**Unit Overview**

**Content Area:** Mathematics

**Unit Title: **Looking for Pythagoras: Investigation #3

**Target Course/Grade Level:** 8th grade Resource Room

**Name: **Kim Fassett

**School: **Lakeside Middle** **

**Date: **July 15, 2010

**Unit Summary**

In this unit students discover the Pythagorean Theorem and explore its implications. However, these students with Specific Learning Disabilities may lack the prior knowledge needed of angle measurement, area of triangles, rules of exponents, and estimating roots. Breaking this unit into smaller chucks allows the students the opportunity to gain a more complete understanding of the Pythagorean Theorem

**Primary interdisciplinary connections**

World History/Physical Education/Music/Geography

**21st century themes**

Global Awareness/ Health Literacy/Creativity and Innovation/Critical thinking/Problem Solving/Communication/Collaboration/Information Literacy/ ICT Literacy/Life/Career Skills

**Unit Rationale**

The**Pythagorean theorem has many applications in the everyday real-world and in mathematical situations. It is an important building block for future work in mathematics.**

**Learning Targets **

**Standards Key Ideas and Details **

8G: Geometry

**Content Statements**

Understand and apply the Pythagorean Theorem

8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

**Unit Essential Questions**

- How can the Pythagorean Theorem be used to make decisions and solve real-world problems?
- What is the relationship in the lengths of the sides of a right triangle?
- How can knowing the slope of a line help solve problems?

**Unit Enduring Understandings**

- The Pythagorean Theorem can be used to make decisions and solve real-world problems by showing the relationship of the lengths of the sides of a right triangle.
- Knowing the slope of a line helps to solve everyday real world and mathematical problems.

**Unit Learning Targets**

*Students will...*

**Explain the relationship between the sides of a right triangle using dot paper and a straight edge to discover the Pythagorean Theorem.**- Begin to work out the Pythagorean Theorem through exploration again showing the relationship between the sides of a right triangle.
- Use the Pythagorean Theorem to find areas of squares drawn on a dot grid or rubber bands on geo-boards.

**Evidence of Learning**

**Summative Assessment (1 day)**

- A formal written assessment includes showing problem solving steps.

**Formative Assessments **

- informal observations
- completion of all activities
- Small group discussions
- exit tickets
- student check list (informal observation)

**Lesson Plans**

Lesson 1

Investigation 3.1

Discovering the Pythagorean Theorem

Assessment Methods: interactive notebooks, informal observations, small group discussions, completion of all activities, formal written assessments

(2) 80 minute days

Lesson 2

Investigation 3.2

Puzzling Through a proof

(2) 80 minute days

Lesson 3

Investigation 3.3

Finding Distances

(2) 80 minute days

Lesson 4

Investigation 3.4

Measuring the Egyptian Way

(4) 80 minute days

**Teacher Notes**

Modifications:

- Place students in homogeneous/heterogeneous cooperative learning groups.
- highlight/underline instructions/content
- Provide additional instruction
- assign peer buddies
- provide alternative grading when necessary
- provide extra set of books/materials for home
- school/home contact daily using student agenda book...teacher/parent signature daily
- provide study guides prior to testing

**Curriculum Development Resources**

Refer to the UDL toolbox at the end of this document.

**Lesson Plan 1**

**Content Area**: Mathematics

**Grade:** 8th Resource Room

**Lesson Title:** Looking for Pythagoras

**Timeframe:** (1-2) 80 minute days

**Lesson Components**

**21st Century Themes**** **

X Global Awareness: Egyptian Pyramids

Financial, Economic, Business, and Entrepreneurial Literacy

Civic Literacy

X Health Literacy: Pythagorean Theorem in baseball

**21st Century Skills**** **

** **x** **Creativity and Innovation- Egyptian Knotted Ropes** **

Media Literacy

x Critical Thinking and Problem Solving- How to create a triangular building after viewing an informational website selected by the teacher.

x ICT Literacy: Smart-Board, document camera, Geo-Sketch Pad software

Communication and Collaboration

x Life and Career Skills: Explore real life careers that use the Pythagorean Theorem

Information Literacy: Carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers of many sorts need to know it. One time I observed people who needed to measure and mark on the ground exactly where the building would go. They had the sides marked, and they had a tape measure to measure the diagonals, and they asked ME what the measure should be, because they couldn't quite remember how to do it. This diagonal check is to ensure that the building is really going to be a rectangle and not a parallelogram. It's not easy to be sure that you have really drawn the two sides in a right angle.

Interdisciplinary Connections: World History of Ancient Greece and Ancient Egypt

Integration of Technology: Using the Smart Board with GeoSketch Pad

Equipment & materials needed: colored pencils, pencils, grid paper, dot paper, calculators. Rulers (straight edge), Connect Math text books pages 27-28. Smart-Board, document camera, Geo-Sketch Pad software, and internet access** **

** **Teacher Resources:** **document camera, student work, colored pencils, pencils, grid paper, dot paper, calculators. Rulers (straight edge), Connect Math text books pages 27-28. Smart-Board, document camera, Geo-Sketch Pad software and internet access.

**Goals/Objectives/CPIs**

Students:

- Work out the Pythagorean Theorem through exploration and use the Pythagorean Theorem to find areas of squares drawn on a dot grid with at least 80% accuracy.
- Will be able to identify that a right triangle 2 legs and a hypotenuse with at least 80% accuracy.

**Learning Activities/Instructional Strategies**

**Engagement:**

- Begin with reviewing the terms square roots and area.
- Next place the text book on the document reader and point out and explain the legs and hypotenuse of a right triangle.
- Now discuss 3.1 on pg.27 discovering the Pythagorean Theorem.
- Introduce the idea of drawing squares on the sides of a right triangle and comparing their areas as shown on page 27 in their text books Investigation 3 the Pythagorean Theorem.
- Now have students draw on dot paper 3 different size right triangles and look for the squares of their triangle and figure out the areas of each.
- http://fractalfoundation.org/category/teachers/
- http://www.homeschoolmath.net/teaching/why_need_square_roots.php

**Exploration:**- Ask that each student make a table.
- Describe the problem and have groups of 2 students work on the problem
- Encourage group members to share the work.
- http://www.historyofscience.com/G2I/timeline/index.php?id=1661
**Explanation:**- Talk about the pattern in the table as a whole class.
- It is known that the Egyptians used a knotted rope as an aid to constructing right angles in their buildings. The rope had 12 evenly spaced knots, which could be formed into a 3-4-5 right triangle, thus giving an angle of exactly 90 degrees. Can you make a rope like this? Now use your knotted rope to check some right angles in your room at school or at home.
- http://www.math.com/school/subject3/lessons/S3U3L4DP.html
**Elaboration**

- Discuss whether the relationship works for triangles that are not right triangles.
- The Egyptians used this triangle for land surveying. Some believe that they also used it to help design their pyramids. Whether they did or not, the 3-4-5 triangle is still used by surveyors. Carpenters and woodworkers also use it to make their corners square.
- http://www.math.com/school/subject3/lessons/S3U3L4DP.html
- http://xaej806.wordpress.com/2008/10/28/pythagorean-in-the-real-world/

**Evaluation**- Complete an exit ticket …Problem 3.1D
- Complete the follow-up for homework.

**Formative Assessment Tasks **

- informal observation…each student will create a quiz and answer key which will be randomly handed out and the students will complete the quiz given to them and then returned to the creator to grade
- completion of all activities and small group discussions
- formal written assessment showing problem solving steps proving the Pythagorean Theorem

Universal Design for Learning Options

**Multiple Means of Representation: **Each student will create a quiz and answer key which will be randomly handed out and the students will complete the quiz given to them and then returned to the creator to grade.

- Perception: http://fractalfoundation.org/category/teachers/
- Language & symbols: http://www.ronblond.com/MathGlossary/
- Comprehension: Cue Card Maker

**Multiple Means of Action and Expression**

- Physical activity: http://www.math.com/school/subject3/lessons/S3U3L4DP.html
- Expressive skills and fluency: http://www.glogster.com/
- Executive functions: http://www.udlcenter.org/sites/udlcenter.org/files/Research_Project_Example_0.jpg

**Multiple Means of Engagement**

- Recruiting interest: Have students create a list of where they see the shape of triangles in the everyday world. (i.e. bridges, roof tops, etc.) http://www.homeschoolmath.net/teaching/why_need_square_roots.php
- Sustaining effort and persistence: http://www.skype.com/intl/en-us/home
- Self-regulation: http://www.teachingenglish.org.uk/think/articles/learner-diaries

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**Resources **

UDL toolbox