### “Mathematics & Navigation” Lesson Plans
Developed by Mathematics and Navigation summer teacher institute participants
in fulfillment of course taught at Michigan Technological University June 28-July 2, 2010
with funding from the Great Lakes Maritime Research Institute and the University of Wisconsin-Madison CFIRE.
MIDDLE SCHOOL
6th grade
**Lesson Overview**
Rates have many real life applications, such as miles per hour, miles per gallon, cost per Lesson of weight, and calories per serving. Students in sixth grade are expected to solve applied problems involving rates such as those listed above. In this activity, students plan a boat trip from Leland to North Manitou Island. Their goal is to compute the travel time. They utilize charts to measure the distance in miles, they are given miles per hour.
6th grade
**Lesson Overview**
With sandy beaches, waterfowl and raptors aplenty, and sand cliffs, kayaking is a natural way to explore North Manitou Island. In this lesson, students have arrived on North Manitou Island (see lesson #1). They are embarking on a trip around the Island via kayak. To get them ready for the adventure, I’ll bring in my sea kayak loaded as if we’re traveling for several days (food, clothing, safety equipment, camping gear). Students will have the opportunity to see what’s involved in the planning process. (My husband and I have circumnavigated the island this way, so I’ll also share my photos of the trip.) Students will calculate distance around the island, determine the speed they will travel (they can use the internet to figure out about how fast they can expect to travel in knots), and finally determine the amount of time it will take to complete their journey. They ought to consider time of year, prevailing winds, camping regulations, and weather for August when charting their circumnavigation (which way will they travel?).
Pre-Algebra, 7th and 8th grade
**Lesson Overview**
Students will determine the amount of time it will take to traverse a given course by using given speeds and distances determined through use of maps.
6th grade
**Lesson Overview**
This lesson will be used after completing lesson 3-6 (Pre-Transition Mathematics) on Angle Measure and Addition in order to show a real life use of a 360 protractor/compass rose.
Students will learn about and practice using a 360-degree protractor. This is important in that a good understanding of these angles makes navigation much easier. Finally students will apply their skills using the compass rose on a navigation chart to draw a line showing the intended course from one location to another.
6th grade
**Lesson Overview**
This lesson will be used following lesson 7-1 “The Rate Model for Division” (Pre-Transition Mathematics) in order to show real life application of concept involving distance/rate/time.
In this lesson, students will use the formula “distance = rate x time” in a variety of activities. The lesson will culminate with the students using the navigation chart (from the previous lesson plan) to determine how long it will take to navigate a boat from point A to point B.
HIGH SCHOOL
Geometry
Lesson Overview
The goal of this lesson is to teach students how to use landmarks to find the visual fix of a location. This lesson connects to many concepts in the current Geometry curriculum, including measurement, parallel and perpendicular lines, Pythagorean Theorem and unit conversion.
Geometry
Lesson Overview
The goal of this lesson is to teach students how to plot a course. This lesson connects to many concepts in the current Geometry curriculum, including measurement, the distance formula and unit conversion.
- Relating boat position, bearing, and heading to polar coordinates by Melanie Harmala
** **Precalculus
Lesson Overview
Students will use their knowledge of a variety of topics to perform several activities related to marine navigation. They will measure distances and angles, determine latitude and longitude of given points, use trigonometry and the Pythagorean Theorem to calculate angles and distances, and relate a marine navigation course to polar coordinates. (I would include this lesson soon after students are comfortable working with polar coordinates.)
- Relating boat position, bearing, and heading to polar vectors by Melanie Harmala
** **Precalculus
Lesson Overview
Students will use their knowledge of a variety of topics to perform several activities related to marine navigation. They will measure distances and angles, determine latitude and longitude of given points, use trigonometry and the Pythagorean Theorem to calculate angles and distances, and relate a marine navigation course to vectors. (I would include this lesson soon after students are comfortable working with vectors.)
- How Far Will We Go and When Will We Get There? by Cathy Hill
** **Geometry
Lesson Overview
The overall goal of this lesson is to calculate the distance and travel time between two points in a navigational setting while incorporating mapping skills, chart reading, construction skills and conversions.
The Keweenaw Waterway runs for almost 25 miles from its southern end at the mouth of the Portage River to its northern end at the Upper Entry in Michigan’s Upper Peninsula. This navigation channel is rich in history from the copper boom of the late 1800’s. Many large ships have traveled through this narrow canal. Navigation through this canal is hazardous due to its many turns and narrowness. Students will begin the first steps of learning how to measure distance in degrees, minutes, seconds on a nautical chart. Then they will convert these distances into nautical miles and then find the time it would take to travel this distance given a constant rate.
- Traveling from the Portage Lake Lift Bridge to the Lower Entry Light by Cathy Hill
** **Geometry
Lesson Overview
This lesson has been designed to calculate the distance from the Portage Lake Lift Bridge to the Lower Entry Light at White City. This lesson will be used to determine the time it takes to travel between these two points using three different methods.
** **Advanced Algebra
Lesson Overview
This lesson is being used primarily as an introduction to GPS and some of its uses and as a graphing exercise. I will conduct this lesson within the first two weeks of the year as a refresher/review exercise in graphing, and as an introduction to a new technology. Since some of the exercise will be conducted out of the classroom, early in the school year works best for weather conditions also.
** **Geometry
Lesson Overview
All students are expected to take Geometry as part of the new Michigan Merit Curriculum. Many concepts required to master basic ideas in geometry are very abstract. Ideas such as lines continue forever, and angles are measured as the number of degrees between two rays are difficult for low ability students to assimilate or make sense of. Maps by definition take relative locations of objects and orient them on a two dimensional plane. The lessons in this Lesson build concepts based on practce with very concrete examples that move to from hands-on to an outdoor activity to a mapping activity using nautical charts.
- Using Laws of Sine and Cosine to Solve Navigation Problems by Susan Romska
** **Geometry
Lesson Overview
Review the Laws of Sine and the Laws of Cosine from section 8.5 in Holt Geometry, use these to solve navigation problems. Students will use their knowledge to solve problems in small groups and also have to make presentations to class.
- Using Vectors in Navigation by Susan Romska
** **Geometry
Lesson Overview
Students will use previous lessons and knowledge on vectors to solve navigation problems related to finding magnitude (speed traveled) and direction (bearing).
Lesson Overview
Students will determine distances (in nautical miles) between two points listed in latitude and longitude in both cardinal directions (N/S, E/W) and straight line (“as the crow flies”) distances. The overall purpose of this lesson is to provide students with an opportunity to apply the Pythagorean theorem in a real world situation as well as introduce students to various types of units in use around the world.
Geometry
Lesson Overview
This is day one of a 2 day lesson for a Freshman Geometry class. In this lesson, students will be introduced to latitude and longitude in D m s and D mm.mm and learn to convert between the two forms. They will be introduced to the definition of a nautical mile and how it relates to latitude. The students will take 2 points located on a photocopy of a local map of a nearby location, estimate their position in latitude and longitude in D m s and D mm.mm using a rolling ruler and divider then find out how far the 2 points are from each other using the changes in latitude and longitude.
Geometry
Lesson Overview
This will be a continuation of the lesson for a Freshman Geometry class from the previous day that focused on introducing latitude, longitude and finding location on a map provided. The activity today will reinforce the skills from day 1, but will add 2 different tools. The students will use the website www.USGS.gov and a handheld GPS to reinforce the topics and introduce error estimations. |