In the "Leaf Raking" Challenge, students have opportunities to employ all five process standards. They will problem solve as they search for pertinent information in the challenge, and decide on a strategy for solving a problem where the answer is not immediately obvious. They reason as they choose the appropriate operations and sequence for working through the problem, and again as they justify their solution. They should have opportunities to communicate about their solution by discussing it with classmates, parents, and teachers. While working this problem and its extensions, students will have opportunities to connect several mathematical topics – money, time, multiplication or repeated addition. Finally, they 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.

The Number and Operations Standard is the content focus of this challenge. However, the harder challenge and the extension activities also focus on Measurement, Data and Statistics and Algebra and Pattern.

About the mathematics involved in this challenge:
On the surface, this challenge seems to require only simple operational and computational skill- knowing what operation to use, (multiplication or repeated addition) then carrying out those operations. However, in reality, this multi-step problem is rich with mathematics. Children will need to know about how time is measured: How many minutes are in a half-hour? They will need to know the value of coins: How much is 30 dimes worth? How do I convert this into dollars? Dealing with all the mathematical knowledge that is required to solve this problem requires flexible thinking, an important component of computational fluency. It is important to let children discuss the thinking that went into their solutions. Talking about their solutions with others helps children:

• Organize and consolidate their thinking
• Communicate their thinking clearly to others
• Analyze and evaluate the thinking and strategies of others
• Use the language of mathematics to express ideas.

Modifications of the problem for different age and ability groups:
Teachers and parents may want to reduce the complexity of the problem for younger learners by:

• Supplying real coins. (It is certainly acceptable for a younger child to give a solution in number of coins without converting these coins into money values.)
• Adjusting the numbers in the problem. For instance, the children could earn a penny a minute for fewer minutes work.

Likewise, older children may enjoy these additional challenges:

• If Aunty paid with coins only, how many quarters would she owe each child? How many dimes? How many nickels?
• Aunty offered to pay the children each 3.oo for the job or 1 penny for the first minute, 2 pennies for the second, 4 pennies for the 3rd. Which would have been a better deal for them if they each worked 10 minutes?

Connecting Activities: Money

Number and Measurement: Trade for a Quarter Game
Children just learning about coin equivalencies might enjoy this simple game.

Materials: I number cube and a collection of pennies, nickels, dimes and quarters.
Goal: To be the first to get a quarter.

Players take turns tossing die and taking that many pennies. When players accumulate 5 pennies, the pennies must be traded for a nickel. When 2 nickels are accumulated, they must be traded for a dime. When a player has 2 dimes and nickel, s/he may trade for a quarter and win the game.

(It is interesting to observe children playing this game. At first, if a child has a nickel and rolls a 5 on the die, s/he will first trade those pennies for a nickel, then take the other nickel and trade for a dime. Eventually, however, children who have a nickel and roll a 5, will just trade the nickel for a dime. This internal trading process cannot be hurried, so it is unproductive to demonstrate this "short cut." It is better to let the child discover this strategy. This happens as a result of playing this game many times, so I recommend that teachers send this game home for parents and children to play together.

(Older players may use 2 number cubes and trade up to a dollar.)

Algebra: What Coins Do I Have?

After children have mastered coin equivalencies, they will enjoy this "mystery game." Put a few coins under a bowl. Give these clues:

I have 4 coins under the bowl. Their total value is 35¢. What are my mystery coins?

It is interesting to observe how children solve these problems. For instance many times children will respond with a coin combination that has a value of 35¢ (perhaps a quarter and a dime) but will forget the other clue that there must be four coins. With time and practice, they will learn to discard combinations that don't fit both clues. This reasoning will prepare them for higher level mathematics.

Algebra (Discrete Mathematics) and Data: Coin Combinations:

• Using coins only, how many different ways could she pay Danny?

Making an organized list is necessary to find all the combinations of this challenging problem.

Number:
According to some sources, a mature healthy tree can produce 200,000 leaves annually. During 60 years of life, such a tree would grow and shed 3600 pounds of leaves. How many pounds might you have to rake annually if you had a mature tree on your property? How many pounds would you have to rake in order to have raked up 1,000,000 leaves?

A few words from Aunt Mathilda about Teaching Money:

Teaching money concepts is the bane of many primary teachers in the United States. It seems that no matter how much class time is spent on it, many children continue to have difficulty understanding the value of coins and how to count coins or make change. There are several reasons for this:

• U.S. coinage is not proportional to its value. (A nickel for example is larger than a dime but worth less.) This system is not sense-making to young children. I recommend teaching money values using proportional models for instruction. For further information regarding simplifying instruction about money, refer to Teaching the Value of Coins by Randell Drum and Wesley Petty, in the Jan. 1999 issue of Teaching Children Mathematics, a National Council of Teachers of Mathematics publication that should be found in every elementary school's teacher resource library.
• Before 2nd grade, many children simply have not practiced working with single units (ones) enough to work efficiently with units other than one (fives, tens, twenty fives), making formal instruction before this time a frustrating and often unproductive experience for teachers and students alike. Counting coins requires that children switch fluently back and forth between counting systems of 1's, 5's, 10's and 25's, something that develops slowly, over time.
• Children in the United States seldom have opportunities to use coins in daily life. (There simply are not many things that can be purchased for a nickel, dime or quarter these days.) Therefore, children should be given opportunities to use real coins* as often as possible in order to solve problems. Activities such as setting up and having children operate a store in a second/third grade classroom, or having students be responsible for collecting milk money, field trip money, etc., will go far in helping children to internalize money concepts. Also money concepts should be presented frequently in a meaningful problem-solving context, rather than as isolated lessons.

*Some teachers will not feel comfortable using real coins in a classroom, and of course, this is an individual teaching decision. I will share that I have always used real coins because I observe that children take their work more seriously when they are using the "real stuff." Here is how I use coins in my classroom. I have a large jar of real coins in my room, with a note on it reading "My granddaughter Sarah's Coins for College." As each year begins, I show my class Sarah's picture, and then I explain that these are not my coins, but that they belong to Sarah, who is saving all her coins so that she will be able to go to college. I ask the children to be very careful with these coins and make sure they all get back in the jar, so I can return them to Sarah for her college fund. (After 18 years of teaching, I must report that I have not lost more than a dollar's worth of coins!) Other teachers have each child bring in a zippered plastic bag with coins in it for classroom use. (2 quarters, 5 dimes, 5 nickels, and 9 pennies are a good assortment.)