs: Compare the equations above.  Let x= the number of hours and
Let y= total cost for parking

Chicago Parking (CP):  y=2x +5.  How? CP costs $2.00 per hour with a starting cost of $5.00.  So, y (total cost) = 2 x (# of hours)  + 5  (starting cost)
Secure Parking (SP): y=3x +2.  How?  SP costs $3.00 per hour, and its start up cost is $2.00. So, y (total cost)= 3 x (# of hours) + 2 (starting cost)

The picture above shows an example graph.  This graph shows the equation y = 2x +3.  We can see the start up cost would be 3 and then it would increase at a rate of $2.00 an hour.
Assignment:
A. Graph the equations using a graphing calculator from this website link:   Holt Mc-Dougal's website
B. What is the "point of intersection" on the graph?  This is where both companies have the same cost.  we
C. Solve the equations using algebra to find the same answer.  Solve the equation below to find x.
      2x + 5 = 3x +2
D. Create your own problem.  Start with 2 equations, or 2 companies that offer slightly different charges.
Share your questions and also some hints for solving the problem.

The Good, The Bad, The Inequality - Triangle Inequalities

              Triangles aren't always triangles, they have a hidden side, an inequality! Whether you're building a ramp or a support the safety of people is a big thing, and triangle inequalities are the hidden suspect. What happens when you know one side, but not the 2 others? Seems like a frightening prospect, but with the concept of Triangle Inequalities you can find the answer in just a moment.

             

             To construct a triangle successfully, the sum of the 2 shorter sides of the triangle must be greater than the largest side of the triangle. It would be impossible to create a triangle if the sum of the 2 shorter sides  are less than or equal to the largest side. 

      The side lengths were 4 units, 8 units, and 2 units. We added 4 and 8 and the sum is 12 which is greater than 2. Then we added 8 and 2 and the sum is 10 which is greater than 4. Next we added 4 and 2 and the sum is 6 which is less than 8. This makes the sides impossible to construct the triangle

           The side lengths were 1 unit, 2 units, and 3 units. We added 3 and 1 and the sum is 4 which is greater than 2. Then we added 3 and 2 and the sum is 5 which is greater than 1. Next we added 1 and 2 and the sum is 3 which is equal to 3. This makes the sides impossible to construct the triangle

Here are some problems for you to work on at home:

Tell whether it is possible to construct a triangle with the following side lengths

1.) 3 in, 2.5 in, 5 in      2.) 8 in, 6 in, 5 in

3.) 1.5 in, 2 in, 5 in

The Plus Side of Percents- Interest Earned

Percent doesn't always have to mean extra money that you have to pay.  When we experience a gain, percentage can be a powerful tool that helps measure our success. Here are some ways percents may work in your favor.

When looking back on your test results from before and after you can find the percent growth.  For instance,  if you had gotten 37 out of 48 questions correct on your first test, and then you went up to get 45 out of 48 points.  What was the percent increase?

Percent increase =  change in amount ÷ original amount.  

                                  45-37  ÷  37

                                      8     ÷  37   = .22  or a 22 % increase.
Check out this neat game using Percent Change, it's called the Rags to Riches Game.  Are you ready to use percents to solve problems and earn the monetary rewards?

Percents can also be used to find out how much interest you can expect  to earn for saving or investing money?   If a person saved $10 a week in "the bank of Dad" for 1 year and each week Dad paid 7% interest on the amount in the account, how much money would they have at the end of the year?

Without interest=   $20 x 52 weeks = $1,040 total 

With interest =  $20 per week X 1.07 per week = $1,111.40 total

Interest earned   1111.40-1040 = $71.40
Make a spreadsheet for your saving goals.  Try changing the amount you save and percent interest earned. 

When spending money percents discounts can be used to find which store offers the best deal. An example may be a new computer that was regularly $700 is now on sale for $550, another store offers a 25% discount and will match the price.  

Regular price $700 - Sale Price $550

Regular price $700 x .75 (percent you pay for the item after 25% off) =  $525

Difference in price $550-$525 = $50
Here's a link from Mathopolis where you can find more information and practice solving percent problems.

What are some experiences you've had with the plus sides of percents?  When the percents are used to tell stories of success, doing the math is fun!   

Ice Cream Cone Math

Have you ever wondered how much ice cream will fit into a cone?   The inside of a cone is also called its volume.  Use the formula below to find the volume of a cone.

• Sam has an ice cream cone that is 5 inches in height, and has a radius of 1.25 inches.  Find the volume of Sam's ice cream cone.  Remember to use cubic inches for labeling the answer :)

• Did you notice that the cone is one-third the volume of a cylinder of equal height?   If the volume of the cone were 8.1 cubic inches, compare the volume of a cylinder with the same radius and height?     

• Compare the volume of different ice cream cones:  Waffle cones, large sugar cones, and small sugar cones.  How much more ice cream do the bigger cones hold?  Is it worth it to buy a bigger cone?

Surface Area of Cylinders

Group Results of Cylinder having Surface Area of 900 cm² 

Radius	Height cm	Volume  cm³	Surface Area cm²	Ratio Surface Area/Volume cm²/cm³
2	70	879.2	904.32	1.03
3	45	1271.7	904.32	0.71
5	24	1884	910.6	0.48
9	7	1780.38	904.32	0.51
10	4	1256	879.2	0.70

1. What things stand out about the data displays?
2. Compare the relationship between radii and heights.  
3. Predict the size other radius/height pairs that fit the pattern. 
4. What are some further questions for this investigation?

Volume Graphs

What is the volume? Did you know volume relates to everything you do? From the food you buy in packages from the books you read, volume is in everything you do! As you read through the page you'll see ways you can find volume for cylinders and prisms!

                                           

What is the relationship between the volume of a cylinder and its radius?

What is the relationship between the volume of a prism and its base?

       Want to learn more?  Check out the videos and activities below :)

Make inflatibles using surface area and volume. Math Interactives Website 

Holiday Fractions

Word Problems

Eli was celebrating the holidays with her family and was getting ready to pick a piece of cake to share with her friends.  If Eli wanted to get the most cake which part should she take 2/5 of chocolate cake or 3/10 of marble cake?  

Marcus sorted the recycling after a holiday gift exchange.  He found that about 1/4 paper products and 1/3 plastics.  Which type of the recyclables has the greatest amount? 

Opinion

Do you think recycling is a good idea, or do you think it is too expensive?   What are the pros and cons of recycling for you?   

Fractions in review

Here is a sheet on adding and subtracting fractions.  Remember to decide on a common denominator.  Then remember to multiply the numerator and denominator by the same number to make equivalent fractions.  

Create one of your own fraction word problems to share.  Add the answers too:)

Answers:  2/5= 4/10,  so 2/5 is a bigger piece than 3/10

1/4= 25%, 1/3=33%,  1/3 is the greater amont

   

Comparing World Populations

We can use population size to compare different countries in our world.  One way to write large numbers such as population is in scientific notation.  Scientific notation is written as a decimal between 1 and 9 multiplied by a power of 10.
For example the population of Turkey is 79,749,000 people, or as 7.9749 * 10^7 in scientific notation.

The country of Mexico has a population of about 114,975,000 people, or 1.14975 X 10^8.  Mexico has approximately 35 million more people than Turkey.  

Another way to compare populations is to find out the population density.  The population density compares the population to the land area.   For example Turkey has a land area of 769,632 square kilometers. When we divide the population by the land area we find the population density.  The population density tells us how many people live per square kilometer of land.  In Turkey the population density is 79,749,000 people /  769,632 square kilometers = 104 people/square kilometer.  Mexico has about 1,943,945 square kilometers of land, and its population density is 59 people per square kilometer. So, even though the population is less in Turkey, it may feel more crowded because of a higher population density.

Population density is used to compare how crowded places are around the world.  The density of a place might suggest a good place to begin a business, or a good place to advertise a product/service.   Others may think that a place with low population density may be a more peaceful place to live with less noise or traffic congestion.  How do you feel, would you prefer to live in place with a high or low population density?

Use the websites below to explore places in the part of the world you live, or places that interest you.  Here are some questions to include as you explore as you compare world populations. 

• How do different cities or countries compare in population ? 
• How would the populations be expressed in Scientific Notation?
• How does the population density compare between the cities or countries?  

US Populations

World Populations

Illinois Facts

World Land Areas

Quiz on Population Density

Here's an example of a Density Chart

Country	Population (people)	Scientific Notation (Population)	Land Area (Square Kilometers)	Population Density Population ÷Land Area
United States	313,847,000	3.13847 X 10^8	9,161,966	34
Mexico	114,975,000	1.14975 X 10^8	1,943,945	59
Turkey	79,749,000	7.9749 X 10^7	769,632	104
Peru	29,550,000	2.9550 X 10^7	1,279,996	23

By looking at Density I learned about the ratio of people to land area.  It was interesting to me that Turkey had the highest population density of the four countries since Turkey had the second lowest population.
 I think that this high density may be caused by it's smaller land area.  Another thing I noticed was the lowest population density was from Peru who had just 32 people per square kilometer.  The population in Peru is therefore more spread out than the other countries which have a higher density than Turkey.

References

1. World By Map, http://world.bymap.org/LandArea.html, December 18, 2012
2. World By Map, http://world.bymap.org/Population.html, December 18, 2012

Geometry and Right Triangles

The diagram of above shows the relationship between the area of the large purple square and the area of the middle tan colored square.  The website Math is Fun tells how how this puzzle fits together.  The right triangles and squares inside the figure create a unique pattern that we use today to find out the length of the sides of right triangles.  

Use this Pythagorean Theorem interactive tool to experiment with how the sum of the squares of the triangle legs is equal to the longest side squared.

Here's a Pythagorean Theorem problem to try. Take a moment also to share your own problem as well.
A ladder stretches from the floor to a shelf diagonally. (c)  The distance from the floor to the base of the ladder is 4 meters (b).  The height of the wall which makes a 90 degree angle with the floor is 3 meters (a).  What is the length of the ladder?

3*3  + 4*4=  ___ * ___

9  + 16  = ___ * ___

25 = ___ * ____           Hint: square root of 25 = ?
How many meters long is the ladder?

Translations, Rotations, and Reflections

Coordinate Graphs use ordered pairs to plot different shapes and designs.  One way we use coordinate graphs is to move shapes around to different parts of the graph through translations, rotations, and slides.

This is used in the real world by computer programmers to make models and animations.   

Try your skill at using translations, rotations, and
slides by clicking on this link for a Khan Academy Simulation.  Transformations Computer Practice  

Transformations are a general term that means that things are being moved around in the coordinate graph.  

Here are some terms that you probably know already:  slides, flips, and turns.

• Slides- when an object is moved without lifting it off the page is a slide.  Another way of saying slide is a translation.  Translations can be found by using an equation like (x + 2, y - 1).  For example, if the original point in an ordered pair (x,y) is (4,1)  then the translation of that point would be (4+2, and 1-1) or  the new point would be at (6,0). 
• Flips-  another word for a flip is a reflection.  The reflection of the object is when it is "flipped" on the opposite side of the x or y axis.   One way to do the reflection of a shape in an ordered pair (x,y) is to multiply either the x or y by negative 1.   Let's say for instance that we want to flip a point over the x axis. Using point (3, 2) we would multiply the x coordinate 3 by -1.  The new point would be at (-3,2) 
• Turns-   turning an object around from a center point is yet another way we can move the object.       A Rotation is another name for a turn because there is a center point that remains the same as the shape rotates on 1 point.  Both the size of the figure and the distance between the points remains the same as the figure rotates to different quadrants.                                                                                                                  

Here's a video link that shows how to rotate a quadrilateral 90 degrees and a way to predict where the new points will be on the coordinate graph.  Rotations of 90 Degrees Video

Online Quiz for Transformations: Translations, Reflections, and Rotations.

Electoral Votes and the United States Election

270 electoral votes are needed to win the presidential election.  What are some different ways that the states' electoral votes could be combined to equal the 270 votes?

For example, using the chart of electoral votes by state some of the top states that would sum up to 270 include: California  55, Pennsylvania 20, New York 29, Florida 29, Michigan 16, Texas 38, Illinois 20, North Carolina  15, Georgia 16, Minnesota 10. Washington 12. and Wisconsin 10.  

The map of electoral votes state by state shows who's in the lead in each state going into election week.
What different combinations of electoral votes would give your candidate the 270 votes needed to win?
If you were running for president in which states would you want to spend your resources?

270 votes is what percent of the total 548 votes needed to win the election?

How can we divide it?

Diagrams like number lines and animations help us show and explain real life situations.  

Websites: 
Interactive number line investigates making equivalent fractions and finding common multiples.  

Math Animation explores finding out how to sort markers and video game prizes into paper bags.

Problems:
1.  If x=5 and y=3 prove that:

6x  - 5 =  5
y

3y-x +9 = 18-5

2. There are two kinds of breakfast bags that are served: Hot or Cold.  If there were 18 hot breakfasts and 27 cold breakfasts available one day,  How boxes of the hot and cold breakfasts could be made that each have the same amount of hot and cold breakfasts?

Hint:  What are the common factors of 18 and 27?
 ____, ____, _____,
 ____, ____, _____