Newtown Math Trail 2009: Fun and Challenging Math Problems for Students

1. Newtown Math Trail 2009 Created by Goodnoe’s Enriched Math Sixth Graders Class of 2015

2. The Class of 2015 is proud to present the 2009 Newtown Math Trail. Brought to you by: Ryan, Amrit, Katie, Andy, Alison, Eric, Harris, David, Dan, Edmond, Laura, Patrick, Aditya, Chloe, Connor, Kathleen, Travis, Joe, Celeste, and Nancy

3. The Starbucks Smoothie Starbucks has a new drink out called the Osoberry smoothie. A dedicated customer named Bob Osoberry comes in on the release date and tries the drink. He loves it! From this point on, every day he buys a smoothie. After 2 weeks, Bob decides it’s time to spread the word of this delicious drink. Every dayafter thathe tells one friend. Every new customer buys one smoothie a day and starts telling one person a day about the drink starting on the second day. After 1 perfect month of exactly 28 days (4 weeks), how many drinks will be purchased on the final day of that month? Katie & Chloe

4. The Starbucks Smoothie Cost: 3 cups osoberries - $2.00 1 gallon ice cream - $3.00 1 gallon milk - $4.00 1 tablespoon raspberry syrup - $0.01 Ice – FREE How much does it cost to make: one smoothie? the whole day’s worth of smoothies on the last day of the month? The cost of a Grande Osoberry is $4.95. After subtracting the cost of all ingredients, what profit is made from all of the Smoothies bought on the final day of the month? Osoberry Smoothie Ingredients: ¾ cup osoberries ¼ cup vanilla ice cream 1 cup milk 2 tablespoons raspberry syrup 1 cup ice Grande size only On that day after 1 month how much of each ingredient will be needed? Katie & Chloe

5. Bookshop Brain-Buster Newtown Bookshop’s store is empty so the owner has been ordering books to get ready for their biggest sale ever. Finally the books arrive in sets of five, but one set is the wrong order. Newtown Bookshop needs to return the books, which must be rewrapped. How much plastic wrap does the store need if each book has: A front of 5 in x 8 in One side of 1 in x 8 in Another side of 1 in x 5 in What amount of plastic does the bookshop need to return the books wrapped correctly? Kathleen

6. Challenging Chandler Chandler Baseball Field has three fields. Bob, who mows the fields, wants to mow the fields from the smallest to the largest field. The problem is that Bob does not know the sizes of the fields. He only knows some facts, which are written as math problems on the next slide. Solve for each variable to determine the field sizes. You only need to find the area of the outfield. For Example: Determine the distance from home to first and double it. Home to first is 1 yd, then it would be a 1yd x 2 yd = 2 yd2 rectangle that Bob needs to mow.

7. Back Field Solve for n to find the distance from first to home: 5n + (-2)= (√100 x.2 + (-5)) x 5 -18 ÷ 2 +24 + 58 Middle Field Solve for y to find the distance from home to second: πy = (25 + 5 ÷ 5 + 4) ÷ 10 x 4 +(122 - √144) – 49.8 ÷ 6 + 12.56 + 9.42 The distance from home to first is ¾ of the distance from home to second. Front Field Solve for z to find the distance from first to home: z = 12/8 x 10 +√400 – 25 x (y/n x 12/9) Edmond

8. Aden and Mika’s Baseball Brain Buster Baseball Batting Frenzy Newtown Math Trail 2009 In Newtown’s Babe Ruth League, there are 90 players. There are 20 outstanding, 50 decent hitters, and 20 mediocre. The best hitters go 4 for 4 every game, except they experience a one day slump every 4 games. The batter’s slump occurs with no hits when they bat.While the decent hitters are 2 for 4 each game. The mediocre hitters are 1 for 4 each game with batting slumps every other game. If there are 12 games for each team and each batter bats 4 times each game, what is the average of all the batters in the league? There are no play-offs. David & Eric

9. Sweet Pea’s Predicament Oh no! There has just been an attempted robbery at the Sweet Pea Ice Cream Parlor! Luckily, nothing has been stolen, but four orders are mixed up. The only clues that the employees have is….. • The orders were a large chocolate cone, a hot fudge sundae, one strawberry smoothie, and a vanilla cup with chocolate chips. • The people that ordered were: Bobby Norris, Amrit, Nancy, and Alison • Amrit does not like smoothies. • Nancy loves any kind of chocolate. • The person’s name who ordered the chocolate cone begins with an “A.” • Bobby likes toppings on his ice cream. • Alison ordered the strawberry smoothie. • Nancy likes warm creamy toppings on her ice cream. Use the chart to help with the orders before the customers take their business to The Zebra Striped Whale. Laura

10. La Stalla’s Bar The La Stalla Restaurant has two televisions at their bar. On a June night, there are many people from New York visiting relatives at La Stalla. At the bar there are 81 people: 2/3 are New Yorkers 1/3 Philadelphians The Phillies are playing the Florida Marlins, the New York Knicks are playing the Boston Celtics, and the Mets are playing the Atlanta Braves. The fans are as follows: 8/9 of the Philadelphians love the Phils 4/9 of the New Yorkers like the Knicks 1/3 of the New Yorkers cheer for the Mets The games listed above begin at the same time. Which two games should La Stalla put on at the bar that would attract more business? Harris

11. (We Need A) Sweet Pea Solution Alison and Nancy just had banana splits at The Sweet Pea Ice Cream Parlor. They decide to burn off the calories by walking to Alison’s house. Before setting out, they want to know how many calories were consumed. The girls ask the owner for the amount of calories in each ingredient, but all they received are a few equations! The equations must be solved in order to find out the amount of calories consumed. The ingredients and their corresponding equations are: 2 Bananas: 75 calories per banana 1 Scoop of Vanilla Ice Cream: 2x(2 bananas)+(1/4 of chocolate syrup) 1 Scoop of Strawberry Ice Cream: (1/5 of √625)+5+(2 scoops of vanilla)÷2 1 Tbsp. Chocolate Syrup: Same caloric value as 10 cherries 1 Cherry: 25 calories per 4 cherries A. How many calories were consumed in total? B. How many calories were consumed excluding fruit? C. How many calories from fruit were consumed? Alison & Nancy

12. Problem with the Popcorn At the Newtown Theater the small popcorn is 1/3 the size of the large popcorn. The medium size is 2/3 of the large. The extra-large is all three sizes combined. If a large size contains 17.85 oz, how many ounces does the extra-large contain? Daniel, Connor, & Travis

13. Toothpick Tiles At Starbucks The floor at Starbucks is made of many tiles. On each side of a tile there is one toothpick. The toothpicks do not overlap. The tiles come in: wholes, halves, thirds, and fourths. The half tiles have 2 half toothpick sides, one third tiles have 2 one third toothpick sides, and one fourth tiles have 2 one fourth toothpick sides. If the numbers shown below are correct then how many toothpicks does it take to outline all of the tiles at Starbucks? 433 wholes 62 halves 39 thirds 16 fourths Ryan Andy

14. Three of the most famous ancient civilizations have taken over the borough of Newtown. The only way you can defeat them is to outsmart them by using the math of these ancients. First you encounter Julius Caesar and his posse of chariots. To defeat these Romans, use Roman Numerals. See the bust of Nero for help. Saving Newtown Roman Numerals are essentially letters that represent numbers. I=1, V= 5, X=10, L= 50, C=100, D= 500 and M=1,000. There is no zero. Example:2009 = MMIX Patrick Decode these: Encode these: MDCLXXXIV 1776 MDCCXXXIX 4111 CDLX + XXVI 411 + 234 CDXXIV x IL 1049 + 696 MMCDLXXXVIII 3419

15. The next person you encounter is Alexander the Great of Greece. He is taking over the houses of worship in Newtown. Use the Pythagorean Theorem to solve the missing lengths. Archimedes will assist you. Saving Newtown The Pythagorean Theorem is a2 + b2 = c2, which involves right triangles. Patrick St. Andrew’s Main Church Old PresbyterianChurch ______ Blocks 9 Blocks 12 Blocks 12 Blocks Quaker Meeting House Shir Ami Presbyterian Church St. Andrew’s Old Church 16 Blocks ______ Blocks

16. Last you encounter Pharaoh Tutankhamen, who has taken all the water from the water tower. To defeat him use Egyptian Fractions. See Ra for help. Saving Newtown Egyptian fractions are simple. Take a fraction which must be written as a unit fraction or the sum of unit fractions. Example: ¾ = ¼ + ½ Patrick 26/32= 71/72= 29/44= 31/64= Congratulations if you solved all of the problems correctly Newtown is saved from these notorious civilizations. Great job!

17. The Starbucks Coffee Caper Jeremy was a regular at the local Starbucks. He loved their delicious coffee and wondered about the store’s weekly income. Jeremy inquired, and the manager (a long time mathematician), decided to have a little fun with Jeremy’s question. Instead of just giving Jeremy the answer, he handed him a paper with this problem: If we add together 1/2+ 1/3+ 1/3 of our daily income in regular coffee we get $3,500. If we make this same income (for regular coffee) everyday of the week except on Mondays when our income in regular coffee is doubled (because everyone is extra tired on Mondays), and add 1/3 of our daily income of decaf coffee you will find our weekly income in all coffee. P.S. Wondering what our daily decaf income is? Here is a ratio for you of decaf coffee to regular: 4:5. Can you help Jeremy figure out the weekly income of Starbucks Coffee? Joe & Aditya

18. Ali’s Igloo Profit At the end of the day on Friday, Ali’s Igloo’s manager decided to count the money that was earned for the last five days. On Monday 1/2 as much was earned as on Friday. While on Wednesday 1/2 as much was earned as on Monday plus Friday’s earnings. $6.25 less than Wednesday was earned on Thursday. The store was closed on Saturday and Sunday due to Mother’s Day Weekend. Calculate Friday’s earnings: KeyFriday’s EarningsKey s= small 1) (s + m) ÷ √4 small= $2.75 m= medium 2) [(L × s) 0.10] ÷ 4 medium= $3.50 l= large 3) 9 × L ÷ 3 large= 4.25 4) m ÷ s × 4 × 1 5)[(3.44 × 1) ÷ 3.44] Add your answers for 1 to 5 above. Then multiply the sum by 15, move the decimal in the product one space to the left and remove the number in the thousandths place, giving you Friday’s earnings. Continue to solve: How much money did Ali’s Igloo earn from Monday to Friday? Celeste

19. Bird in Hand “Oh, no!!” During a fight over Hessian prisoners, a glass mirror has been broken. Now, this is no ordinary mirror, but a contraption allowing the rebel jailers of Hessian soldiers to look outside and observe the goings on without appearing vulnerable. The glass in question is at the Bird in Hand, the rebel’s headquarters located in Newtown, Pennsylvania. Word is that the Hessians are planning an attack to retrieve their prisoners. Without the glass, the rebels are completely defenseless. Can you help? The glass fragments were too small to repair, but they managed to find the measurements of the original intricate glass mirror (Listed on the next slide but incomplete, so infer). Only use a pencil, compass, calculator, and a ruler to make the mirror. The glass is layered with a rectangle at the bottom, then a circle, and finally a triangle. Now that you have all of the information that you need, draw out a plan of the glass and help save the revolution! Amrit

20. Bird in Hand Measurements: Circle: • Circumference + 4 = 22.84 inches • Radius times 4 = 12 inches • Diameter ________________ • Rectangle: • One side length: 6 inches • Area – 9 = 33 • Other side length: ________ • Triangle: • Type: isosceles • One side length divided by 2 = 1.5 inches • Area= 10.5 square inches • Other side length _______________ Amrit

21. Scary Harry now lives in a painting in the Half Moon Inn. In his time, the Inn was known as the Court Inn. Scary Harry was once alive and well in the 1700’s. He also once had a major math problem that he needed help with…. “Argh! Ugh!” grunted Harry. He heaved himself against the heavy library door. “Darn,” he thought, “All I wanted to do was get a book to read while I whoop everybody at piquet downstairs, but now I’m locked in the library, and I only have five minutes before the game starts!!” Harry wallowed in self-pity until a glowing, glittery, purple light appeared. “Wha?” he exclaimed, dazed and blinded by the razzle- dazzle light. A large wriggling octopus in an Elvis suit emerged and handed Harry a note in one of his squishy tentacles; then vanished into thin air. The note said… Take the grid and equation below, You’ll get what you need to know. Solve the problem and plot this number: 5, The grid is of a shelf on the front, to your right. Can you help Harry with his predicament? The equation and grid are on the next slide. Use your superior brain skills and knowledge to solve it. Scary Harry’s Horrendously Horrible Library Lockup Laura & Amrit

22. Grid Y=2X

23. You may think that Harry’s free, but that is where you are wrong. • “Good. Now I have the place. Now to find the book.” • Harry ran to the shelf and grabbed the correct book, Poor Richard’s Almanac. When he opened it, there was a hollowed out rectangle with a sealed envelope in it. He eagerly ripped open the envelope and began to read… • I see that you’re in trouble, • You need to unlock a door. • In order to help you out, • I give you these clues four. • The 2nd and 3rd digit added together = the 1st digit • The 2nd digit + 4th digit = the 5th digit • The 1st digit + 1 = the last digit • The last digit – 2 = the 4th digit • Scary Harry unlocked the door and proceeded to win $500 playing piquet. As a memory of that historical game, a picture was painted of him and hung above the table where he had played piquet! Can you outsmart Scary Harry and beat the clock? Solve the problem and challenge a friend to piquet!! Laura & Amrit

24. A special thank you to my sixth graders, Class of 2015 for a third successful Math Trail. Best wishes at Newtown Middle School! Always follow the math trail to find future success.