Notes to Teachers and Parents Regarding
"Apple Picking Time" Challenge

PROCESS STANDARDS*	CONTENT STANDARDS*
Problem Solving	Number and Operations
Reasoning and Proof	Patterns, Functions, Algebra
Communication	Geometry and Spatial Sense
Connections	Measurement
Representations	Data Analysis, Statistics, Probability

*According to the N.C.T.M.'s Principles and Standards for School Mathematics (PSSM) 2000 document.

In the “Apple-Picking” 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.
• reason as they think about and justify their solution as the only possible correct answer.
• communicate about their thinking by discussing it with classmates, parents, and teachers, as well as by posting their written solution on the message board, and communicating via the message board with other students from around the world.
• make connections between the mathematical operations of addition and multiplication, and connect mathematics to real world situations of apple picking and apple-packing.
• 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 mathematical content in this challenge:
This challenge focuses on the content area of number and operations (using addition and/or multiplication to arrive at a total number of apples or bushels. )

About the Challenge:
This challenge encourages children to think about unitizing, which is necessary for place value understanding. In order to solve the problem, the student must be able to recognize that the apples remain single units and become parts of greater units AT THE SAME TIME. For instance in the original challenge, each box has 4 groups of 4 apples, or 16 apples. Younger children who do not understand this “duality” concept, may draw the problem and count it in the following manner:

Extensions of the Problem:
Number - Base Ten Understanding:
Teachers could redesign the “apple-picking” situation as the catalyst for a project that would support the development of place value understanding. Set up an “apple- packing “ plant, and have children pack “”apples”* into bags which must contain ten apples. Ten bags of apples would then be packed into boxes, which contain 100 apples. Orders could come in daily to be filled. For instance if the Perfect Pie company orders 45 apples, they would be packaged in the following manner:

*Any handy manipulative, such as beans, unifix cubes, counters, or even apple-shaped Elison™‘cut-outs’ will work great. (I have a flowering Crabapple tree which produces cherry sized “apples” at this time of year, which I use for this project.)

Doing projects such as this will help young children understand the idea of grouping by tens better than using base ten blocks, because it is the child who is continually composing the groups.

Number
Making up Problems About Apples
Provide the following facts about apples and challenge children to use one or more of these facts to make up interesting math problems for their classmates to solve:

Two pounds of apples make one 9-inch pie. It takes about 36 apples to create one gallon of apple cider. A medium apple is about 80 calories.
	2500 varieties of apples are grown in the United States. 7500 varieties of apples are grown throughout the world. 100 varieties of apples are grown commercially in the United States. Apples are grown commercially in 36 states.
The largest U. S. apple crop was 277.3 million cartons in 1998. In 2001 United States consumers ate an average of 45.2 pounds of fresh apples and processed apple products. Americans eat 19.6 pounds or about 65 fresh apples every year. 61 percent of United States apples are eaten as fresh fruit. Apple trees take four to five years to produce their first fruit.
	Some apple trees will grow over forty feet high and live over a hundred years. America's longest-lived apple tree was reportedly planted in 1647 by Peter Stuyvesant in his Manhattan orchard and was still bearing fruit when a derailed train struck it in 1866.
Apples harvested from an average tree can fill 20 boxes that weigh 42 pounds each. 25 percent of an apple's volume is air. That is why they float. A peck of apples weight 10.5 pounds. A bushel of apples weights about 42 pounds and will yield 20-24 quarts of applesauce.

Measurement and Math/Science Connections:
Ask students to bring their favorite kind of apple. (Tell them to make sure they know what variety they have.*) Make a bar graph listing each apple variety that was represented in the activity and how many times each variety was chosen as someone’s favorite.

*This month’s Family Circle Magazine, found in grocery stores, has a colorful pull out poster of apple varieties.

Do some of the following activities, depending on the math curriculum for your grade level:

• Increase students’ powers of observation by having the students carefully observe, feel, smell, their apple for one minute. Then put all apples in the middle of a circle. Have children sit around circle, and, in turn, find their own apple from the bunch.

• Order apples by size (Apple varieties range in size from a little larger than a cherry to as large as a grapefruit.)

• Have children use a balance scale with gram weights to weigh apples, then order by weight. (The largest apple picked weighed three pounds.)

• Use string or a centimeter measuring tape to measure the circumference of the apples at the widest point.

• Have an apple-peeling contest using apple parers brought from home. (The world's longest apple peel was created by 16 year old, Kathy Wafler Madison on October 16, 1976, in Rochester, NY. It was 172 feet, 4 inches long.)

• Investigate the number of seeds inside each apple. (Is it true or false that every apple has exactly 5 seeds inside?)

For more information about apples, visit the website http://www.urbanext.uiuc.edu/apples .

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