Summer Math Program 2013

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Question 1 I went bug watching after school every day for a week. On Monday, I saw 1 bug. On Tuesday, I saw 3 bugs. On Wednesday, I saw 5 bugs... On Sunday, after I went bug watching I said, “Wow! It’s a pattern!” How many bugs did I see that week?

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Question 2 On a new planet the astronauts discovered unusual creatures. Each had 3 eyes, 1 horn, 4 legs and 6 arms. If the astronauts saw 6 creatures in all, how many eyes, horns, legs and arms did they see in all?

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Question 3 How many tentacles are on 1 octopus? How many tentacles are on 2 octopuses? How many tentacles are on 3 octopuses? How many tentacles are on 5 octopuses? How many tentacles are on 10 octopuses?

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Question 4 The pan has hot chocolate in it. It takes 3 full ladles to fill each mug. How many ladles will it take to fill mugs for 4 children that are coming in from sledding? 5 children? 6 children? 10 children?

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Question 5 We have 12 chicken eggs in our classroom incubator. If the 1st egg hatches today at 11:00, and the next one hatches at 11:20, and then next one at 11:40, and this rate continues, what time will it be when all 12 eggs hatch? Show how you figured this out.

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Question 6 Ida the Iguanodon and Stan the Stegosaurus are having a Valentine’s Day party. They have invited the following friends: • Trisha and the 3 Triceratops • The 3 Little Diplodocons • Mr. Saltopus and Ms. Compsognathus • Archaeopteryx and the 7 Sauropods • Ann the Ankylosaurus How many guests will they invite in all? They are planning a sit down meal for their party. They have square tables that will fit 1 guest on each side. How many tables will they need to set up?

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Question 7 We are planning a 100th birthday celebration in our classroom. We have 100 balloons with which to decorate the tables in our classroom. There are 6 tables and 20 students in our class. What is the best number of students to seat at each table, and how many balloons should decorate each table?

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Question 8 There are 50 chips in the bag. The chips are green, blue or yellow. I am going to pull out 10 chips, and we are going to record the order and the color of those chips: 1. Green 2. Green 3. Blue 4. Green 5. Yellow 6. Blue 7. Blue 8. Green 9. Green 10. Yellow What do you think will be the next color? Why? (Predict) What color(s) is most of the chips in the bag? What makes you think so? What color(s) is the least amount of chips in the bag? What makes you think so? Based on the first 10, how many of each color total should be found in the bag.? Green? Yellow? Blue?

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Question 9 There were 3 students that made 6 clay pots. If each student made at least 1 pot, how how many did each student make? Be sure to show all of the possibilities.

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Question 10 For Easter, the Hershey’s Chocolate Company makes Kisses® wrapped in colored foil. The Easter Bunny has a bag containing the following candies: Light Pink Dark Pink Silver Green Blue The Easter Bunny is making up Easter baskets for Flopsy, Mopsy and Peter Cottontail. If he does his best to share the Kisses® evenly among the 3 bunnies, show what each of their baskets will contain.

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Question 11 I have a miniature collection of farm animals and I would like to fence in a rectangular area to put them in. I have only 20 centimeters of fencing. Find all the different rectangles that you can that have a perimeter of 20 centimeters. Write or tell about any patterns you see about your rectangles. Write or tell why you think you found all the ways to make the different rectangles.

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Question 12 Beavers are very strong animals. They have sharp teeth that they use to build dams that are an average of 65 feet long. The Vermont Fish and Wildlife Group has reported that the teeth and jaws of a beaver are so powerful they can cut down a tree 20 inches thick in 15 minutes.If a family of 4 beavers was building a dam and they worked for 1 hour, how many inches of tree could they use for their dam? Remember to show your work.

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Question 13 How many different snow people can you draw with a red or green hat and a blue or orange scarf? What if there are 3 different color hats? What if there are 4 different color hats?

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Question 14 I went to the store to buy a muffin. Muffins cost 25 cents each. I had a lot of change in my coin purse. How many ways could I pay for the muffin?

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Question 15 Itâ&#x20AC;&#x2122;s Saturday morning and you need to practice the piano 2 hours every week before your lesson on Friday afternoon. Your teacher says not to work more than 20 minutes at a time. Show what your practice schedule might look like (how many minutes each day?).

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Question 16 Tim and Lisa are having friends over for a pizza party. There will be 5 children, which includes Lisa and Tim. Each child wants 3 slices of pizza. How many whole pizzas do they need? How will the pizzas be cut into fractional parts?

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Question 17 On a ski slope there are 9 tracks. If each person went down once, how many skiers and snow boarders were there? What are all of the possibilities?

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Question 18 Choose a common household object. Collect data on the different colors that you find that object in. Make a bar graph showing your results. Choose another way in which to describe your object and make a bar graph of those results. What are the most common characteristics of your object?

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Question 19 There is an apartment building called The Ten Feet Apartment Building. The owner allows people and pets to rent apartments in the building, but each family (including pets) can only have a total of 10 feet living in its apartment. Find the different combinations of people and pets that equal 10 feet. Draw pictures and write or tell about your family.

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Question 20 Only 3 people and their supplies can fit in the canoe at one time. How many trips will it take a Native American family of 8 to cross a river using 1 canoe? How many trips will it take a Native American family of 9 to cross a river using 1 canoe? How many trips will it take a Native American family of 10 to cross a river using 1 canoe? How many trips will it take a Native American family of 11 to cross a river using 1 canoe? How many trips will it take a Native American family of 12 to cross a river using 1 canoe?

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Question 1 An ant is at the bottom of a 5-foot deep well and is trying to get to the top. During the day he climbs up 4 Â˝ inches, but at night he slides back down 2 Âž inches. How long does it take for him to get out of the well?

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Question 2 A group of your friends have invited you to join them on Saturday to go to a matinee movie, get some lunch and play some video games. Your parents say you can go, but only if you do all your chores first. Your chores list includes doing your homework which will take you about 45 minutes, collecting and taking out all the trash which takes about 15 minutes, 30 minutes of folding laundry, and cleaning your room which usually takes about 45 minutes. Your friends are getting together at 11:00 a.m. and you have a lot to do before your mom and dad will let you go. What time do you need to wake up? Are there other things you need to do before you leave the house?

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Question 3 Cornelius was born on February 20, 1988, at 11:05 a.m. His birthday falls on a Friday this year, but he will be celebrating it with a party on Saturday, February 21, 1998, at 3:00 p.m. On the birthday cake she made, his mom wants to write the exact age he will be at the start of his party. How could she write it?

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Question 4 For Mr. Mitchellâ&#x20AC;&#x2122;s birthday, he received some wonderful birthday cakes! There was one cake that had many different flavors all in one cake! 4 12 of the cake was chocolate, and the rest was carrot and yellow cake, but not in equal amounts. What could this deluxe birthday cake look like? How do you know?

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Question 5 Olivia had fun baking lots of cookies. She left them on a plate to cool while she went shopping with her dad. Her brother saw the cookies and took ½ of them to his Scout meeting. Her sister took ⅔ of the remaining cookies to share with her friends. Finally, her mom took ½ of the remaining cookies to her Book Club meeting. When Olivia and her dad got home, there were only 5 cookies left on the plate. How many cookies had Olivia baked?

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Question 6 Om is trying to buy enough origami paper to make his class Valentineâ&#x20AC;&#x2122;s Day cards. He has figured out that there are 17 students in his class. He knows that he needs 1 3-inch-square piece to make 1 card, but he can only find paper in 9-inch-square size pieces. Help Om to determine the number of sheets of 9-inch square paper he needs to buy. Remember that he does not want to buy too much and waste this expensive paper!

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Question 7 Kwanza is not a religious holiday. It is a festive celebration spent with family and friends that lasts 7 days. It is a time when African-Americans and others join together to honor traditions of their ancestors and focus on the year to come and how to make themselves better people and a better community. One part of the traditional Kwanza celebration includes lighting candles at night. There are 7 candles called the mishama saba placed in a special candle holder called a kinara. Each candle stands for 1 of the 7 principles of Kwanza. The black candle is placed in the center of the kinara and represents unity. This is lit on the first night of Kwanza as well as every other night of the celebration. The 3 red candles are placed to the left of the black candle and represent purpose, creativity and faith. The 3 green candles represent self-determination, collective work and responsibility and cooperative economics. They are placed to the right of the black candle. The mishama saba are lit for 7 nights. The 1st night only the center, black candle is lit. The 2nd night, the black candle and a red candle are lit starting at the left side of the kinara. Each night, the candles that were burned are replaced and lit again, increasing the number of candles lit each night by 1. At the end of the Kwanza celebration, how many candles of each color have been used? How many all together?

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Question 8 You are a part of a reading club at the school library. The librarian let you choose your own teams. Each team had to read as many books as they could in a month. The team that worked the hardest gets a chance to go to the town bookstore and pick out a book for free. The librarian has a problem. She noticed that the teams have a different number of members. This is going to make it hard to decide which team worked the hardest. The librarian wants it to be fair, so she has asked you to be the judge. You have to come up with a mathematically fair way to pick the hardest working team. Please write a letter explaining exactly why you picked a certain team and why you think it is a fair decision. Team 1 Name John Clare Beth CiCi Jeanne

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Books 5 5 2 4 3

Team 2 Name Alicia David Cindy

Books 4 4 6

Team 3 Name Peter Wendy Mary Terri

Books 4 3 4 3

Question 9 Miss Guy has a very energetic puppy. The puppy loves to play outdoors, so Miss Guy decided to build a pen to allow her pet to be outside while she is at school. She just happens to have 50 feet of fencing in her basement that she can use for the pen. What are some of the ways she can set up the pen that uses all the fencing? What are the dimensions of the rectangular pen with the most space available for the puppy to play?

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Question 10 You have just won a contest! You must choose between 2 prizes. You may choose \$.75 a day for 15 days or you may choose a penny the 1st day, which doubles every day for 15 days. Which prize would you choose?

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Question 11 Last week I brought the stamps my class made down to the post office to hang up. Karen, the postmaster, said she had other things to display and wanted to know how much space was needed to hang up all the stamps. If the stamps are 8.5 cm x 11 cm and there are 17 of them, what would you tell her? Are there other possibilities?

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Question 12 Stamford is having its annual Holiday Fair. Items are sold for either 25 cents or 50 cents. Each child in our class can spend \$2.00 at the fair. Assuming that every child spends all of her/his money, how many 25-cent and 50-cent gifts could have been bought altogether by our class of 11 students?

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Question 13 There are 13 keys on a xylophone (C, D, E, F, G, A, B, C, D, E, F, G and A, in that order). The 1st key (C) is 7.5 inches long. The 2nd key (D) is 7.25 inches long. The 3rd key (E) is 7 inches long. ...and so on continuing this pattern in size. Zeno the xylophone maker was about to begin construction on a new xylophone when he realized he had run out of metal stripping used to make the keys. This metal stripping only comes in 1-yard pieces. How many 1-yard pieces does Zeno need to buy in order to make 1 xylophone?

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Question 14 You have been asked to supply milk for the medieval feast in our class. When you go to the store to buy some, you discover you have many choices. You can buy milk in pints, quarts, Â˝ gallons and gallons. You know you have to buy 16 servings, and ideally each person should get a 6-ounce serving. You also realize you should not waste money or food, so you only want to buy as much as we need. What should you buy and why?

MILK \$2.39 Gallon

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MILK

MILK MILK

\$1.59 Half Gallon

\$1.19 Quart

\$0.65 Pint

Question 15 When I was watching our bird feeder at school, I noticed that Monday a chickadee took 1 sunflower seed from our bird feeder. On Tuesday, he took 4 seeds. On Wednesday, he took 7 seeds. The day after, he took 10 seeds. If this pattern continues, how many sunflower seeds would the chickadee eat on the 10th day? Day 25? Day 47? Day 63? Day 100?

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Question 16 Ollie the Owl is opening a new food court in his local mall that will entice his fellow owl buddies. He is planning on having 7 restaurants in his food court, and he already has 7 restaurant owners ready to move in. The problem is that Ollie now has to decide how much rent to charge each of the restaurant owners for the space they have in his court. McMole’s owner said he used to pay \$1,100.00 a month for a similarly sized space in a mall close by, so Ollie agreed to that price. If Ollie charges each of the other restaurant owners a rent in proportion to McMole’s, how much should each restaurant owner pay each month for rent?

House of Mouse Shrew Burgers

McMole’s

Deer Mouse Delights

Creeping Vole’s Deli

Townshend Eatery

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Bendire Shrew’s Italian Cuisine

Question 17 Mrs. Smith and Mrs. Jones both volunteer in their children’s classrooms. Mrs. Smith volunteers every 3rd school day for ⅓ of a day. Mrs. Jones volunteers every 5th day of school for ½ of a day. In a given month, which parent spends more time volunteering?

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Question 18 Ms. Gammon was bumped from an airplane flight. The reason the airline gave her was because of the overall weight of the plane. Apparently, an airplane can only hold 10,000 pounds in addition to the weight of the airplane itself. According to the list below, how much luggage and how many people would equal exactly 10,000 pounds? Determine some possible and reasonable combinations.

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People Weights

Luggage Weights

Child: 40 pounds Teenager: 85 pounds Adult (female): 135 pounds Adult (male): 170 pounds

Small bag: 25 pounds Medium bag: 50 pounds Large bag: 75 pounds

Question 19 Brianâ&#x20AC;&#x2122;s family was moving to a new town to be closer to his grandparents. All his friends organized a goodbye party for him with lots of games. Everyone liked the block-and-can game the most, where they had to throw balls to knock down a large block on one table and a small can on another table. The thrower got 15 points for every large block knocked down and 30 points for every small can knocked down. Brian scored 210 points, which made him the thrower with the highest score. He won a camera and took many pictures of his friends so he would remember them well. What are all the ways Brian could have scored 210 points?

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Question 20 Bridges are made of trusses. Examples of Warren Trusses are shown below: 1 Truss: 5 meters long

2 Trusses: 10 meters long

3 Trusses: 15 meters long

A new bridge is being built across the Lake Champlain Islands. The bridge will be 1 kilometer long. Part 1: If the engineers build the bridge using Warren Trusses, how many trusses will they need? Part 2: How many beams will this new bridge require?

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Question 1 List all possible digits that occur in the units place of the square of a positive integer. Use that list to determine whether 5233 is an integer.

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Question 2 Find an example where |a-b|>|a|-|b| and another example where |a-b|=|a|-|b|. Then, prove that |a-b|â&#x2030;Ľ|a|-|b| for all a and b.

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Question 3 Consider |u+v| and |u|+|v|, where uâ&#x2030; 0 and vâ&#x2030; 0. a. Are the values of the expression always equal? If not, what conditions make them not equal? b. If the two expressions are not equal for certain values of u and v, is one of the expressions always greater than the other? Fully explain your thought process.

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Question 4 Find the area of the shaded region in each figure. Fully simplify your final answer. a. 2x + 6

b.

x+4 2x

x

8x

12x

c. x+6

3x + 10 6x

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9x

Question 5 Graphs can help us visualize relationships between 2 objects, they can also be used to mislead people. The graphs below show the same data points.

Company profits

Company profits

50 40 30 20 10 0

J M M J S N

Month

a. Which of the graphs might be used to mislead people? b. What situations can you think of that might make it helpful for someone to deceive people with a graph?

34. 4 34.0 33.6 33.2 32.8 32.4 32.0 J M M J S N

Month

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Question 6 Mrs. McGowan loves to play with marbles. She and Mr. McGowan have created a probability game using marbles and a triangular prism whose base looks like this (there is no cover on the top of the prism so they can throw the marbles into it):

x+4 x x+2

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4 (x + 2) x

If Mrs. McGowan can throw the marble into the prism and have it land in the shaded region, she will win. Find the probability that Mrs. McGowan will win the game.

Question 7 Mr. McGowan wants to make a rectangular prism for them to use in their marble game. He is going to make the prism (again with no cover on it so they can throw the marbles into it) from a piece of cardboard that is 18cm by 26cm and looks like this: He will cut out squares of length x from the corners of the cardboard (as detailed in his blueprint to the left).

x

26 - 2x

18 - 2x

x

x

18cm x

26cm

The final product should look like the image below:

x

18 - 2x 26 - 2x

a. Find the volume of the prism in terms of x (meaning your final answer should still have an x in it). b. Find the volume when x=1, x=2, and x=3. c. (Challenge) What should the value of x be such that the prism will have the maximum volume?

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Question 8 The volume V (in cubic inches) of the rectangular prism shown below, can be modeled by the equation: V=2x3+x2-8x-4 where x is measured in inches. a. Find an expression for the surface area of the rectangular prism. b. Then, find the surface area if x=6 inches. 2x + 1

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Question 9 Mr. McGowan likes to explore linear equations that are of the form ax+b=0. a. What is the sign (positive or negative) of the value of x if ab>0? b. What is the sign (positive or negative) of the value of x if ab<0? Explain your reasoning for each part.

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Question 10 Mrs. McGowan was watching the Tottenham Hotspurs game recently and her favorite player, Clint Dempsey, passed the ball from a point that was 50 yards from the sideline and 42 yards from the endline. Gareth Bale received the pass 18 yards from the endline and 12 yards from the same sideline, where he shot and scored. How long was Clint Dempseyâ&#x20AC;&#x2122;s pass? (A diagram of the play is below):

Distance (in yards)

50

(50,42)

40 30 20 10

(12,18) 10 20 30 40 50 60 Distance (in yards)

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Question 11 When the polynomial -x3+3x2+2x-1 is subtracted from an unknown polynomial, the difference is 5x2+8. If it is possible, find the unknown polynomial. If not, explain why it is not possible.

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Question 12 Find the degree of the sum of two polynomials of degrees m and n if m<n. Give an example that fully explains your reasoning.

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Question 13 Find the degree of the product of two polynomials of degrees m and n if m<n. Give an example that fully explains your reasoning.

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Question 14 For the expression below: 5(x6+1)4 (6x5)(3x+2)3+3(3x+2)2 (3)(x6+1)5 a. Fully expand the expression. b. Give the degree of the fully expanded polynomial.

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Question 15 Geometry The Volume V of concrete used to make the cylindrical concrete storage tanks shown in the figure is V = inside radius, and h is the height of the storage tank. R

h r

R2h - r2h, where R is the outside radius, r is the

a. Factor the expression for the volume b. From the result of part (a), show that the volume of concrete is 2 (average radius)(thickness of the tank)h. c. An 80-pound bag of concrete mix yields â&#x2026;&#x2014; cubic foot of concrete. Find the number of bags required to construct a concrete storage tank having the following dimensions Outside radius, R = 4 feet Inside radius, r =3â&#x2026;&#x201D; feet Height, h feet

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Question 16 a. Rewrite the expression u6-v6 as the difference of two squares. b. Find a formula for completely factoring u6-v6 c. Factor x6-1 completely (it might be helpful to use your answer from part b). d. Factor x6-64 completely (it might be helpful to use your answer from part b).

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Question 17 Prove that

(

2x1+x2 2y1+y2 , 3 3

)

is one of the points of trisection of the line segment joining (x1,y1) and (x2,y2). Find the midpoint of the line segment joining

(

)

2x1+x2 2y1+y2 , 3 3 and (x2,y2) to find the second point of trisection.

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Question 18 Prove that the diagonals of the parallelogram in the figure below intersect at their midpoints: y

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(b,c)

(a + b,c)

(0,0)

(a,0)

x

Question 19 Use the plot of the point (x0,y0) in the figure. Match the transformation of the points (labeled a, b, c, and d) with the correct plot labeled (i), (ii), (iii), and (iv). Explain your reasoning. a. (x0,-y0)

y

b. (-2x0,y0)

(x0,-y0)

(i)

x

y

(ii)

c. (x0,Â˝y0)

x

(iii)

y

x

(iv)

x

d. (-x0,-y0)

y

y

x

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Question 20 Every week Mr. McGowan, Mrs. McGowan and 3 of their friends get together for a card game. They used a table with 6 chairs. Eventually, they realized that they had chosen a different seating arrangement each week and had exhausted every possibility. How long had they all played together? If they started their card games the first day of the new year, on what date would they have exhausted the possibilities?

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Stamford American International School 279 Upper Serangoon Road Singapore 347691 www.sais.edu.sg

Stamford Summer Math Problems

Stamford Summer Math Problems