Rational or Irrational (Day 1 of 2)
Lesson 6 of 8
Objective: SWBAT differentiate between rational and irrational numbers.
For this activity, I pair up students and ask each pair to answer a prompt that I have written on the board:
Write everything you can remember about the following types of numbers.
- Whole numbers
- Rational numbers
I ask students to record their answer on a sheet of notebook paper. Students always ask if they can write examples. I allow some examples to be written, but I encourage students to write explanations as well.
Once they are done brainstorming, I have volunteer groups share what they wrote. I divide the board into 3 columns (Whole Numbers, Integers, Rational Numbers) and students go up and share their ideas.
Common mistake to look out for: Students usually think that 0 is not part of the whole number set or the set of integers. I always look out for this and restate that 0 is in both sets of numbers.
After discussing student's responses, and if not already written on the board, I add the following to the 3 columns.
Whole numbers: Sometimes called Counting numbers
Integers: All the whole numbers and their opposites
Rational numbers: Comes from the word "ratio" A number that can be written as a ratio of integers
Although they have worked with them for several years, many students struggle with the meaning of rational numbers. I've found an approach to defining rational numbers that helps most students to internalize the meaning of "a rational number":
I begin by writing the numbers below on the board and saying, let's use these four examples to define a rational number:
½ = 0.50000…
3.25 = 3.2500000…
1/6 = 1.6666666…..
25/99 = 0.2525252525….
Then, I ask the students to infer a definition from the evidence. In most cases, students conclude that a rational number is a repeating decimal. I usually have to add the idea that numbers in which the repeating digit is 0 are called terminating and can be written without the repeating string of 0's.
At this point I hand each pair a Real number Venn activity sheet. Students should answer the questions first, then complete the diagram writing the words Real, Whole, Integers, and Rational on the lines. They should fill in the parenthesis with examples of numbers that correspond with their sets. I motivate them to use different values than the ones on the board. Students may not be able to complete the right hand side of the venn (irrational numbers), or they may place incorrect values in. I tell them it's ok, they will complete it or make corrections during Day 2 of this lesson.
Teacher's Note: Some students choose to complete the diagram before answering the questions. This may actually make it easier to answer some of the questions.
Question 3 leads to the most confusion. I find that students often argue that there is a pattern in the numbers at 3a and 3c. When discussing Question 3, I focus on the following two points:
- The decimal expansion of a rational number repeats the same finite sequence of digits over and over.
- These two numbers cannot be expressed as a ratio of two integers.
As an extension of the conversation, I show students this Khan Academy video describing how numbers with repeating digits are converted into fractions:
To end Day 1 of the lesson, I call on students to share their answers to the questions on the activity sheet. During this discussion, student questions on the this section usually pop up. It's important that doubts are cleared up before Day 2 of the lesson.
I finally project or draw the Venn Diagram on the board, whichever is easier, and ask student volunteers to go up and complete the diagram as they did in the activity sheet. Students will ask about the right hand side of the Venn Diagram. Many may even already know what belongs there. The next part of the lesson will address this.
As class ends, I ask students to make corrections on their the activity sheet and then to save the worksheet for use in tomorrow's lesson.