Problem Solving Tasks DatabaseMelissa AldyiSummit 2014Ch Ch Changes…Grade Level

Task Solution

5-6 Caleb always shows his pigs at the county fair. This year, he was asked to help get the animal shelter ready before the fair started. There are seven types of animals that need separate pens, and each pen should have about the same amount of space. The building is rectangular and is 72 yards long and 48 yards wide. Each pen needs to have a 2-yard walkway on all sides in order for fair goers to be able to see the animals up close.

Draw a layout of the building and show how much space each type of animal has in their pen. Show the area of each pen in square yards.

4-5 At the fair, there are three different games in which you can win fish- the ring toss, ping-pong ball toss, and coin toss.  Juan really wants to win a black fish.  He wants to figure out which game will give him the best chance of winning a black fish.

At the ring toss game, there are 11 yellow fish, 9 silver fish, and 10 black fish. At the ping-pong ball toss game, there are 13 yellow fish, 7 silver fish, and 12

black fish. At the coin toss game, there are 8 yellow fish, 6 silver fish, and 10 black fish.

Explain what game Juan should play in order to most likely end up with a black fish. How do you know that that game gives him the best chance?

ring= 10/30, which is 33.3% chance (16/48)ping-pong= 12/32, which is 35.3% chance (18/48)coin= 10/24, which is 41.7% chance (20/48)

4-6 Penny bakes pies to sell at the fair.  All her pies are the same size, but she makes four different kinds: strawberry, cherry, apple, and blueberry.  The strawberry pies are cut into 4 equal slices, the cherry pies are cut into 3 equal slices, the apple pies are cut into 6 equal slices, and the blueberry pies are cut into 8 equal slices.

Alli, Sam, Jose, and Carrie each bought a slice of a different kind of pie.

Alli bought the biggest slice but only ate 1/3 of her piece Sam bought the smallest slice and ate 2/3 of his piece Jose’s slice was bigger than Carrie’s, and he ate 1/2 of his piece Carrie ate 3/4 of her piece of pie

Which kind of pie did each person buy? Put the students in order to show who ate the least amount of pie to who ate the greatest amount.

Sam- blueberry pie (2/24 = 1/12) Alli- cherry pie (1/9) (Jose and Carrie ate the same amount of pie) Jose- strawberry pie  (1/8) Carrie- apple pie (3/24 = 1/8)

5-6 Mallory is participating in a bake sale at the fair to raise money for her school.  She is going to make cookies to sell using this recipe:

2 1/4 cups flour 1 teaspoon baking soda 1 teaspoon salt 1 cup butter 3/4 cup sugar 3/4 cup brown sugar 1 teaspoon vanilla 2 large eggs 2 cups chocolate chips

This recipe makes 60 cookies.  Mallory told her teacher that she would make 50 baggies filled with 3 cookies each.  How much of each ingredient will Mallory need in order to make exactly enough cookies for the bake sale?  Describe how you found your answer.

Mallory needs 150 cookies, which is 2 1/2 times the original recipe.

2 1/4 + 2 1/4 + 1 1/8 = 5 5/8 cups of flour 1 + 1 + 1/2 = 2 1/2 teaspoons baking soda 1 + 1 + 1/2 = 2 1/2 teaspoons salt 1 + 1 + 1/2 = 2 1/2 cups butter 3/4 + 3/4 + 3/8 = 1 7/8 cups sugar 3/4 + 3/4 + 3/8 = 1 7/8 cups brown sugar 1 + 1 + 1/2 = 2 1/2 teaspoons vanilla 2 + 2 + 1 = 5 large eggs 2 + 2 + 1 = 5 cups chocolate chips

4-6 The tent used for selling elephant ears is in bad shape.  It has some rips and tears, and the fabric needs to be replaced.  How much fabric will be needed to cover the tent frame?

2 sides look like this:

the other 2 sides are 12 feet long One side of the tent needs an opening that is 4 feet wide and 7 feet tall

Explain how you would find about how much fabric is needed (don't worry about leaving extra fabric for the seams).

The 2 sides in the picture are compound shapes with a 7x7 square on the bottom and a triangle on top with a base of 7 and height of 4.  (49 square feet + 14 square feet = 63 square feet on each end)

The other sides are 12 feet long, and there are 2 rectangles on top of each other.  Up to the "roof line" would be a 12x7 rectangle, and from the "roof line" to the top of the tent would be a 12x5 rectangle.  (84 square feet + 60 square feet = 144 square feet on each side)

144 + 144 + 63 + 63 = 414 square feet of fabric

The opening is a 4x7 rectangle (28 square feet)

414 - 28 = 386 square feet of fabric

5-6 A popular music artist is coming to the fair this year.  100 tickets have been sold so far, and the ticket booth has collected $748.  How many youth, adult, and senior tickets have been sold?  Show all your work.  Describe the strategy you used to solve the problem. Is there only one right answer? Try to find alternatives to your original solution.

Youth (12 and under) tickets: $5.00 Adult tickets: $8.50 Senior (55+) tickets: $6.50

12 youth tickets (12 x 5) = $60 58 adult tickets (58 x 8.50) = $493 30 senior tickets (30 x 6.50) = $195

(possible solution)

Either the number of adult and senior tickets both has to be even or both need to be odd in order to have a whole dollar amount in the end.  If one is even and the other is odd, the amount of money collected with have 50 cents in change.  I used guess and check.  If you sold all 100 tickets as youth tickets, you would have $500.  If all 100 were adult, the ticket booth would have $850, and if all 100 tickets were senior tickets it would equal $650.  At first, my dollar amount was too low, so I knew I needed more adult tickets because they cost the most money and less youth tickets.  At one point, I ended up with an amount that was $2 short, so I traded one senior ticket for an adult ticket since the difference between the two is $2.

4-6 Nine people participated in the tractor pull at the fair.  The longest that each tractor can go is 100 meters, which is known as a "full pull."  Once the tractor pull was over, the distances pulled were rounded to the nearest whole meter and listed from shortest to longest.  The average (mathematical mean) number of meters pulled was 55 meters.  The distance in the middle (median) of the list was 64 meters.  The distance on the list the most times (mode) was 68 meters pulled.

Come up with two lists of distances that would fit these criteria.  Describe how you came up with your solution.

1. 20 meters2. 33 meters3. 49 meters4. 58 meters5. 64 meters6. 66 meters7. 68 meters8. 68 meters9. 69 meters(possible solution)

4-5 Avery went to the fair and bought 15 tickets.  They came connected in one strip of tickets.  There are 15 fair rides, and each ride costs a different number of tickets from 1 to 15.  She knows that she won’t be able to ride every ride, but how could Avery separate her ticket strip into 4 differently-sized sections that will allow her to pay for any ride she chooses without separating the tickets again?

1 ticket, 2 tickets, 4 tickets, and 8 tickets

1= 1 ticket 2= 2 tickets 3= 2 ticket + 1 ticket 4= 4 tickets 5= 4 tickets + 1 ticket

6= 4 tickets + 2 tickets 7= 4 tickets + 2 tickets + 1 ticket 8= 8 tickets 9= 8 tickets + 1 ticket 10= 8 tickets + 2 tickets 11= 8 tickets + 2 tickets + 1 ticket 12= 8 tickets + 4 tickets 13= 8 tickets + 4 tickets + 1 ticket 14= 8 tickets + 4 tickets + 2 tickets

15= 8 tickets + 4 tickets + 2 tickets + 1 ticket4-5 Lincoln paid 21 tickets to ride 5 different rides: Tilt-a-Whirl, Zipper, Ferris

Wheel, Octopus, and Bumper Cars.

Ferris Wheel + Octopus = 7 tickets Tilt-a-Whirl + Zipper = 6 tickets Octopus + Bumper Cars = 9 tickets Zipper + Ferris Wheel = 10 tickets

How much did Lincoln pay for each ride? Explain how you found the solution to this problem.

2 tickets- Tilt-a-Whirl 4 tickets- Zipper 6 tickets- Ferris Wheel 1 ticket- Octopus 8 tickets- Bumper Cars

5-6 Andrew won a raffle at the fair, and the prize was a large amount of tickets for rides and games.

He spent two thirds of his tickets paying for his 23 friends and himself to ride on The Drop Tower.

Then he spent two thirds of what he had left on taking his 3 brothers and himself through The Fun House.

After that, he spent two thirds of what he had left for his best friend Joe and himself to ride The Skycoaster.

He spent his last 4 tickets playing the ring toss game.

How many tickets did Andrew win from the raffle?  How many did he spend for each activity?  How many tickets does it take for one person to ride on The Drop Tower, go through The Fun House, and ride The Skycoaster? Describe or show how you found the solutions to these questions.

Andrew won 108 tickets total. He spent 72 tickets on The Drop Tower. He spent 24 tickets on The Fun House. He spent 8 tickets on The Skycoaster. Drop Tower = 3 tickets per person Fun House = 6 tickets per person Skycoaster = 4 tickets per person

5-6 Landon is in charge of clearing a rectangular area that is 200 feet wide and 600 feet long for a new dance club. He uses a bulldozer-type machine to clear loose rocks and smooth out the surface. The bulldozer covers an area 20 feet across. If Landon begins at an outside corner, starts going around the edge of the rectangle, and works his way into the center with each lap, on which trip around will he have cleared half of the area?

Solution = during the 3rd trip around Area = 120,000 sq. ft. ½ area = 60,000 square ft. After 1st trip around = 89,600 sq. ft. remaining (560 x 160) After 2nd trip = 62,400 sq. ft. remaining (520 x 120) Landon needs to clear an addition 2,400 sq. feet during his 3rd lap

to get half way done4-5 Professor Orion and Professor Polaris are working every night this month at the Mt.

Olympus Observatory. On Monday night, Prof. Orion notices 12 new stars. He finds 18 new stars on Tuesday, 30 on Wednesday, 36 on Thursday and 48 on Friday. Prof. Polaris notices 18 new stars on Monday, 26 on Tuesday, 44 on Wednesday, 52 on Thursday, and 70 on Friday. If the professors continue to spot new stars at the same rate, during what day of the week will their combined total equal the number of days in a year?

Wednesday

5-6 Business was booming for the Sensational Events Company. During the pat year they supplied 1295 entertainers for parties. They sent three times as many magicians as clowns, 25% as many fire-eaters as clowns, one third as many people in gorilla suits as magicians, and a number of jugglers equal to the combined number of clowns and magicians. How many of each type of entertainer did Sensational Events supply for parties during the past year?

clowns 140 magicians 420 fire-eaters 35 people in gorilla suits 140 jugglers 560

4-6 Yesterday afternoon, Jason fed all of the fish in his fish tanks. He has five different fish, and each one is on a different feeding schedule. He feeds the glass fish every 2 days, the black molly every 3 days, and the catfish every 5 days. He feeds the eel every 2 ½ days, and the angel fish every 1 ½ days. On how many days in the next 30 days will Jason feed at least 4 fish?

4 days (15th, 18th, 20th, and 30th)

4-6 It’s the 5th Annual Fernwick Frog-Jumping Contest. Each frog gets a certain number of tries. A try is made up of three jumps, and the score for the try is the total of the scores for those three jumps. Today, Clementine jumped a total of 12 feet on each of her tries. Her shortest jump was 1 foot, her longest jump was 8 feet, and she jumped a whole number of feet on each jump. How many different ways could Clementine have scored 12 feet in three jumps? (remember that, in this case, 2, 3, 7 is different from 3, 7, 2)

46 different ways

5-6 Annie has the Saturday delivery schedule for chicken chunks fro Rooster-on-the-Run. One Saturday, she set out in the truck and left 3/5 of her orders at the college. She went back to the restaurant and picked up 6 orders. Her second delivery was for 3/7 of the total amount that she had in the truck. She headed back to pick up 4 more orders and then delivered ¾ of her total. She picked up 6 more before heading to the local businesses to deliver 1/3 of her orders. She went back for 2 more orders, and delivered ½ of her total amount. She returned for 2 orders and made her last delivery of 6 orders at the hospital. How many orders did she deliver all together?

40 orders all together

5-6 Andy and Ellen began baking early at the Muffin Mansion, which supplied muffins for local businesses. Today they baked oatmeal bran muffins, which cost $2.31 for one dozen, banana yogurt muffins, which cost $4.52 for one and a half dozen, an chocolate raisin muffins, which cost $7.66 for two and a half dozen. During the day, they sold a total of 642 muffins for $153.21. How many of each kind of muffin did they sell?

Banana yogurt- 216 muffins (18 dozen = $54.24) Bran oatmeal- 156 muffins (13 dozen = $30.03) Chocolate raisin- 270 muffins (22.5 dozen = $68.94)

4-6 A fierce army of ants marches forth to battle. The general makes up the whole first row. Two colonels make up the second row. Three lieutenant colonels form the third row. Four majors make up the fourth row. Five soldiers form the fifth row, and so on, row after row. The army consists of 525 ants altogether. Make a prediction of how many rows you think there will have to be before you start working out the problem. How many rows are there, and is the last row full? Was your prediction too big or too small, and why do you think your answer was different than your prediction?

32 rows, the last row is not full

(most students, I think, would predict a larger number of rows for 525 ants)

4-6 Jana, Garth, Ian, Elsa, Hak, and Adriana are all employed in these careers: bionic engineer, used space shuttle sales, robot repair, intergalactic weather analyst, laser technician, and robot relations counselor. Jana, Elsa, and Adriana did not take robot relations courses in college. The bionic engineer and laser technician often ride the bus with Ian, Hak, and Adriana. The used shuttle salesperson, weather analyst, and bionic engineer often have lunch with Jana, Garth, and Adriana. Hak and the weather analyst work on different floors in the same building. Can you match each person with a job?

Jana- laser technician Elsa- bionic engineer Garth- robot relations Ian- weather analyst Hak- shuttle sales Adriana- robot repairs

5-6 Sarah’s job is to clean fireplaces. She uses a super vacuum cleaner to remove the soot and ashes from each chimney. Sarah doesn’t want to take the noisy cleaner inside a customer’s home, so she leaves the cleaner outdoors and runs a hose from it to the fireplace. The distance from the cleaner to the fireplace is different at each home, so she has sections of hose that she can snap together to make the length of hose she needs. She has 3 sections each of these lengths: 8 feet, 10 feet, 12 feet, 16 feet, and 24 feet. Sarah is ready to clean a fireplace and needs a 100-feet hose. How many different combinations of hose sections could she use?

20 different combinations

4-6 Otto is responsible for feeding the otters at the Sea Circus. He has to feed them one pound of fish each day. He can feed them smelt, perch, sardines, and squid. The smelt weigh 3 ounces each, the perch weigh 4 ounces each, the sardines weigh 1 ounce each, and the baby squid weigh 1 ½ ounces each. How many different combinations of fish can Otto use to feed the otters?

46 different combinations

4-6 On the day of his grand opening, Toofer’s Tape Store opened at 8:00 am. In the first hour, Toofer sold 4 tapes. In the second hour he sold 8 tapes, twice as many as the first hour. In the third hour customers bought 16 tapes, twice as many as the previous hour. In the next hour, Tom sold three-fourths as many as in the previous hour, or 12 tapes. Tom kept selling tapes in the same way all morning: twice as many as the previous hour, twice as many as the previous hour, then three-fourths as many as the previous hour. Also, Tom gave customers one free tape for every tape they bought. During the last hour Toofer’s was open, customers left the store with 1,296 tapes. How late did the store stay open?

Toofer’s closed at 10:00 pm

4-5 Imagine you have 25 beads. You have to make a three-digit number on an abacus and use all 25 beads for each number you make. How many different three-digit numbers can you make? Write them in order.

You can make 6 different numbers: 799 889 898 979 988 997

4-6 The pages of Jack’s book are numbered starting with 1. The page numbers have a total of 555 digits. How many pages does the book have? How many of the digits are a 5?

221 pages42 of the digits are a 5

4-6 At Julie’s café, here is what food costs: 1 bowl of soup + 1 cup of tea = $4 2 bowls of soup + 2 pieces of pie = $9 1 piece of pie + 2 cups of tea = $2

What do you have to pay in total for 1 bowl of soup, 1 piece of cake, and 1 cup of tea? How much does each item cost on its own?

all three = $5.00

soup = $3.50 pie = $1 tea = 50 cents

4-5 36 people live in the eight houses in Albert Square. Each house has a different number of people living in it. Each line of three houses has 15 people living in it. How many people live in each house?

Possible solution:

4-5 Each shape stands for a number. The numbers shown are the totals of the line of four numbers in the row or column. Find the remaining totals.

circle- 5 triangle- 8 club- 6

4-6 Would you rather work seven days at $20 per day or be paid $2 for the first day and have your salary double every day for a week?

If you are paid $20 per day for seven days, then you earn $20 x 7 or $140. If you are paid $2 the first day and your salary doubles every day for the next six days, then you earn $2 + $4 + $8 + $16 + $32 + $64 + $128, or $254. The second scheme earns you more money by the end of the week.

6-8 A combination lock uses three numbers from 0 to 39. It opens when these numbers are dialed in a particular order: right, left, right. How many possible combinations are there?

The lock in the challenge requires that you choose from 40 different numbers,0–39,three different times. Therefore, there are 40 x 40 x 40 or 64,000 combinations.

Alg. 1 During the 100 meter dash in the 1988 Olympic Games in Seoul, Florence Griffith-Joyner was timed at 0.91 seconds for 10 meters. At that speed, could she pass a car traveling 15 miles per hour in a school zone? One method for converting between measures is called dimensional

analysis. The conversions between measures are written as fractions so the common units cancel out. Her rate is about 24.6 miles per hour, and she could easily pass a car going at a rate of 15 miles per hours.

7-8, Alg. 1 Bill and Marcus wanted to make a rectangular box out of plywood. At the lumber yard, they realized that they had forgotten the dimensions of the box. Bill remembered that one of the sides had an area of 96 square centimeters. Marcus remembered that the other sides had areas of 252 and 168 square centimeters. What are the dimensions of the box?

8 cm by 12 cm by 21 cm. Since the dimensions are a by b by c, then ab = 96, bc = 252, and ac = 168. Multiply the first two equations: ab^2c = 96 × 252. Dividing both sides of the equation by ac gives the value of b^2 which is 144 so b = 12, a = 8, and c = 21.

6-8 A leaky faucet drips at the rate of 1 drip every 3 seconds, and 165 drips will fill one eight-ounce cup. Could one swim in all the water that has dripped in one year?

About 4,000 gallons will have dripped in one year--enough to swim in. Set up a series of ratios to find the number of drips in a year: 365 days/year × 24 hours/day × 60 minutes/hour × 60 seconds/minute × 1 drip/3 seconds = 10,512,000 drips in 1 year. Another series of ratios will convert drips to gallons: 10,512,000 drips/year × 1 cup/165 drips × 1 gallon/16 cups = 3,981.8 gallons.

7-8, Alg. 1 Tommy has a 1.69 ounce bag of M&M's. The bag contains 56 M&M's. He decides to share his candy by giving Lucy two-thirds of three-fourths of his M&M's. Lucy decides to share her M&M's with Melissa by giving Melissa two-thirds of three-fourths of her M&M's. If half of Melissa's M&M's are red, how many red M&M's does she have?

7 red M&M's. To solve, 3/4 of 56 = 3/4 × 56 = 42; 2/3 of 42 = 2/3 × 42 = 28, or 2/3 × 3/4 × 56 = 28. Tommy gives Lucy 28 M&M's. To continue, 3/4 of 28 = 3/4 × 28 = 21; 2/3 of 21 = 2/3 × 21 = 14, or 2/3 × 3/4 × 21 = 14. Lucy gives Melissa 14 M&M's. If half are red, then Melissa has 7 red M&M's. Also, 2/3 × 3/4 = 6/12, or 1/2. Therefore, 2/3 of 3/4 is equivalent to 1/2.

7-9 How many different four-digit PINs are possible using the telephone keypad, assuming that 0 and 1 are not used and that all four digits are different?

1680 The keypad had 8 possibilities for the first digit if 0 and 1 are excluded, 7 for the second digit, 6 for the third, and 5 for the fourth. By the multiplication principle, 8 × 7 × 6 × 5, or 1680, PINs are possible.

7-8 A campus vendor charges $3.50 for one slice of pizza and one medium soda and $9.15 for three slices of pizza and two medium sodas. Determine the price of one medium soda and the price of one slice of pizza.

p + s = 3.50 -2(p + s = 3.50) -2p - 2s = -7.00 3p +2s = 9.15 3p +2s= 9.15 3p + 2s = 9.15 p = $2.15 and s = $1.35

6-8

Given four sizes of cups, small, medium, large, and extra large, how many ways can you order a hot chocolate at Dunkin Donuts considering size, variety, and flavors? Show your reasoning, so that others can understand how you found the total number of possibilities.

7-8 The three seventh grade classes at Sunview Middle School collected the most boxtops for a school fundraiser, and so they won a $600 prize to share among them. Mr. Roberts’ class collected 3,760 box tops, Mrs. Mays’ class collected 2,301, and Mr. Canyon’s class collected 1,855. How should they divide the money so that each class gets the same fraction of the prize money as the fraction of the box tops that they collected?

7916 box tops were collected. Mr. Roberts’ class collected 3760/7916. So multiply 3760/7916*600 to get the part of the money for that class. For Mays: 2301/7916*600, Canyon: 1855/7916*600.Rounding to the nearest cent $284.99 should go to Mr. Roberts’ class, $174.41 should go to Mrs. Mays’ class, and $140.60 should go to Mr. Canyon’s class.

7-8 There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5:4. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4:5. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance is girls?

Colfax: 5x+4x=270 9x=270 x=30 so 5x=150 boys, 4x=120 girls Winthrop:4x+5x=180 9x=180 x=20 so 4x=80 boys, 5x=100 girls 120+100=220 girls; total people is 270+180=450 so girls to total is 220:450. So the fraction of students at the dance who are girls is 220/450=22/45.

6-8 A friend has won the grand prize in a radio sweepstakes. His gifts will be delivered over 10 days. On the first day, he gets a kitten. On the second day, he gets two hummingbirds and a kitten. On the third day, he receives three rings, two hummingbirds, and a kitten. This pattern continues for seven more days! The remaining gifts, in order of appearance, include books, boxes of candy, bracelets, bouquets, cologne, CDs, and puppies. After 10 days, which gift did he receive most?

6-8 Angelo has $35 to spend on music CDs. He and his friend, Alex, decide to check out the two music stores in the mall. At Mike’s CD Mart, each CD is $14.00, but is on sale for1/5 off the original price. At CD Circus, each CD is $16.00, but is on sale for1/4 off of its original price. At this mall, tax is 6% of the bill. Angelo wants to purchase two CDs for himself and one for Alex. Alex asserts that Angelo will not have enough money for three CDs at either store. Is Alex being reasonable? Justify your answer.

3 *$14 =$42 since the CDs are 1/5 off the cost is only 4/5 of the price so 4/5 *42 is $33.60. 6% of 33.60 is $2.02 so the cost at Mike's CD Mart is $35.62 3*$16 =$48 but will pay only 3/4 of the cost so 3/4*48 is $36 and that's before tax. Alex is being reasonable. Angelo doesn't have enough money to buy CDs at either store.

6-8 Mrs. Smith’s classroom can hold at the most 30 students. Her first period class had a ratio of 4 boys for every 3 girls. When 3 boys and 1 girl were added to the class, the new ratio became 3 boys to 2 girls. How many total students are now in the class? Justify your answer.

6-8 Carey decided to have a lemonade stand one afternoon. She found some lemonade mix in the kitchen pantry.

The directions printed on the mix:

For one glass of lemonade, combine 3 tablespoons of lemonade mix and 6 ounces of water

Carey realizes she can only locate a teaspoon-measuring spoon, some 4-ounce plastic cups, and some 8-ounce plastic cups.

Can Carey make the lemonade mix according to the package directions and still have a lemonade stand? Explain why or why not.

Carey will have the lemonade stand. Carey should realize that the ratio of mix to ounces is one tablespoon for two ounces. Carey needs to convert one tablespoon to teaspoons to continue to solve the problem. It takes three teaspoons to make one tablespoon. The ratio of mix to water is 3 tsp. to 2 ounces. Carey can make cups of 4 ounce and 8 ounce size?

6-8 Report cards were published today. Unfortunately, you dropped your cell phone and broke its screen today, too. Since this is the second time you broke your phone screen, you need some good news for your parents. You decide looking at the report card just is not going to be enough. You want to impress your parents by interpreting the data for them. Impress your parents!

The student can impress her parents with information about percent of increase and decrease in her grades, as well as information about the grades holding steady.Other options include finding the percent of As, Bs, Cs, finding the ratios of As to Bs, Bs to Cs, and finding her mean and median averages.

6-8 Sam is a carnival ride enthusiast and looks forward to the yearly state fair. He would like to pay for his parents, his little brother, and himself to attend the fair and ride all the rides. Sam works twenty hours per week, earning $6 an hour. For the past year, he has been saving 15% of his paycheck for the fair trip. Admission to the fair is $10 per person and each of the fifty rides costs $2.50 per person. Is it possible for Sam to treat his family to this outing? Explain why or why not.

20 • 6 = 120 per week • 15% = 18 a week • 52 weeks = $936 saved

Admission is $40 for family

50 rides • 2.50 = 125 • 4 = 500 + 40 for admission = $540 for the outing

Yes, Sam has enough money to pay for the outing.

6-8 You are the new manager of Buddy’s Buffet. The restaurant has been losing money. The restaurant charges $9.99 for one buffet dinner, $19.98 for two, $28.98 for three, $36.98 for four, $43.98 for five and so one. The buffet costs the restaurant $6.99 a person. Explain, in depth, three ways Buddy’s Buffet could earn a profit without losing its customer base.

Possible Answers:1. charge each customer the same price2. limit the size of the groups to 93. have the pattern of prices change to something other than the current pattern

6-8 It’s Game Show Friday in math class. You are trying to win a homework pass that is hidden in one of three cardboard boxes. After you choose a box, but before it is uncovered, the teacher uncovers one of the other boxes, in which there is no homework pass. You can then switch to the remaining covered box or stay with your choice. Should you stay or switch in this game? Explain.

Initially, the probability P(pass)=1/3 but after the reveal the P(pass)=1/2. There would be no increased probability switching to the other box.

6-8 You have researched patio designs. Your favorite designs continue the pattern shown below. You have at most 52 bricks to use. Can you use them all to make one of your favorite designs?

No, you could not make a design with 52 bricks. The patio designs shown are perfect squares. 52 is not a perfect square.

6-8 You and your little brother mow your neighbors yard. She pays $40 and includes a 15% tip. You mow 2/3 of the yard and your brother mows 1/3 of the yard. What is your part of the payment? Your brother’s part? Explain your reasoning.

40(.15) = 6--so $46 in all for the job. Since $46 isn’t a multiple of 3, dividing the money out evenly isn’t quite so easy. The students will have to calculate 2/3 of $46 and 1/3 of $46-- which is $30.67 and $15.33. It will be interesting to see if a student is willing to round down to $30 and up to $16 for his/her brother--another option is split the money in half.

7-8 Connor ordered a gross of $18.00 sunglasses for his store. He marked them up 50% when he sold them in his store. After nine weeks, he put them on sale for 75% off the selling price. Connor sold 2/3 of his sunglasses at the regular price and 1/3 at the sale price. Did Connor make a profit on his sunglasses?

Cost of sunglasses: $18 • 144(gross) = $2592

Selling Price: $27(.5•18 = 9; 18 + 9 = 27)

Sold at regular price: 2/3 of 144 = 96; 96 • 27 = $2592 Connor breaks even at this point

Sale price: 27 • .25 = 6.75

Sold at sale price: 48 • 6.75 = $324

Connor made a profit on his sunglasses. His profit was $324, which is $2.25 per pair.

5-6 A zoo wants to double the number of lions that is has in the African Safari Exhibit. There are currently 3 lions in an exhibit that is 3,500 sq yards. Each lion must be given enough area to roam to maintain a healthy lifestyle. The zoo was able to purchase a plot of land and building that was adjacent to the current exhibit – see the measurement below. The zookeepers are looking for a recommendation from you about acquiring three additional lions. What do you recommend and why?

The recommendation of the student should be that the zoo does not have enough room in the exhibit with the newly purchased land to house the three additional lions. By converting the measurements to yards the enclosure is 80 yards by 50 yards with a total of 4000 square yards including the building. The student should then subtract the yardage taken up by the shelter, which is 525 square yards. Therefore there is actually only 3475 yards, which does not meet the requirements of 3,500 sq yards.

5-6 There is a new employee at the zoo, he is having a very rough day and the last task he has is to feed the baby elephants their afternoon snack. He throws a 25-pound (400 ounces) bag of herbivore pellets (made of ground up grasses, grains and vegetables) on the 4-wheeler and grabs a measuring cup from the counter. When he gets to the elephant exhibit he discovers that the measuring cup can only measure two cups at a time. He has to split the snack evenly between the three baby elephants. How much food, in pounds, does each baby elephant end up receiving? How many scoops of food does each elephant receive?

First, convert the amount in the bag to cups so 400 ounces divided by 8 tells us that there are 50 cups of food in the bag. Then I would divide the amount of cups by three so that I can count how many cups each elephants will receive. Each elephant will receive 16 2/3 cups of food which when converted back to ounces equals 133 1/3 ounces of food for each elephant. The question wants to know how much in pounds so dividing the 133 1/3 ounces by 16 will give the amount of pounds an elephant will eat, which is 8 1/3. For the second question I know that each elephant is going to receive 16 2/3 cups of food, since the scoop is equal to 2 cups, this means 8 full scoops and then one scoop that contains 2/3s of a cup of food.

240 ft 75 ft

45 ft

45 ft

45 ft

30 ft

Shelter and Storage (not included in yardage)

5-6 There was an issue with the food order for the pygmy goats at the zoo this week. There was a volunteer who was given the task of opening the containers of hay for the goats as well as some of the food containers for the other animals. She misunderstood the directions given to her and she ended up breaking each bale of hay in half before placing it in the storage area. Each time the goats get fed, they are supposed to have .35 of a whole bale of hay. The volunteer says that you should be able to just take .35 of each half of a bale and that will be equivalent to .35 of the whole bale of hay. The zookeeper who takes care of the goats isn’t so sure that the volunteer is correct and needs you to prove or disprove her theory. Please confirm or deny that she is correct and make sure to explain to the zookeeper how he can find the correct amount of hay for the goats if this happens in the future again

The volunteer is not correct. There can be a variety of ways that the students can disprove the volunteer’s theory. One example may be that they diagram the halves broken up into hundredths and shading in .35 of each. Then adding the two shaded portions together to show that it actually equals .70 of the whole bale. For future reference for the zookeeper the students could create a diagram showing that the goats will be getting .70 of a half of bale.

5-6 Jack, Art, Fran, and Megan work as volunteers at the local zoo. Jack brushes the goats every 4 days; Art cleans out the parrots’ cages every 6 days; Fran supervises the petting zoo every 2 days; and Megan helps in the reptile house every 3 days. How many times in 12 weeks will all 4 helpers be at the zoo on the same day?

The students can create a table to keep track of when each volunteer is at the zoo. The students could also write an expression for each volunteer’s pattern of work and use that in the table. There are 7 times that they will all be at the zoo on the same day.

5 Clyde is bored sitting in math class, daily his teacher picks a student to be the messenger for the day. Clyde decides to find a way to make sure he is always the messenger to get out of class as much as possible. His plan is that the whole class sits in a circle. Each student counts off one at a time, beginning with the person tot he left of the teacher (1, 2, 3, 4,...). Once everyone has a number, the teacher moves to the middle of the circle and begins sending children to their seats by skipping number1, sending number 2, skipping number 3, sending number 4, and so on until she goes around the circle completely. She doesn't stop, however, and continues skipping everyother student until one is left. This lucky person gets to be the messenger. If there are ten students in the class, the number 5 will be chosen. Where should Clyde sit if there are eleven students? Twelve students? Fifteen students? Twenty students? Thirty students? Can you find a way that Clyde will always know where to sit in the cirlce no matter how many students are in the class?

The lucky spot is always an odd number. The students should set up a table or chart to keep track of their trials. Their descriptions and conclusions will vary based on the number of students in each of their trials.

4-6 John Turnkey, the prison warden decided to free his prisoners for good behavior. The cells were numbered from 1 to 25. Each had a lock that opened when you turned it once and locked when it was turned again, and so on. One night when the prisoners were sleeping, he quietly turned allt he locks once, opening all the cells. He bagan to worry that he may have freed too many prisoners so he went back and turned every second lock (2, 4, 6, 8, ...24) which locked half the cells. Thinking that there still might be too many prisoners freed, he gave every third lock a turn (3, 6, 9, 12, ... 24), then every fourth, fifth, sixth, seventh, eighth, and so on all the way to the every twenty-fifth lock. Who got out of jail in the morning?

The students may set up a chart, or in some way collect the data about which locks are turned how many times. They will come to the conclusion that the prisoners in cells 1, 4, 9, 16 and 25 will be freed.

7-8 Mom gave me $45 to buy a new outfit (one shirt and a pair of jeans). I found 2 identical outfits at 2 different stores, each with its own sale. Which combination would be the best deal? After our state’s 7% sales tax, how much will I have left for lunch?

Store A: 20% off of $20 or more, 25% off of $30 or more (Shirt: $20.99, Jeans: $25.99)Store B: 40% off all regular priced items (Shirt: $22.99; Jeans: $34.99)

Store A: 20% off when you spend $20 or more, 25% off when you spend $30 or moreShirt Tag says $20.99; Jeans Tag on says $25.99(20.99)(.80)= $16.79 for shirt only(25.99)(.80)= $20.79 for jeans only(20.99+25.99)(.75)= $35.24 for both shirt and jeans togetherStore B: 40% off all regular priced itemsShirt Tag says $22.99; Jeans Tag says $34.99(22.99)(.60)=$13.79 for shirt only(34.99)(.60)= $20.99 for jeans only(22.99+34.99)(.60)= $34.79 for both shirt and jeans togetherSummary of Costs: A jeans + A shirt = $35.24 B jeans + B shirt = $34.79A jeans + B shirt = $20.79 + $13.79 = $34.58 B jeans + A shirt = $20.99 + $16.79 = $37.78Best deal: Store B shirt, Store A jeans22.99(.60)= 13.79413.79(1.07)= $14.76 total cost for shirt at Store B25.99(.80)=20.79220.79(1.07)= $22.25 total cost for jeans at Store A

14.76+22.25= $37.01 total spent on complete outfit45-37.01= $7.99 left for lunch

6-8 Your friend has called you to find out how to find the area of this figure. How would you explain this process?

Rectangle: A = LW Half-Circle: A = ½∏r2 Triangle: A = ½ bh A = 36 (12) A = ½∏ (12/2)2 A = ½ (12)(49-6-36) A = 432 cm2 A = ½∏ (6) 2 A = ½ (12)(7) A = 56.5 cm2 A = 42 cm2Full Figure: 432 + 56.5 + 42 = 530.5 cm2Separate the figure into three shapes: a triangle, a rectangle, and a half-circle. Find the areas of each of those shapes then add them together for the area of the entire figure. The length and width of the rectangle are given, just make sure you don’t use the length of the entire figure. The width of the figure is the same as the diameter of the circle, so half of that would be the length of the radius. You have to include multiplying by ½ in the circle formula since you’re only finding the area of a half-circle and not a whole circle. The base of the triangle is the same as the width of the rectangle. To find the height of the triangle, you need to subtract the radius of the circle and the length of the rectangle from the total length of the figure.

7-8 Dan drew the following graph, but he forgot to label the y-axis. Write a scenario that could account for such a graph and fill in the missing label. What does the rate of change represent?

(Possible Solution) The local book fair started with around 120 books and sold out after the 12th day. After day 2, there were 100 books left, and there were only 40 books left after day 8. Using these two points, (2,100) and (8,40), in the slope formula (y2-y1/x2-x1), to find the rate of change (40-100)/(8-2) = -60/6 = -10, shows that the book fair sold around 10 books each day. You could also use points (0,120) or (12,0). The -10 rate of change shows the number of books that sold each day, so the y-axis should be labeled as “Number of Books.”

7-8 Write algebraic expressions to represent the information requested for this problem. Then evaluate the expressions using the order of operations.

Two friends and I go shopping on the second Saturday of every month. We limit the amount we spend to 20% of our previous month’s paycheck, which we earn by working on the weekends. Given the information below on our last month’s earnings, find how much we are each able to spend. Then find the maximum amount we could spend if we decided to combine an equal amount of our earnings to purchase one item to share.

Lauren works as a waitress at the local diner, and makes $5.25 per hour. She also gets tips, which amounted to $138, but she has to give her boss 25% of her tip money to split between the kitchen crew and the bussers. She worked a total of 20 hours.

Rylie works as a cashier at the local fast-food restaurant making an hourly wage of $7.15. She was scheduled to work 30 hours, but due to being sick, she missed 5 hours.

I work as a bagger at the local grocery store making $1.70 more per hour than Lauren and worked a total of 7 hours more than Rylie. Last month, our boss gave all front-end employees a $20 bonus for our improvement in efficiency and friendliness.

Lauren Rylie I5.25h + 138(1 - .25) 7.15h (1.70 + L)h + 205.25(20) + 138(.75) 7.15(30 - 5) (1.70 + 5.25)(7 + R) + 20105 + 103.5 7.15(25) (1.70 + 5.25)(7 + 25) + 20$208.50 $178.75 6.95(32) + 20 = 222.4 + 20 $242.40How much they can each spend208.50(.20) 178.75(.20) 242.40(.20)$41.70 $35.75 $48.48

Maximum amount they can spend if they combine their money35.75(3) = $107.25

6-8 The first three terms of a particular arithmetic sequence are 1/2, 1, and 3/2. Change one of these first three terms to create the first three terms of a sequence that would be geometric.

Changing the term 3/2 into 2 will create the following geometric sequence: 1/2, 1, 2,…

6-8 On Monday, Ted was hired as a clerk at a hardware store. He did such a good job on Tuesday that he received a 7% raise. On Wednesday, he made a mistake with inventory and his boss decided to cut his salary by 7%. When Ted told his mother about the situation, she was upset about Ted's new pay and said he should have just kept his original salary. Explain mathematically what Ted's mother is angry about.

On Tuesday, Ted's salary rose to 107% of what it was on Monday. When he got a pay cut on Wednesday, his salary dropped by 7% of 107%, which is 99.51% of the original salary. Ted now makes less than he did when he was hired.

6-8 A chef has a round cake pan in the shape of a cylinder. The radius of the bottom of the pan is 4.5 inches, and the height of the pan is 2 inches. He wants to cover the inside of the pan as neatly as possible with parchment paper before filling it with batter, so he cuts a circle with a radius of 6.5 inches out of the paper. Is the parchment paper circle the same size and shape as the inside of the cake pan? Prove your answer.

No. The parchment paper is in the shape of a circle with an area of approximately 132.7 square inches. The inside of the cake pan is composed of a circle with an area of approximately 63.6 square inches and a rectangle with an area of approximately 56.5 square inches, for a total of 120.1 square inches.

6-86- Place one term of the arithmetic sequence -3/5, -1/5, 1/5, 3/5,… in each of the squares below in such a way that each row, column, and diagonal will give a sum of 3.

8 Fred has moles digging tunnels all over his backyard. The exterminator told him he would have to place a mole trap every ten feet along the main mole burrow. The mole burrow runs along the main diagonal through Fred's rectangular backyard. Fred knows that his backyard measures 50 by 75 feet. How many mole traps will he need?

By the Pythagorean Theorem, the diagonal measures approximately 90.1 feet. Starting at the end, Fred should place 10 mole traps along the diagonal.

7 A four-inch cube is painted blue and then cut into one-inch cubes. How many one-inch cubes will have paint on one side? Two sides? Three sides? No sides?

There are 64 total one-inch cubes arranged in four layers of 16 cubes each. 24 cubes have paint on one side, 24 have paint on two sides, 8 have paint on three sides, and 8 have no paint.

8 There is an Olympic-sized swimming pool that will hold 500,000 gallons of water. It is being filled at a rate of 250 gallons per hour. The pool was 1/4 full at 10:45 p.m. on February 29th. Find the moment when the pool began filling with water. Find the moment when the pool is full.

The pool gains 6000 gallons of water per day, which means the pool started filling up at 2:45 on February 9th. It will take 83 total days and 8 hours for the pool to fill completely, which will be 10:45 a.m. on May 2nd.

8 Benjamin just received a chain letter in the mail with a list of 15 names and addresses attached. It says he should send $1 to the name at the top of the list, cross it out, and then add his name and address at the bottom. Next, he is supposed to send the letter and new list to 10 of his friends. If every recipient follows the instructions, how much money will Benjamin receive when his name is at the top of the list?

The list of names works in powers of 10 following the pattern 100, 101, 102, 103,… and so on until Benjamin's name is reached. Benjamin is the sixteenth name on the list, so Benjamin will theoretically receive $1015, or $1,000,000,000,000,000.

Alg. 1 Without using equations for horizontal or vertical lines, write the equations of four lines that form a square.

Students need to find pairs of equations with opposite slopes and opposite y-intercepts, while also having pairs of equations with the same slopes and the same y-intercepts. For example, y=x+2, y=x-2, y=-x+2, and y =-x-2.

7-Alg. 1 Andy started his rose garden by planting five bushes in the fall. Each year thereafter, he planted three bushes in the spring and three bushes in the fall. He now has forty-seven bushes. For how many years has he maintained his rose garden?

The equation 5 + 6x = 47, where x stands for the number of years, can be solved to find that the rose garden has been maintained for 7 years.

6-8 Could an NFL football team have an eleven-player defensive starting lineup that weighted at least a ton? A metric ton? A long ton?

All three weights are possible. A ton is 2000 pounds, and a metric ton is 1000 kilograms. A team that weighs a ton would have an average weight of about 182 pounds per player. A team that weighs a metric ton would average 90.9 kilograms, or just over 200 pounds per player. A long ton is 2240 pounds, so each player would weigh about 204 pounds each.

7-8 You plan to spend the summer with your grandparents and would like to download movies while you visit. One online movie site charges a membership fee of $15, then rents each movie for 25 cents. A second movie site rents movies for $1 each with no membership fee. You expect to rent about 3 movies per week, and your visit will last from June 10 through August 1 this summer. Which site is the best deal for you?

Using the equation 15 + 0.25x = x, the break even point for the two websites is 16 movies. Since your visit will last almost 9 weeks, it will be cheaper to use the first website that charges the $15 membership fee.

5-8 If you spend $1 each second, how long would it take to spend $1 billion? Answer in appropriate units.

You would spend $86,400 in one day. If there are 365.25 days in one year, you would spend $31,557,600 per year. This means it would take about 31.69 years to spend $1 billion.

6-7 A large cube has a surface area of 216 square centimeters. Two small cubes have edges that are exactly half the length of the edges of the large cube. What is the total surface area of the two small cubes?

If the large cube has a surface area of 216 square centimeters, each face must have an area of 36 square centimeters, and each edge must be 6 centimeters in length. The smaller cubes have edges 3 centimeters in length, so the area of each face is 9 square centimeters. The surface area of each small cube is 54 square centimeters, so their combined surface areas would be 108 square centimeters.

5-8 Five years ago, I planted a four-foot-tall tree in my front yard. It has grown the same amount each year, and last year it was twice as tall as it was at the end of the first year. How tall is the tree now?

Let x feet be the growth each year. At the end of the first year, the tree was 4 + x feet tall and at the end of the fourth year it was 4 + 4x feet tall. Since the height at the end of the fourth year was twice the height at the end of the first year, we can set up this equation: 4 + 4x = 2(4 + x). Solving the equation gives x = 2 feet. Since the tree grows 2 feet each year, it has grown 10 feet in 5 years and is now 14 feet tall.

5-6 The Sole D'Italia Pizzeria sells small, medium, and large pizzas. A small is 9 inches in diameter, a medium is 12 inches in diameter, and a large is 15 inches in diameter. Prices for cheese pizzas are $6.00 for small, $9.00 for medium, and $12.00 for large. Which measurement--radius, diameter, circumference, or area---seems most closely related to price? Explain your answer.

This problem is intended to be given to students before they have area and circumference formulas at their disposal. Answers will vary. Most students will say that the diameter is most closely related to the price because, as the diameter changes by 3 inches, the price changes by $3.00.

Alg. 1 After finishing a very boring book, Fred noticed that the page numbers contained exactly 29 zeros and exactly 137 ones. The book started with page 1. How many pages were in the book?

196 pages.Start by counting the number of times the numeral 1 appears. In the first 99 pages, a 1 occurs in the units digit 10 times and in the tens digit 10 times. Therefore, the numeral 1 appears 20 times. From pages 100 through 199 a 1 also appears in the hundreds digit, giving us 120 additional 1s, for a total of 140 occurrences of the numeral 1 by page 199. Since the numeral 1 appeared only 137 times, we know the book must have ended three pages earlier. Therefore, there are 196 pages in the book.