Higher+Level+Math+Problems


 * Multi-Step Math Problems (Grades 4-5 depending on math level)

The Bicycle Race** Sam, Tim, Billy, Harry and Paige are having a bicycle race after school tomorrow. Sam’s bike has a flat tire so she only bikes at 1 meter per second. Tim has been training all year for the race and bikes at 5 meters per second. Billy gets a head start of 5 meters because he is riding a tricycle and only bikes at 2 meters per second. Harry never pays attention so does not hear the starting whistle so starts 3 seconds after everyone else, he then pedals his legs off at 7 meters per second. Paige bikes at 8 meters per second for the first 10 meters of the race but then gets tired and slows to 2 meters per second for the remainder of the race. The first race will be 50 meters.

1. What order will the racers finish for the 50 meter race? 2. How long will it take it racers to finish the 50 meter race? 3. Would the racers’ order still be the same if the race was lengthened to 75m? Why or why not?


 * You Won the Lottery!**

You just won the lottery. You have two options to choose from. The first option is to receive a check for one million dollars. The second option is to receive one penny the first day, double the that penny the next day, then double that amount the next day and so on for a month. Which option would you choose and why?

Jessica and Paul joined the cross-country running team. Jessica was able to run for 20 minutes the first day. Paul was only able to run for 9 minutes and 30 seconds. The coach wants them to increase their time every day, to be able to run for 30 minutes without stopping, within the next month. Jessica wants to take it slow and only increase her time by 15 seconds every day, Paul feels he needs to catch up to Jessica and will increase his running time by 30 seconds every day.
 * Cross-country running team**

1. Will they be able to meet the coaches' requirements? How long will it take them to be able to run 30 min? 2. If they cannot meet the requirements, what time increase would they need to meet the coach's requirements? 3. When will Paul's run time catch up to Jessica's run time?


 * A good problem: MMMMM!**

I believe that the problem that I have created would make a great pattern block problem.

It is your birthday and you decide to have a pizza party for you and five of your best friends. You have enough money to buy 2 pizzas for the six of you, but you are at a dilemma as to which deal to go for. The cost is the same for 2 deals, but the pizzas are different, not by size, but by shape and cuts. You want to get the deal that will allow you and your five friends to get the most pizza each. The first deal gets you 2 rectangular pizzas cut into 24 pieces each. The second deal gets you 2 circular pizzas cut into 8 pieces each. Without dividing the precut pieces, which deal would give you and your friends the biggest fraction of a pizza each? (In other words, any pieces that are left over after distributing the pieces evenly are not allowed to be divided into smaller pieces. Left-over pieces are essentially thrown out.) Explain your answer, including what fraction of a pizza each person would receive with each deal.


 * No Field Trips Necessary!**

Mrs. Tenney is taking 160 sixth grade students, 240 seventh grade students, and 200 eighth grade students to a national mathematics conference for middle school students. At the conference, the students will be divided up for brainstorming groups. All groups must be equal in size and will include sixth, seventh, and eighth graders. All groups will have an equal number of sixth, seventh, and eighth graders. There will be as many groups as possible. How many brainstorming groups will there be in all? How many sixth, seventh, and eighth graders will there be in each group?


 * 6th Grade Dance**

Mrs. Kempf needed help with the 6th grade dance so she posted a sign asking for volunteers. She was thrilled when every student in the 6th grade signed up to help. The students signed up to help as follows:

Committee: Part of Students

Decorating: 30% Food: 2/5 Entertainment : 0.25

Additionally, six students volunteered to help clean up after the dance.

Please help Mrs. Kempf find the number of students that signed up for each committee and the total number of students in the 6th grade.

You have been asked to work at a fundraiser for a local sick child chosen annually. When you arrive, the person in charge explains that every 10th person that enters the fundraiser will get a free pretzel, and every 25th person will get a free soft drink and hotdog.

On the day of the fundraiser, 398 people attend. How many people received a free pretzel? How many, of the people received a free soft drink and hotdog? How many people received both a free pretzel and a free soft drink and hotdog?


 * Birthday Money**

In the creation of this problem, the mathematical concept I wanted to stress was determining factors. I wanted to emphasize to students that when a whole number is divisible by a second whole number, the second number is a factor of the first. I also wanted to be able to adjust the values to accommodate the different abilities within my classroom. This problem could be used to introduce or reinforce the concept of factors. The birthday scenario was chosen because my sixth graders still bring treats to the junior high to celebrate their birthday with friends and peers. The next day is usually followed by a detailed description of the cake, candles, and presents.

Bubba received a $100 check from his grandparents for his twelfth birthday. When Bubba cashed the check at the bank, the teller gave him one hundred dollars in bills that were all the same denomination. What bills did the teller give Bubba? Use words, pictures, and numbers to explain how you got your answer(s) and why you think your answer(s) makes sense and is correct.


 * On the Job**

You have just begun your first job as director of sporting activities for the local recreation commission. It is your responsibility to set up the soccer league and 180 participants have signed up to play. The league will consist of 24 eight-year-olds, 96 nine-year-olds, and 60 ten-year-olds. You want to divide the players into teams that have the same number of eight-year olds, nine-year-olds, and ten-year-olds. What is the greatest number of teams that can be formed? How many ten-year-olds will be on the team?

You work at a hardware store and are assigned to stack different types of concrete stepping stones in the storage room. You have three different kinds of stepping stone bricks that are 3 inches, 4 inches and 5 inches thick. You want to make three stacks. Each stack must contain just one type of stepping stone. The stacks must be the same height. The height must be as small as possible for safety reasons. How many stones will there be in each stack?

The solution to this problem incorporates examples of number theory listed by Billstein. Can you find the number of stones using these theories??


 * Lazy Larry**

Larry’s mother asked him to mow their lawn while she was at work. He mowed 1/3 of the lawn, watched a Scooby doo cartoon on television, mowed 1/4 of the lawn, lay in the hammock for an hour, mowed 1/12 of the lawn, and then went for a swim in his friend’s pool. His mother arrived home from work to find he had not finished the lawn. How much was left not mowed?


 * Tailgating:**

You and your parents decide to host a big tailgate party before the big game Friday night. You estimate that there will be a total of 75 people attending the party. Hotdogs are a must at every tailgate party. You go to the store and see that hotdogs are sold in packages of 10 while the buns are sold in packages of 8. What is the least amount of packages of each you should buy if you want to have 2 hotdogs for each person and have an equal amount of hotdogs and buns?

Can someone actually tell me why hotdogs are sold in packages of 10 and the buns are sold in packages of 8? It is one of life's great mysteries. Date Modified: 2 Oct 08 7:05 PM MST

You have two part time jobs. You earn $6 per hour babysitting and $5 per hour walking dogs. You can work a total of 10 hours this weekend and hope to earn at least $55. If you spend the same amount of time at each job will you meet your goal? Can you meet your goal by working all 10 hours at only one job? How long will it take to earn enough money to buy an ipod nano at $125?

You are planning a birthday party. At this party you plan on giving a gift bag to each guest. You have 240 balloons, 200 gel pens, and 280 pieces of candy. Using everything that you have, what is the largest number of people that you can invite to your party so that every guest gets the same items in their bag? How many balloons, gel pens, and pieces of candy will each guest receive?


 * Boston Run**

Our district runs a half-mile foot race each fall called the Boston Run. Schools in the district compete against each other in the following categories: 4th grade boys, 4th grade girls, 5th grade boys, 5th grade girls, 6th grade boys, 6th grade girls. Today was the Boston Run. The scores I recorded were as followed:

4th grade girls: 90 4th grade boys: 91

5th grade girls: 88 5th grade boys: 65

6th grade girls: 41 6th grade boys: 56

Points are awarded as follows: 1st—1 point, 2nd—2 points, 3rd—3 points, 4th --4 points, etc. The first five runners for each team are scored.

1. What are two possible score combinations for the 6th grade girls? Given that the highest runner got 3rd. 2. What are two possible score combinations for the 6th grade boys? Given that the highest runner got 4th. 3. What was the overall girls total score? 4. What was the overall boys total score? 5. What was the 4th grade total score? 6. What was the 5th grade total score? 7. What was the 6th grade total score? 8. Challenge Question: The 6th grade girls won by 11 points. Given an example of what the number two team’s girls scores would look like. 9. Challenge Question: The 6th grade boys lost by 4 points. Give an example of what the number two team’s boys scores would look like.


 * Video Deals**

Jose and his two brothers all enjoy playing video games and want to purchase five new games priced at $60 each. After combining their savings, they have $325 to spend. Vernon’s Video Store sells all video games 10% off year round. For the holidays, there is a special: “Buy 2, Get 1 50% Off.” Jose says that the holiday special is the better deal. Juan says that the usual discount is the better deal. Pedro says that both offers are the same. Who is correct? Why do you think so?


 * Fun Math Problem**

There are 7 girls in a bus

Each girl has 7 backpacks

In each backpack, there are 7 big cats

For every big cat there are 7 little cats

Question: How many legs are there in the bus?

For this problem, you will examine the perimeter and area of similar rectangles that increase in size by a given ratio. After completing the activity, be prepared to share and discuss your findings with your classmates.
 * Geometry Problems:**
 * Perimeter and Area:**

1. Determine the perimeter and area of a rectangle that has a width of 6 inches and a length of 8 inches. Make a sketch of this shape and label the dimensions of perimeter and area.

2. Using a ratio of 1:2, draw a similar rectangle that is double the width and length of the first shape. Determine the perimeter and area of this new rectangle and label the measurements on your sketch.

3. Draw one more rectangle using a ratio of 1:3 (a rectangle that is triple the width and length of the original shape). Determine and label the perimeter and area.

Compare the measurements of the similar rectangles. What patterns do you notice about the perimeters and areas of the rectangles as they increase in size? What generalizations can you make? How might you find the perimeter and area of another similar rectangle that is increased by a ratio of 1:n?

For this problem, you will explore three-dimensional shapes and how proportional increases in the dimensions affect surface area and volume. Complete the following problem: 1. Determine the surface area and volume of a rectangular prism that has a width of 6 inches, a length of 8 inches, and a height of 2 inches. Make a sketch of this three-dimensional shape. Calculate the surface area and volume and label these dimensions on your sketch. 2. Using a ratio of 1:2, draw a similar rectangular prism that is double the width, length, and height of the first shape. Determine the surface area and volume of this new rectangular prism and label the measurements on your sketch. 3. Draw one more rectangular prism using a ratio of 1:3 (a prism that is triple the width, length, and height of the original three-dimensional shape). Determine and record the corresponding surface area and volume.
 * Surface Area and Volume:**

Write a description of your solution to this problem, including comparisons of the measurements of the similar rectangular prisms. Describe the patterns you noticed about the surface areas and volumes of the prisms as they increase in size. What generalizations can you make? Explain how you could find the surface area and volume of another similar prism that is increased by a ratio of 1:n.