Lesson 1

The purpose of this activity is for students to see why an expression that contains dividing by zero can't be evaluated. They use their understanding of related multiplication and division equations to make sense of this.

Arrange students in groups of 2. Tell students to consider the statements and try to find a value for \(x\) that makes the second statement true.

Give students 1 minute of quiet think time followed by 1 minute to share their thinking with their partner. Finish with a whole-class discussion.

Select 2–3 groups to share their conclusions about \(x\).

As a result of this discussion, we want students to understand that any expression where a number is divided by zero can't be evaluated. Therefore, we can state that there is no value for \(x\) that makes both equations true.

The purpose of this activity is to introduce the idea of input-output rules. One student chooses inputs to tell a partner who uses a rule written on a card only they can see to respond with the corresponding output. The first student then 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:

• Appear to have a strategy for choosing which numbers to give, such as always starting with 0 or 1, or choosing a sequence of consecutive whole numbers.
• Notice the difference between the result for odd numbers and the result for even numbers for the card with a different rule for each.

Students may have difficulty with the last rule because it involves conditional statements. Encourage them to focus on a single value at a time to help them understand how this kind of rule works.

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 the player without a card chose, or the number placed in the 'input' box in the applet.)
• “Where in the activity was the rule applied?” (The rule was applied when the player with the card 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 was the output?” (The output is the number the player said after applying the rule to the given input, or the number the black box in the applet gave.)

Select previously identified students who appeared to have a strategy for figuring out a rule to share it. Sequence students starting with the most common strategies to the least. Make connections between the successful aspects of each strategy (for example, if a student does not say it, stating clearly why using a sequence of integers or using 0 and 1 are helpful for determining the rule).

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 representing a rule 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: \(\frac34\) 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 indicates if the rule is "reasonable but not my rule", "correct! how did you know", or "does not match data."

Arrange students in groups of 2. Display the following diagram for all to see:

Tell students that this diagram is one way to think about input-output rules. For example, if the rule was "multiply by 2" and the input \(\frac32\), then the output would be 3. Tell students they will use diagrams like this one during the activity to complete tables of input and output values.

Give students 3–5 minutes of quiet work time to complete the first three tables. Then give them time to share their responses with their partner and to resolve any differences. 

Give partners 1–2 minutes of quiet work time for the final rule followed by a whole-class discussion. Depending on time, have students add only one additional input-output pair instead of two.

Students may have trouble thinking of “write 7” as a rule. Emphasize that a rule can be anything that produces a well-defined output, even if it ignores the value of the input. Students who know about infinite decimal expansions might wonder about the second rule, because \(0.999 \dots = 1.0\), so the same number could have two outputs. If this comes up, discuss how you might refine the rule in this case.

The purpose of this discussion is for students to see rules as more than arithmetic operations on numbers and 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 you to divide by 0, or when the input must be non-negative, such as a side length of a square when you know the area.)
• “How does using a variable \(x\) to denote the input and \(\frac 1x\) to denote the output help us understand the function rule?” (You can clearly see the relationship between the input and output.)