In "Last Fling" challenge, students have opportunities to employ all five process standards. They will problem solve as they decide on a strategy for solving a problem where the answer is not immediately obvious. They reason as they think about and justify their solution. They should have opportunities to communicate about their thinking by discussing it with classmates, parents, and teachers. Children will be able to make connections between mathematical concepts (patterns and number) in order to solve the problem and do the extension activities. Finally, all students will have opportunities to create and use representations (drawings, charts, words, equations, manipulatives, etc.) to provide a record of their efforts to understand the mathematics of this challenge and make their understanding available to others.

About the mathematics involved in this challenge:
This problem focuses on number and operations. Its solution is basic - counting two sets and comparing these sets (figuring out how many are in line and how many will fit on the Ferris Wheel and comparing these numbers). However, more sophisticated counting strategies, such as counting by 3's or having fluency with multiplication facts makes the problem easier to solve. A common error is to forget to add the three children to the line count. The problem becomes algebraic when older children try to solve the modification of the challenge which asks them to figure a way to determine the wait time for any person in line, because this requires seeing a pattern and making a generalization about the situation.

Modifications of the Challenge:
Modifications can usually be made by allowing younger learners to use physical models and illustrations to help understand the situation. Very young children can solve the problem if it is modified so that the numbers are smaller. (The Ferris Wheel could have 6 chairs, each seating 2 people, and the challenge could be to figure out how many could ride each turn.) To make the problem more challenging for older learners, make it more complex. For instance, ask "If a turn on the Ferris Wheel takes 5 minutes, can you discover a way to accurately predict the wait time for a person in any place in the line?"

How To Present Challenges to Children – Some Suggestions from Aunty Math
Teachers often ask me for suggestions on how to present the challenges to children. Since this is the beginning of a new school year here in the United States, I thought I would share how I do this in my classrooms. (I have taught grades K-5 in this same fashion.) I follow George Polya's 4 steps for Problem Solving: Understand, Plan, Carry Out, and Look Back.

First I present the challenge and we discuss it. I even let the children talk a bit about personal experiences that the problem recalls - for instance the current problem is about attending a summer fest. Children might discuss which summer fests they attended over the summer. I find this "math talk" is important so the children can make a connection with their lives, and better understand the problem. (In reading, we call this the Text to Life connection.) We also discuss if we have ever seen a problem like this before.

Next we discuss the problem itself - not how to solve it, but just to see whether they understand the situation, or whether there are any assumptions that are not stated in the problem (ex: For instance in this problem, there is the assumption that everyone in line will be riding the Ferris Wheel and from personal experience, the children may know that this is not always the case. They may have to repose the question: "Assuming all people in line will ride and no one cuts into line unfairly"… The children can also ask any questions at this time, other than "how do you do it?"

Next I remind them of the different ways (strategies) they have used to solve problems. I usually list them as the children name them. (Act out, use models, draw a picture, guess and check, make a chart, etc.) Then we discuss which ones we think might help in solving the problem. Usually there are 2 or 3. I ask the children to make a decision about a strategy they will try first. (I encourage early finishers to try to solve the problem using a different strategy and determine which way was more efficient.)

With younger children, who are new at problem solving, I ask what things they might use to solve problems. I allow the children to use anything in my room including calculators, paper/pencil, chalkboards, blocks, markers, chips, chart paper, etc.

Next I give the children the option to work alone or with a partner. The children then begin to work on the problem using their first strategy. If they find it doesn't work, they choose another one. Occasionally they get stuck and ask for help (although I find this happens less than you would imagine if I have followed the previous steps.) I might ask a pertinent question: What are you trying to figure out? Where do you think you should start? Occasionally I have to lead a bit more. Do you think drawing a picture would help? etc.

Finally (usually after about 15-20 minutes depending on the challenge) I ask for volunteers to demonstrate their method of solving and defend their answer. (There are usually many volunteers!) The children not demonstrating must show respect at this time. When the demonstration is over, the audience has two options: They may affirm the student(s) " I never thought of doing it that way!", "Your explanation was so clear!" OR they may ask a question. This is the option often played when someone disagrees with the answer. (I don't allow for criticism - only questions.) For instance, a child may say " I don't understand why you didn't consider the fact that...could you talk a bit about that?" or " I wonder if you considered subtracting the amount instead of adding it?" Sometimes a child really doesn't get something and will ask for clarification, which the presenters must answer. Sometimes they need a little help because it is not clear to them what is not understood. Sometimes I help, but most often I ask for a volunteer to help clear up the confusion.

After the first demonstration, I ask if someone has a SIGNIFICANTLY different solution. My children are so eager to demonstrate that we will see the same thing over and over if I don't put this criterion on it. Sometimes I have to ask "How is yours significantly different than John's? Soon the children learn to evaluate their solutions in this light. Often they will just show the different part- not the whole piece. (This sophistication comes with time. It doesn't happen overnight!) I also have children who will say, "I messed up" when they get an incorrect answer. Because my classroom atmosphere is risk free, I will often ask them to show their mistake, stating that I often learn more from mistakes than correct answers. After a while I find the children are as eager to share their "detour on the road to understanding" as eagerly as a correct answer because they know they are contributing. (I am careful to affirm these "great mistakes" in thinking!) I have a sign in my classroom that sums up this attitude about mistakes:

Quite often, depending on a child's interpretation of a problem, we will have more than one answer that is reasonable. Often this is because I didn't pose the problem clearly enough. However, I have learned that teachers in Japan often deliberately pose vague or incorrectly stated problems, hoping to encourage critical thinking and reasoning in their students. While I am determined to sharpen my problem posing skills, I realize that if I do make a mistake, not only will alert students bring it to my attention but also such a poorly posed problem may reap an interesting harvest! (Once I asked children to find all the ways I could spend 5 dollars for pumpkins if large pumpkins cost $2, medium pumpkins cost $1 and little pumpkins cost 2 for $1. Most of the children answered 12 combinations, but one answer was illuminating:

All reasonable methods and answers are accepted.

Another part of looking back is evaluating efficient ways of solving and asking children which strategy they would use if they ever saw a similar problem. I also ask what they would like to remember about this problem, or what was particularly tricky about the problem. This is a vital step because, according to research, this "feed back" time is characteristic of good problem solvers. They connect new problems to previous ones they have seen.

As you can probably figure out, one problem can last an entire class period. I do this about once per week and have never regretted the time I spend on them. My students return year after year telling me how easy State Performance Assessments of Problem Solving are for them, and how much they enjoy solving problems and thinking of new problems. (They even help me write these challenges on occasion! For example, I have included some extensions of the Last Fling challenge that some of my students wrote.) I hope these suggestions are helpful as you start off a new school year!

Extensions of the Last Fling Challenge:
Logical Reasoning
Barney was between Danny and Gina. The girl was not last. What order were the children in line?

Estimation and Measurement
Estimate, then measure how long a single file line of 20 children would be. Estimate how much longer the line would be if it were made up of adults only. How did you figure this out?

Measurement – Money/ Multiplication
The first Ferris Wheel debuted at the 1893 Columbian Exposition in Chicago, Ill. * Designed by George Washington Gale Ferris, the wheel had 36 cars, each holding 60 passengers. Tickets were 50¢. What were the possible total receipts made from a ride on this Ferris Wheel?

The present Ferris Wheel on Navy Pier in Chicago has 40 cars or "gondolas" which each hold up to 6 persons. Tickets are now $6.00 per person. What are the possible total receipts made from a ride on this Ferris Wheel? Which wheel, when full, made the most money per ride?

The London Eye, in England measures 135 meters high and was the largest until one was built in China which measures 162 meters high. How much higher is the "Star of Nanchang" than the "London Eye"? The Texas Star is the largest Ferris Wheel in the U.S. It is 50 feet lower than the highest Ferris Wheel. How many meters high is the Texas Star?