K-12 Math (US): 3-5 Data Analysis K-12 Math Tables Input-output tables

Grade 3–5 Math Study Guide: Input-Output Tables

1. The Driving Question

"If you put 3 into a machine and get 7 out, then put 5 in and get 9 out—what’s the secret rule? And how can you write it down so someone else can predict what comes out when they put in 10, or 100, or even a number they haven’t tried yet?"

2. The Core Idea — Built, Not Listed

Imagine a lemonade stand where every cup costs $2, but there’s a mystery fee of $1 for the cup itself. If you order 1 cup, you pay $3. If you order 2 cups, you pay $5. If you order 3 cups, you pay $7. The stand owner doesn’t tell you the rule—just gives you a table like this:

To crack the rule, you look for a pattern: How does the input turn into the output? Here, the rule is "Multiply the input by 2, then add 1." That’s the "secret code" of the table. Once you know it, you can predict the cost for any number of cups—even 100!

Key Vocabulary:
- Input: The number you start with (e.g., cups ordered).
- Example: In a table tracking how many push-ups you do each day, the input is the day number (Day 1, Day 2, etc.).
- Output: The number the rule produces (e.g., total cost).
- Example: In a table showing how many pages you read per hour, the output is the total pages read.
- Rule: The operation (like +, –, ×, ÷) that turns input into output.
- Example: A rule like "input × 3 – 2" could describe how many balloons you blow up per minute (if you start with 2 already blown up).
- Pattern: The consistent way inputs and outputs relate (e.g., "output is always 4 more than input").
- Example: A table tracking how many legs are on a group of spiders (8 legs each) follows the pattern "input × 8."

3. Assessment Translation

How This Appears in Classroom Assessments:
- Exit Tickets: "Here’s an input-output table. Fill in the missing output for input = 6, and write the rule in words." - Short Constructed Response: "Explain how you found the rule for this table. Use numbers and words." - Show-Your-Work Problems: "Create a table for the rule ‘input ÷ 2 + 5.’ Show at least 3 input-output pairs."

Proficient vs. Developing Responses:
| Proficient | Developing | |----------------|----------------| | Table: Input: 2 → Output: 7; Input: 4 → Output: 9; Input: 6 → Output: 11. Rule: "Add 5 to the input." Explanation: "I saw that 2 + 5 = 7, 4 + 5 = 9, so the rule is +5." | Table: Filled in correctly but Rule: "The numbers go up by 2." Explanation: "I don’t know how to write the rule." (Loses credit for not connecting the pattern to the operation.) | | Missing Output: For input = 8, output = 13. Work: "8 + 5 = 13." | Missing Output: 14. Work: "I guessed." (No credit for no reasoning.) |

Model Proficient Response:
Prompt: "Find the rule for this table and fill in the missing output for input = 10."

Student Response: "The rule is ‘input × 3 + 1.’ I figured it out because 1 × 3 + 1 = 4, 3 × 3 + 1 = 10, and 5 × 3 + 1 = 16. For input = 10, 10 × 3 + 1 = 31, so the missing output is 31."

4. Mistake Taxonomy

Mistake 1: Misidentifying the Operation
- Prompt: "What’s the rule for this table?"

• Common Wrong Response: "The rule is +4 because 2 + 4 = 6."
• Why It Loses Credit: The student added instead of multiplying. The pattern holds for the first row but fails for the others (4 + 4 ≠ 12).
• Correct Approach:
• Test addition: 2 + 4 = 6 ✔️, but 4 + 4 = 8 ≠ 12 ❌.
• Test multiplication: 2 × 3 = 6 ✔️, 4 × 3 = 12 ✔️, 6 × 3 = 18 ✔️.
• Rule: "Input × 3."

Mistake 2: Ignoring the Starting Point
- Prompt: "Fill in the missing output for input = 5."

• Common Wrong Response: "15, because 5 × 3 = 15."
• Why It Loses Credit: The student multiplied but missed the +2 in the rule (input × 2 + 3).
• Correct Approach:
• Find the pattern: 1 → 5 (+4), 2 → 7 (+5), 3 → 9 (+6). The added number increases by 1 each time, but that’s not the rule.
• Look for a consistent operation: 1 × 2 + 3 = 5, 2 × 2 + 3 = 7, 3 × 2 + 3 = 9.
• Rule: "Input × 2 + 3." For input = 5: 5 × 2 + 3 = 13.

Mistake 3: Incomplete Explanation
- Prompt: "Explain how you found the rule for this table."

• Common Wrong Response: "The rule is ×3 because 1 × 3 = 3."
• Why It Loses Credit: The student didn’t test the rule for all inputs (e.g., 0 × 3 = 0 ✔️, but they didn’t show it).
• Correct Approach:
• Test the rule for every input: 0 × 3 = 0 ✔️, 1 × 3 = 3 ✔️, 2 × 3 = 6 ✔️, 3 × 3 = 9 ✔️.
• Write: "I tried multiplying each input by 3, and it matched every output. The rule is ‘input × 3.’"

5. Connection Layer

1. Within Math: Input-output tables → algebraic expressions

2. Why it matters: A table’s rule (e.g., input × 2 + 1) is the same as an expression like 2x + 1. Understanding tables makes algebra feel like cracking a code instead of memorizing symbols.

3. Across Subjects: Input-output tables → science experiments

4. Why it matters: In science, you might measure how the input (e.g., minutes of sunlight) affects the output (e.g., plant height). The table is your data, and the rule is your hypothesis.

5. Outside School: Input-output tables → video game leveling

6. Why it matters: In games like Minecraft, your input (e.g., 10 wood blocks) turns into an output (e.g., 4 planks). The rule is the crafting recipe—and if you figure it out, you can predict how much wood you need for 100 planks.

6. The Stretch Question

"What if the rule changes halfway through the table? For example:

How would you describe the rule now? Can you write it as two separate rules, or is there one rule that covers the whole table?"

Pointer Toward the Answer:
The first three rows follow "input × 2 + 1," but the last two don’t. Look for a pattern in the differences between outputs (e.g., 3 to 5 is +2, 5 to 7 is +2, but 7 to 10 is +3). Maybe the rule changes after input = 3—or maybe there’s a hidden pattern, like "input × 2 + (input – 2)." Try testing both ideas!