Sorting It All Out Mathematical Topics Subsets of real numbers and the relationships between these subsets Similarities and differences between rational and irrational numbers Convert equivalent forms of rational numbers Venn diagrams Terminating decimals and their fractional equivalents Repeating decimals and their fractional equivalents Ordering real numbers Required Materials Calculator Overhead of the given Venn diagram

STANDARD/ELEMENTS M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation. Elements: a. Find square roots of perfect squares. b. Recognize the (positive) square root of a number as a length of a side of a square with a given area. c. Recognize square roots as points and as lengths on a number line. d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign. e. Recognize and use the radical symbol to denote the positive square root of a positive number. f. Estimate square roots of positive numbers. g. Simplify, add, subtract, multiply, and divide expressions containing square roots. h. Distinguish between rational and irrational numbers. i. Simplify expressions containing integer exponents. j. Express and use numbers in scientific notation.

Work Period Complete odd numbers on 2-1 changing fractions to a decimal and decimal to a fraction

Sponge Rational or Irrational numbers in fractional form   Rational or Irrational numbers in decimal form 1/3 3/8 5/11 What pattern do you see?

Home Work 1Rational Numbers have a decimal expansion that a.) terminates or b.) doesn’t terminate a.) 3/4 b.) 2/3 True or False Every integer is a rational number Every rational number is a whole number Every natural number is a whole number d.) 3 is an element of the rational numbers Express the following rational numbers as decimals: a.) 5/11 b.) 10/11 c). -19/10000

Closing Determine which numbers in the set are natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

WORK PERIOD Use long division, without a calculator, to write each of the following rational numbers as a decimal. a. b. c. 9. Describe any patterns you observe in Question 8.

Sorting it All Out Prerequisite Knowledge Equivalent representations of simple fractions and decimals Knowledge of Meaning of the square root symbol Basic idea of Venn diagrams Basic equation-solving with one variable Concept of sets and subsets Definition of an even number Number lines

VENN DIAGRAM

Introduction Your teacher is going to use a Venn diagram like the one shown below to help you understand the relationships between different types of numbers. Your goal is to figure out the description for each of the circles and regions of the diagram. You will gain information by suggesting numbers and seeing their correct placement in the Venn diagram.

Begin by calling out a number for your teacher to place into the Venn diagram. Listen carefully to the number suggestions given by your classmates and try to determine how your teacher decides where to place each number in the diagram. As this activity progresses, see if you can guess where a number will be placed in the Venn diagram before your teacher shows the class. As an extra challenge, try to suggest a number that will be placed in a region of the Venn diagram that does not yet have many (or any) numbers. Think about what each region of the Venn diagram might represent. Be ready to give a description for each region in the diagram. Throughout this Unit, you will discover which of your predictions and descriptions are correct, and you will learn to identify and describe different sets of numbers.

Homework Rational Numbers have a decimal expansion that a.) terminates or b.) doesn’t terminate 3/4 2/3 EXAMPLE: True or False Every integer is a rational number Every rational number is a whole number Every natural number is a whole number d.) 3 is an element of the rational numbers

WORKPERIOD

Closing Which number is rational A.

Records remain that show that many ancient cultures used different systems of writing numbers. Most of these cultures used representations for the numbers 1, 2, 3, 4, 5, . . . This set of numbers is named the set of natural numbers or set of counting numbers and is often represented by the symbol N. 1. Why do you think the set of numbers {1, 2, 3, 4, 5, …} is named the set of natural or counting numbers?

Ancient cultures did not have a concept of the number 0 Ancient cultures did not have a concept of the number 0. One of the first cultures to have the full use of the concept of 0 was the Mayan culture. If you add the number 0 to the set of natural numbers, you form the set of whole numbers. The set of whole numbers is often represented by the symbol W.

2. Think about the set of whole numbers. and 2. Think about the set of whole numbers and the set of natural numbers. a. What are some similarities between the set of whole numbers and the set of natural numbers? b. What are some differences between the set of whole numbers and the set of natural numbers?

The numbers –3 and 3 are examples of opposites The numbers –3 and 3 are examples of opposites. On the number line, these two numbers are the same distance from 0, in opposite directions.

One of the greatest accomplishments of ancient Chinese mathematicians was recorded about 2000 years ago. The ancient Chinese mathematicians are noted for their use of negative numbers. The Chinese performed computations by manipulating counting rods – short rods approximately 10 centimeters long – on a table or counting board. Red rods represented positive numbers and black rods represented opposite or negative numbers.

3. When the set of whole numbers is combined 3. When the set of whole numbers is combined with the numbers’ opposites, the new set of numbers formed is made up of integers. The symbol Z is often used to represent the set of integers. a. What is the opposite of 52? b. What is the opposite of −312? c. What is the opposite of 0?

In the Venn diagram in the Introduction, the rectangle shown by region D contains what are referred to as the set of rational numbers. These numbers can all be written as a fraction or ratio of two integers. The set of rational numbers is often represented by the symbol Q.

Explain why natural numbers, whole numbers, and integers are all subsets of the set of rational numbers. Look back at the numbers your teacher placed in the original Venn diagram. a. List the numbers that were placed in region D, but not in regions A, B, or C. b. Describe those numbers you listed in Part (a).

Some of the numbers written in region D may have been expressed as the ratio of two integers and are rational numbers. Other numbers in region D may have been written as decimals.

7. Some of the decimals written in region D may have been terminating (or ending) decimals. In order for these decimals to be rational numbers, it must be possible to express these numbers as the ratio of two integers. a. Write 0.35 as a ratio of two integers. Express your answer in lowest terms. b. Write 2.004 as a ratio of two integers. Express your answer in lowest terms. c. Can all terminating (ending) decimals be written as fractions? Explain and give examples.

REVIEW

WORK PERIOD 10. Based on the patterns you listed in Question 9, complete the following table and then use a calculator to verify your results.

Because the repeating decimals shown in the table above can be expressed as fractions or the ratio of two integers, these repeating decimals are rational numbers. The following questions will demonstrate how a repeating decimal can be changed to a fraction.

11. Let a. Complete the equations below.   10x =   x = b. Subtract x from 10x and show the results from both sides of the equations from Part (a). c. Solve the resulting equation in Part (b) for x.

12. Let a. Complete the equations below. 100x = x = b 12. Let a. Complete the equations below.  100x = x = b. Subtract x from 100x and show the results from both sides of the equations from Part (a). c. Solve the resulting equation in Part (b) for x. 13. Do you think that all repeating decimals can be written as fractions? Explain your reasoning.

Part III We know that all natural numbers, whole numbers, integers, fractions, terminating decimals, and repeating decimals can be expressed as the ratio of two integers and thus are part of the set of rational numbers. The ancient Greeks once believed that all numbers were rational numbers. They also believed that rational numbers would fill in all of the points on the number line.

14. Use the number line to plot each rational number given below. SPONGE a. 1.5 b. -6/4 c. 0.2 d. .5          .

15. Find the coordinate of point A by. finding the length of the 15. Find the coordinate of point A by finding the length of the hypotenuse of the right triangle.

In Question 15, you found that the coordinate of point A is In Question 15, you found that the coordinate of point A is . The number cannot be written as the ratio of two integers. This means that there are more numbers than the rational numbers represented by points on the number line. The Pythagoreans, the students and followers of Pythagoras, were among the first to prove that is not rational.

The proof caused quite a crisis in the world of mathematicians in ancient Greece, as it had been thought that all numbers were rational. The Pythagoreans used a method called reductio ad absurdum, proof by contradiction, to argue that could not be written as the ratio of two integers.

16. First assume that is a rational number 16. First assume that is a rational number. Thus can be written as the ratio of two integers, , where a and b are integers that share no common factors. In other words, is a reduced fraction. a. Square both sides of the equation and write the result below. b. Multiply both sides of the resulting equation from Part (a) by the denominator of the fraction. What is the result?

c. Because , a2 must be an even number. Explain why this is true. d. Because a is an even number, for some integer p. Explain why this is true.

e. In Part (a), you determined that e. In Part (a), you determined that . Substitute 2p for a in this equation. f. Use the resulting equation from Part (e) to show that b must be an even number. Show your work. g. Why does the fact that both a and b are even numbers contradict the original statement in Question 16? Explain your reasoning.

Because a contradiction to the fact that a and b share no common factors was reached, the original assumption that is a rational number must be false. The number is not a rational number. It belongs to a set of numbers known as the set of irrational numbers. The set of irrational numbers is often represented by the symbol . 17. Can there be a terminating or repeating decimal representation of ? Explain why or why not. 18. Give some additional examples of irrational numbers.

The set of rational numbers, along with the set of irrational numbers, complete the number line. The set containing all of the rational and irrational numbers is called the set of real numbers. The set of real numbers is often represented by the symbol R.

19. The Venn diagram below represents all real numbers 19. The Venn diagram below represents all real numbers. Correctly label each region of the diagram with the symbol or name of a subset of the set of real numbers.

20. Use the Venn diagram above to help complete the table below by placing a check-mark under the name of each set of numbers to which the given number belongs.  

Sponge Aug 20 21. Name one number for each of the following criteria. a. A whole number but not a natural number b. An integer but not a whole number c. An integer and a natural number d. An irrational number e. A rational number but not an integer

Scoring Criteria Problem solving Exceed Expectation Meet Did not meet Gives multiple correct examples for each set of criteria. Gives correct examples for at least four of the five sets of criteria. Gives correct examples for fewer than four of the five sets of criteria.