As the Crow Flies Name:___________________dnwestonstudentteachingportfolio.weebly.com/.../115682… · Web viewNote that the theorem and its converse have been proven in numerous

—Launch, Explore, Summarize Lesson Plan 1—“Pythagorean Theorem”

o Description of the Lesson

Subject: Applying the Pythagorean Theorem and Pythagorean Triples Grade: 8th Timing (How long will the lesson take?): 60 minutes Reference for the lesson: http://illuminations.nctm.org/LessonDetail.aspx?ID=L683

2. Learning Goals and Objectives

What is the essential content of the instruction? (What are the big mathematical ideas of the investigation?)

o Students explore real world objects using the Pythagorean Theorem. Using measurement, students will apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

CCSS Standard (What do I want kids to know when this investigation is finished?)8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

What are my specific and measurable objectives for the lesson?

o Measure and determine the lengths of right triangles.o Use the Pythagorean Theorem to determine unknown side lengths of any given

right triangle.

Consider student background, knowledge and experience. What prior knowledge do the students have concerning this topic?

Students will be able to describe a relationship between the lengths of legs of right triangles and the hypotenuse of right triangles.

What mathematical vocabulary does this investigation bring out?

Pythagorean Theorem- a²+b²=c²Legs- sides that are adjacent to the right angleHypotenuse- the longest side of a right trianglePythagorean Triples- consists of three positive integers a, b, and c, such that a2 + b2 = c2

Right Triangle-a triangle with one 90 degree angle

http://en.wikipedia.org/wiki/Integer

http://illuminations.nctm.org/LessonDetail.aspx?ID=L683

What misconceptions might arise?

o Students don't identify the variables as different legs on a right triangle (c is hypotenuse, a & b are legs). There is no connection made between the formula and the definition of Pythagorean Theorem.

o Students generalize that the Pythagorean Theorem applies to all triangles. (definition of the Pythagorean Theorem includes the conditional statement that you have a right triangle is restricted to Euclidean Geometry)

Curriculum adaptations and instructional modifications: (modifications to meet the needs of diverse populations and students with various ability levels)

For students with visual impairment: Allow the use of rulers with larger centimeter marks.For students with learning disabilities in writing: Have another student record information and/or take notes.

3. Resources, Materials, and Preparation for Instruction (What facilities, resources, and tools will be needed and how are they to be used?)

Collaborators (Other teachers, outside visitors, or technical staff): n/a Technological tools for the teacher (demonstration):n/a (look for SmartBoard

adaptation) Technological tools for the student: n/a Manipulatives: ruler/yardstick, calculator, instruction sheet, recording sheet Handouts and recording sheets (see attached)

4. Instructional Method and Procedures

a. Launch (5 minutes)Teacher

Considerations(Before)

Description of Learning Activities

Anticipated Student Responses

Teacher Guidance(During)

Engage prior knowledge by having students discuss what characteristics they know about right triangles and objects that they see in everyday life that are or contain right triangles.

Address misconceptions, but don’t spend too much time correcting definitions until end of lesson.

At end of lesson bring into perspective how students might see this on a standardized test.

Begin by asking students what they know about right triangles and what types of objects they see on an everyday bases that are right triangles.

Ask students to think about the following question: Does the Pythagorean Theorem only be applied to right triangles?What makes a triangle Pythagorean Triple?

Hand out and read aloud the instruction sheet. Provide each student with a recording sheet and divide students into five different groups.

One angle has to be 90 degrees.

Two angles can be the same degree or all three different degrees.

Half a sandwich, bookshelf, TV, window, smartboard, etc.

Yes/No

Triangle with lengths 3,4,5

I don’t know

Triangle whose lengths of the two sides legs squared add up to be the length of the hypotenuse squared

Draw right triangle.

Record student responses of different characteristics of right triangle on board.

Call on different students about different objects that may contain right triangles.

b. Explore (35-45 minutes)

This is where students work individually or in small groups to solve the problem. This is their chance “to get messy with the math.”

Teacher Considerations

(Before)




Students will work in four different groups

Materials:Instruction sheet; Recording sheet; ruler/yardstick; calculator

For this exploration, students will measure different objects around the classroom to determine unknown lengths of right triangles. Based on the handout, students will measure each length of the given object and use the Pythagorean Theorem to determine the unknown length. Students will rotate through five different stations around the classrooms featuring four different objects. For the first four stations, once the student has measured the given lengths, students will use the Pythagorean Theorem to determine the unknown side length and record their measurements and findings on the

Correct solutions rounded to the nearest inch (measurements will not be exact)Station 1-Desk Tabletop:Given lengths:Hypotenuse- 16 inchesHeight- 18 inchesUnknown length:Width- 24 inchesStation 2-Smartboard Screen:Given lengths:Width-65inchesHeight-50 inchesUnknown length:Hypotenuse- 82 inchesStation 3-Locker Door:Given lengths:Width-12 inchesHypotenuse-19 inchesUnknown length:Height- 15 inchesStation 4-TV Screen:Given lengths:Hypotenuse- 26 inchesHeight-13 inchesUnknown length:Width- 23 inchesStation 5-Stop SignUse the Pythagorean Theorem to determine the side lengths of a triangle in a stop sign. Is it possible? No

Questions to promote student thinking (note – not necessary to ask all students all questions): What kind of

patterns did you notice when measuring each triangle?

Were all of your measurements whole numbers or were there some decimals? Did you estimate? How will this affect your findings?

How could you determine if a triangle was a Pythagorean Triple given two lengths of measure without having to measure the third side length?

What kind of questions or comments did you come upon while participating in this activity?

Which side of a right triangle must be the longest? Why?

recording sheet along with any comments they may have after completing the station. For the fifth station, students will need to determine if a right triangle can be deciphered from the object and if the Pythagorean Theorem can be used. Students will need to identify which lengths were given on their recording sheet, making sure to label each measurement with the appropriate name of length of measurement used.

c. Summarize (15-20 minutes)The purpose of the Summarize section is to bring groups back together and have students explain their solutions while assessing how students are progressing towards the goals of the lesson. The teacher’s role is to guide students to the big ideas, to make sure that they have nailed the mathematics. Use the discussion to help you determine whether additional teaching and/or additional exploration by students is needed before they go on to the next lessons.


(Before)




Build from student answers to questions during explore to organize whole class discussion.

Focus on patterns ad student responses to draw out generalizations about which triangles are Pythagorean Triples.

Students might need more practice to determine lengths of right triangles using the Pythagorean Theorem.

Start by asking question asked during launch: Does the Pythagorean Theorem only be applied to right triangles?What makes a triangle Pythagorean Triple?

What connections did you find between the Pythagorean Theorem and different objects that you measured?

What generalizations did you make?

Yes

Triangle whose lengths of the two sides legs squared add up to be the length of the hypotenuse squared.

The Pythagorean Theorem can only be used when working with right triangles.

The Pythagorean Theorem can only be used when working with right triangles.

Not all side lengths of right triangles are Pythagorean Triples.

Why do you think this is the case?

Can you provide me some examples of lengths for Pythagorean Triples?

What did you learn about the Pythagorean Theorem and right triangles from doing this activity?

What is special about a right triangle that allows for the Pythagorean Theorem to be applied?

If students are not able to make conjectures about Pythagorean Triple, provide some more examples with all three lengths of sides given and have students use the Pythagorean Theorem

Right triangles are present in many objects that we see on an everyday basis.

to determine if lengths are a Pythagorean Triple.

Push students to be observant when they leave class to see if they can determine objects where the Pythagorean Theorem could be used to determine lengths of right triangles.

5. Assessment of Student LearningPlan for assessment (both formative and summative) to assess student learning. Some guiding questions to consider:

What questions are appropriate for my students to do after the investigation? What are the goals of the homework/class work assignment? How will students be supported in completing the assignment? Do I provide

information and support for students and parents?

As an assessment for this activity, I would have students do a small reflection at the end of the activity. This reflection should contain what students learned about the Pythagorean Theorem, Pythagorean Triples, and how prevalent the Pythagorean Theorem is in everyday life. This reflection will provide students the opportunity to provide questions and/or concerns that they still have about the Pythagorean Theorem. Students will turn this reflection in at the end of the class.

Student Instructional Sheet

Station 1

At your station, you have been given directions to take the following two measurements of the

DESK TABLETOP

1. Hypotenuse

2. Height

Your group’s goal is to find the unknown side length by using the application of the

Pythagorean Theorem. You have until I blow the whistle (10 min) to determine the unknown

side length. If you need help, you may ask other groups for advice or raise your hand and I

will come help you. Be sure to label using the appropriate measurements! Round your

measurements to the nearest inch.

Station 2


SMARTBOARD SCREEN

1. Width

2. Height






Station 3


LOCKER DOOR

1. Width

2. Hypotenuse




will come help you. Be sure to label using the appropriate measurements!

Round your measurements to the nearest inch.

Station 4


TV SCREEN

1. Hypotenuse

2. Height






Station 5

STOP SIGN

Do right triangles exist in a stop sign? If so, measure the lengths of the 2 legs and use the

Pythagorean Theorem be used to determine the length of the hypotenuse. Tell which sides you

measured and which side you used the Pythagorean Theorem to determine the length. If not, tell

me why the Pythagorean Theorem can’t be used. What connections have you made between the

Pythagorean Theorem and right triangles? Round your measurements to the nearest inch.

Student Recording Sheet

Station 1-Desk Tabletop

Hypotenuse: Height: Unknown side length:

Show work here:

Station 2- Smartboard Screen

Width: Height: Unknown side length:

Show work here:

Station 3-Locker Door

Width: Hypotenuse: Unknown side length:

Show work here:

Station 4- TV Screen

Hypotenuse: Height: Unknown side length:

Show work here:

Station 5-Stop Sign

Unknown side length 1: Unknown side length 2: Unknown side length 3:

Show work here or comments:



Subject: Proving the Pythagorean Theorem Grade: 8th Timing (How long will the lesson take?): 60 minutes References for this lesson:

o Muschla, J. A., Muschla, G. R., & Muschla, E. (2012).Teaching the common core math standards with hands-on activities. (1 ed., pp. 228-229). San Francisco: Jossey-Bass.



o Students will explore the National Library of Virtual Manipulatives. Usingthe website http://nlvm.usu.edu/en/nav/vlibrary/html, students will then virtually prove the Pythagorean Theorem.

CCSS Standard (What do I want kids to know when this investigation is finished?)8.G.6 Explain a proof of the Pythagorean Theorem and its converse.


o Prove the Pythagorean Theorem using deductive arguments.o Students will be able to sketch their findings on graph paper.


Students will be able to explain the relationship between the different side lengths of a triangle and why the Pythagorean Theorem can be applied.


Pythagorean Theorem- a²+b²=c²Geometric Proof- an argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the thing you’re trying to proveCongruence- having identical shapes so that all parts correspond

http://nlvm.usu.edu/en/nav/vlibrary/html

What misconceptions might arise?o The slogan “a squared plus b squared equals c squared” is an incomplete

statement of the Pythagorean theorem because there is no reference to a right triangle nor identification of the meaning of the variables

o Only one proof correctly proves the Pythagorean Theorem


For students with visual impairment: A computer screen and keyboard that accommodates the child’s visual need needs to be available for use.For students with learning disability in writing: Allow student to work with another student who can sketch the figures for both students.

3. Resources, Materials, and Preparation for Instruction (What facilities, resources, and tools will be needed and how are they to be used?) Collaborators (Other teachers, outside visitors, or technical staff): Computer

technician (if necessary) Technological tools for the teacher (demonstration): Smartboard and Internet access Technological tools for the student: Computer and Internet Access Manipulatives: graph paper Handouts and recording sheets: n/a

4. Instructional Method and Proceduresa. Launch (5 minutes)


(Before)




Engage prior knowledge by having students explain what they already know or have already learned about the Pythagorean Theorem.

Address misconceptions, but don’t spend

Begin by asking students what they have previously learned about the Pythagorean Theorem.

Ask about special conditions needed in order for the Pythagorean to work.

Ask students to make

a²+b²=c²

used for right triangles

lengths can be Pythagorean Triples

Yes/No

Draw obtuse, right, and acute triangles on the board for students to identify which triangle the Pythagorean Theorem can be used for

Ask students what some of the special conditions necessary in order for the Pythagorean Theorem to be used

too much time correcting definitions until end of lesson.

At end of lesson bring into perspective how students might see this on standardized test.

Explain to students that good behavior is necessary in the computer lab if privilege of using computers in class does not want to be lost.

a conjecture about the following question:Can the Pythagorean Theorem be used using shapes that are not congruent?

Take students to computer lab where each student will have their own computer to work on.

Engage thought process about the question asked.

b. Explore (35-45 minutes)This is where students work individually or in small groups to solve the problem. This is their chance “to get messy with the math.”


(Before)

Description of Learning Activities Anticipated Student

Responses


Students will work individually on their own computer for the activity

Materials: Computer and internet access; graph paper; pencil

For this exploration, students will use the website of the National Library of Virtual Manipulatives to prove the Pythagorean Theorem by completeing two puzzles. Each puzzle requires students to fill two congruent regions by rotating and translating congruent triangles and squares. Students will go to the website http://nlvm.usu.edu/en/nav/vlibrary.html. Students should click on “Geometry”, grades 6-8, scroll down to the Pythagorean Theorem and click the link. The first puzzle should appear on the screen. Directions for moving and rotating the figure are included on the right-hand side of the screen. Students

Possible answer for Puzzle 1: The white region on the left has an area of c². The white region on the right has an area of a²+b². As each white region is filled with the same figure, a²+b²=c².

Possible answer for

Questions to promote student thinking (note – not necessary to ask all students all questions): Why does

the white region on the left in Puzzle 1 have an area of c²?

Why does

http://nlvm.usu.edu/en/nav/vlibrary.html

will then be instructed to complete the first puzzle by filling each white region the congruent triangles and squares that are below each area. Completing the puzzle correctly illustrates the Pythagorean Theorem. After completing the first puzzle, students should sketch the positions of the triangles and square that they placed in each white region. Students should also write an explanation of how their sketch illustrates the Pythagorean Theorem. After students have finished the first puzzle, they should click on the link for Puzzle 2 and follow the same procedure as they did for Puzzle 1. Note that the figures students will be using to fill the white region vary from the figures in Puzzle 1.

Puzzle 2: The two white regions are the same. The white region on the left is filled with four congruent triangles and a light green square whose area is c². The white region on the right is filled with the same four congruent triangles and a light blue square whose area is b² and a royal blue square whose area is a². If the four congruent triangles are removed from each white region, a²+b²=c².

the white region on the right in Puzzle 1 have an area of a²+b²?

Why is it necessary to translate and rotate the figures in the puzzles?

Would this activity work if you had to translate and rotate triangles and squares that were not congruent? Why or why not?



(Before)





Students might ask to see another example of a proof of the Pythagorean Theorem.

Start by asking question asked during launch:Can the Pythagorean Theorem be used using shapes that are not congruent?

What did you notice about each puzzle as you were completing them? What were some similarities and differences?

How does your sketch prove the Pythagorean Theorem?

Given the activity and the sketches, no.

Both puzzles asked to use the same shapes.

Each puzzle could be solved in multiple ways.

Both puzzles represented the Pythagorean Theorem.

The white region on the left and the white region on the right represents a²+b²=c².

Why do you think that is the case?

Students might ask if other shapes besides squares and triangles can be used to virtually prove the Pythagorean Theorem. Either ask class for an explanation or explain it yourself.

If not evident by the completion of this interactive activity, ask students to use Tangrams to use and prove the Pythagorean Theorem.


i. What questions are appropriate for my students to do after the investigation?ii. What are the goals of the homework/class work assignment?

iii. How will students be supported in completing the assignment? Do I provide information and support for students and parents?

As an assessment for this activity, I would have students turn in their sketches of the puzzles as well as their explanations for why each puzzle represents the Pythagorean Theorem. For homework, I would ask students to consider and research some other possible proofs for proving the Pythagorean Theorem to bring to the next class. This will help students with the next lesson and next activity about the converse of the Pythagorean Theorem.



Subject: Explaining the converse of the Pythagorean Theorem Grade: 8th Timing (How long will the lesson take?): 60 minutes References for this lesson:

o Muschla, J. A., Muschla, G. R., & Muschla, E. (2012).Teaching the common core math standards with hands-on activities. (1 ed., pp. 227-228). San Francisco: Jossey-Bass.



o Students will use their research on other possible proofs for the Pythagorean Theorem to find the converses of each proof to be able to explain both the proof and its converse.

CCSS Standard (What do I want kids to know when this investigation is finished?)8.G.6 Explain a proof of the Pythagorean Theorem and its converse.


o Prove the Pythagorean Theorem and its converse using deductive arguments.o Be able to explain proofs and its converse of Pythagorean Theorem


Students can explain the relationship between the different side lengths of a triangle and why the Pythagorean Theorem can be applied.


Pythagorean Theorem- a²+b²=c²Geometric Proof- an argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the thing you’re trying to proveConverse- an if-then statement in which the hypothesis and the conclusion are switched

What misconceptions might arise?o The converse of a statement can sometimes be true.


For students with speech impairment: Student can write their research and explanations rather than explain it to a partner. For students with visual impairment: A computer screen and keyboard that accommodates the child’s visual need needs to be available for use.For students with learning disability in writing: Allow partner to write down notes and explanations for student.

3. Resources, Materials, and Preparation for Instruction (What facilities, resources, and tools will be needed and how are they to be used?) Collaborators (Other teachers, outside visitors, or technical staff): Computer

Technician (if necessary) Technological tools for the teacher (demonstration): Smartboard access, Internet

access Technological tools for the student: Computer and Internet Access Manipulatives: notebook paper, pencil; Optional: graph paper, rulers, protractors,

scissors, math reference books Handouts and recording sheets: n/a



(Before)




Engage prior knowledge by having students discuss characteristics of the Pythagorean Theorem in general as well as proving the theorem.

Address misconceptions, but don’t spend too much time correcting

Begin by asking students what it means to be a converse of a geometric proof.

Ask students to make a conjecture about the following question:Is there only one

The shoe

Inverse of the proof

Opposite of what the proof says

Switching the hypothesis and conclusion

Yes/No

Provide examples of a proof and its converse

Engage thought process about the question asked

definitions until end of lesson.

At end of the lesson bring into perspective how students might see this on standardized test.

Make sure students use a verified Pythagorean Theorem proof and its converse.

proof and converse that can be used to correctly prove the Pythagorean Theorem?

Take students to computer lab where each student will have their own computer to work on.

b. Explore (35-45 minutes) This is where students work individually or in small groups to solve the problem. This is their chance “to get messy with the math.” Teacher

Considerations

(Before)




Students can work in pairs

Materials:Computer and Internet access; notebook paper; pencil

Optional Materials: graph paper, rulers, protractors, scissors, math reference

Begin by explaining to students that they will be working in pairs to explain proofs of the Pythagorean Theorem and its converse. Discuss the Pythagorean Theorem and its converse with your students. Note that the theorem and its converse have been proven in numerous ways by different mathematicians. Many of these proofs are available online as well as in math texts or math reference books. Suggest that students consult math reference books (if available) or do a Web search for “Proofs of the Pythagorean Theorem”. They may also visit the following Web Sites:

Student responses will vary depending on the proof that they chose to investigate. Some possible proofs that students might have researched and explained are:

-Euclid’s two proofs-proof by Quang Tuan Bai-Parallelogram proof by John Molokach-Bhaskara's Second Proof of

Questions to promote student thinking (note – not necessary to ask all students all questions): What are some

conditions that were stated in the proof that you researched?

Were there any proofs and its converse that did not prove the Pythagorean Theorem?

books o http://www.mathis fun.com/Pythagoras.html

o www.cut-the-knot.org/pythagoras/index.shtml

o http://www.davis-inc.com/pythagor/index.shtml

Instruct the students to select a proof of the theorem and its converse or use the proof that they found the night before and find its converse. They are to explain this proof and its converse to their partner. Partners should help each other clarify their explanations by asking questions or providing suggestions. Encourage the students to use graph paper, rulers, protractors, and scissors if their proof requires sketches or drawings. After students have discussed their proofs and converses with their partners, each student is to write an explanation of his or her proof and its converse.

the Pythagorean Theorem

How could you determine the converse of a Pythagorean Theorem Proof using only steps and figures and not use an explanation?

What is your conjecture?



(Before)





Students might

Start by asking question asked during the launch: Is there only one proof and converse that can be used to correctly

Based on the research that we did on Pythagorean Theorem proofs and the converses of those proofs, no

Why is this true?

http://www.davis-inc.com/pythagor/index.shtml



http://www.cut-the-knot.org/pythagoras/index.shtml

http://www.cut-the-knot.org/pythagoras/index.shtml

need to see more proofs and their converses to gain a better understanding

prove the Pythagorean Theorem?

Were all the converses of each Pythagorean Theorem proof consistent and correctly state the appropriate converse?

What did all the Pythagorean Theorems and converses have in common?

Yes.

Each converse correctly stated the converse of the cooperating Pythagorean Theorem proof.

They all said the same thing.

Each proof and its converse correctly proved the Pythagorean Theorem as a²+b²=c².

Some students might disagree with this. If so, have students explain their proof and its converse to the whole class. If the student still doesn’t understand, explain it yourself.

If this is not evident through the activity and student explanation, ask students if they can find a proof and its converse that doesn’t correctly prove the Pythagorean Theorem.




As an assessment for this activity, I would have students turn in their explanations of the proof that they researched. The explanation should include why the proof and its converse is appropriate to use in proving the Pythagorean Theorem. This assessment should show me that the students understand how to derive a converse from its cooperating proof and if they understand the meaning of and how to prove the Pythagorean Theorem.

—Launch, Explore, Summarize Lesson Plan 4—“Pythagorean Theorem”1. Description of the Lesson

Subject: The Distance Formula Grade: 8th Timing (How long will the lesson take?): 60 minutes References for this lesson:

o http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM- Task/Crow.pdf



1. Develop an understanding of the meaning of the formula

CCSS Standard (What do I want kids to know when this investigation is finished?)8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.


1. Students will analyze, reflect, solve, and connect problems based on the Distance Formula

2. Students will develop an understanding of the meaning of the Distance Formula3. Students will be able to use the Distance Formula in appropriate situations4. Students will be able to find the distance between two points in the coordinate

plane

Consider student background, knowledge and experience. What prior knowledge do the students have concerning this topic?Students will be familiar with the Pythagorean Theorem and coordinate systems.


Pythagorean Theorem- a²+b²=c²Cartesian Coordinate Plane- a plane spanned by the x-axis and y-axis inwhich the coordinates of a point are itsdistances from two intersecting perpendicularaxesDistance Formula- used to determine distance, d, between two points

http://www.corestandards.org/Math/Content/8/G/B/8

http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/Crow.pdf

http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/Crow.pdf

Equidistant- distant by equal amounts from two or more placesMidpoint- The point of a line segment that divides it into two parts of the same lengthLeg- sides that are adjacent to the right angleHypotenuse- the longest side of a right triangleSlope- The rate at which an ordinate of a point of a line on a coordinate plane changes; rise over run

What misconceptions might arise?o Students will see that the distance formula and the Pythagorean Theorem are

actually two similar formulas with one common goal, however, the distance formula is used with coordinates


For students with visual impairment: Provide a visual graph with larger coordinates

3. Resources, Materials, and Preparation for Instruction (What facilities, resources, and tools will be needed and how are they to be used?) Collaborators (Other teachers, outside visitors, or technical staff): n/a Technological tools for the teacher (demonstration): n/a Technological tools for the student: n/a Manipulatives: Grid Paper, Student activity sheet Handouts and recording sheets: As the Crow Flies activity sheet: see attached



(Before)




Engage prior knowledge by having students discuss the differences between rural and urban areas and explain to students who may not live in rural areas that streets are usually laid out in a grid format


At end of lesson bring in aspect of how might see on standardized test.

Be sure to not make any reference distance formula at this point

Begin by asking students how far away their house is from the school.

Ask students to make a conjecture about the following question:How can we calculate the distance along the straight-line path from A to B?

Hand out student activity sheet and read over it with students

This answer will vary depending on where each student lives in the area-some might be really close or some might be really far away from the school

Yes/No

Make sure students understand that in this context, a block is a unit of length.

Ask students to try and find the driving distance between their house and the school.

Students will work and teacher will pick a few students to present their strategies to the class

Encourage students to show a “simple” path

b. Explore (35-45 minutes) This is where students work individually or in small groups to solve the problem. This is their chance “to get messy with the math.” Teacher

Considerations

(Before)




Students can work individually or in pairs.

Materials: Grid Paper,

Student activity sheet

Ask the students to find the driving distance from their house to their friend’s house, as described in question 1 on the activity sheet. You could have them work either individually or in groups on this question. Afterward, in a whole-class discussion, gather a range of student ideas about how to find this distance. As different students present their solutions, ask, “Why do you think that will work? Could you draw a diagram that will make this clearer?” Although encouraging a variety of solutions is helpful, you should also be sure to show a “simple” path. Next, ask your students to find the distance that they would travel if they could fly by helicopter—the “straight line” distance often described as distance “as the crow flies,” discussed in question 2 on the activity sheet. As before, students could work either individually or in groups. If necessary, you might ask the students questions such as, “What will the direct path look like on your grid? If we include the path taken when driving, do you see a familiar shape? Do we know a theorem that might be helpful?” In the subsequent whole-class discussion, be sure that students see the usefulness of the Pythagorean theorem in finding the direct path.Once the students

Student responses will vary depending on where they place their points on the coordinate plane-students will label a point for their house, the school, and the movie theatre.

Questions to promote student thinking (note – not necessary to ask all students all questions): “What will the

direct path look like on your grid? If we include the path taken when driving, do you see a familiar shape?

Do we know a theorem that might be helpful?”

What distances did we find to help us calculate the direct distance?

How could we determine the horizontal and vertical distance if we were given only the coordinates?

How would we use these distances to find the length of the direct

see the connection to the Pythagorean theorem, you might ask them to “coordinatize” the situation, as described in questions 3 and 4 on the activity sheet. Again, you might have the students work individually or in pairs on these two items.Following a whole-class discussion of how coordinates might be useful, guide your students in developing a generalization of the situation, which should result in the emergence of the distance formula. Ask questions such as, “What distances did we find to help us calculate the direct distance? How could we determine the horizontal and vertical distance if we were given only the coordinates? How would we use these distances to find the length of the direct path?”You might then ask the students to do an additional example with the ordered pairs (2, –5) and (–3, –7) to be sure that they understand the general pattern. You could then pose the general case of the distance between (x1, y1) and (x2, y2) to lead to the distance formula.

path?”



(Before)




Build from student answers to

Have groups explain a problem to the whole

The distance formula is similar to the

To help them make this connection, you

questions during explore to build class discussion.

To help your students move toward the general formula, you might ask them to compute the distances between other pairs of points without drawing a picture.

Note that students may interpret distance differently since different question refer to distance in different ways

class (how they used the distance formula, connections that they made, etc.)

Start by asking question asked during launch:How can we calculate the distance along the straight-line path from A to B?

Pythagorean Theorem since the horizontal and vertical distances, along with the direct path, form a right triangle.

Use the Pythagorean theorem

The straight-line distance is the length of the line segment joining A and B. This segment is the hypotenuse of a right triangle whose legs lie on a street and an avenue.

might ask:• Can we form a

familiar shape by considering the horizontal and vertical distances?

• How do we know that a right angle is formed?”




As an assessment, students will use the knowledge that they gained from doing the class activity to write a brief and clear explanation of the distance formula and why it works. This will be used to help the teacher better understand if the student grasped the concept of the distance formula and why it is used. Students will be given ten minutes at the end of the class period to ensure that some enriched thought is put towards doing this short assignment. Students will bring their explanations to the next class and we will discuss different student explanations at the beginning of the next class period.

As the Crow Flies Name:___________________

Student Activity Sheet

Suppose that the city in which you live has a system of evenly spaced perpendicular streets, forming square city blocks. On the map below, label your school, house, and your friend’s house on any point on the graph. You will use these points to answer the questions below. Be sure to label which points represent which location.

As the Crow FliesStudent Activity Sheet Continued

1. How many blocks would you have to drive to get from your house to your friend’s house? Draw a path that you would drive, and calculate the distance. Show your work.

2. What if you could use a helicopter to fly straight from your house to your friend’s house? Draw the path that you would take. How could you find the distance “as the crow flies”?

3. Establish a coordinate-axis system, using the school as the origin. What would the coordinates be for your house? For your friend’s house?

As the Crow FliesStudent Activity Sheet Continued

4. How could you use the coordinates to calculate the distance “as the crow flies” from your house to your friend’s house?

5. Suppose that your uncle lives two blocks east and one block south of the school and that you decide to stop by his house on the way home from your friend’s house. Compute the round-trip distance from your house to your friend’s house, to your uncle’s house, and then back to your own house.

a. Draw a picture to show your solution.b. Show how you could find the round-trip distance by using only coordinates.

—Launch, Explore, Summarize Lesson Plan 5—“Pythagorean Theorem”1. Description of the Lesson

Subject: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system

Grade: 8th Timing (How long will the lesson take?): 60 minutes References for this lesson:

o http://www.pdesas.org/module/content/resources/20553/view.ashx



1. Connections between the Pythagorean Theorem and the distance formula will be made

2. Different Methods will be used to determine the unknown measures of a right triangle

3. Recognition of instances in which the distance formula and the Pythagorean Theorem will be made

CCSS Standard (What do I want kids to know when this investigation is finished?)8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.


1. Students will be able to find the shortest distance between two points on a map.

2. Students will be able to relate the distance formula to the Pythagorean Theorem.

3. Students will be able to find the shortest distance between two points on a

coordinate plane.

Consider student background, knowledge and experience. What prior knowledge do the students have concerning this topic?Students will be able to recall the Pythagorean Theorem and the Distance Formula.Students will be able to plot points on a coordinate plane.

http://www.corestandards.org/Math/Content/8/G/B/8

http://www.pdesas.org/module/content/resources/20553/view.ashx


Pythagorean Theorem- a²+b²=c²Cartesian Coordinate Plane- a plane spanned by the x-axis and y-axis inwhich the coordinates of a point are itsdistances from two intersecting perpendicularaxesDistance Formula- used to determine distance, d, between two points Equidistant- distant by equal amounts from two or more placesMidpoint- The point of a line segment that divides it into two parts of the same lengthLeg- sides that are adjacent to the right angleHypotenuse- the longest side of a right triangleSlope- The rate at which an ordinate of a point of a line on a coordinate plane changes; rise over run

What misconceptions might arise?o Students may assume that both formulas are the sameo Students may assume that for any size triangle, both the Pythagorean Theorem

and the Distance Formula may be used instead of just right triangles


For ELL students: It may be difficult for an ELL student to read and understand a map of an American city. Have this student use a map that they might be familiar with-maybe of the city they are from or the use of the type of maps that are used where the student is from.

3. Resources, Materials, and Preparation for Instruction (What facilities, resources, and tools will be needed and how are they to be used?) Collaborators (Other teachers, outside visitors, or technical staff): Technological tools for the teacher (demonstration): Technological tools for the student: Manipulatives: Minneapolis Map, Distance Worksheet. Exit Ticket, rulers Handouts and recording sheets: Minneapolis Map, Distance Worksheet, Exit Ticket


Teacher Considerations(Before)




Engage prior knowledge by asking students what they have already learned about the Pythagorean Theorem and the Distance Formula


At the end of the lesson, bring in perspective of how this might be seen on standardized test.

Begin by asking students what the Pythagorean Theorem is; the Distance Formula

Ask students what they have already studied about the Pythagorean Theorem and Distance Formula

Ask to make a conjecture about the following question?How does the Pythagorean Theorem relate to the Distance Formula?

Hand out Minneapolis Map, Distance Worksheet, and rulers

a²+b²=c²

the square of the length of thehypotenuse of a right triangle is equal to the sum of the squaresof the lengths of the other sides

They are related

Both formulas are used to find the distance between two points

Both are used to find distance

I don’t know

Draw an example of a problem where the Pythagorean Theorem would be usedDraw an example of a problem where the Distance Formula would be used

Have students refer to both formulas to make connections

Have students mentally think about their explanation to the question

b. Explore (35-45 minutes) This is where students work individually or in small groups to solve the problem. This is their chance “to get messy with the math.”


(Before)




Students can work in pairs

Materials: Minneapolis Map, Distance Worksheet, rulers

Each pair of students will be given the Minneapolis Map. Each pair of students should choose an intersection on the map and mark it with a highlighter so they can see it better. Then, each pair should highlight the path they would drive to get from one student’s intersection (point) to the intersection (point) their partner chooses. Have each pair calculate the distance (in blocks) between the two intersections. Next, have each group consider the shortest distance from one point to the other if they were restricted to staying on the road or going around obstacles such as buildings. Have them highlight this route. Ask students: “How can we find the shortest distance between your starting and ending places?” “What shape have you highlighted on your map?” “What theorem do you know that relates the lengths of the sides of a right triangle?” Have students measure the distance between the two points using rulers, and then measure the distance it would take to get to that point by going down the streets. If students measure the sides of the triangle, have them compare and see that for their measurements, a²+b²=c². Demonstrate this to the class on the board. Mark a point on the board (point A), and then go up three inches and mark another point (point B); then go right

For the Minneapolis Map, responses will vary depending on the points that students graph and the routes that students highlight.

For the Distance Worksheet: (round to the nearest tenth)A. 5B. 5.3C. 9.5D. 6.4

Questions to promote student thinking (note – not necessary to ask all students all questions): How can we

find the shortest distance between your starting and ending places?

How does the distance formula relate to the Pythagorean Theorem?

What connections can be made between the types of methods used to determine distance between two points?

How can you determine distance without seeing it on a map?

(forming a 90 degree angle in the process) 4 inches and mark another point (point C). Measure the distance between A and C and show the class that it is 5 inches; note to students that this is a Pythagorean Triple. Show students that for this example, a²+b²=c² or 3²+4²=5². Now challenge students to go back to their maps and see how many right angles they can find, and have them measure the straight-line distance versus the distance going down the legs of the triangle. After giving students a few minutes exploring the exercise, have students gather back together, and remind students of the Pythagorean Theorem, taking care to stress that a and b represent the lengths of the legs and c represents the length of the hypotenuse. Have each pair of students find the shortest distance between their two points using the Pythagorean Theorem. Select a few groups to present their maps. Have students explain what points they selected and why, the street route they highlighted and found that distance and the shortest route and how they found that distance. Make sure it is clear to students by the end of the lesson how to use the Pythagorean Theorem to find the distance between two points. Distribute the Distance Worksheet to students. Instruct students to graph the points, draw the street route between the points, draw the shortest route between the points and use the Pythagorean Theorem to find the distance between the points. If

students have difficulty determining the lengths of the sides of the right triangles on the Distance Worksheet, ask students to examine the two x-coordinates and determine how far it is from one x-coordinate to the other x-coordinate and illustrate that the distance between the x-coordinates is simply the length of one of the legs. Repeat this with the y-coordinates. After students have finished the Distance Worksheet, write the distance formula on the board. Ask students “How does the distance formula relate to the Pythagorean Theorem?” Students should note the difference between the two and discuss how the two are, algebraically, the same formula. For example, the distance formula has a square root in it, and the Pythagorean Theorem does not; however, solving the Pythagorean Theorem for c (rather than c²) results in a square root. Depending on the class’s algebra skills, the algebraic relationship between the two may be explored in depth. Show this example: Draw a right triangle with a right angle on the origin (0,0) and two vertices (0,4) and (-3,0). Show students that the subtraction of x2 from x1 is the same as 0 – (-3). Then the subtraction of y2 from y1 is the same as 4-0, which is 4. Now show that 32 + 42 is the same as a2+b2 from the Pythagorean Theorem. As an exit activity, have students write an explanation of how to find the shortest distance between the two points.



(Before)




Build from student answers to questions during explore to organize whole class discussion about relationship between Pythagorean Theorem and the Distance Formula

Focus on connections to draw generalizations about the Pythagorean Theorem and the Distance Formula

Students may need to see another example to further emphasize the relationship between the two formulas

Start by asking question that was asked during the Launch: How does the Pythagorean Theorem relate to the Distance Formula?

What did you notice about the similarities in the two distance formulas?

What is ultimate goal of both the Pythagorean Theorem and the Distance Formula?

What generalization can be made about their relationship?

They are practically the same formula

Formulas are very much alike

Both formulas find the distance between two points

To find the distance between two different points on a coordinate plane

The Pythagorean Theorem and the Distance Formula both find the distance between two points on a coordinate plane

How do you know? What evidence supports this?

Encourage students to be specific about what the Pythagorean Theorem and the distance formula are find the distance between-2 points on a coordinate plane

If this is not evident by student engagement and discussion during the class activity, have students complete another problem and explain to the whole class how both the Pythagorean Theorem and the Distance Formula could have been used

Be sure to encourage students to be able to support their justifications




As an assessment for this lesson, have students turn in an exit ticket. After completing the class activity, students will be given about ten minutes at the end of class to give an explanation of how to find the shortest distance between two points on a coordinate plane. Students will complete this in class and turn in at the end of the class period.

Map of Minneapolis, MN, courtesy of Google Maps

For each pair of points:

1. Graph each point on the graph paper.2. Draw the “street route” between the two points.3. Draw the shortest route between the two points.4. Calculate the distance between the two points using the Pythagorean theorem.

A. (4, 2) and (0, 5) B. (7, 3) and (1, 1)

C. (−6, 2) and (3, 5) D. (3, −8) and (−1, −3)

Exit TicketExplain in your own words how to find the shortest distance between two points. Give this paper to your teacher as you leave class.

Topic Rationale

The Pythagorean Theorem is an important part of the 8th grade curriculum in the Common

Core. Students may have already been exposed to the concept of the Pythagorean Theorem before

they come into your classroom. However, more than likely, these students will not anymore about

the theorem except the formula that is used to define its purpose. Teaching the Pythagorean

Theorem is such an important task for students to learn because not only can it be applied to

everyday life, but the Pythagorean Theorem is also an important task that will be used throughout a

student’s mathematics career.

The Pythagorean Theorem is introduced to students during the middle school years, but

becomes increasingly important when students reach the high school level. It is not enough to

simply state the algebraic formula for the Pythagorean Theorem. Students need to see the

geometric connections as well. The teaching and learning of the Pythagorean Theorem can be

enriched and enhanced through the use of dot paper, geoboards, paper folding, Smartboard

technology, as well as many other instructional materials. Through the use of manipulatives and

other educational resources, the Pythagorean Theorem can mean much more to students than just

the formula and plugging in numbers into that formula.

It cannot be stressed enough that students need to understand the geometric concepts

behind the theorem as well as its algebraic representation. An example of understanding the

geometric concepts behind the theorem is formulating the distance formula. The Pythagorean

Theorem is the basis in which the distance formula is derived. When students learn the

Pythagorean Theorem, students can then make connections to other geometric concepts and what

role the Pythagorean Theorem has in these other concepts. An important idea to remember when

teaching the Pythagorean Theorem to middle grades students is to make use or real world

applications of the Theorem. Students need to be exposed to situations where the Pythagorean

Theroem might be useful to use, whether it’s to determine how far away the top of a tree is from

a house in case it fell during a storm or the theorem is used when surveyors survey land. When

students reach a higher level Geometry class, students will not only know what the Pythagorean

Theorem is, but they will understand the reasoning behind it and why the theorem works. The

sooner that each student is introduced to the Pythagorean Theorem, the more prepared that these

students are going to be in their future math classes and in the jobs that students will take on the

future.

Content Map

Pythagorean

Theorem

Lesson 2

-Prior Knowledge- explain the relationship between the different side lengths of a triangle and why the Pythagorean Theorem can be applied-Vocabulary: Pythagorean Theorem, Geometric Proof, Congruence-Address Misconceptions-Demonstrations involving proving the Pythagorean Theorem through words and visuals

Lesson 1

-Prior Knowledge-describe relationships between lengths and hypotenuse of right triangles-Vocabulary: Pythagorean Theorem, Legs, Hypotenuse, Pythagorean Triples, Right Triangle-Address Misconceptions-Understand connections and make patterns

Lesson 5

-Prior Knowledge-recall the Distance Formula and be able to plot points on a coordinate plane-Vocabulary: Pythagorean Theorem, Cartesian Coordinate Plane, Distance Formula, Equidistant, Midpoint, Leg, Hypotenuse, Slope-Address Misconceptions-Know how to find the shortest distance between two places using the Distance Formula while making connections to the Pythagorean Theorem

Lesson 3

-Prior Knowledge-explain the relationship between the different side lengths of a triangle and why the Pythagorean Theorem can be applied-Vocabulary: Pythagorean Theorem, Geometric Proof, Converse-Address Misconceptions-Be knowledgeable and talk about the many proofs and their converses for the Pythagorean Theorem

Lesson 4

-Prior Knowledge-familiar with Pythagorean Theorem and coordinate systems-Vocabulary: Pythagorean Theorem, Cartesian Coordinate Plane, Distance Formula, Equidistant, Midpoint, Leg, Hypotenuse, Slope-Address Misconceptions-Know about Distance Formula, be able to derive Distance Formula using Pythagorean Theorem, know relationship between two formulas

Summative Assessment

Pythagorean Theorem Projects

Students will be divided into groups of 4 or 5, depending on the number of students in the class.

With your group, choose one of the following:1. Create a scale model of the Great Pyramid of Giza. The pyramid is 480 feet tall and

the base is 756 feet on each side.2. You’re a builder starting a new house, but you forgot your square. How can you use

the Pythagorean Theorem to create a right angle for the corner of your wall? Create a model showing your solution.

3. You want to buy a new flat screen TV, but you can’t afford a new entertainment center to hold it. If your current entertainment center is 24"(W) x 18"(D) x 60"(H), what is the biggest TV you can purchase and still have it fit in the cabinet? Hint: TV measurements are given as the length of the diagonal across the screen.

4. The Wheel of Theodorus (See page 46 in your book) starts with an isosceles triangle with legs that are 1 unit long and winds around counter clockwise. Each new triangle is drawn using the hypotenuse of the previous triangle as the new leg. Create a Wheel of Theodorus neatly and in color. Label each hypotenuse with its exact length.

5. Bosque needs to build a wheel chair accessible ramp to Portable 2. You plan to build a wooden ramp with closed sides. How much wood will you need to make the ramp? Allow 15% extra for cutting and connecting.

6. A circus tent has sidewalls 3.5 meters tall, a peak height of 10.5 meters and a diameter of 33 meters. (Hint: A circus tent is a cylinder with a cone on top.) How much fabric do you need to manufacture the tent? Allow 15% extra for cutting and stitching.

7. A roof is 5.5 meters high. (See the figure.) The owner would like to lay new copper roofing to stop the leaks. Calculate how many square meters of copper paneling he will need in order to cover the entire roof, allowing for 15% extra for cutting and connecting the metal?

Each group needs to:

Come up with a list of additional questions/information needed to solve the problem. Research the answer to these questions. Create a model or diagram Do the calculations. Remember to:

Define your variables! Use the vertical format aligning the equal signs and do one step at a time! WPWA!

Create a poster of your findings that includes the following: Title The problem Any additional information or research you did. Tell where you got the

information. If you did not need to do any additional work, say so! Example- Knowing surface area of a triangular prism

A model or drawing Calculations correctly formatted – define variables, show and write out each step WRITTEN EXPLANATIONS of why you performed the calculations that you

did. Describe what you did step by step. For example – “First we found the area of the pizza. To do this we used the formula a=πr2

and substituted in the height and diameter from the problem. Here are our calculations:”

Your poster will be graded as follows:

Section Points Title – neat and easy to read 2Problem – neat, easy to read 3Additional Research – Stated the question asked Gave source (MLA not necessary!) Gave the answer found

10

Model/Drawing – neat, large enough to see, labeled appropriately, easy to understand, accurate

15

Calculations – Variables definedEach step of calculations shownAccurate WPWA

15

Written Explanation of your process -What were you thinking?Why did you do what you did?Why did you do each calculation?Give Step-by-step process.

10

Overall – neat, creative, thorough 5Total 60

References:

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB4QFjAA&url=http%3A%2F%2Fteacherweb.com%2FNM%2FBosqueSchool%2FKimLestermath%2FPythagoreanTheoremProjects.doc&ei=9maFUO3yJISm8gTswIGoDg&usg=AFQjCNE443WvFqWArSHF8EDCnUM93MGStQ




Documents

As the Crow Flies Name:___________________dnwestonstudentteachingportfolio.weebly.com/.../115682… · Web viewNote that the theorem and its converse have been proven in numerous