Exploring Irrational Numbers on a Geoboard TASEL-M August 2006 Institute Pre-Algebra Lesson Mark Ellis, CSUF

CA Content Standards Geometry & Measurement 3.3: Know and understand the Pythagorean Theorem and its converse and use it to find the length of the missing side of a right triangle. Number Sense 1.4: Differentiate between Rational and Irrational Numbers Number Sense 2.3: Use the inverse relationship between raising to a power and extracting the root of a perfect square integer; for an integer that is not square, determine without a calculator the two integers between which its square root lies and explain why.

Goals and Objectives Goals  Students will learn to apply the Pythagorean Theorem to find a missing side length of a right triangle (distance between two pegs on a geoboard)  Students will learn to identify the square root of a perfect square as rational and the square root of a non-perfect square as irrational.  Students will better understand the concept of square root by visualizing and estimating various square root measures as a distance between two pegs on a geoboard. Objectives  Given a segment on dot paper or a geoboard that spans two points, students can find its length and state whether it is rational or irrational.  Given a right triangle and the lengths of two legs, students can find the length of the hypotenuse using the Pythagorean Theorem.  Given an irrational number that is a square root, students can estimate its value between two integers.

Warm Up 1. Find the length of the hypotenuse for each of the right triangles shown. 2. Find the length of 1 side of a square with an area of 36 square units.

Warm Up Note: A line segment must start and end on a peg, not in between pegs. 1. Make a line segment that touches 3 pegs. What is the length of this line segment? 2. Make another line segment that touches 3 pegs, but with a different length. Explain how you know whether this new line segment is longer or shorter than the one in #1. 3. Find the shortest line segment that can be made on this board. 4. Find the longest line segment. How do you know it’s the longest?

Lesson Introduction On your own, write two rational and two irrational numbers, each one on a sheet of Post-It paper. Discuss with a partner why you selected the numbers and how you know whether they are rational or irrational. Stick your numbers on either the Rational or Irrational number poster. What do we know about rational and irrational numbers?

Task Description Find the length of all unique line segments that can be made on a 5-peg by 5-peg geoboard.  What are some initial thoughts you have about this? Discuss with a partner.

Using the Geoboard What is a Geoboard?  Set of pegs in a square pattern on which rubber bands are placed to explore geometric concepts and properties Use tools properly!  The owner of a rubber band pistol was recently arrested because it was a W.M.D. WEAPON OF MATH DISRUPTION

Activity Directions On a 5 peg by 5 peg geoboard find all unique line segments. Record the lengths of each line segment on the Activity Sheet, including a picture. For non-integer lengths, estimate their lengths as best you can. Can you find an exact value for these? Put the lengths in order from shortest to longest.

Sharing Strategies Virtual Geoboard How can the Pythagorean Theorem help?

Follow Up Questions 1. How many unique lengths are there? 2. Which of the lengths are rational? 3. Which are irrational? How do you know this? 4. Which of the irrational lengths are “related” to each other? How are they related? 5. Write an equivalent expression for: a. b. c. How can you prove mathematically that your expressions are equivalent?

Closure Find the lengths of the two line segments to the right. Give an exact value and an estimation. Using the terms “rational” and “irrational” write two sentences about the lengths of the two line segments.

Homework 1. Using a 10 by 10 geoboard grid, create a quilt or stained glass window design with at least 5 intersecting line segments. For each shape in the design, name it with a letter (e.g., shape A) and find the lengths of its sides and its perimeter (keep irrational values as square roots). 2. Make a square with an area of 4 square units on the geoboard or dot paper. What is the length of 1 side of this square? Prove that you are correct. 3. Make a square with an area of 8 square units on the geoboard or dot paper. What is the length of 1 side of this square? Prove that you are correct.