Unit Title: Looking for Pythagoras Suggested Time: 14 Days (75 – 90 minute block)

Enduring understanding (Big Idea): Calculating the distance between two points in the plane; Finding areas of figures drawn on a coordinate grid with whole-number vertices; Understanding square roots as lengths of sides of squares; Understanding the Pythagorean Theorem and how it relates the areas of the squares on the sides of a right triangle; Using Pythagorean Theorem to solve problems; Investigate rational numbers written as decimals; Understanding irrational numbers as non-terminating, non-repeating decimals; and Understanding slope relationships of perpendicular and parallel lines.

Essential Questions: Is it appropriate and useful to use the Pythagorean Theorem in this situation? How do I know this? Do I need to find the distance between two points? How are irrational numbers and areas of squares related? How can I estimate the square root of a number? How can I find the length of something without directly measuring it?

Unit Plans

Common Core Standards Alignment

Connection to 2003 Standards

Investigation 1
Coordinate Grids
Problems 1.1,1.2, 1.3
Math Reflections

Apply and extend previous understandings of numbers to the system of rational numbers. Review 6.NS.6- Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates

Goal 3.02, 3.03

Investigation 2
Squaring Off
Problems 2.1, 2.2, 2.3
Math Reflections

Expressions and Equations Work with radicals and integer exponents. 8.EE.2- Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Goal 3.02

Investigation 3
The Pythagorean Theorem
Problems 3.1, 3.2, 3.3, 3.4
Math Reflections

Understand and apply the Pythagorean Theorem 8.EE.2-Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Understand and apply the Pythagorean Theorem 8.G.6- Explain a proof of the Pythagorean Theorem and its converse.

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

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 8.G.9- Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (ACE Problems)

Goal 3.01, 3.02

Investigation 4
Using the Pythagorean Theorem
Problems 4.1, 4.2, 4.3, 4.4
Math Reflections
Looking Back and Looking Ahead

Know that there are numbers that are not rational, and approximate them by rational numbers. 8NS.1- Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

8.NS.2.- Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2).

Understand and apply the Pythagorean Theorem 8.EE.2-Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Understand and apply the Pythagorean Theorem. 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.

8.G.9 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (ACE problems)

Goal 1.01, 3.02

Prior Knowledge: Measuring lengths; working with coordinates; measuring areas of polygons and irregular figures and surface area of three-dimensional shapes; applying the formula for area of a square; formulating, reading, and interpreting symbolic rules, solving problems in geometric and algebraic contexts; understanding fractions and decimals; and representing fractions as decimals and decimals as fractions.

Mathematical Practices Standards for Common Core
1-Make sense of problems and persevere in solving them 2-Reason abstractly and quantitatively 3-Construct viable arguments and critique the reasoning of others 4-Model with mathematics 5-Use appropriate tools strategically 6-Attend to precision 7-Look for and make use of structure 8-Look for and express regularity in repeated reasoning

Essential Terms Developed in This Unit

Useful Terms Referenced in This Unit

Terms Developed in Previous Unit

Hypotenuse
Legs
Pythagorean Theorem
Real numbers
Square roots

Looking for Pythagoras is filled with investigations that develop a fundamentally important relationship connecting geometry to algebra: the Pythagorean Theorem. Students are not merely introduced to a meaningless formula. To encourage deep understanding of what the theorem means, students explore squares created with various lengths, their areas and how these relate to the side lengths of right triangles. Students also explore square roots and strategies for estimating square roots. Irrational numbers are introduced and students are expected to be able to estimate where these occur on a number line.

For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Looking for Pythagoras.

Goals of the Unit • Relate the area of a square to the side length • Estimate the values of square roots of whole numbers • Locate irrational numbers on a number line • Develop strategies for finding the distance between two points on a coordinate grid • Understand and apply the Pythagorean Theorem • Use the Pythagorean Theorem to solve everyday problems

Unit Title: Looking for PythagorasSuggested Time: 14 Days (75 – 90 minute block)Enduring understanding (Big Idea):Calculating the distance between two points in the plane; Finding areas of figures drawn on a coordinate grid with whole-number vertices; Understanding square roots as lengths of sides of squares; Understanding the Pythagorean Theorem and how it relates the areas of the squares on the sides of a right triangle; Using Pythagorean Theorem to solve problems; Investigate rational numbers written as decimals; Understanding irrational numbers as non-terminating, non-repeating decimals; and Understanding slope relationships of perpendicular and parallel lines.Essential Questions:Is it appropriate and useful to use the Pythagorean Theorem in this situation? How do I know this? Do I need to find the distance between two points? How are irrational numbers and areas of squares related? How can I estimate the square root of a number? How can I find the length of something without directly measuring it?Unit PlansCommon Core Standards AlignmentConnection to 2003 StandardsInvestigation 1Coordinate Grids

Problems 1.1,1.2, 1.3

Math Reflections

Apply and extend previous understandings of numbers to the system of rational numbers.Review 6.NS.6- Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinatesInvestigation 2Squaring Off

Problems 2.1, 2.2, 2.3

Math Reflections

Expressions and Equations Work with radicals and integer exponents.8.EE.2- Use square root and cube root symbols to represent solutions to equations of the formx2 =pandx3 = p, wherepis a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.Investigation 3The Pythagorean Theorem

Problems 3.1, 3.2, 3.3, 3.4

Math Reflections

Understand and apply the Pythagorean Theorem8.EE.2-Use square root and cube root symbols to represent solutions to equations of the formx2 =pandx3 = p, wherepis a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.Understand and apply the Pythagorean Theorem 8.G.6- Explain a proof of the Pythagorean Theorem and its converse.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==8.G.8- Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.==

## Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 8.G.9- Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (ACE Problems)

Investigation 4Using the Pythagorean Theorem

Problems 4.1, 4.2, 4.3, 4.4

Math Reflections

Looking Back and Looking Ahead

Know that there are numbers that are not rational, and approximate them by rational numbers.8NS.1-Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.8.NS.2.-Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2).Understand and apply the Pythagorean Theorem8.EE.2-Use square root and cube root symbols to represent solutions to equations of the formx2 =pandx3 = p, wherepis a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.Understand and apply the Pythagorean Theorem.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.8.G.9 -Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (ACE problems)Prior Knowledge: Measuring lengths; working with coordinates; measuring areas of polygons and irregular figures and surface area of three-dimensional shapes; applying the formula for area of a square; formulating, reading, and interpreting symbolic rules, solving problems in geometric and algebraic contexts; understanding fractions and decimals; and representing fractions as decimals and decimals as fractions.Mathematical Practices Standards for Common Core1-Make sense of problems and persevere in solving them 2-Reason abstractly and quantitatively 3-Construct viable arguments and critique the reasoning of others 4-Model with mathematics 5-Use appropriate tools strategically 6-Attend to precision 7-Look for and make use of structure 8-Look for and express regularity in repeated reasoning

Essential Terms Developed in This UnitUseful Terms Referenced in This UnitTerms Developed in Previous UnitLegs

Pythagorean Theorem

Real numbers

Square roots

Irrational number

Isosceles triangle

Rational number

Repeating decimal

Terminating decimal

congruent perpendicular lines

coordinate grid quadrilateral

coordinates ratio

equilateral triangle right angle

length right triangle

parallel lines square

parallelogram symmetry

## Looking for Pythagoras

Pythagorean TheoremLooking for Pythagorasis filled with investigations that develop a fundamentally important relationship connecting geometry to algebra: the Pythagorean Theorem. Students are not merely introduced to a meaningless formula. To encourage deep understanding of what the theorem means, students explore squares created with various lengths, their areas and how these relate to the side lengths of right triangles. Students also explore square roots and strategies for estimating square roots. Irrational numbers are introduced and students are expected to be able to estimate where these occur on a number line.## For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Looking for Pythagoras.

Online resources for*Looking for Pythagoras## Perfect squares

## Square roots

## Proof of the theorem

## Exploring the theorem

## Proof without words

Other online resources## General Homework Tips

## Homework examples from Looking for Pythagoras

ResourcesLab-Sheet

Additional Practice/Skills Worksheets

CMP2 Website –online & technology resources

Formal Assessment

Check-Ups

Assessment Options

Notebook check

Parent Guide-Unit Letters

Spanish Assessment Resources

TeacherExpress CD-ROM

LessonLab Online Courses

Unit Technology Tips

Goals of the Unit• Relate the area of a square to the side length• Estimate the values of square roots of wholenumbers• Locate irrational numbers on a number line• Develop strategies for finding the distancebetween two points on a coordinate grid• Understand and apply the PythagoreanTheorem• Use the Pythagorean Theorem to solveeveryday problemsHomework Help