Lesson 2

Rule Machines

Est. Class Sessions: 2

Developing the Lesson

Part 2: Other Rules on the Rule Machine

Introduce Other Rules. Use the display of the Rule Machine Master to help students learn to apply other rules to numbers and write number sentences for the function. Some possible rules are: add 10, subtract 7, and subtract 20. Have students learn to develop rules from data and also to generate data from rules.

Work through one or two examples as a class. Begin with some easy rules such as add 1 or subtract 2 and as a class fill out the rule machine data table for that rule.

Ask:

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  • My rule is add 1. If my input number is 2, what is my output? (3)
  • What is the number sentence? (2 + 1 =  3)
  • If my output is 5, what is my input? (4)
  • What is the number sentence? ( 4 + 1 = 5)
  • If you have the output number and you have to find the input number, think: What number plus one equals 5?

See Figure 6 for a sample +1 rule machine with number sentences.

Find the Rule. Use the display of the Rule Machine Master to provide a data table with some completed lines and ask students to find the rule for the machine. Some students will probably want to guess your rule right away, without any data. Encourage students to make guesses based on the data you provide. Ask students to look at the data and determine the rules that might fit the rule machine. See Figure 7 for a sample data table.

Ask:

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  • What is the rule for this machine?
  • Look for a pattern to get from the input number to the output.
  • What is the number sentence for the input and output numbers?

As students suggest rules that might fit the input-output numbers, write the rule in the box above the input and output numbers and write the number sentences in the third column. Have students test the rules with all the numbers on the data table. Students can adjust their guesses and eliminate rules that do not fit. Remind students that a rule has to work for all the input numbers on the table. It may also be necessary to adjust the number sentences after students discover the rule that fits all the numbers. For example, for the sample data table in Figure 7, if a student thought the rule was to subtract 3 and suggested the number sentence 6 − 3 = 3, the number sentence should be adjusted to 6 ÷ 2 = 3 to fit the halving rule. Use the Sample Dialog as a guide for this discussion.

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Testing possible rules is a way to improve students’ abilities to identify the pattern and rule between the input and output. Ask students to think about all the ways to get from an input to an output and then test each possible way. A rule will work for all input numbers. Some students will have trouble identifying rules to test but will eagerly test rules. This testing will develop students’ abilities to find, identify, and extend patterns in a data set. This is sometimes called a guess-and-check strategy.

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Use this sample dialog to guide your discussion for determining the rule for the table in Figure 7.

Teacher: Look at the first row of this data table. Can you guess what the rule is?

Faith: I think the rule is to subtract 3. If your input is 6 and the output is 3, you can subtract 3 to get 6. My number sentence is 6 − 3 = 3.

Javier: I think the rule is halving. My number sentence is
6 ÷ 2 = 3.

Darius: I looked at the second row and I think the rule is to subtract 10. My number sentence is 20 − 10 = 10.

Teacher: Remember that the rule has to fit all the numbers. Let’s write those rules on the chart and see if they fit all the input and output numbers. Test the rules and see if they fit all the numbers.

Faith: I tested my subtract 3 rule on the numbers 20 and 10. If my input is 20 and I subtract 3, my output is not 10, so my rule doesn’t work.

Darius: I tested my rule, too. My rule is to subtract 10 and I tried it on the first row of numbers, 6 and 3 and it doesn’t work.

Teacher: Let’s cross out the rules that don’t work. Did anyone test the halving rule? Choose different input numbers and see if the rule works.

Javier: I tried the rule on 20 and 10 and it works.

Sarah: I tried it on the third row. If you find half of 10, the answer is 5.

Jana: I tried it on 40 and 20 and it works for that row, also.

Teacher: Since it works for all the numbers, halving must be the rule for this rule machine. Let’s make sure all the number sentences reflect this rule.

Adventure Book:
Student Activity Book:
Master:
Using the guess-and-check method to find half of 132
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Sample rule machine for the rule Add 1
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Data table with completed lines
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