Function Launch Lesson Plan Lesson Plan Algebra 1 • Unit 3: Functions
Function Launch: What is a Function?
Pacing: 35 Minutes
Target Grade: 8th / 9th Grade
NYS Next Generation Mathematics Standards
AI-F.IF.1: Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range. Represent relations as ordered pairs, tables, mapping diagrams, and graphs.
SMPs: MP.2 (Abstract Reasoning), MP.3 (Construct Arguments), MP.6 (Precision)
Student Learning Objectives
Define relation , function , domain , and range .
Determine if a relation is a function from sets, tables, mappings, and graphs.
Apply the Vertical Line Test (VLT) to graphs.
Critical Misconception Alert
Students often assume outputs cannot repeat. Emphasize: Multiple inputs CAN yield the same output (e.g., \( (2, 5) \) and \( (3, 5) \) is a function!). A function only fails when one input yields two different outputs .
Academic Vocabulary & Key Phrases
Input / Domain (\(x\)) : Independent values.Output / Range (\(y\)) : Dependent values.Function Rule : "Every input has exactly one output."
Pacing Guide & Instructional Delivery (Part 1 of 2)
Phase 1: Bell Ringer & Concept Hook (5 Minutes)
0:00 – 0:05
Activity: Students analyze "The Broken Vending Machine" on page 1 of the student packet.
Teacher Facilitation: "If you press button B3 and get a Sprite, press B3 again and get a Sprite, is the machine working? Yes. But what if you press B3 and get a Diet Coke? Can one button reliably give two different drinks?"
Key Takeaway: A function is a reliable mathematical machine. Each button (input) must lead to an exact, predictable drink (output).
Phase 2: Direct Instruction & Step-by-Step Modeling (10 Minutes)
0:05 – 0:15
Concept Walkthrough: Connect vending machines to mathematical relations across 4 key formats.
1. Ordered Pairs & Tables: Model inspecting \(x\)-values first. If all \(x\)-coordinates are distinct, it is automatically a function. If an \(x\) repeats, compare the \(y\)-coordinates.
2. Mapping Diagrams: Teach students to count arrows leaving each input circle. Exactly 1 arrow leaving every input = Function. More than 1 arrow from an input = NOT a function. Multiple arrows arriving at an output is completely valid!
3. Graphs & Vertical Line Test (VLT): Demonstrate placing a ruler or pencil vertically across the graph. If any vertical line intersects the curve more than once, it fails because that single \(x\)-value corresponds to multiple \(y\)-values.
Algebra 1 • Function Launch Lesson Plan Page 1 of 2
Algebra 1 • Unit 3: Functions
Instructional Delivery & Assessment Protocol
35-Minute Fast-Paced Period
Phase 3: Active Independent & Partner Practice (13 Minutes)
0:15 – 0:28
Students complete Problems 1–6 on page 2 of the student packet.
Teacher Circulation Checklist:
Are students writing down explicit reasoning, not just guessing "Yes/No"?
Spot-check Problem 2 (identical \(y\)-values): verify students recognize this IS a function.
Have students physically draw vertical lines on Problems 5 and 6.
Prompting Questions:
"Which column represents your inputs? Are any inputs used twice?"
"If I give you an \(x\)-value of 4, do you know with 100% certainty what \(y\) is?"
"Where does your vertical pencil hit two points simultaneously?"
Phase 4: Diagnostic Exit Ticket & Wrap-Up (7 Minutes)
0:28 – 0:35
0:28 – 0:33 (5 min): Independent, silent completion of the tear-off Exit Ticket at the bottom of Page 2. Measures mastery of (1) table evaluation with justification and (2) graphic evaluation using VLT.
0:33 – 0:35 (2 min): Collect tickets at the door. Have students recite the daily takeaway unison chant: "Every input has exactly one output!"
Targeted Differentiation & Scaffolding
Support / Intervention
Provide highlighters to mark all \(x\)-values in yellow. If yellow repeats with different values, immediately circle "NOT a function." Use clear acetate sheets with a vertical line for graphs.
MLL / ELL Supports
Anchor chart cognates: Función (Function) , Dominio (Domain) , Rango (Range) . Pair with sentence stems: "This relation is / is not a function because the input ___ has ___ outputs."
Extension / Early Finishers
Assign Error Analysis (Marcus's claim). Challenge students: "Can a vertical line on a graph ever be a function? Why or why not?" Introduce the Horizontal Line Test preview (one-to-one functions).
Exit Ticket Diagnostic Criteria (2 Points Total)
Item 1 (Table Evaluation): Student correctly identifies it is NOT a function AND specifically notes that input \( x = 3 \) maps to two different outputs (7 and 12).
Item 2 (Graph & VLT): Student correctly states whether the circle graph is a function (No) and explains that a vertical line passes through two points simultaneously.
Algebra 1 • Function Launch Lesson Plan Page 2 of 2
Function Machine Student Packet Algebra 1 • Unit 3: Functions (NYS AI-F.IF.1)
Lesson 1: Introduction to Functions
Name:
Date:
Period:
Part 1: Bell Ringer — The Vending Machine Dilemma (5 Min)
Warm-Up
Imagine you visit a drink vending machine in the school cafeteria. You press a button (Input) and receive a can (Output).
Scenario A: Machine #1
Press B2 → Grape Soda. Press B2 again → Grape Soda. Press C1 → Water.
Is this machine functioning properly?
Yes No
Scenario B: Machine #2
Press B4 → Lemonade. Press B4 again → Diet Cola. Press B4 again → Iced Tea.
Is this machine functioning properly?
Yes No
Key Question: In your own words, what makes a machine "work correctly"?
Part 2: Core Concept & Mathematical Rule
Relation
Any set of ordered pairs \( (x, y) \). A relationship between inputs and outputs.
Domain & Range
Domain = Set of all inputs (\(x\)).Range = Set of all outputs (\(y\)).
THE FUNCTION RULE
Every input (\(x\)) must have EXACTLY ONE output (\(y\)).
Part 3: Guided Instruction — Step-by-Step Model Examples
Teacher-Led Instruction
Model 1: Ordered Pairs & Tables Test: Check Inputs (\(x\))
Relation A:
Relation B:
Step-by-Step Decision:
Rel A: Function? YES (All \(x\)-values unique).
Rel B: Function? NO (Input \(x=2\) gives two outputs: 4 and 9).
Model 2: Mapping Diagrams Count Output Arrows
Domain (x) 1 2 3 Range (y) 6 8
Is this a function? YES!
Every input has exactly 1 arrow leaving it. Both 1 and 2 map to 6, which is allowed!
Model 3: Vertical Line Test (VLT) Pencil Test
Passes VLT (1 hit)
Fails VLT (2 hits)
Rule: If a vertical line touches the graph at more than one point, it is NOT a function.
Algebra 1 • NYS AI-F.IF.1 Standards Page 1 of 2
Algebra 1 • Unit 3: Lesson 1
Part 4: Independent Practice (13 Min)
Determine whether each relation is a function. Justify your reasoning!
1. Relation as a Table [Table Check]
Function? Yes No
Why / Why not?
2. Set of Ordered Pairs [Set Notation]
\{ (-2, 4), (-1, 4), (0, 4), (3, 4) \}
Function? Yes No
Explain the output (\(y\)) values:
3. Mapping Diagram [Arrows]
Input 4 7 9 Output 10 15
Function? Yes No
Justification:
4. Parameter Challenge Critical Thinking
Given relation: \( \{ (1, 6), (4, 9), (k, 12) \} \)
a) Choose a value of \(k\) so the relation is NOT a function: \(k =\) _______ because:
b) Choose a value of \(k\) so the relation IS a function: \(k =\) _______
5. Graph A
Function Not Function
VLT Explanation:
6. Graph B
Function Not Function
VLT Explanation:
Error Analysis: Critique Marcus's Reasoning
Marcus looks at the table \( \{ (1, 5), (2, 8), (3, 5), (4, 9) \} \) and says: "This is NOT a function because the number 5 appears twice."
Is Marcus correct? Correct his misunderstanding:
Exit Ticket: 5-Minute Function Checkpoint
Detach & Hand In
Name:
Period:
Score: ___ / 2
1. Is this table a function?
Yes No
Explain:
2. Is this graph a function?
Yes No
Reason (VLT):
Algebra 1 • Function Machine Packet Page 2 of 2
Function Machine Answer Key Teacher Answer Key Algebra 1 • Unit 3: Lesson 1
Introduction to Functions: Complete Solutions & Annotations
NYS AI-F.IF.1
Part 1: Bell Ringer Solutions & Teacher Debrief (5 Min)
Solutions
Scenario A: Machine #1
Press B2 → Grape Soda. Press B2 again → Grape Soda.
Functioning properly? YES
Reasoning: Every time button B2 is pressed, you get the exact same predicted outcome.
Scenario B: Machine #2
Press B4 → Lemonade, Diet Cola, Iced Tea.
Functioning properly? NO (Broken)
Reasoning: One button produces multiple unpredictable drinks. A user cannot rely on it!
Key Question Debrief: A machine "works correctly" when each button (input) guarantees exactly one specific item (output). In mathematics, this reliability is the foundational definition of a function .
Part 2: Essential Vocabulary Check
Relation: Any pairing of \(x\) and \(y\).
Domain / Range: Domain = Inputs (\(x\)); Range = Outputs (\(y\)).
Function Rule: Every single input has exactly one output.
Part 3: Guided Instruction Solutions & Step-by-Step Pedagogy
Teacher Reference
Model 1: Ordered Pairs & Tables
Relation A: FUNCTION: YES
Inputs: \(\{1, 2, 3, 4\}\). Outputs: \(\{5, 10, 15, 20\}\).
Explanation: Every \(x\)-value is distinct. No input is paired with more than one output.
Relation B: FUNCTION: NO
Inputs: \(\{2, 3, 2, 5\}\). Outputs: \(\{4, 7, 9, 11\}\).
Explanation: The input \(x = 2\) is associated with two different outputs (\(y = 4\) and \(y = 9\)).
Model 2: Mapping Diagrams FUNCTION: YES
Domain: \(\{1, 2, 3\}\) | Range: \(\{6, 8\}\)
Why it passes: Exactly one arrow leaves each number in the domain (1→6, 2→6, 3→8). Multiple inputs pointing to output 6 is completely allowed!
Model 3: Vertical Line Test Pencil Test
Graph A (Parabola): Function (Yes). Any vertical line intersects at most 1 point.
Graph B (Circle): Not a Function (No). Vertical lines intersect at 2 points.
Algebra 1 • Function Machine Answer Key Page 1 of 2
Teacher Answer Key & Diagnostic Guide
Part 4: Independent Practice & Exit Ticket Solutions
Unit 3 • Lesson 1
1. Table Relation FUNCTION: YES
Domain: \(\{-3, 0, 2, 5\}\) | Range: \(\{1, 4, 9\}\)
Accepted Justification: All inputs (\(x\)-values) are distinct. Even though \(y = 1\) occurs twice (for \(x = 0\) and \(x = 2\)), each input has exactly one output.
2. Ordered Pairs \(\{ (-2,4), (-1,4), (0,4), (3,4) \}\) FUNCTION: YES
Domain: \(\{-2, -1, 0, 3\}\) | Range: \(\{4\}\)
Accepted Justification: Every \(x\)-value is different. This is a constant horizontal line function (\(y = 4\)). Repeating \(y\)-values does NOT violate the definition of a function.
3. Mapping Diagram FUNCTION: NO
Domain: \(\{4, 7, 9\}\) | Range: \(\{10, 15\}\)
Accepted Justification: Input \(x = 4\) has two arrows leaving it (it maps to both 10 and 15). One input cannot have two outputs.
4. Parameter Challenge ALGEBRAIC REASONING
a) NOT a function: \(k = 1\) or \(k = 4\)Reason: If \(k = 1\), the input 1 has outputs 6 and 12. If \(k = 4\), input 4 has outputs 9 and 12.
b) IS a function: Any number except 1 or 4 (e.g., \(k = 0, 2, 3, 5\)).
5. Graph A (Absolute Value) FUNCTION: YES
Passes the Vertical Line Test.
Accepted Justification: Any vertical line drawn across the coordinate plane will intersect the V-shaped curve at no more than one point.
6. Graph B (Sideways Parabola) FUNCTION: NO
Fails the Vertical Line Test.
Accepted Justification: A vertical line drawn through \(x > 0\) intersects the curve at two distinct points, meaning that input \(x\) has two outputs.
Part 5: Error Analysis Solution — Marcus's Claim
Correction
Marcus is INCORRECT. The relation \( \{ (1, 5), (2, 8), (3, 5), (4, 9) \} \) IS a function.input (\(x\)-value) repeats with different outputs.
Part 6: Exit Ticket Solutions & Diagnostic Scoring Rubric
Total: 2 Points
1. Table Question Answer: NO
1 Point Criteria: Student circles "No" AND writes that input \(x = 3\) has two different outputs (7 and 12).
Remediation: If student circled "Yes," remind them to scan the \(x\)-row for repeated numbers first.
2. Circle Graph Answer: NO
1 Point Criteria: Student circles "No" AND explains that a vertical line passes through the circle at two points (fails VLT).
Remediation: Demonstrate drawing a vertical line right through the center of the circle to highlight the two intersections.
Algebra 1 • Function Machine Answer Key Page 2 of 2
