Lesson 2

Rule Machines

Est. Class Sessions: 2

Developing the Lesson

Part 1: Doubling and Halving on the Rule Machine

Build a Doubling Machine. In Unit 1, students were introduced to the doubles and doubles +1 rule machine. Briefly review how rule machines work by telling students to imagine a machine that takes in numbers (input), does something to the numbers (rule), and gives you the results (output). Show students the doubles rule machine you prepared. Point out the rule on the machine.

Fold one of the rule machine rows you prepared in half. Write 4 on the left side or “input” side and ask:

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  • If 4 is the input, what is the output? (8)

As students say the output numbers, write the double on the right side or “output” side and hang the row on the display to make a rule machine. Repeat the procedure with other numbers as needed.

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Functions. The concept of function is an important one. Students will study it formally when they study algebra in later years. We introduce the basic concept of a function in an informal way, without ever giving students a definition. One way of defining a function is to say that it is a rule that assigns to every input number, exactly one number, the output number. At this level, we call them “rule” machines, but in later grades they will be referred to as “function“ machines.

Tell students that it is their turn to try. Give each student one of the rule machine rows you prepared showing an input that they will double. See Materials Preparation. Have connecting cubes readily available. Encourage students to think of a variety of strategies for doubling the number.

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When building and analyzing patterns in tables and rule machines, students often have trouble focusing on the relationship between the input and the output. By breaking the table into rows, students are forced to focus on this relationship rather than the relationship between the rows.

Observe students’ strategies for finding doubles and choose a few students to share their strategies. As students complete their rows, have them add their rows to the rule machine you already started.

Strategies for Doubling. Upon completion, have students share their strategies for doubling numbers with the class. See Figure 3 for sample strategies.

Strategies for Halving. Use the Doubles Rule Machine you prepared to demonstrate finding half of a number. Show students a row with 10 as the output number.

Ask:

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  • Now I want you to work backward. If you know the output, can you figure out the input? If 10 is the output number, what is the input number? (5)
  • How did you find the answer? (Possible response:
    I know my doubles: 5 + 5 = 10.)
  • If we write 5 under the “input” column, does the output number 10 follow the rule of doubling? (Yes, 10 is double the number 5.)
  • What did you have to do to 10 to find the input number? (Possible response: Instead of doubling the number, I had to find half of 10.)
  • If the rule is to double the number, why are we finding half of a number? (Possible response: Since we have the output, we have to think, “What number can I double to find the output 10?”)
  • If 20 is the output, what is the input? (10)
  • How did you find the answer? (Possible response:
    I know that if I add 10 + 10, the answer is 20.)

Give each student one of the rule machine rows you prepared showing an output that they will halve. See Materials Preparation. Have connecting cubes readily available. Encourage students to think of a variety of strategies for finding half of a number.

Observe students’ strategies for finding halves. Finding half of a number will be more difficult for students than doubling a number. Choose a few students to share their strategies. As students complete their rows, have them add their rows to the rule machine.

Upon completion, have students share strategies for finding half of a number. See Figure 4 for sample student strategies for finding half of a number.

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For students who are having difficulty finding half of a number, provide connecting cubes or encourage them to use the guess-and-check method. For this method, students try an input number and adjust their responses higher or lower until they find the answer. For example, if the output number is 132, the student can try 60 + 60 = 120. The input number needs to be higher. The student continues adjusting his guesses until he finds the answer. See Figure 5.

Adventure Book:
Student Activity Book:
Master:
Sample student strategies for doubling a number
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Sample student strategies for halving a number
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