Assisting Teacher Handoff Sheet

Session 2.2 - Domain of Linear, Quadratic, and Radical Functions | Week 2, Thursday

FOR THE ASSISTING TEACHER: Read the information below before the instructor leaves at 2:00. You are supervising worksheet time from 2:00-3:00. Students work independently from 3:00-3:30.

What Students Just Learned (30-second summary)

Students learned three domain rules: Linear functions (like 3x + 2) have domain = all real numbers. Quadratic functions (like x² - 5x + 6) have domain = all real numbers. Radical functions (like √(x - 3)) have a restriction: the expression under the square root must be ≥ 0. To find the domain of a radical, set up an inequality and solve.

Three-Second Checklist:
Q: "Is there a square root?"
If NO → domain is all real numbers: (-∞, ∞)
If YES → set radicand ≥ 0, solve for x, write domain in interval notation.

The Worksheet (40 points, 9 problems across 4 parts)

Part A (Story): Identifying function types (linear, quadratic, radical) from real-world contexts. Students must distinguish mathematical domain from contextual (real-world) domain. 10 points.

Part B (Visual): Using Desmos graphing tool (https://www.desmos.com/calculator) to observe where graphs exist and where they have gaps. Students write domains based on visual observations. 8 points. Students need computer access to Desmos.

Part C (Traditional): Finding domain algebraically. For linear and quadratic: state domain = all reals. For radicals: set up inequality, solve, write interval notation. This part includes a tricky section (Problem 6) with quadratics under radicals that require factoring. 10 points.

Part D (Synthesis): Summary table and written explanation comparing the three types. 7 points. THIS IS AN EVEN SESSION: Problem 9 is a Weekly Reflection (5 points) asking students to integrate learning from Sessions 2.1 and 2.2. This is required.

Top 3 Mistakes to Watch For

1. Using ≠ instead of ≥ for radical domain.
Student writes: "f(x) = √(x - 5), so x ≠ 5."
Correction: "Remember: √0 = 0, which works fine! We need x - 5 ≥ 0, not ≠ 0. This is an inequality, not an exclusion. What is the domain?" (Guide them to x ≥ 5.)
Why it happens: Students confuse square roots with fractions (where you DO exclude the point).

2. Failing to factor when solving Problem 6 (quadratics under radicals).
Example: g(x) = √(x² - 9). Student writes "x ≥ 9" instead of solving x² - 9 ≥ 0 properly.
Correction: "This is a quadratic inequality. Let's factor: (x - 3)(x + 3) ≥ 0. When is a product of two numbers positive? When both are positive OR both are negative. Test x = 0: (0-3)(0+3) = -9. Is that ≥ 0? No, so x = 0 is out. Test x = 5: (5-3)(5+3) = 16. Yes. Test x = -5: (-5-3)(-5+3) = 16. Yes. So both sides work: x ≤ -3 or x ≥ 3. Domain: (-∞, -3] ∪ [3, ∞)."
Why it happens: Students treat it like a simple radical instead of solving a compound inequality.

3. Confusing bracket notation: [ ] vs ( ).
Student writes: f(x) = √(x - 5), domain = (5, ∞) instead of [5, ∞).
Correction: Point to the Quick Reference box at the top of the worksheet. "See the table? [ ] means includes the endpoint. ( ) means excludes. Since √0 is defined, we INCLUDE 5. Use [."
Why it happens: Abstract notation without context.

The Quick Reference Box (at the top of the worksheet)

The worksheet starts with a detailed Quick Reference that includes:

The three domain rules spelled out clearly
A table of interval notation with examples
Key explanations of brackets vs. parentheses

Encourage students to use this table liberally. They do NOT need to memorize interval notation. During grading, we evaluate understanding of domain concepts, not memorization of notation.

If a Student Is Stuck

Tier 1 (Quick help): Ask: "Does this function have a square root? If yes, what must be true about what's under the root?" Guide them to set up the inequality. Then say: "Solve for x."

Tier 2 (Graphical help): For any problem, have them type the function into Desmos. "Look at the graph. Where does it exist? Where does it start or stop? That's the domain."

Tier 3 (Conceptual reset): If they're stuck on Problem 6 (quadratics under radicals), say: "This is a product of factors. When is (x-3)(x+3) positive? Let me test a few values with you." Use x = -5, 0, 5 to show the pattern.

Tier 4 (Check the video): Direct them to the Reading file. It links to Organic Chemistry Tutor's 19-minute video "Domain of Square Root, Rational, Polynomial Functions." If they're confused on a specific type, this video may clarify.

Priority Problems If Running Low on Time

If students have only 30 minutes in the 2:00-3:00 block:

Must do: Problem 1 (Part A), Problem 3 (Part B with Desmos), Problem 5 (Part C), and Problem 9 (Weekly Reflection). These hit all four learning modes and the reflection is required on even sessions.
Nice to do: Problem 6 (quadratic under radical) - this is the most challenging and best for differentiation.
Skip if necessary: Detailed interval notation practice (Problem 4) and the full summary table (Problem 7 Part C). These can be reviewed later.

Weekly Reflection (Problem 9) - Required

Because this is an EVEN session, Problem 9 (Weekly Reflection) is REQUIRED. It asks students to integrate learning from Sessions 2.1 and 2.2 by explaining, in 3-5 sentences using math vocabulary, why linear and quadratic functions differ from radical functions in terms of domain.

If a student skips the reflection: Remind them: "The reflection is required today. Write 3-5 sentences explaining the difference between the domain of linear functions and radical functions. Use words like 'domain,' 'restriction,' 'radicand,' and 'inequality.'"

What to Watch Students Doing

Observable Behaviors During Worksheet Time
Behavior	What It Might Mean	Your Response
Student labeling all domains as "all real numbers"	They understand linear and quadratic but haven't grasped that radicals are different.	Ask: "Does this have a square root? What must be true about what's under it?"
Student repeatedly using ≠ instead of ≥	Confusion between division by zero (next session) and square root restrictions.	Show them: "√0 = 0. Zero works! We need ≥, not ≠. Why?"
Student skipping Desmos entirely	They prefer algebra and may not realize visual verification is powerful.	Say: "Let's type this into Desmos quickly. I want to show you something." Then graph it and point out the domain visually.
Student getting Problem 6 wrong	Standard struggle with factoring or solving compound inequalities.	Walk through factoring and testing values together.
Student leaving blank answers on problems	Could be confusion, time pressure, or just skipping.	Gently ask: "What's the question asking?" and guide them to start.

Session Handoff Notes for the Instructor (if checking in)

If Kaa Shaayi checks in during the 2:00-3:00 block, you can report:

How many students are on track to finish vs. struggling
Whether the biggest issue is understanding or algebra mechanics (factoring, inequality solving)
If anyone is noticeably ahead and ready for Problem 6 challenge
Whether Desmos access is working for all students

Common Success Indicators

Students showing understanding:

They quickly state "all real numbers" for linear and quadratic without overthinking
They set up the radicand ≥ 0 inequality automatically for radicals
They verify their algebraic answer by looking at the Desmos graph
They use the Quick Reference table confidently for interval notation
They attempt Problem 6 and show factoring/testing work even if the final answer isn't quite right