In the Cold Weather-Warm Hearts 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 (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.
  
• connect mathematics to a real world situation –making valentines. This problem also connects the mathematics of number and operations to the mathematics of algebra because the problem deals with unknowns.
  
• 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:
Solving a problem, such as the one described in this challenge, falls under the math content standards of Number and Operations and Patterns, Functions, and Algebra.

About the Challenge: This challenge offers an opportunity to present the problem solving strategy "work backwards". Starting with what is left and working backwards is an effective way of solving such a problem. However it is not the only way to solve it. Other ways include "Guess and Check", incorporating making a list to keep track of the initial guesses, and "Guess and Check" using manipulatives. Even when using working backwards, it may be necessary to model this kind of thinking using a model such as unifix cubes or small valentine stickers.

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 www.nctm.org/focal points/

The Cold Weather – Warm Hearts challenge addresses the following recently released NCTM focal points:

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 non-routine problems.

Grade 2
Children add and subtract to solve a variety of problems, including applications involving measurement, geometry, and data, as well as non-routine problems

Grade 3
Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.

Extensions of the problem:
Math Art Connections:
Making valentines provides an excellent opportunity to investigate "mirror" symmetry with primary students. See instructions below. Even though valentines may vary in length or width when constructed this way, they will all have one vertical line of symmetry. For a wonderful web lesson on symmetry go to

http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/4_Line_Symmetry/index.html

Measurement of Area for older students

Andrew wants to buy his mother a chocolate heart for Valentine’s Day. He has a dollar to spend. At the store he sees two different chocolate hearts, both wrapped in red foil lying on paper behind a glass case as shown in the drawing below. They have no markings on them indicating weight, but they look to be about the same thickness. Without removing them from the case, is there a way Andrew can figure out which chocolate heart has more chocolate?

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