In Gina's Quilt 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.

• make connections between geometry and spatial patterns, square numbers, and number patterns. They will also connect mathematics to a part of American culture – quilt making.

• 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 standard of Geometry. When looking at this quilt in order to find all the squares that are contained within its borders, the child has to clarify for him/herself the characteristics and properties of this two-dimensional shape. It is not enough that it have 4 straight sides and four right angles. Each side has to be the same length. While solving this challenge, students will investigate the results of putting together and taking apart sections of this large square quilt composed of 16 smaller squares and will develop mathematical arguments about the geometric relationships they discover.

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/

Gina's Quilt Challenge challenge addresses the following recently released NCTM focal points:

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Geometry: Describing shapes and space
Children interpret the physical world with geometric ideas (e.g., shape, orientation, spatial relations) and describe it with corresponding vocabulary. They identify, name, and describe a variety of shapes, such as squares, triangles, circles, rectangles, (regular) hexagons, and (isosceles) trapezoids presented in a variety of ways (e.g., with different sizes or orientations), as well as such three-dimensional shapes as spheres, cubes, and cylinders. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

Grade 1
Geometry: Composing and decomposing geometric shapes
Children compose and decompose plane and solid figures (e.g., by putting 4 smaller congruent squares together to make a larger square), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry.

Grade 3
Number and Operations and Algebra: Developing understandings of multiplication and division and strategies for basic multiplication facts and related division facts. Students understand the meanings of multiplication and division of whole numbers through the use of representations (e.g., equal-sized groups, arrays, area models, and equal "jumps" on number lines for multiplication, and successive subtraction, partitioning, and sharing for division).

Number and Operations: Developing an understanding of fractions and fraction equivalence
Students develop an understanding of the meanings and uses of fractions to represent parts of a whole, parts of a set, or points or distances on a number line. They understand that the size of a fractional part is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less than, or greater than 1. They solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or common numerators or denominators. They understand and use models, including the number line, to identify equivalent fractions. (In this example, they use a grid model.)

Grade 4
Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes
Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g.decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle.

About the Challenge:
The quilt challenge is differentiated for varying levels of mathematical sophistication. Younger learners may be able to focus only on the individual squares and possibly be led to see the one large quilt square as well for a total of 17 squares. One way to help students see the squares within the large square is to use wooden cubes to model this situation.

Once this example is given, it would be wise to refrain from giving further help and let the children investigate this problem. (This is the DOING of mathematics themselves that will move children forward in math more than LISTENING to mathematics being done by adults.)

Older learners may be ready to take on this more difficult geometric and spatial challenge: keeping in mind the characteristics of a square, visually separate the large square into squares smaller than the large square, but larger than the individual squares. It is not as important that the children find all 26 squares as it is that they find some and be able to discuss their findings with others. This is a problem that can be revisited from time to time. Given time and a spirit of cooperation within a classroom, it is highly possible that all squares will be found.

Extension for More Mature Learners:
What if Gina's quilt had 5 rows of with 5 fabric squares in each row. How many different squares could be found within this quilt?

Geometry/Number Connections: the Area model of multiplication – 3rd Grade Focal Point
Connect the quilt rows idea to the concept of multiplication. For example, If Gina's quilt had 3 rows of 4 squares, she would have 12 squares. She would also have 12 squares if she had a quilt with 4 rows of 3 squares in each row. (Commutative property of multiplication is vividly illustrated with this area or array model).

This would be an appropriate time to teach students that the symbol X which represents the operation of multiplication means "groups of".

If Gina's quilt had 5 rows and each row had 2 squares, how many small squares?
If Gina's quilt had 2 rows and each row had 5 squares, how many small squares?
(Be sure to emphasize that in these problems we are looking only for small squares. Looking for additional squares within a 5x2 quilt would make a nice enrichment activity, however.)

Geometry/Number/Art Connections - Fractions – 3rd Grade Focal Point

"What fractional part of this quilt square is red?"

After showing students a variety of traditional quilt patterns, (such as those found in the book Eight Hands Around: A Patchwork Alphabet by Ann Whitford Paul and Jeanette Winter or from samples found on the internet), provide 16 block grid paper to students and have them make their own quilt squares using crayons or markers. (It is wise to limit the use of colors to 2 plus white.) Then have the student determine the fractional part (or percentage) each color is of the design. For example in the above quilt square, ¼ of the design is red, ¼ is green, and ½ is white.

You may want to visit the site below to make copies of traditional quilt squares for this activity as well.
http://www.quilt.com/ColoringBook/QuiltColoringBook.html

Geometry/Measurement/Number
Math/Art Connections: Area – 4th Grade Focal Point

"If the area of this quilt piece is 16 square units, what is the combined area of the red shapes in this quilt piece?" Students can design and then figure out the area of one color in their design based up the total area of the square.

Math/Literature Connections:
Quilts make an excellent topic for an interdisciplinary, thematic unit that integrates math, art, social studies and reading. The site listed below has a comprehensive list of books with a quilting theme suitable to all ages.

http://eduscapes.com/ladders/themes/quilts.htm

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