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

Learning Objectives

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

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?

Do these have single formulas? Or something else?

Procedural Definitions: Steps Are Valid Functions

A procedure — a series of steps — 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 just requires: one output per input. Procedures can do this.

Phone Plan: A Procedural Function

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

Procedure:

1. Start with base = $30
2. If data > 2 GB, add $10 × (data − 2) to base
3. Add $0.05 × (number of texts) to running total
4. Result is monthly cost

Each input (data, texts) gives exactly one output (cost).

Quick Check: Is This a Function?

The phone plan procedure 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 in step 2)
• When does the conditional in step 2 apply? When does it not?

A function just needs: one output per input.

GPA: No Single Formula Needed

Scenario: GPA calculation with variable courses and credits.

Procedure:

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

Different students have different numbers of courses — the procedure handles all cases.

Choosing the Right Representation

Which form fits which situation?

• 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.

Translation: Explicit Recursive (Linear)

For linear (arithmetic) functions, translation is direct:

• Slope becomes the additive step
• -intercept becomes the initial value

Linear: common difference slope

Example: Linear Translation Both Directions

Explicit → Recursive:

Recursive → Explicit:

Verify: ✓, ✓, ✓

Translation: Explicit Recursive (Exponential)

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

• Base becomes the multiplicative step
• Initial value becomes the starting point

Exponential: common ratio base

Verification: Both Forms Agree

Explicit:
Recursive: ,

Values must match exactly — if they don't, recheck the translation.

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.

Quick Check: Which Form to Use?

Decide 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?

Match the question to the right tool.

Practice: Translation Problems

Given one form, write the other. Then verify with 3 terms.

1. Explicit: — write the recursive form
2. Recursive: , — write explicit
3. Explicit: — write recursive
4. Recursive: , — write explicit

Work all four, then advance for answers.

Answers

1. → ,

2. , →

3. → ,

4. , →

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

Key Takeaways

✓ 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

What's Next

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

You now have three ways to define a function. Next: combining them.