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Explicit and Recursive Expressions | Lesson 2 of 2

Procedures, Choices, and Translation

Lesson 2 of 2: When and How to Switch Forms

In this lesson:

  • Write procedural descriptions for complex calculations
  • Translate between explicit and recursive forms
  • Verify translations by computing matching terms
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Lesson 2 Learning Objectives Overview

Lesson 2 focus — by the end of this lesson, you will be able to:

  1. Write a step-by-step procedure for situations with complex or conditional logic
  2. Explain why procedures are valid function definitions
  3. Translate explicit ↔ recursive for linear and exponential functions
  4. Verify translations by computing matching term values
  5. Choose between explicit, recursive, and procedural forms based on context
  6. Connect explicit and recursive through common difference and common ratio
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Can Every Function Be a Formula?

You know functions like and .

But what about:

  • A phone plan with flat fee + tiered data + per-text charges?
  • GPA calculation across different courses with different credit hours?
  • Income tax with multiple brackets?
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Procedural Definitions: Steps Are Valid Functions

A procedure is a valid function definition when:

  • The situation involves conditional logic (if-then)
  • Multiple sub-calculations combine for the final output
  • A table lookup or rule applies differently in different ranges

A function requires: one output per input. Procedures do this.

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Phone Plan: A Procedural Function

A plan charges $30 for up to 2 GB, $10/extra GB, plus $0.05/text.

  1. Start: base = $30
  2. If data > 2 GB, add $10 × (data − 2) to base
  3. Add $0.05 × (texts) to total
  4. Result is monthly cost
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Quick Check: Is This a Function?

The phone plan assigns one cost to each (data, texts) pair.

  • Is it a function? Why or why not?
  • Could it be written as a single formula? (Hint: think about the conditional)
  • When does the conditional in step 2 apply? When does it not?
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

GPA: No Single Formula Needed

GPA calculation with variable courses and credits:

  1. For each course: multiply grade points by credit hours
  2. Sum all the products
  3. Sum all credit hours
  4. Divide total grade points by total credit hours
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Matching the Representation to the Question Asked

Decision table: explicit vs. recursive vs. procedural — when to use each

  • Explicit: "Find the 100th value directly"
  • Recursive: "Describe how each value builds on the last"
  • Procedural: "Multi-step or conditional calculation"

Match the form to the question.

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Translation: Explicit ↔ Recursive (Linear)

For linear (arithmetic) functions, translation is direct:

Explicit Recursive
,
  • Slope → additive step in recursive
  • -intercept → initial value
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Example: Linear Translation Both Directions

Explicit → Recursive:

Recursive → Explicit:

Verify: ✓, ✓, ✓

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Translation: Explicit ↔ Recursive (Exponential)

For exponential (geometric) functions, translation is equally direct:

Explicit Recursive
,
  • Base → multiplicative step in recursive
  • Initial value → starting point
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Mismatched Values Signal a Translation Error

Explicit:
Recursive: ,

Side-by-side table: explicit and recursive values for n = 0 to 4

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Guided Practice: Recursive to Explicit

Given: ,

Write the explicit form:

  • What is the common difference? ___
  • What is ? ___
  • Explicit: $a(n) = $ ___

Then verify: compute both ways.

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Quick Check: Which Form to Use?

Which form is more efficient for each question?

  • "What is the population after 50 years?" → explicit or recursive?
  • "How does the temperature drop from minute to minute?" → explicit or recursive?
  • "What is the tax on a $75,000 income with brackets?" → any of the three?
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Practice: Translate Between Explicit and Recursive Forms

Given one form, write the other. Verify with 3 terms.

  1. → write recursive
  2. , → write explicit
  3. → write recursive
  4. , → write explicit
Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Translation Practice Answers with Verification

  1. → ,

  2. , →

  3. → ,

  4. , →

Verify problem 4: recursive = 20, 28, 36, 44. Explicit: . ✓

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Lesson 2 Key Takeaways and Watch-Outs

✓ Procedures are valid functions: steps that assign one output per input

✓ Explicit → Recursive: slope becomes common difference; base becomes common ratio

✓ Recursive → Explicit: extract initial value and constant step, write direct formula

✓ Verify every translation with 2-3 matching terms

⚠️ Watch out: Some recursive functions have no simple explicit form — linear and exponential are special; not all recursions can be "solved"

⚠️ Watch out: A procedure IS a function — no single formula is required

Grade 9 Functions | HSF.BF.A.1.a
Explicit and Recursive Expressions | Lesson 2 of 2

Coming Up: Combining Standard Function Types

Coming up: HSF.BF.A.1.b — Combining Standard Function Types

  • Add, subtract, multiply, and divide functions
  • Build models with multiple components (like: revenue minus cost equals profit)
  • Identify and decompose complex real-world models
Grade 9 Functions | HSF.BF.A.1.a