| 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 "Gingerbread Men " 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. They reason as they think about and justify their solution. They should have opportunities to 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. Children will be able to make connections between mathematical concepts in order to solve the problem and do the extension activities. Finally, all students 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.
About the math content in this problem:
The math contained in t his challenge falls into the Number and Operations category. However, solving a problem like the Gingerbread Men challenge requires more than just the ability to add or multiply, subtract or divide, although these skills are essential to the final solution. Children must also recognize what the question is - and this requires some logic. While there may be enough chips to make 15 noses, will there be enough of the other ingredients to make 15 gingerbread men, according to the directions? The extension problem that asks how many different ways a 30¢ gingerbread man can be purchased using coins, is an example of discrete mathematics and falls under the content strand of functions, patterns and algebra.
About the challenge:
This problem requires careful reading and an understanding that some of the ingredients will not be needed. This may surprise some students who are used to math problems "coming out even." Older children might divide the total number of any ingredient by the number needed for one cookie. However, it will still take some logic to come up with an answer to the question asked. Younger children may actually have an easier time solving it because they will probably choose to draw a picture or use manipulatives - thus seeing the problem of not having enough ingredients to finish cookies.
The Gingerbread Men challenge is a problem that can be easily solved by drawing a picture. Besides making a problem more available to younger students, the "draw a picture" strategy has many benefits. As Mary Jane Moran and Jennifer Jarvis explain in their article "Helping Children Develop Higher Order Thinking" (Young Children, Sept. 2001), when children represent their understanding using drawings, they involve their mental and their motor capabilities to recall and recreate the situation, thus actively constructing their knowledge of the solution. Drawing helps children:
* Recall and reconstruct the details of the problem and develop ideas for solving it
* Represent and study the problem and its solution more carefully.
* Establish connections and make predictions about possible solutions.
In the example below, it is clear how drawing helped a first grade child represent, then solve the problem. The child first drew all the ingredients. (Recalling and reconstructing details of the problem.) Then the child drew one gingerbread man as illustrated above and crossed off the ingredients used for that gingerbread man. (Representing and studying the problem and its solution more carefully.) She made a tally mark to represent that gingerbread man. "I was going to draw lots of gingerbread men, but I thought I could just make a mark for each one instead", she said. She then explained how she marked off the rest of the ingredients. "For each gingerbread man mark, I had to "x" two candies, one chip and four raisins. I thought I would use everything but I didn't use everything." (Establishing connections and making predictions about possible solutions.)
Modifications of the Problem
Younger children can think about building gingerbread men too, but the number of ingredients will probably need to be modified. For example, I would suggest presenting this problem to children if you think the original one is too difficult.
"If Gina has 10 candies how many cookies can she make if each cookie has 2 eyes."
Extensions of the Problem
Measurement (Money)
Children who are studying the value of coins and coin equivalences may enjoy working on the following problem:
I went to the store to buy a gingerbread man. I found out that gingerbread men cost 30¢ each. I had a lot of change in my coin purse. I couldn't decide how to pay for the gingerbread man. How many different ways could I pay for the gingerbread man?
This is an open-ended problem because most children can solve part of it and some children can solve all of it. The problem lends itself to bulletin board display, where children can add combinations as they think of them, and determine if there are duplicates (for instance 1 dime, 1 nickel and 15 pennies is the same combination as 1 nickel, 15 pennies and 1 dime.)
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Aunty Math problems, copyright 2003, Angela G. Andrews. You may download, print and make copies of "Aunt Mathilda's Math Challenges" for use in your classroom provided that you include the copyright notice shown on that page with all copies.
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