Lesson 5 — Functions: How One Quantity Depends on Another

Summary

Learners define a function as a rule that assigns each input exactly one output, and learn to read and evaluate function notation f(x). They move between a rule, a table, a graph, and a story, and name the independent and dependent variables.

Objectives

• Define, evaluate, and compare functions, and use function notation to describe how one quantity depends on another. (D05.S2.08.01)

Connection

How far you travel depends on how long you move; what a basket of rice costs depends on how much you buy; how tall a seedling stands depends on the days of sun and water it has had. Each is a rule: put in one quantity, get out exactly one other. Mathematicians name that rule a function and write it compactly — “f of x” — so anyone can predict the output before it happens. A market price, a crop’s growth, a medicine’s dose: functions are how dependence becomes a language.

Materials

• Function machine sheet
• Function cards

Preparation

• Copy the function machine sheet and function cards.
• Have worked examples ready: f(x) = 2x + 1; evaluate f(3) = 7; a table (0,1), (1,3), (2,5), (3,7); a story “a flat fee of 1 plus 2 per item.”
• Recall from Lesson 1 and Grade 7: y = kx is a special function (k times x).

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a function is a rule that gives each input exactly one output; (2) function notation f(x) reads “f of x” and means the output when the input is x — f(3) = 7 says “input 3 → output 7”; (3) the input is the independent variable and the output the dependent variable, because the output depends on the input. Because reading and evaluating f(x) is a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of reading f(x) as “f times x” and of thinking a function can give two outputs for one input (it cannot, by definition). The technology lens: a graphing tool makes the rule-table-graph connection visible in one move — but the understanding is in the learner, not the tool. The egalitarian lens: a function is a compact, portable truth — once you can read f(x), you can check any claim that says “this depends on that,” whoever makes it (philosophy §4). The global lens: describing how one quantity depends on another appears across traditions (e.g., the proportional “per one” reasoning of the Rhind Papyrus, S-288) and grows into the modern universal form. Retrieval: y = kx (Grade 7) is the special function where the output is a constant multiple of the input.

Procedure

1. Gather (5 min). Last time you held the very large and very small. Today you describe how one thing depends on another.
2. Meet the function (10 min). A function is a rule that gives each input exactly one output. The input is the independent variable (x); the output is the dependent variable (f(x)) — it depends on x.
3. Read the notation (10 min). f(x) = 2x + 1 reads “f of x equals 2x + 1.” Evaluate: f(3) = 2(3) + 1 = 7. Build the table: (0,1), (1,3), (2,5), (3,7).
4. Match rule, table, graph, story (15 min). With a partner, match each function card’s rule to its table, graph, and story. Say why they belong together, then check with another pair.
5. Close (10 min). Share one function you can now read and evaluate. Remember: a function says how one quantity depends on another — one input in, exactly one output out.

Differentiation

• Support: Use a single rule with whole-number inputs and a pre-built table; describe each step aloud for learners with low vision; read f(x) as “f of x” every time.
• Accessibility: Offer the function machine as a tactile model (counters in and out through a cup or box) and describe each step aloud; for learners with dyscalculia, pre-fill the table and use counters for the arithmetic so the focus stays on input → output.
• Extension: Compare f(x) = 2x and g(x) = 2x + 1 — which output is larger for each x, and how do their graphs differ (parallel lines)?

Assessment

• Formative (observation/self): Can the learner define a function, evaluate f(x) for given inputs, and match a rule to its table, graph, and story?
• Self-check: The learner asks, “Did I read f(x) as ‘f of x’? Does each input have exactly one output? Did I substitute the input correctly and name which variable depends on which?”

Home connection

At home, find a rule where one quantity depends on another (cost and amount, distance and time, water and plant growth), write it as a function f(x), and evaluate it for one real input.

Resources

• Functions and function notation are standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019) (S-026). The note that “per one” proportional reasoning appears in the Rhind Papyrus (Egypt, c. 1650 BCE) is documented history (MacTutor, S-288); treating all traditions as worthy is a value (docs/philosophy.md §4). On explicit instruction for novice skills: Kirschner, Sweller & Clark (2006), S-011.