Assignment #4

Stage 3 - Understanding by Design

Charles Kelley

Title of Unit The Pythagorean Theorem and its

uses

Grade Level 6-8

Essential Questions

Overarching Questions:When is it appropriate to use the Pythagorean theorem and for what purpose?

How does using variables to represent unknown amounts help solve problems?

Topical Questions: What types of triangles can the

Pythagorean Theorem be used with? What side lengths can be calculated using

the Pythagorean Theorem? What do the three variables (a, b, c)

represent? How do you apply the solution to the

theorem? How are the variables matched with a

value? What does the area of the sides have to do

with the length of the hypotenuse?

Stage 3

W Where are your students headed and Why?What is required of them?

State each goal for the unit and address the relationship between the goal and the essential questions to establish where the unit is going.

To address why we are learning the content highlight practical/ real-world application of the theorem through performance tasks

Address what is required of the students by briefly introducing the rubrics for each assignment and stressing the expectations for those assignments. Present to the students previous examples of Target work.

H How will the students be hooked and held in this unit?

Hook the interests of the students by providing a story of how I have personally used the Pythagorean Theorem while building my garage.

Ongoing hold of the unit will be addressed in the performance tasks by challenging the students in role-play situations. Also allow students a choice in content in performance task #2.

E What activities, instruction, and guidance will be provided to enable and equip students to explore and experience the important ideas in this unit?

I will equip the students for learning by breaking down the required skills needed to perform the Pythagorean Theorem in the first week of instruction through a mixture of direct instruction, experiential tasks (problem-based, construction/measurement) and project-based performance tasks.

Students will explore their newly learned content through 2 performance tasks and the presentation of these tasks.

R What activities, products and performances will be designed to provide students with the opportunity to reflect, rethink, and revise?

Through the use of a self-assessment rubric, I will allow students to reflect upon their level of understanding and mastery. By having the students take this assessment at the beginning of the third week, I have allocated some time to revisit and rethink those concepts that the students believe they need more work on.

Students’ performance activities will be peer reviewed with standardized rubrics. Students who get unacceptable grades will have the input from the other students and a chance to refine their projects until it reaches an acceptable level.

E What self-assessments and self-evaluations will students participate in to allow for reflection and transfer of learning?

To encourage self-evaluation I have included introductory and closing activities that are designed to evoke self-reflection and evaluation in the students.

T How will instruction and activities be tailored to provide for the various learning needs, styles, knowledge and interests of students?

I have tailored the assignments in the unit to accommodate different learning styles and levels of prior knowledge by differentiating the type and scope of the assignments. The Performance Tasks are fairly open-ended and can be achieved through many different means. I have provided a rubric for what has to be in the assignment but have not limited the assignment in any other way. The class assignments are a mixture of hands-on construction assignments, written work and problem solving which offers many different types of learners some likeable facet. Also, the rubric/ grading is set up in a way where there are concrete goals for the student in each activity and a chance to correct unacceptable work.

OHow will learning experiences be organized/sequenced to provide for greatest acquisition/understanding

I have chosen to organize the unit from a mastery-approach point of

view. The students receive a week of basic concepts and prerequisite skill building before the higher concepts are put into place. The student then attempt to master the higher concepts of the unit before they put the concepts into practice in the performance tasks.

Week One- Learning SequenceMonday:a. Exploration of variables and their

uses (vocabulary)i. Pre-assessment

ii. Vocabulary introduction (variables, triangle, square, square roots, Isosceles, Scalene, Angles, Sides, Hypotenuse, Equation) . Have students explore the definitions of these words so that they will become familiar with them and are able to utilize them correctly in the unit.

iii. Closing Activity- Students quickly write a sentence or two on “What questions do I still have about the vocabulary words?”

Tech. Integration: Students utilize outside of class. http://www.geom.uiuc.edu/~demo5337/Group3/prereq.html

Tuesday:b. Balancing equations

i. Students will be given several unbalanced equations and asked to use the principle of equality to balance the equation. The student must isolate the variable and perform equal functions on each side of the equation. Students will be asked to justify the steps taken when balancing the equations. (Experiential Entry)

Tech. Integration: Students utilize outside of class.

http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html

Wednesday:c. Squares and square roots

Sponge Activity- Students quickly answer, “What are your strengths in balancing equations?”

i. Students will be assigned a page in their workbook on squares and square roots. Finding squares and their square roots

ii. Student will explore the inverse relationships of the two functions and will be asked to create several examples of this relationship on their own.

iii. Students write a short paragraph on the patterns that they experienced today and what it meant to them. (Experiential Entry)

Thursday:d. Different

triangles and their properties (measurements)

EQ: What types of triangles can the Pythagorean Theorem be used with?

i. Students will indentify Right, Isosceles, and Scalene triangles from examples they find in magazines. The students will then be asked to find the

Friday:e. Indentifying

parts of the triangle/ Square construction

i. Students are assigned a section in their workbook that illustrates Side A, B Hypotenuse(C) and Right angle in a right triangle. Students will complete the assignment in the book and then create right triangles on their own and label the parts of the

angle measurements and side measurements of the triangles that they have found. (Experiential Entry)

triangle. Students are then asked to create the squares that can be created by each side.

ii. Closing Activity- “How has what you learned this week changed your thinking about the Pythagorean Theorem?” (Foundational Entry)

Week Two- Learning Sequence

Monday:a. Matching values to the

Theorem EQ: What side lengths can

be calculated using the Pythagorean Theorem?

Tuesday:b. Measuring and placing

Values into the Theorem EQ: What side lengths can

be calculated using the Pythagorean Theorem?

i. Students will take the

Wednesday:c. Placing Values into

the Theorem with missing Values (Variables)

EQ: How does using variables to represent unknown amounts help

Thursday:d. Performance

Task # 1

EQ: When is it appropriate to use the Pythagorean theorem and for what purpose?

Friday:e. Complete and

Present Performance Task #1

i. Students will complete the first

Sponge Activity- Students answer, “How could you improve on the skills you learned last week?”

i. The students will practice matching values from a right triangle to the corresponding variable in the Pythagorean Theorem. Students will have several right triangles with different measurements. The students must take the measurements and insert them into the Theorem.

Tech Integration: Student supplemental aid http://www.regentsprep.org/regents/Math/fpyth/Pythag.htm

measurements of several right triangles and insert the values into the theorem. The students will then solve the equation with no variables and should get an equal amount on each side. (Experiential Entry)

Tech Integration: Student supplemental aid http://shodor.org/interactivate/activities/pyth2/index.html

solve problems?

i. The students will repeat the previous assignment now with missing variables.

ii. Closing Activity- “How difficult was using variables instead of actual numbers?” (Aesthetic Entry)

i. (Logical/ Quantitative Entry)

Your task is to give an estimate for a roof replacement. You are a roofer that has just started their business and on the first job. Your client is a homeowner looking to have the roof over their garage replaced with a new roof. The homeowner needs a quick estimate and you do not have enough time to get out the scaffolding needed to access the roof. The garage is square and is 20 feet long and 20 feet wide. You need to develop an estimate for the homeowner quickly that is as close to the overall cost as possible. The project cost you $3.00 a square foot. You can’t charge too much because the homeowner will go with another company. You can’t charge too little because you need to make money on the job. Use the principles discussed in the unit to come up with the square footage of the roof and give the homeowner an estimate for the job. Before coming up with an answer think about how

performance task and prepare to present the information to their peers. Students will receive rubrics to grade peers’ presentations. (Narrational Entry) (See attached Rubric)

roofers make a profit from their work. You must give the homeowner a fair price estimate for the roof that also makes a profit for you. In order for you to accomplish this goal the estimate must be as close as possible. If the price is higher than the rest of the prices in the class your bid will fail. If your bid is lower than the actual cost you will lose money.

1 2 3Performance task 1 (Counts twice)

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete. Roofing estimate was grossly inaccurate.

Task was completed with an outcome that showed a basic understanding of the concepts needs to estimate the roof cost. The estimate was somewhat high or low indicating minor inaccuracies in the performance of the task.

Task was completed with an outcome that showed a thorough and complete understandings of the concepts needed. The estimate was perfect indicating an insightful real-world application of the concepts.

Week Three- Learning Sequence

Monday:a.Self Evaluation

i. Students complete self-evaluation (see attached) and review some of

Tuesday:b. Performance Task Day 1

i. Students are given performance task #2

EQ: How do you apply the solution to the theorem?

Wednesday:c. Performance Task Day

2 Presenti. Students finish

work on Performance task #2 and prepare to

Thursday:d. Review through

“Student Teach”

ii.Assign each student team with a section of the unit to teach

Friday:e. Formal assessment

i. Standardized formal assessment broken down by specific skill-sets that allows me to

the most common misunderstandings of the unit content. (Foundational Entry)

Wiki presentation:

i. As a group, the students will make a presentation Wiki on Pbworks that demonstrates how to find the distance between two locations in their hometown using a map and the Pythagorean theorem. The group will be asked to accept the role of civil engineer given the task to find the length of pipe needed to connect the water plant to a new water treatment facility. There is a dense forest in between the two locations so there is no direct way of measuring the length. There are 2 roads, one that leads to the water plant and the other to the treatment plant. They intersect at a right angle. Use the concepts explored in this unit to come up with a plan to measure the distance between the two plants. Create your own distances and link the processes in the project with the ideas in the unit. Each step in your process must be illustrated and justified with supporting evidence from the unit. Include in your presentation why being a Civil Engineer would be a tough job and why it would be rewarding. Students in other group

present. Students present the Wiki while their peers evaluate using attached Rubric.

to the other students. Have groups study the material and come up with a “teaching plan” the first half of class and then have the groups rotate and teach their unit to another group. (Narrational Entry)

track student mastery.

ii. Bonus Activity- Students write a short paragraph on, “What accomplishment or activity were you most proud of in the unit and why?” (Aesthetic Entry)

may then post comments and suggestions at the bottom of the page. All rubric scores will be posted on the site. (Experiential Entry) (Narrational Entry)

Wiki Site:

Pbworks.comhttp://mrkelleystudents.pbworks.com/

RubricSelf Assessment Pythagorean Theorem

Component 1 2 3 Section Scores

EffortI did not participate in most activities and did not complete most tasks. My work that I did complete was turned in past the deadline. I was not engaged in class discussion nor whole-group exercises.

I participated in most activities and completed all tasks but not always in the allotted time. I was not always fully engaged in class discussion and whole-group exercises.

I fully participated in all activities and completed the assigned tasks in the time allotted. I was engaged in class discussion and whole-group exercises.

performance

My performance is ineffective. The evidence is produced in a fashion that indicates a complete misunderstanding of the content.

My performance is effective. There are some problems with clarity and thoroughness but task are accomplished.

My performance is highly effective. The tasks are accomplished in a clear and thorough manner that demonstrates a mindful application of content.

General knowledge and understanding

Has little apparent understanding of the relevant concepts in the unit. Problems with process reveal major misunderstandings with

Has a solid though somewhat naïve understanding of relevant concepts from the unit. Is able to comprehend content but needs to check the

Has a sophisticated understanding of the concepts included in the unit. The justifications for the methods used are appropriate for the material.

material. information for accuracy. Total Scores

/ 9

Performance Task #2 Rubric1 2 3

Performance Task 2 (Counts twice)

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete. Presentation of materials showed a lack awareness of the material.

Task was completed with an outcome that showed a basic understanding of the concepts needs to find the distance from the two plants. The distance was not quite correct indicating possible inaccuracies in the functions of the equation. Presentation of the material was adequate and somewhat insightful.

Task was completed with an outcome that showed a thorough and complete understanding of the concepts needs to find the distance from the two plants. The distance was exactly correct indicating precise use of the functions of the equation. Presentation of the material was fluent and effective and showed great insight into the content of the unit.

Technological Resources:Foundational Knowledge-http://en.wikipedia.org/wiki/Pythagorean_theoremhttp://www.cut-the-knot.org/pythagoras/Multimedia-http://player.discoveryeducation.com/index.cfm?guidAssetId=49D6C767-1084-474F-BCDD-ED06B6BF8185&blnFromSearch=1&productcode=US

RubricWeek 1

Component 1 2 3 Section Scores

Exploration of variables and their uses (vocabulary)

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete. Variables not appropriately used.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result.

Student shows a thorough and complete understanding of variables and other vocabulary.

Balancing Equations

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of balancing equations.

Student shows a thorough and complete understanding of balancing equations. Appropriately applied knowledge of equations.

Squares and Square roots

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of squares and square roots

Student shows a thorough and complete understanding of squares and square roots. Problems completed with total accuracy.

Different triangles and their properties (measurements)

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of different triangles and their properties.

Student shows a thorough and complete understanding of different triangles and their properties. Problems completed with total accuracy.

Indentifying parts of the triangle/ Square construction

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of identifying parts of a triangles and constructing squares.

Student shows a thorough and complete understanding of Identifying the parts of a triangle and successfully constructed squares. Problems completed with total accuracy.

Total Scores / 15

Score Rating13-15 Target10-12 Acceptable10 and below Unacceptable

*** Unacceptable performance for the week will result in the need for modification of assignments outside of school and resubmission.

RubricWeek 2

Component 1 2 3 Section Scores

Matching values with Triangle parts

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of matching values with triangle parts.

Student shows a thorough and complete understanding of matching values with triangle parts. Problems completed with total accuracy.

Pythagorean with no

variables

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of applying the Pythagorean theorem with no variables.

Student shows a thorough and complete understanding of applying the Pythagorean theorem with no variables. Problems completed with total accuracy.

Pythagorean with variables

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete.

Work partially completed but generally accurate with minor inaccuracies that do not affect the overall result. Somewhat effective use of the concepts of applying

Student shows a thorough and complete understanding of applying the Pythagorean theorem with variables. Problems completed with total

the Pythagorean theorem with variables.

accuracy.

Performance task 1 (Counts twice)

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete. Roofing estimate was grossly inaccurate.

Task was completed with an outcome that showed a basic understanding of the concepts needs to estimate the roof cost. The estimate was somewhat high or low indicating minor inaccuracies in the performance of the task.

Task was completed with an outcome that showed a thorough and complete understandings of the concepts needed. The estimate was perfect indicating an insightful real-world application of the concepts.

Performance Task 2 (Counts twice)

Significant errors throughout the assignments with a general pattern of misunderstanding. Work partial or incomplete. Presentation of materials showed a lack awareness of the material.

Task was completed with an outcome that showed a basic understanding of the concepts needs to find the distance from the two plants. The distance was not quite correct indicating possible inaccuracies in the functions of the equation. Presentation of the material was adequate and somewhat insightful.

Task was completed with an outcome that showed a thorough and complete understanding of the concepts needs to find the distance from the two plants. The distance was exactly correct indicating precise use of the functions of the equation. Presentation of the material was fluent and effective and showed great insight into the content of the unit.

Total Scores / 21

Score Rating19-21 Target16-18 Acceptable17 and below Unacceptable

*** Unacceptable performance for the week will result in the need for modification of assignments outside of school and resubmission.

Pre-assessment- Pythagorean TheoremFind the square root of the following numbers

1. √42. √163. √254. √495. √10

Solve the following equations6. xa+2=67. 3+b=98. 19+b=229. 25+ b= 100

Indentify the following triangles10.

11.

12.

Activities ExamplesSeveral Worksheets will be constructed off of this basic form to allow students to see the progression of their skills and to keep a link from the previous day’s activity to the next. The numbering and the shapes will stay the same but the tasks will increasingly become more difficult and more in line with what needs to ultimately be accomplished to implement the theorem.

Pythagorean Theorem and its values

#1 Label the different parts of the triangle (a, b, c, right angle) that correspond to the two legs (a and b) the Hypotenuse ( c ) and the 90 degree angle (right).

#2 Label the different parts of the triangle (a, b, c) and insert the values in the triangles into the Pythagorean TheoremPythagorean Theorem:a2+b2=c2

#3 Measure the different parts of the triangles and insert the values into the Pythagorean theorem.

#4 Measure the different parts of the triangles and insert the values into the Pythagorean theorem. Then use the Theorem to check the length of the hypotenuse against your measurement.

1.

2.

3.