## How Irrational!

In this lesson, students use geoboards to construct non-traditional, "tilted" squares whose side lengths are irrational numbers. This lesson addresses standards in both Number Sense and Measurement.

Begin by having students evaluate expressions involving square roots. Review the notation for square root and the geometric meaning of square root, which is the length of a side of a square with a given area.

Construct these triangles on a virtual geoboard. Provide students with the formula for the area of a triangle () and ask them to determine the areas of the displayed triangles. [12.5 units^{2} and 4.5 units^{2}.]

Have a student summarize the previous day's lesson, *In Search of Perfect Squares*.

Tell students that there are more squares that fit on a geoboard than those found during the previous lesson. Today's lesson will focus on finding some atypical squares using geoboards.

Project the How Irrational! Overhead. Discuss the questions on the overhead. Ask, "How can we
determine the area of the square?" [One possible method is to construct
a square that surrounds this square; determine the area of the four
corner triangles, and subtract the area of the large, outer square
minus the area of the four triangles.] Students may need a review on
finding the area of a triangle. Also, introduce the term *irrational number*.
An irrational number can be understood as the side length of a square
whose area is not a perfect square. For example, 5 is not a perfect
square. Therefore, the square root of 5 is an irrational number. In other words, there is no rational number that, when squared, is equal to 5.

Distribute the How Irrational Activity Sheet. Have students work in groups of four to find more squares with non-horizontal, non-vertical sides using geoboards. Although the students have their own geoboards, group interaction will promote communication and provide peer support for this challenging mathematical concept.

Show students this method for constructing a non-traditional square whose side has a slope of 3/2 on a geoboard. Place a rubber band on a geoboard peg. Stretch the rubber band "north" 3 units and "east" 2 units and hook the rubber band. Then rotate the geoboard 90° clockwise; and repeat 3 times.

For groups that have difficulty, consider providing pictures of atypical squares drawn on dot paper. Students could use the pictures to help them construct squares on their geoboards.

Circulate with a geoboard to check for understanding and provide additional support. As students complete the activity sheet, assign each group a specific type of square, categorized by the slope of its side, to present at the end of class. Each group should create a poster and include:

- A drawing of their assigned square.
- The area of the square, including their reasoning.
- The side length using square root notation.
- The side length rounded to the nearest whole number, including their reasoning.

As a class summary, groups of students should present their posters to the whole class. As each group presents their poster, ask questions to emphasize key concepts, such as, "How do you know your figure is a square?", "How did you measure the area of your square?" and "Explain how you estimated the side length of your square to the nearest whole number."

As a conclusion to the lesson, present the final square. (See *Family of Squares Whose Side has Slope of *below.)
This
is the first example we have seen of a square with non-vertical sides
that is perfectly square both geometrically and numerically. That is,
the side length of this atypical square is a whole number!

### Twenty-Seven Geoboard Squares, Organized by Type

A typical 11 × 11 geoboard (shown below) has 11 pegs in both the horizontal and vertical directions. On this type of geoboard, 27 different squares can be formed. These squares are categorized below, based on the slope of their sides.

On a 10 x 10 geoboard, there are 10 squares whose side lengths are whole numbers.

The areas of these squares are called *perfect squares*, because their side lengths are whole number measures.

The areas of the 10 squares, in ascending order, are as follows:

1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |

##### Family of Squares Where the Side has a Slope of 1

These squares are grouped together because the slope of their segments is 1. The grey square's side has a slope of 1, because the ratio of rise to run is .

Area | 2 | 8 | 18 | 32 | 50 |
---|---|---|---|---|---|

Side length | |||||

Whole number estimate | 1 | 3 | 4 | 6 | 7 |

##### Family of Squares Where the Side has a Slope of 2

These squares are grouped together because the slope of their segments is 2. The gray square's side has a slope of 2, because the ratio of rise to run is .

Area | 5 | 20 | 45 |
---|---|---|---|

Side length | |||

Whole number estimate | 2 | 4 | 7 |

##### Family of Squares Where the Side has a Slope of 3

These squares are grouped together because the slope of their segments is 3. The grey square's side has a slope of 3, because the ratio of rise to run is .

Area | 10 | 40 |
---|---|---|

Side length | ||

Whole number estimate | 3 | 6 |

##### Family of Squares Where the Side has a Slope of 4

These squares are grouped together because the slope of their segments is 4. The grey square's side has a slope of 4, because the ratio of rise to run is .

Area | 17 | 68 |
---|---|---|

Side length | ||

Whole number estimate | 4 | 8 |

##### Family of Squares Where the Side has a Slope of

These squares are grouped together because the slope of their segments is 3. The grey square's side has a slope of , because the ratio of rise to run is .

Area | 13 | 52 |
---|---|---|

Side length | ||

Whole number estimate | 4 | 7 |

##### Family of Squares Where the Side has a Slope of

The grey square's side has a slope of , because the ratio of rise to run is .

Area | 29 |
---|---|

Side length | |

Whole number estimate | 5 |

##### Family of Squares Where the Side has a Slope of

The grey square's side has a slope of , because the ratio of rise to run is .

Area | 34 |
---|---|

Side length | |

Whole number estimate | 6 |

##### Family of Squares Where the Side has a Slope of

The grey square's side has a slope of , because the ratio of rise to run is .

Area | 41 |
---|---|

Side length | |

Whole number estimate | 6 |

##### Family of Squares Where the Side has a Slope of [Reserve for Teacher Presentation]

The grey square's side has a slope of , because the ratio of rise to run is .

Area | 25 |
---|---|

Side length | |

Whole number measure (exact) | 5 |

### Reference

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., and Phillips, E. D. (2005). Looking for Pythagoras (Connected Mathematics 2). Developed as part of the Connected Mathematics Project. Upper Saddle River, NJ: Pearson Prentice Hall.

- 10 × 10 geoboards with elastics
- Internet access to a virtual geoboard or overhead geoboard
- How Irrational! Activity Sheet
- How Irrational! Overhead
- Dot Paper
- Chart or Poster Paper

**Assessment Options**

- Journal Entry: Have students compare and contrast the "perfect squares" they explored in the previous lesson with squares having irrational side lengths.
- Have students explain why the area of a square is a whole number, the side length is not necessarily a whole number.
[It is possible for a square to have an area of 2 square units, but the side length is not a whole number. The reason is that no whole number, multiplied by itself, equals 2. In other words, the square root of 2 is an irrational number.]

- Use a "One-Question Quiz" as a formative assessment and ask students to estimate the square root of 28 to the nearest whole number.
- As an entry in their notebooks, students define the term "irrational number" and include an example based on what they learned in class today.

**Extensions**

- Students interested in the history of mathematics could research the origins of irrational numbers and research why the discovery of irrational numbers disturbed the ancient Greeks.
- Investigate right triangle lengths without introducing the Pythagorean theorem, but instead, considering the hypotenuse as one side of a "non-traditional" square.

**Questions for Students**

[Break the square into parts.]

[Take the square root of the area.]

[Answers will vary. Encourage answers such as: the square root of 15 rounds to 4, since 15 is between 9 (3^{2}) and 16 (4^{2}), but closer to 16. Therefore, the square root of 15 rounds to 4.]

**Teacher Reflection**

- What parts of the lesson do you feel went well? Why?
- Did students become confused at any point in the lesson? When and why did this occur?
- Who was disengaged during the class discussion time? How could you involve more students in the future?
- Did you notice students making shapes on their geoboards that were not squares? What part of the "launch" or set-up phase could have been done differently?

### In Search of Perfect Squares

### Learning Objectives

Students will:

- Estimate square roots to the nearest whole number without a calculator.
- Relate squaring a number and finding square root as inverse operations.
- Define irrational number and give examples.

### NCTM Standards and Expectations

- Develop and analyze algorithms for computing with fractions, decimals, and integers and develop flue

- Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

- Use geometric models to represent and explain numerical and algebraic relationships.

- Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

### Common Core State Standards – Mathematics

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.A.2

Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = 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.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.