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Multiple Choice Quiz
1. Which of the following characteristics best defines an irrational number according to the document’s implied definitions?
- A number that can always be written as a fraction of two integers.
- A number whose decimal representation is terminating or repeating.
- A number whose decimal representation is non-terminating and non-repeating.
- Any real number that is not a positive integer.
View Answer & Explanation
Correct Answer: A number whose decimal representation is non-terminating and non-repeating.
Explanation: Page T-3, in the ‘Common issues’ table, clarifies that ‘An irrational number can never be written as a fraction of integers. It always has a non-repeating non-terminating decimal.’ This directly points to the decimal representation as the distinguishing characteristic.
2. When assessing students’ initial individual work on the ‘Rational or Irrational?’ task, what does the teacher guide strongly recommend not doing?
- Collect students’ written work for review.
- Make informal notes on student difficulties.
- Score students’ work with grades.
- Formulate questions to help students improve their solutions.
View Answer & Explanation
Correct Answer: Score students’ work with grades.
Explanation: Page T-2, under ‘Assessing students’ responses,’ explicitly states: ‘We strongly suggest that you do not score students’ work, as research shows that this is counterproductive.‘
3. What is required to conclusively establish that a statement is ‘Sometimes True’ during the collaborative task?
- A formal mathematical proof demonstrating its conditional nature.
- Finding at least one example for which the statement is true and at least one example for which it is false.
- Observing that the statement holds for most, but not all, of the examples tried.
- A majority vote within the small group, supported by various examples.
View Answer & Explanation
Correct Answer: Finding at least one example for which the statement is true and at least one example for which it is false.
Explanation: Page T-4, in the ‘Suggested Lesson Outline’ under ‘Now explore the reasoning involved in the task,’ explicitly details: ‘What did you need to do to show the statement was sometimes true? [Find an example for which the statement is true and an example for which it is false. This is established with certainty once there is one true example and one false example.]‘
4. A student tests the statement ‘The sum of two irrational numbers is always irrational.’ They use (irrational) and (irrational). Based on these examples, the student concludes the statement is ‘Always True.’ What common issue does this student’s reasoning demonstrate?
- Does not properly distinguish between rational and irrational numbers.
- Does not identify relevant formulas to use.
- Empirical reasoning.
- Does not provide examples.
View Answer & Explanation
Correct Answer: Empirical reasoning.
Explanation: Page T-3, in the ‘Common issues’ table, describes ‘Empirical reasoning’ as when a student ‘provides examples that show the statement is not true and concludes that there are no values of a, b that make the statement true.’ While the given examples for sum of irrationals don’t show a ‘false’ case, the underlying issue is concluding ‘Always True’ based on a limited number of examples, rather than requiring a general proof, which is a hallmark of empirical reasoning in this context. For example, , which is rational, proving the statement is only ‘Sometimes True’.
5. What is the primary content standard from the Common Core State Standards that this lesson unit addresses?
- F-IF: Analyze functions using different representations.
- N-RN: Use properties of rational and irrational numbers.
- A-CED: Create equations that describe numbers or relationships.
- G-SRT: Understand similarity in terms of similarity transformations.
View Answer & Explanation
Correct Answer: N-RN: Use properties of rational and irrational numbers.
Explanation: Page T-1, under ‘COMMON CORE STATE STANDARDS,’ clearly states: ‘This lesson relates to the following Standards for Mathematical Content… N-RN: Use properties of rational and irrational numbers.’
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