World's Fair Math Explanation Example

Surface Area and Volume of a Building

The Harold Washington Library is in the shape of a rectangular prism for the main building.  I want to find the surface area and volume of the building.  The surface area of is the total area of all the faces.  Surface area is measured in square units (cm²) because like area it is the amount of material that covers a flat surface and can be put into square units of measure.  The other measurement that I want to show is the volume which is the space inside the figure.  Volume is measured in cubic units (cm³) because it is measuring the units inside of a 3 dimensional figure. 

The HWL is approximately 50 feet long, 25 feet wide and 30 feet tall.

Surface area = Sum of all 6 of the rectangular faces 

SA= 2 (lxw) + 2 (lxh) +2 (wxh)

SA= 2 (50x25) + 2 (50x30) + 2(25x30)

SA = 2 (1,250) + 2 (1500) + 2 (750)

SA = 2,500 + 3000 + 1500

SA = 5,500 + 1500

Surface Area = 7,000 square feet  

I found the surface area of the rectangular prism by measuring the area of the 6 faces of the prism.  I knew there were 6 faces from the net drawing of a rectangular prism.  First I knew the building was 50 ft. length, 25 ft. width, and 30 ft. height and put them into the equation.  SA= 2 (lxw) + 2(lxh) + 2(wxh). I multiplied inside parentheses first.  For example, 50x25=1,250. Then I knew that the 2 on the outside meant to multiply since there was a number next to a parentheses.   So I multiplied 2 times 1,250 to get 2,500 using mental math. I multiplied by two because I knew the top and the bottom of the prism were the same size. The front and back sides were each 750 square feet, or 25 times 30, and the top and bottom were each 1,250 which was found by multiplying 50 by 25.  Then I doubled the front to find the total area of the front and back sides since they were both 25 by 30. 1500 times 2 equaled 3,000.  The combined surface area was found by adding all these areas together 2,500 + 3,000 + 1,500 was 7,000 square feet.  I checked my answer using a calculator and found the surface area of the rectangular prism would be 7,000 square feet.

Volume of a rectangular prism= Length x Width x Height

Volume = 50 x 25 x 30 =  37,500 cubic feet

The volume of the figure was found by using an equation.  I knew that the base of the figure was 50 by 25 feet.   So, I multiplied 50 times 25 to get 1,250 square feet.  The height of the rectangular prism was 30 feet, so I multiplied the base times the height to find out how much volume spans up the height of the prism.  I found that 1,250 times 30 was 37,500 cubic feet.  The volume of the rectangular prism is 37,500 cubic feet.  

Reading Projects

Create a culminating project about the novel that you have read.  In your project write about the themes and main topics that were addressed.  Include the vocabulary terms that you learned, character analysis, and conflict/resolutions.

Format: PowerPoint Presentation, Poster Board, or Book Blog.
Grouping Preference: Group vs. Individual Projects
Due Date: December 10th

• Group Projects or Individual Projects:  Group projects should include multiple aspects of the book, the projects should describe more chapters, or aspects of the book (tone of the book, questions/answers, polls, and other features that you learned from your book)  
• Individual projects should be focused on a central theme or topic of the book.  The individual project can include some information about a favorite chapter or specific aspect of the book.  (tone, questions/answers, polls and other features)

Triangle Properties

Angles

What is the sum of the interior angles of any triangle?
If angles 1 and 2 are 40 degrees and 60 degrees what is the missing interior angle? What is the measure of the exterior angle?

Sides

What is the relationship between the sides of any triangle?   If side a is 8 centimeters, what is the length of sides b and c?  When we add two sides of the triangle how does the sum compare to the remaining side?

Tax Problems

Janice went to Jewel Food Store and bought a box of Orange Popsicle for $3.30.  The tax rate for groceries was 4%. How much would 4 boxes of Orange Popsicles cost? (Include tax)

Step 1:  $3.30 X 4 boxes = $13.20

Step 2: $13.20 X 1.04 (4% tax) = 13.728      (1.04  = 1 + .04   (1 to get the cost, and .04 for 4/100 tax)

The total with tax would be $13.73 (Rounded to the nearest cent)

Assignment: Write your own tax problem.  Include the cost of the product.  Find out how much the item would cost after a 4% tax is included.                       

Riddles and Limericks

What riddles do you know?   Halloween riddles from kidspot.com.au

A. What's a ghost's favorite dessert?  

B. What movie do ghosts love to watch?

C. What did the boy ghost say to the girl ghost?

D. What kind of roads do ghosts haunt?

E. What did the polite ghost say to her son?

F. Who was the most famous French skeleton?

And a Limerick...  From briandehumor.com
          Mosquitoes

Mosquitoes are what a bat eats,

For him they are edible treats,

He's known to devour,

A thousand an hour, 

We're happy when he overeats. 

A limerick is... a 5 line poem.  It has rhyme pattern on Lines 1, 2, + 5, Lines 3+4.  The limerick has 36 syllables- 8 each on lines 1,2, + 5, and 6 each on 3+4.  The last line usually has a punch line, or play on words.  

Write a limerick  

1. Start by brainstorming a funny/silly idea. (make a question, + answer it)

2. Write lines 1 and 2 end with a rhyming word.  (Need help with rhymes try- rhymezone.com) 

3. Add lines 3 and 4 (just 6 syllables each)

4. Finish with your punchline. (answer to question)

        

     Freedom (No Turning Back)                

                         by, Mr. Hull  

     Fresh air rushes into my lungs,
     New life appears like a bell rung.
     My old burdens escape,
     Tears spill free from my face.
     Hope grows for a new life begun.  

Riddle Answers  A. I-scream  B. High S-ghoul Musical  C. You look boo-tiful tonight

D. Dead Ends  E. Don't spook until your spooken to   F. Napoleon Bone-A Part 

Book Club Discussions

Book Club Discussion Starters

Who are the main characters in the story, what challenges do they face?

Describe the setting or background of the story.  (Time period, cultures, family structure)

What is the tone of the story so far?  (Does it feel tense, exciting, sad?  Support with examples)

Mystery Number Questions

Create a mystery number problem using integers.   Use these ideas for developing your clues.  Begin by picking a number between -100 and 100

• Odd or even?
• Is it a positive or negative number?
• Is it a square number
• Can you name some of the factors?
• More or less than a number?

Here's an example....

I'm thinking of a number that is less than -10

It is an even number

The number is a multiple of 4 and 9

When you add the absolute value of the digits together you get 9.

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