What Is a Function? Input, Output, and the One-to-One Rule

What Is a Function? Input, Output, and the One-to-One Rule

Grade 8 · Math · NYS NY-8.F.1 · 55 Minutes

Math Teacher Review Pilot: This resource is eligible for review by a NYS-certified Mathematics teacher.

NYS-Aligned Standard

NY-8.F.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. NYS Next Generation Mathematics Learning Standards (2017)

Learning Objectives — “I Can” Statements

• I can explain what a function is using the “one input → one output” rule.
• I can identify whether a table, graph, or set of ordered pairs represents a function.
• I can plot ordered pairs from a function and describe the graph.
• I can use the vertical line test on a graph to determine if it is a function.

Essential Question

What makes a relationship a function — and why does the “one output per input” rule matter?

Lesson Sequence

Hook / Warm-Up (8 minutes)

1. “Vending Machine” analogy: Show a vending machine image. “If I press A3, I get exactly one item. Can pressing A3 give me TWO different items? What if it gave chips sometimes and soda other times?”
2. “That would be a BAD machine. A function is like a RELIABLE vending machine — one input always gives EXACTLY one output.”
3. Present two tables: Function vs. Not-a-Function (one input maps to two outputs). Let students spot the difference.

Direct Instruction (12 minutes)

1. Define: Function = a rule where each input has exactly one output
2. Representations: table, ordered pairs, mapping diagram, graph
3. Mapping diagram: draw arrows from input to output. “If any arrow goes from one input to TWO outputs, it’s NOT a function.”
4. Ordered pairs: {(1,3), (2,5), (3,7), (2,8)} — is this a function? No! Input 2 maps to both 5 and 8.
5. Graph and vertical line test: draw a vertical line — if it crosses the graph MORE than once, not a function.

Guided Practice (12 minutes)

1. Give students 4 mapping diagrams; they determine function or not-a-function, explain why.
2. Convert a table of values to ordered pairs; plot on coordinate grid.
3. Apply vertical line test to 3 pre-drawn graphs.

Independent Practice (14 minutes)

Students complete the Functions Practice Worksheet: determine function/not-a-function from tables, ordered pairs, mapping diagrams, and graphs; write ordered pairs from a rule (y = x + 3).

Closure (9 minutes)

1. Return to vending machine: “Can you think of a real-world relationship that is NOT a function? Why?”
2. Exit ticket: “Given {(2,5), (4,5), (6,7)} — is this a function? Explain.”

SDI & Differentiation Block

Supports for MLLs/ELLs

Entering/Emerging (NYSESLAT Levels 1–2):

• Vocabulary card: function, input, output, rule, ordered pair, graph — with visual examples
• Focus on mapping diagrams only (most visual representation); arrows make the one-to-one relationship visually clear
• Sentence frame: “This ___ (is / is not) a function because each input has ___ output(s).”
• Use physical arrows or yarn to build mapping diagrams as a manipulative

Transitioning/Expanding (NYSESLAT Levels 3–4):

• Pre-teach: input/output using the vending machine analogy in writing
• Academic vocabulary: mapping, diagram, ordered pair, vertical line test
• Allow student to complete verbal explanation before written response

Supports for Students with IEPs

SDI Adaptation Dimensions: content, methodology, delivery

• Content: Focus on function identification from tables only (most structured); reduce ordered pairs task to 3 items; skip vertical line test if graphing is a barrier
• Methodology: Provide a function checker card: “Does any input repeat with a different output? YES = not a function. NO = function.” Use color-coding: highlight same inputs in yellow to check
• Delivery: Allow extended time; provide pre-drawn coordinate grid with labeled axes; allow verbal explanation; reduce number of problems

Suggested Placement: ICT, Resource Room

Extensions for Advanced Learners

• Connect to function notation: f(x) = x + 3; evaluate for f(2), f(5)
• Research: Why does a circle graph fail the vertical line test? Explain using the definition.
• Explore NY-8.F.2: Compare properties of two functions represented in different ways

Answer Key

Exit ticket: {(2,5), (4,5), (6,7)} — YES, this IS a function. Each input (2, 4, 6) maps to exactly one output. Having the same output for different inputs is allowed — it’s having the same input for different outputs that breaks the rule.

Alignment Record