The purpose of this activity is to introduce the idea of input-output rules. Player 1 uses a rule written on a card that only they can see. Player 2 chooses inputs to tell Player 1 so that Player 1 can respond with the corresponding output. Player 2 guesses the rule on the card once they think they have enough input-output pairs to know what it is. Partners then reverse roles.

Monitor for students who use different strategies to determine the rule, such as:

• Choosing which numbers to give, such as always starting with 0 or 1, or choosing a sequence of consecutive whole numbers.
• Looking for a difference between the outputs that correspond to inputs that are even numbers and the outputs that correspond to inputs that are odd numbers.

The goal of this discussion is for students to understand what an input-output rule is and share strategies they used for figuring out the rule. Tell students we start with a number, called the input, and apply a rule to that number, which results in a number called the output. We say the output corresponds to that input.

To highlight these words, ask:

• “What is an example of an input from this activity?” (The input is the number that Player 2 chose, or it is the number placed in the ‘input’ box in the applet.)
• “Where in the activity was the rule applied?” (The rule was applied when Player 1 applied the rule to the input their partner told them, or it’s what happened in the black box in the applet.)
• “Where in the activity is the output?” (The output is the number that Player 1 said after applying the rule to the given input, or it is the number given in the black box in the applet.)

Invite previously identified students to briefly describe their strategy for figuring out one of the rules, recording their process for all to see. For example, for the rule “add 7,” they may describe the table they used to organize their input-output pairs and that their last row is and . Use Compare and Connect to help students compare, contrast, and connect the different strategies. Here are some questions for discussion:

• “What do the strategies have in common? How are they different?”

• “Did anyone have a similar strategy but would explain it differently?”

• “Are there any benefits or drawbacks to one strategy compared to another?”

Students might think the last rule isn't allowable because there were two “different” rules. Explain that a rule can be anything that always produces an output for a given input. Consider the rule “flip,” where the input is “coin.” The output may be heads or tails. We will consider several different types of rules in the following activities and lessons.

The purpose of this activity is for students to think of rules more broadly than simple arithmetic operations in preparation for the more abstract idea of a function, which is introduced in the next lesson.

Each problem begins with a diagram that represents a rule and is followed by a table for students to complete with input-output pairs that follow the rule. Monitor for students who notice that even though the rules are different, each one starts out with the same input-output pair: and 7. An important conclusion is that different rules can determine the same input-output pair.

If using digital activities, there is a rule generator as an extension. Students give an input, and the generator gives an output. After a few inputs, students can guess a potential rule. The generator then gives feedback about the rule: “reasonable but not my rule,” “Correct! How did you know?” or “does not match data.”

The purpose of this discussion is for students to see rules as more than arithmetic operations on numbers and to consider how sometimes not all inputs are possible.

Display a rule diagram with input 2, output 6, and a blank space for the rule for all to see.

Select 2–3 previously identified students, and ask what the rule for the input-output pair might be. Display these possibilities next to the diagram for all to see. For example, students may suggest the rules such as “Add 4,” “Multiply by 3,” or “Add 1, then multiply by 2.”

The last rule, “1 divided by the input,” calls back to the Warm-up. Explain to students that not all inputs are possible for a rule. To highlight this idea, ask:

• “Why was 0 not a valid input for the last rule?” (1 divided by 0 does not exist.)
• “What are some other situations when a rule might not have a valid input?” (Any time an operation requires us to divide by 0, or when the input must be non-negative, such as a side length of a square.)
• “How does using a variable to denote the input and to denote the output help us understand the function rule?” (We can clearly see the relationship between the input and output.)