Two sixty-minute class periods
Why do sailboats have triangular sails? Students will learn about sailboat design and how the shape of sails affects their movement. Students will apply what they learn about sailboat design to their math lesson. This lesson introduces the basic postulate of right triangle trigonometry, the Pythagorean Theorem, and is a hands-on way to show students that the Pythagorean Theorem can be tested and proved. They will have an opportunity to actually explore and prove the Pythagorean Theorem and better understand and remember the theorem. Students will have a better grasp of how to take accurate measurements and read a standard ruler. They will also exhibit some creativity in designing/decorating their sailboats.
recognize reasoning and proof as fundamental aspects of mathematics
make and investigate mathematical conjectures
compute fluently and make reasonable estimates
represent and analyze mathematical situations and structures using algebraic symbols
analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
specify locations and describe spatial relationships using coordinate geometry and other representational systems
understand measurable attributes of objects and the units, systems, and processes of measurement
apply appropriate techniques, tools, and formulas to determine measurements
build new mathematical knowledge through problem solving
solve problems that arise in mathematics and in other contexts
apply and adapt a variety of appropriate strategies to solve problems
communicate their mathematical thinking coherently and clearly to peers, teachers, and others
recognize and use connections among mathematical ideas
understand how mathematical ideas interconnect and build on one another to produce a coherent whole
recognize and apply mathematics in contexts outside of mathematics
Common Core Standards
Anchors for Reading:
Key Ideas and Details:
CCSS.ELA-LITERACY.CCRA.R.1 Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text.
CCSS.ELA-LITERACY.CCRA.R.2 Determine central ideas or themes of a text and analyze their development; summarize the key supporting details and ideas.
Craft and Structure:
CCSS.ELA-LITERACY.CCRA.R.4 Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone.
Integration of Knowledge and Ideas:
CCSS.ELA-LITERACY.CCRA.R.7 Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words.
CCSS.ELA-LITERACY.CCRA.R.8 Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance and sufficiency of the evidence.
Range of Reading and Level of Text Complexity:
CCSS.ELA-LITERACY.CCRA.R.10 Read and comprehend complex literary and informational texts independently and proficiently.
Anchor standards for Speaking and Listening:
Comprehension and Collaboration:
CCSS.ELA-LITERACY.CCRA.SL.1 Prepare for and participate effectively in a range of conversations and collaborations with diverse partners, building on others' ideas and expressing their own clearly and persuasively.
CCSS.ELA-LITERACY.CCRA.SL.2 Integrate and evaluate information presented in diverse media and formats, including visually, quantitatively, and orally.
CCSS.ELA-LITERACY.CCRA.SL.3 Evaluate a speaker's point of view, reasoning, and use of evidence and rhetoric.
Presentation of Knowledge and Ideas:
CCSS.ELA-LITERACY.CCRA.SL.4 Present information, findings, and supporting evidence such that listeners can follow the line of reasoning and the organization, development, and style are appropriate to task, purpose, and audience.
CCSS.ELA-LITERACY.CCRA.SL.6 Adapt speech to a variety of contexts and communicative tasks, demonstrating command of formal English when indicated or appropriate.
Anchor standards for Language:
Conventions of Standard English:
CCSS.ELA-LITERACY.CCRA.L.1 Demonstrate command of the conventions of standard English grammar and usage when writing or speaking.
Vocabulary Acquisition and Use:
CCSS.ELA-LITERACY.CCRA.L.4 Determine or clarify the meaning of unknown and multiple-meaning words and phrases by using context clues, analyzing meaningful word parts, and consulting general and specialized reference materials, as appropriate.
CCSS.ELA-LITERACY.CCRA.L.6 Acquire and use accurately a range of general academic and domain-specific words and phrases sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when encountering an unknown term important to comprehension or expression.
- investigate the design of sailboat sails
- take accurate measurements using a ruler
- identify right triangles based on given measurements of sides
- find the missing side of right triangle
- solve word problems involving right triangles
- draw diagrams to show a visual representation of a written problem
- make connections to sailboat design
- make connections between the theorem and real world situations
- colored Pencils
- push pins/thumb tacks or magnets
Right triangle–a polygon with three vertices and three straight line segment sides (a triangle) that has one 90-degree angle.
Hypotenuse–the side opposite the 90 degree angle and the longest side of a right triangle.
Leg–a side of the right triangle that is not the hypotenuse.
Pythagorean Theorem–a2 + b2 = c2 or the sum of the squares of two sides of a right triangle are equal to the square of the hypotenuse.
Begin a class discussion and ask the students if they have ever seen a sailboat. Assuming there are some “yes” responses in the group, ask the students what they noticed about the design of the sailboat. Hopefully, someone will say their sails are triangular. Ask students why they think the sails have this shape. Explain that the sails are designed in a way that allows the boat to take advantage of winds at 90 degree angles by way of “tacking.” The sail design enables the boat to move in previously inconceivable ways. Pass out a copy of “How does a Sailboat move upwind?” http://www.physlink.com/Education/AskExperts/ae438.cfm
to each student and hold a class discussion about how the sail design works. Write key points on the board.
• Define a right triangle and identify the legs and the hypotenuse of the triangle.
• Provide the formula for the Pythagorean Theorem, a2 + b2 = c2, and identify a and b as the legs and c as the hypotenuse.
• Distribute the Pythagorean Theorem Handout and walk the students through each of the four examples.
• Give the students about ten-fifteen minutes to work on the Student Practice problems.
• Invite a few students to put their answers to the problems on the board and explain their work to the class.
• Ask the class if they agree with the work on the board and if they solved the problems in a similar fashion
• Answer any remaining questions about the problems.
• Tell the students that they will be using the information they learned today to design sailboat sails in the next lesson.
Introduce the lesson as a follow-up to yesterday.
"Today we will do an 'experiment' to see if the Pythagorean Theorem really works. We have learned about sailboat design and learned how to calculate the missing side of a right triangle, but how can we be sure this is true?"
• Distribute Sailboat Handout, rulers, and two post-its to each student.
• Instruct each student to measure 2 sides of each of the triangular sails on the handout and record their answers on their post-its.
• Students calculate the measure of the third side of the triangle using the Pythagorean Theorem.
• Students then measure the third side of the triangle and compare their answer to the one they got in the previous step.
• Students record and summarize what they notice.
• Students decorate their sailboats as they choose and write a brief summary on why their sailboat sail design works.
Depending on your classroom, allow students to display their work for a “gallery walk” where they are able to see what other students created.
Check the measurements and subsequent calculations, paying special attention to the substitution in the formula for a, b, and c. I would also pay special attention to the summary portion of the activity. Is the sailboat decorated creatively? In the summary of how the sail on a sailboat works, did the student grasp why the shape and design of the sail help the boat move across distances?
Enrichment Extension Activities
The distance formula is basically the Pythagorean Theorem reorganized. You will need a sheet of graph paper, a transparency sheet of graph paper, and a map (preferably of your town/area with some landmarks). Students place the transparency over the map and plot the locations of two places (they need to be on corners). Students are instructed to draw a right triangle, drawing a vertical and horizontal line to complete this task. They then use the Pythagorean Theorem to find the distance between these points or the hypotenuse of the triangle. Introduce the distance formula and show how its pieces are derived from the Pythagorean Theorem. This could also be used with any map activity.
The students were very successful and seemed to enjoy the activity. When I do it again, I will be sure to spend a few minutes with the whole class reading the ruler accurately. Students did not see the purpose of the activity at first, but it seemed to make more sense to them once they were complete as evidenced by their responses/observations. It may have been helpful to have a variety of “sailboats” such that there were different "answers" in the class.