Lesson 1Inputs and Outputs

Learning Goal

Let’s make some rules.

Learning Targets

  • I can write rules when I know input-output pairs.

  • I know how an input-output diagram represents a rule.

Warm Up: Dividing by 0

Problem 1

Study the statements carefully.

  • because

  • because

What value can be used in place of to create true statements? Explain your reasoning.

Activity 1: Guess My Rule

Problem 1

Try to figure out what’s happening in the “black box.”

Note: You must hit enter or return before you click GO. 

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Keep the rule cards face down. Decide who will go first.

  1. Player 1 picks up a card and silently reads the rule without showing it to Player 2.

  2. Player 2 chooses an integer and asks Player 1 for the result of applying the rule to that number.

  3. Player 1 gives the result, without saying how they got it.

  4. Keep going until Player 2 correctly guesses the rule.

After each round, the players switch roles.

Are you ready for more?

Problem 1

If you have a rule, you can apply it several times in a row and look for patterns. For example, if your rule was “add 1” and you started with the number 5, then by applying that rule over and over again you would get 6, then 7, then 8, etc., forming an obvious pattern.

Try this for the rules in this activity. That is, start with the number 5 and apply each of the rules a few times. Do you notice any patterns? What if you start with a different starting number?

Activity 2: Making Tables

Problem 1

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.

  1. An input output diagram. Three-fourths is the input, the rule is add 1 then multiply by 4. The output is 7. Inputs in table are 2.35 and 42.

    input

    output

  2. An input output table. Three-fourths is the input, the rule is name the digit in the tenths place. The output is 7. Inputs in table are 2.35 and 42.

    input

    output

  3. An input output table. Three-fourths is the input, the rule is write 7. The output is 7. Inputs in table are 2.35 and 42.

    input

    output

    Pause here until your teacher directs you to the last rule.

  4. An input output table. Three-sevenths is the input, the rule is divide 1 by the input. The output is seven thirds. Inputs in table are 1 and 0.

    input

    output

Lesson Summary

An input-output rule is a rule that takes an allowable input and uses it to determine an output.

An input output diagram.

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.

An input output diagram. Three-fourths is the input, the rule is add 1 then multiply by 4. The output is 7. Inputs in table are 2.35 and 42.

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:

An input output table. The input is 2, the rule is divide 6 by 3 more than the input and the output is 1.2

But if the input is -3, we would need to evaluate to get the output.

An input output table. The input is -3, the rule is divide 6 by 3 more than the input and no output is shown

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.