In the "Aunty’s Birthday Money" 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 several branches of mathematics, including number and data, number and measurement (money), number and probability, number and pattern. Children will also connect mathematics to a common life situation- that of receiving birthday money.

• 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:
"Aunty’s Birthday Money" challenge focuses on 3 content standards: Number and Operations (addition and the concept of doubling), Patterns, Functions and Algebra, and Data, Statistics and Probability. Students will need to keep an organized running total of both options and make note of the growing patterns in order to determine whether Aunty made a wise choice. Older students may take on the additional challenge of using the collected data to determine at what point each choice became the wisest option.

About the N.C.T.M. 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/.

Aunty’s Birthday Money Challenge addresses the following recently released NCTM focal points:

First Grade
Number and Operations: Develop an understanding of whole number relationships, including grouping in tens and ones
Algebra: Identify, describe, and apply number patterns

2nd Grade
Number and Operations: Develop an understanding of the base-ten numeration system and place-value concepts

Number and Operations: Develop fluency with multidigit addition, subtraction and multiplicative situations,
Algebra: Use number patterns to extend knowledge of properties of numbers and operations.

3rd Grade
Number and Operations and Algebra: Develop understandings of multiplication and division and strategies for basic multiplication facts and related division facts
Algebra: Understand properties of multiplication.
Data Analysis: Construct and analyze frequency tables, bar graphs, picture graphs, and line plots and use to solve problems that involve the operations of addition, subtraction, multiplication and division.

4th Grade
Number and Operations: Apply understanding of models for multiplication to develop, discuss, and use efficient, accurate, and generalizable methods to multiply multi-digit whole numbers.
Algebra: Identify, describe, and extend numeric patterns involving all operations and nonnumeric growing or repeating patterns. Develop understanding of the use of a rule to describe a sequence of numbers or objects.

5th Grade
Data Analysis: apply understanding of whole numbers, fractions, and decimals to construct and analyze double-bar and line graphs and use ordered pairs on coordinate grids.

About the challenge:
In this problem, students get to investigate the "explosive power of doubling." How quickly numbers increase when doubled is a challenging idea for most students and "can be a springboard for discussing, dramatizing, drafting and drawing children’s interpretations of the powers of 2."* According to Fran Curcio and Myra Zarnowski, "Even children who have encountered problems that illustrate the explosive power of doubling, may still not comprehend how fast numbers grow when doubled, which is one reason by similar problems should be repeated. It is through repeated, similar and refined experiences, that children learn to observe, and reflect and generalize a solution." (See extensions for other examples of doubling problems.)

*From Revisiting the Powers of Two by Fran Curcio and Myra Zarnowski, Teaching Children Mathematics, the journal of the National Council of Teachers of Mathematics, Jan. 1996.

Modifications of the problem
Some students may not understand the concept of Aunty getting "twice as many" dimes. One concrete way to illustrate this notion is to use a mirror and real dimes. "Aunty was trying to think how many dimes she would get. She got one this year. How many will she get next year?" Place the mirror next to the dime and count the dime AND its reflection. "Yes, she will get double- she will get 2 dimes.

Another difficulty of the regular challenge for younger children is that they may be unable to handle adding dimes and relating dimes to dollars. This may be a perfect time to use unifix cubes to act out the solution to this problem. If children understand that ten dimes are equal in value to a dollar, then unifix cubes can be doubled until there are ten, which are then stacked into a "ten train" and which is equal in value to a dollar. These "ten trains" can be traded in for dollars if desired during or at the end of the investigation. (Be prepared! It will take a lot of unifix cubes!)

Other children may not be have the necessary skills to handle the calculations necessary to complete this challenge. Despite this difficulty, younger children can benefit from exposure to this growing pattern problem. For this reason, calculator use would be appropriate for this challenge. Many children will enjoy this challenge, and should not be held back from the thinking simply because they have not yet mastered multi-digit addition.

Extensions of the problem
Revisiting concepts, such as doubling, should be an integral part of teaching. It is not simply repetition. Instead it gives new experiences that allow them to move beyond their current understandings. Examples of literature that illustrate the power of doubling numbers include:

A Grain of Rice by Helena Clare Pittman – NY: Bantam Skylark Books, 1992

The King’s Chessboard by David Birch – NY: Dial Books, 1988

The Rajah’s Rice by David Barry – NY: Freeman and Co., 1994

Anno’s Mysterious Multiplying Jar by Masaichiro and Mitsumasa Anno NY: Philomel Books, 1983

Anno’s Magic Seeds by Mitsumasa Anno NY: Philomel Books 1995.

Older children may enjoy generating and illustrating their own stories, which show the power of doubling. For instance, one of my fourth grader wrote a story about having a nightmare that she found a library book she had lost 30 years ago and being given a choice of paying a library fine at a rate of 10¢ per day over the course of 30 years, or a fine of 1¢ doubled each day for 30 days. Aunty would love to read any such stories. Send to:

Aunty Math
c/o The DuPage Children’s Museum
301 N. Washington St.
Naperville, IL 60540

Who knows? Maybe your story will be the basis of a future "Aunty Math" problem someday!