While solving the "Valentine Making Party" Challenge, students have opportunities to employ all five process standards.* They will problem solve, because in this situation, the solution is not immediately obvious. They reason as they choose an appropriate strategy and again as they justify their solution. They should have opportunities to communicate about their solution 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. The challenge is closely connected to a popular children’s holiday celebration, where they can solve a real world problem with mathematics. Also, in the extension activities, they can connect two math ideas (discrete math and measurement.) Finally, they 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 Problem:
This problem is a combinatory mathematics problem, which is a type of discrete mathematics. Discrete mathematics, which deals, in part, with how many solutions (combinations) may exist for problems, fits under the categories of Data Analysis and Statistics, and Pattern, Functions and Algebra. Although discrete mathematics is not traditionally identified as a topic for the primary/elementary grades, these types of problems can be appropriately investigated in the context of familiar tasks.

According to Lyn English,("Problem Solving with Combinations" in Arithmetic Teacher. Volume 40, #2, Oct. 1992.), primary children enjoy working on such problems, but have little or no self-monitoring strategies. It will be difficult for them to know if and when they have found all the possible combinations. It is helpful therefore, to encourage children to record their combinations in a way that allows them to revisit the problem and think about it over an extended period of time. (My kindergarten class kept a similar problem up for weeks, and continually studied all the recorded possibilities. When a new combination was found, the child eagerly posted the representation of it on the board for all to see.) Teachers may want children to record all possible ways in a math journal, so that they can revisit the problem and add to the journal when a new solution is found.

English notes that elementary children tend to approach combination problems in three ways. Some do the problem, and check later to see if they found all combinations. Some make a tentative plan for finding all solutions, then solve, and finally check back. More mature students make an organized plan, and constantly check to make sure their system is working in order to find all combinations. Teachers will want to observe which methods their students use as they solve this problem.

Modifications of the Challenge:

For Younger Learners:
Using fewer choices, such as 2 colors of paper and 2 kinds of doilies will allow younger children the opportunity to solve this problem. Using paper cutouts of all the different choices may also help children solve this challenge.

For Older Learners:
Older learners enjoy solving combination problems. You may decide to add a few more colors of paper or a few more doily choices. Be careful – the number of possibilities grows quickly

Extensions:

Measurement/Money
For students studying how to add money amounts, teachers could assign values to ingredients used in valentines then have students make valentines and then compute their cost.

Students could also solve problems about valentine making. Assign values to the materials (and add a few of your own materials also.)

(For younger children, you may wish to use the symbol for cents. Older children can compute using the decimal representations of money. These symbols, or course, are never combined in this form .05¢)

Sample Problems:

Beginners:
If Gina made a valentine using red paper and a square doily, how much would it cost?

Apprentices:
If Gina used 1 sheet of paper and 2 doilies to make a valentine, how much would it cost?
How many different ways could you pay for a valentine costing this much?
(This is also a discrete math problem, since the possible combinations are limited to a discrete number.)

Experts:
If Gina made one of each combination in the problem, how much would it cost?

Aunty Math problems, copyright 2001, 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.