Using Functions to Model Real Change - Grade 1 Math

Learn how to use functions to model real changes with visual exercises and step-by-step solutions.

Function Exercises 1 to 3
1 Plant Growth Function
Problem 1
A plant grows 2 inches every day. If it starts at 3 inches tall, how tall will it be after 4 days?
Definition:

Function: A rule that takes an input and gives an output. Input: days, Output: height.

Function method:
  1. Identify the input (days)
  2. Identify the rule (add 2 each day)
  3. Apply the rule to the input
  4. Get the output (height)
Days
→
Height
Grow 2/day
Step 1: Start with initial height

At day 0: 3 inches

Step 2: Apply growth for each day

Day 1: 3 + 2 = 5 inches

Day 2: 5 + 2 = 7 inches

Day 3: 7 + 2 = 9 inches

Day 4: 9 + 2 = 11 inches

After 4 days, the plant will be 11 inches tall
Final answer:

After 4 days, the plant will be 11 inches tall

Applied rules:

• Input/Output: Days in, height out

• Constant growth: Add 2 inches each day

• Sequential change: Each day builds on the previous

2 Savings Function
Problem 2
Emma saves $1 on Monday, $2 on Tuesday, $3 on Wednesday. If this pattern continues, how much will she save on Friday?
Definition:

Pattern function: Each day's savings depend on the day number.

Day
→
Amount
Step 1: Identify the pattern

Day 1: $1, Day 2: $2, Day 3: $3

Step 2: Extend the pattern

Day 4 (Thursday): $4

Day 5 (Friday): $5

On Friday, Emma will save $5
Final answer:

On Friday, Emma will save $5

Applied rules:

• Sequential pattern: Amount equals day number

• Consistent rule: Add $1 each day

• Continuation: Follow the established pattern

3 Cookie Sharing Function
Problem 3
For every child at a party, 3 cookies are needed. If there are 5 children, how many cookies are needed?
Definition:

Ratio function: Input: number of children, Output: number of cookies.

Children
→
Cookies
×3
Step 1: Identify the relationship

Each child needs 3 cookies

Step 2: Apply the rule

For 5 children: 5 × 3 = 15 cookies

15 cookies are needed for 5 children
Final answer:

15 cookies are needed for 5 children

Applied rules:

• Multiplicative relationship: Cookies = Children × 3

• Direct proportion: More children = more cookies

• Scaling: Multiply input by the same factor

Function Concepts and Models
Output = Rule(Input)
Function Rule
Input
x
What goes in
Rule
f(x)
What we do
Output
y
What comes out
2 → 4
Add 2
3 → 6
Double
5 → 8
Add 3
Input → Output: Every input has exactly one output.
Consistent Rule: The same rule applies to all inputs.
Relationship: Changes in input cause predictable changes in output.
Detailed Summary and Key Concepts
4 Function Fundamentals
Key Definitions:

Function: A relationship where each input has exactly one output

Input: The starting value (like number of days)

Output: The result after applying the rule

Input
x
Rule
f(x)
Output
y
Step 1: Identify the input

Determine what information you start with

Step 2: Determine the rule

Figure out what operation transforms the input

Step 3: Apply the rule

Perform the operation on the input

Functions model how one quantity changes based on another
Key concept:

Functions describe predictable relationships between quantities

5 Common Mistakes
Common errors:

Mistake 1: Not identifying the correct input/output relationship

Mistake 2: Changing the rule partway through

Mistake 3: Not applying the rule consistently

Correct Process
Input → Rule → Output
Wrong Process
Input → Change Rule → Wrong Output
Step 1: Keep the same rule throughout

Don't change the operation partway through

Step 2: Apply the rule to each input

Be consistent with all calculations

Step 3: Verify your outputs

Check that outputs follow the expected pattern

Always verify: Does the same rule apply to all inputs?
Remember:

Consistency is key - apply the same rule to each input!

Function Laws, Methods, and Definitions
f(x) = ax + b
Linear Function
Key definitions:

Function: A relationship where each input has exactly one output

Input (Domain): The set of all possible starting values

Output (Range): The set of all possible results

Function problem solving method:
  1. Identify the input: What information are you given?
  2. Determine the rule: What operation connects input to output?
  3. Apply the rule: Perform the operation on the input
  4. Get the output: Calculate the result
  5. Verify: Check that the output makes sense
  6. Generalize: Apply the same rule to other inputs
Tip 1: Always identify what goes in and what comes out.
Tip 2: The same rule must apply to all inputs.
Tip 3: Look for patterns in how inputs and outputs relate.
Tip 4: Draw pictures to visualize the function machine.
Common errors: Changing rules mid-process, not identifying inputs correctly, inconsistent application.
Exam preparation: Practice identifying functions in real-world scenarios.
Function rules to remember:

• Each input has exactly one output

• The same rule applies to all inputs

• Functions model predictable relationships

• Outputs change in a predictable way based on inputs

• Always verify that outputs follow the expected pattern

Questions & Answers

Question: I don't understand what a function is. Can you explain it simply?

Answer: A function is like a magic machine that takes something in and changes it to something else!

  • Think of it like a vending machine: you put in money (input), press a button (rule), and get a snack (output)
  • Or like a cookie cutter: you put in dough (input), use the shape (rule), and get shaped cookies (output)
  • Every time you put the same thing in, you get the same thing out

For example: If the rule is "add 2", then 3 goes in and 5 comes out every time!

It's a way to describe how one thing changes into another!

Question: How can I help my child understand functions using real-life examples?

Answer: Use everyday examples that show input-output relationships:

  • Age → Height: As years increase, height increases
  • Hours worked → Money earned: More hours = more pay
  • Temperature → Ice cream sales: Hotter days = more ice cream sold
  • Number of guests → Pizza slices needed: More guests = more food

Ask your child: "If I do this, what happens?"

Focus on the predictable relationship between two things!

Question: How do I know if I found the right rule for a function?

Answer: Check your rule with these steps:

  • Does the same rule work for ALL the examples given?
  • If I put in 2 and get 4, does the same rule work for other numbers?
  • Does my rule make sense in the real world?
  • Can I use the same rule to predict future values?

For example: If 1→3, 2→6, 3→9, the rule is "multiply by 3"

Check: 1×3=3, 2×3=6, 3×3=9. Yes, it works!