In the wrapping presents challenge, students have opportunities to employ four of five process standards. They will:

• problem solve as they decide on a strategy for solving a problem where the answer (and even the question!) is not immediately obvious.

• reason as they think about and justify their solution, considering all possible variables.

• communicate about their thinking by discussing it with classmates, parents, and teachers.

• use representations (drawings, charts, words, models, 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:
Problems like this one may seem uncomfortably familiar to those of us who struggled through algebra, because it seems to require solving simultaneous equations containing three unknowns. (We don’t know how many presents any of the three children wrapped.) Children can solve this problem, using many different strategies. In fact teachers of older children might decide to assign teams to try to solve it using different strategies: using manipulatives, guess and check, looking for a pattern, making a chart or diagram, drawing a picture, acting it out, or visualizing it. The teams could then present their answer and their strategy to the whole class, and discuss the pros and cons of each strategy.

About the Focal Points addressed in this challenge:
The National Council of Teachers of Mathematics (NCTM) recently released a document titled Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. These "Curriculum focal points are the most important mathematical topics for each grade level. They comprise related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning. They are the topics that should be considered as the basis for decisions about curriculum development. (For more information about these focal points, visit http://www.nctm.org/focal points/

The Wrapping Presents challenge addresses the following recently released NCTM focal points:

Kdg
Number and Operations: Representing, comparing, and ordering whole numbers and joining and separating sets

Grade 1
Number and Operations and Algebra: Children develop strategies for adding and subtracting whole numbers on the basis of their earlier work with small numbers.
They use a variety of models, including discrete objects, length-based models (e.g., lengths of connecting cubes), and number lines, to model "part-whole," "adding to," "taking away from," and "comparing" situations to develop an understanding of the meanings of addition and subtraction

Number and Operations and Algebra: Children solve both routine and nonroutine problems.

Grade 3
Number and Operations: Developing an understanding of fractions and fraction equivalence
Students develop an understanding of the meanings and uses of fractions to represent parts of a whole, parts of a set, or points or distances on a number line. They understand that the size of a fractional part is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less than, or greater than 1. They solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators. They understand and use models, including the number line, to identify equivalent fractions.

Grade 5
Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions. (Harder challenge and extension problems)
Students apply their understandings of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency with standard procedures for adding and subtracting fractions and decimals. They make reasonable estimates of fraction and decimal sums and differences. Students add and subtract fractions and decimals to solve problems, including problems involving measurement.

Extensions and Modifications Around the NCTM Focal Points
Estimation of Length
Children love to wrap presents, so teachers and parents can connect this seasonal activity to many different math concepts. Children can estimate how much ribbon and paper they need to wrap certain presents, or they can estimate how many bows or presents they can wrap using a spool of ribbon or a roll of paper.

Measurement:
NCTM 2nd grade focal point-Developing an understanding of linear measurement and facility in measuring lengths.
Children can also practice using metric or standard measurement tools to measure out paper and ribbon.

NCTM 3rd grade focal point-Solving problems involving perimeter.

NCTM 4th grade focal point-Developing an understanding of area and determining the areas of two dimensional shapes.
Older children can use box tops of presents to measure and compare perimeter and area.

Geometry:
NCTM Kdg. focal point – Identify, Name and Describe shapes and space
Younger children might want to reinforce their understanding of geometric shapes by making square, round, triangular or rectangular cookie shapes and naming the shapes of wrapped boxes.

NCTM 1st grade focal point – Recognize shapes from different perspectives and orientations, describe their geometric attributes and properties.
While tying ribbons around packages, the geometric math terms of horizontal, vertical and diagonal can be introduced. Which method of tying ribbon takes the most ribbon? The least ribbon?

Fractions:
NCTM 3rd grade focal point - Developing an understanding of fractions.
Since cookies are an integral part of this problem, children might try their hand at baking several different varieties, practicing mathematics as they halve, double, or triple recipes.

Here is the recipe for Giant Snickerdoodles.

Giant Snickerdoodles

1 cup butter or margarine, softened
1 1/2 cups plus 2 tablespoons sugar
2 eggs
2 3/4 cups all purpose flour
2 teaspoons cream of tarter
1 teaspoon baking soda
1/4 teaspoons salt
2 teaspoons cinnamon

In large mixer bowl with mixer at medium speed, cream butter or margarine, 1/1/2 C sugar and eggs until light and fluffy. In a separate bowl combine flour, cream of tarter, baking soda, and salt. Add to cream mixture until well blended. Refrigerate for 30 minutes.

Preheat oven to 375 degrees. Combine remaining 2 tablespoons sugar and 2 teaspoons cinnamon in a flat open bowl. Shape dough into 2-inch balls and roll in cinnamon-sugar. Place 3 inches apart on ungreased cookie sheets. Bake 12 to 15 minutes until golden brown. (Snickerdoodles will puff up at first, then flatten out to giant size during baking.) Makes 2 dozen.

Fractions:
5th Grade focal point - Developing an understanding of and fluency with addition and subtraction of fractions.
If aunty made 4 dozen Giant Snickerdoodle cookies, how much of each ingredient did she use?

If she only wanted to make 1 dozen, how much of each ingredient would she use?

Multiplication and Division:
3rd grade focal point - Developing understandings of multiplication and division.
The average buyer will purchase 3 rolls of wrapping paper this holiday season. If everyone in your class purchased this amount, how many rolls of paper would this be?

It takes about 9 feet of ribbon to make a nice bow. If aunty has 12 packages to wrap and ribbon is sold by the yard, how many yards does she need to buy?

There are 12 cookies in a dozen. If aunty makes 4 dozen cookies while the children wrap presents, how many cookies did she make?

A word from Aunt Mathilda about problem solving in the classroom:

Hopefully children will have opportunities to share their different strategies with the group. Teachers should model and encourage good listening behavior when this occurs and encourage the class to comment positively or ask questions about the strategies. However, no one should be allowed to make negative or comparative comments, such as "That's a dumb way!" or "My way is better!" At no time should there be an atmosphere which implies that there is one correct way to solve problems – because, in fact, this is not true. The best way is the one that make sense for the children.

P.S. from Aunty Math:

I would love to see samples of your children's work on this problem and hear any comments you might have. Although I would not be able to return them, I will send a personal thank you to your class and I may mention this work in a later posting on the site. You may send this to me care of:

Aunty Math
Dupage Children's Museum
301 N. Washington St.
Naperville, IL 60540