1.1: Dividing by 0
Warm-up: 5 minutes
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.
Study the statements carefully.
- $12 \div 3 = 4$ because $12=4 \boldcdot 3$
- $6 \div 0 = x$ because $6=x \boldcdot 0$
What value can be used in place of $x$ to create true statements? Explain your reasoning.
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.
1.2: Guess My Rule
Activity: 15 minutes
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.
Arrange students in groups of 2.
For students using the print version: Have a student act as your partner and demonstrate the game using a simple rule that does not match one of the cards (like “divide by 2” or “subtract 4”).
Ask groups to decide who will be Player 1 and who will be Player 2. Give each group the four rule cards, making sure that the simplest rules are at the top of the deck when face down. Tell students to be careful not to let their partner see what the rule is as they pick up the rule cards. If necessary, tell students that all numbers are allowed, including negative numbers.
For students using the digital version: Demonstrate how to use the applet by choosing one input value, clicking the appropriate button, and noting where the output is recorded. Prompt partners to discuss which input values to select for the rule and alternate who guesses the rule.
- Eliminate Barriers. Allow students to use calculators to ensure inclusive participation in the activity.
- Processing Time. Check in with individual students, as needed, to assess for comprehension during each step of the activity.
Try to figure out what's happening in the “black box.”
Note: You must hit enter or return before you click GO.
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.
1.3: Making Tables
Activity: 15 minutes
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.
For each input-output rule, fill in the table with the outputs that go with a given input. Add two more input-output pairs to the table.
input output $\frac34$ 7 2.35 42
input output $\frac34$ 7 2.35 42
input output $\frac34$ 7 2.35 42
Pause here until your teacher directs you to the last rule.
input output $\frac37$ $\frac73$ 1 0
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.)
1.4: What's the Rule?
Cool-down: 5 minutes
Fill in the table for this input-output rule:
Student Lesson Summary
An input-output rule is a rule that takes an allowable input and uses it to determine an output. For example, the following diagram represents the rule that takes any number as an input, then adds 1, multiplies by 4, and gives the resulting number as an output.
In some cases, not all inputs are allowable, and the rule must specify which inputs will work. For example, this rule is fine when the input is 2:
But if the input is -3, we would need to evaluate $6 \div 0$ to get the output.
So, when we say that the rule is “divide 6 by 3 more than the input,” we also have to say that -3 is not allowed as an input.