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* *Reflection: Vertical Alignment
The Real Number System - Section 1: Warm up and Homework Review

These warm ups are a reinforcement that this lesson was necessary for my class this year. As seen in all six samples, taken from students who exhibit a variety of confidence levels, they were missing many of the concepts foundational to understanding imaginary numbers. This is partially due to the transition to Common Core, but I think it also has to do with individual student knowledge and experience. It has been two years since my students saw number systems which is plenty of time to lose mastery of a subject. It is always important to ensure that students have the adequate background although the balance must be maintained between this and keeping to the Algebra 2 standards.

*Vertical Alignment: The Need for Scaffolding Lessons*

# The Real Number System

Lesson 1 of 12

## Objective: Students will be able to classify numbers in the real number system.

*45 minutes*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’s Warm Up- Real Number System is meant to access prior knowledge on number systems. We will use this as a spring board for the day's lesson.

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This lesson is a foundation topic meant to prepare students to enter the complex number system. The real number system was addressed in prior years but I feel that it is important enough to look at again to ensure that all students will be coming into the next couple of lessons from the right place.

My hook for this activity is a video from teacher tube on the history of numbers. It is about 7 minutes and provides a nice survey of the development of numbers into our current number systems.

Next, we build a diagram that shows the connections between the different sets of numbers. As we build this diagram, we relate the numbers back to the historical perspectives in the video as well as the students' experiences learning numbers as they grew up.

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#### Classifying Numbers Activity

*20 min*

Each pair receive a set of cards with the names of each number system: natural, whole, integer, rational, irrational, and real; each of which have been printed on separate colored paper. As I pull up the numbers on the PowerPoint, the pairs will hold up the number systems that apply to that number (**Math Practice 2**). I will then call on a student to explain a portion of their solution. For example, if the number is 12/4, I may ask a student why they chose to put this in natural numbers. Please watch the Video Narrative for additional thoughts on this activity.

An alternate method is to make this a race where the first 3 or 5 groups get a point for holding up the correct systems.

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#### Exit Ticket

*3 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's lesson asks students to determine a number that is a rational number but not an integer.

#### Resources

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Today’s homework allows the students to reaffirm their knowledge of the separate number systems within Real Numbers. The first problem gives them several numbers to categorize within the number systems. The remainder of the problems asks the students to come up with numbers that fit into certain systems but not others (**Math Practice 1**).

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: The Real Number System
- LESSON 2: Simplifying Square Roots
- LESSON 3: Imaginary Numbers Day 1 of 2
- LESSON 4: Imaginary Numbers Day 2 of 2
- LESSON 5: Complex Numbers Day 1 of 2
- LESSON 6: Complex Numbers Day 2 of 2
- LESSON 7: Completing the Square Day 1 of 2
- LESSON 8: Completing the Square Day 2 of 2
- LESSON 9: Real and Complex Number System Quiz
- LESSON 10: Quadratic Formula
- LESSON 11: Complex Numbers and Quadratic Equations Review
- LESSON 12: Complex Numbers and Quadratic Equations Test