Connections

Process Standard

The goals of the NCTM's connections process standard are that "in grades 9–12, the mathematics curriculum should include investigation of the connections and interplay among various mathematical topics and their applications so that all students can—

• recognize equivalent representations of the same concept;
• relate procedures in one representation to procedures in an equivalent representation;
• use and value the connections among mathematical topics;
• use and value the connections between mathematics and other disciplines."
  (NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 146)

Video Overview

In excerpts from eight classrooms, students make connections within mathematics, to other disciplines, and to situations in the real world. In some lessons, they use graphing calculators to investigate relationships between functions and graphs; and in one lesson, they use a Calculator–Based Laboratory^TM to find the relationship between real phenomena, data, and a graph. Specifically, students make connections when:

• applying multiple methods to solve two equations
• gathering and analyzing data from a bungee jump model
• using tree diagrams and matrices to determine probabilities of a vehicle's destinations
• using inductive and deductive approaches to geometry
• writing about what they learned in a scientific notation lesson
• gathering and analyzing data from an experiment on gravitational acceleration
• determining how to maximize business profits
• applying linear programming to find optimal populations

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A Pre–Viewing Exploration for Teacher Workshops

Right Triangle Relationships across the Curriculum and Beyond

Objective: To explore how right–triangle side relationships and the Pythagorean Theorem can be connected to other mathematics, to other disciplines, and to real life.
Materials: Calculator, straight edge, paper, and pencil for each small group.

Activity
Do this activity in small groups.
Many teachers use the Pythagorean Theorem to demonstrate the relationship among the sides in a right triangle.

How could you apply the Pythagorean Theorem to another mathematical topic? Give one example for each subject below and share it with your group. List each person's examples and share the list with the other groups.

1. algebra
2. geometry
3. statistics
4. trigonometry

Exploring Content

1. How could you use the Pythagorean Theorem to find out if a 7" pencil can be placed inside a 6" x 4" box?
2. How might the Pythagorean Theorem be applied to other high school subjects or to situations in real life?

Exploring Teaching Issues

1. Discuss other high school or middle school mathematical topics that could be applied to other areas of mathematics.

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Exploration Answers

Activity
Examples could include the following topics.

1. Algebra: solving literal and quadratic equations dealing with lengths and right triangles; coordinate plane relationships such as distance formula, equations of circles, and other relations that are based on right triangles; equations of spirals; the length of vectors.
2. Geometry: geometric mean, indirect measures, patterning in Fibonacci numbers with spirals, angle relationships in right triangles, similarity, relationships in polygons.
3. Statistics: line and curve fitting.
4. Trigonometry: indirect measures, angle measures, development of trigonometric relationships, law of sines and cosines, polar coordinates and graphs, complex numbers.

Exploring Content

1. You could use the Pythagorean Theorem to find the diagonal of the box. The diagonal of the box is greater than the length of the pencil, so the pencil will fit diagonally.
2. Possible answers are: finding lengths in various contexts; finding lengths by indirect measures in fields such as physics, biology, and astronomy; architectural or structural design and construction; length, distance, area, and volume situations in other fields such as aviation, navigation, engineering, surveying, and art.

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Topics for Discussion

Context

• How does a context help students establish, analyze, and interpret mathematical ideas?
• Describe how some of the lessons using real–life problems introduced the concepts of "messy " numbers (such as results that are nonintegral or are nonterminating decimals) and complex relationships between variables. What do you need to consider when planning lessons involving these two concepts?
• How might the historical perspective of Thales help students understand the importance of the difference between conjecture and proof?

Methods

• Cite examples in which modeling and multiple representations enhanced student understanding.
• Explain how learning multiple problem–solving approaches accommodates various learning styles.
• Explain how the calculator helps students make connections within mathematics. Give examples from the video and your own classroom.
• Describe how finding multiple solutions can deepen students' understanding of mathematics.

Curriculum

• What are the essential characteristics of an integrated mathematics curriculum?
• Consider how your current curriculum does or does not make connections within mathematics, to other disciplines, and to the real world. What changes could you make to increase students' awareness of these types of connections?
• How might your curriculum be affected by regular use of graphing calculators to make connections within mathematics?

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