Title
Square Areas (Dorothy Gieck)
Goals
Students will work with the concept of sequencing. Students will work in small groups in order to find a formula that will calculate the amount of shaded area in the given problem. They will be able to explain and justify, how they arrived at this method to the rest of the class.
Abstract
This activity, which is set in the context of finding areas of squares with shaded areas, asks students to determine probability and looking for a recursive pattern. Students are asked to form small groups to discuss the problem and come to a method for finding the pattern. This activity could be used with a probability unit or in conjunction with recursive and explicit formulas.
Problem Statement
The sides of a square are 16 inches in length. The square is divided into nine smaller squares and the center square is shaded. Each of the eight unshaded squares is then divided into nine smaller squares and the center square of each is shaded.
Instructor Suggestions
1. Set the stage by discussing the "Problem Statement" (see above) with your students.
2. Distribute the "Square Areas" activity sheet and allow the students to individually read the activity and answer the first for questions.
3. Have the students form small-groups and complete the questions on the activity sheet.
4. When the small groups are finished, have a spokesperson for each group share the group answers for specific parts of the activity and explain their method and reasoning involved in arriving at their probability and their formulas.
5. Discuss the students work as it relates to probability and formulas.
Materials
"Square Areas" activity sheet, overhead transparencies of the pictures of the squares, overhead pens, blank overhead transparency sheets.
Time
Introduction of Problem Statement (5 min.), small group work (20 min.), presentation of small group work and large group discussion (15 min.), extension questions (5 min.)
Mathematics Concepts
Discrete Mathematics Concepts
Recursion, Finite differences, Closed-form solutions, Arithmetic and Geometric series
Related Mathematics Concepts
Ratios, Common Ratios, Sequences
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Algebra, Geometry, Discrete Mathematics
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into any class that is studying sequences. It would fit into an algebra or geometry class during the study of sequences and patterns.
Further Investigation
What if each square were divided into 16 squares with the middle four shaded in?
Variations/Comments
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W. H. Freeman and Company.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
Colorado Model Content Standards for Mathematics (1995)
Larson, R., Hostetler, R. & Edwards, E. (1993), Precalculus, a graphing approach. Lexington, MA: D. C. Heath and Company.