In Gina's Present for the Teacher 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.
  
• communicate about their thinking by discussing it with classmates, parents, and teachers.
  
• make connections between math and art, and different math content areas, such as number and pattern.
  
• 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:
This challenge falls under the general content standards of Pattern, Functions and Algebra and Number and Operations. Children must use their understanding of repeating patterns, and at the same time, be able to keep a running total of the cost of the beads.

About the challenge:
This challenge is a good example of an open-ended question. Open-ended problems are designed to support different students' varying levels of mathematical knowledge, fluency, flexibility, originality and elegance. Open-ended problems are based on significant mathematics. They are not immediately solvable. They are designed to elicit varied solution paths and have multiple correct answers. Most children can solve part of it, and some children can solve all of it.

In the "Present for the Teacher" challenge, younger learners may be able to think about making a patterned necklace, but be unready to do the math necessary to figure the cost of the beads. Other learners will be able to assign values to the beads, and make patterns, but find it quite challenging to make a repeating pattern and spending ALL the money without going over. Another characteristic of this open-ended problem is that there are many correct solutions. The level of mathematical maturity (as well as the artistic leanings) of each child will determine the complexity of the necklace pattern. As the child works on a design, discoveries about reversing a pattern in the necklace's center may be discovered as well as challenges in getting the total cost of beads to balance with the desired design. (Which children will use their understanding of line symmetry to help solve this problem?)

"Each side of the "fold" line is worth 50¢"

Much "trial and error" may occur before a final design that fits the criteria is arrived at. Some learners may not be able to keep all the criteria in mind. For instance, they may spend the exact amount, but find that they do not have a majority of purple beads in the design. Others may find it very difficult to end with pink beads on either end. While working on this problem, children will be practicing adding money amounts, counting by 5s and 10s, and keeping a running total.

Extensions:
Working on Multiplication:

If the white beads cost 5¢ each and the green beads cost 7¢ each, how much did the beads in this necklace cost altogether? How can you change the pattern so that the necklace would cost less, but use the same number of beads?

Working on Unknowns

These beads cost 78¢. The circle beads cost 20¢ each. How much did the octagon bead cost? (adjust the numbers to fit the level of your children.)