Function machine: guess the rule

What the function machine does

A function machine is a picture of a rule: a number goes in, the machine does something to it, and a new number comes out. Choose an input from −10 to 20 and press Put it in. Each input and output is added to the table, so patterns build up.

There are three modes. Mystery rule hides the rule; you can choose one-step rules (such as + 7 or × 4), two-step rules (such as × 3, then − 2) or rules with squares. When you think you know it, press I know the rule! and pick from four choices. Some wrong choices fit your first row on purpose, so testing more inputs pays off. My rule lets you build a one- or two-step rule and see it written in algebra. Work backward shows the rule and the output and asks what went in.

Why guess-the-rule games work

Finding a rule from a table is the heart of early algebra: noticing that outputs go up by 3 each time, then checking whether + 3 really works for every input. Typing a prediction before each input turns a guess into a test. Comparing × 2, then + 3 with + 3, then × 2 shows that the order of steps matters, which is the idea behind the order of operations. Work backward introduces inverse operations, the same thinking used to solve equations.

Common mix-ups

• Checking only one row. 3 goes to 6 for both + 3 and × 2, so test another input before deciding.
• Looking only at how the outputs change. If outputs go up by 2 when inputs go up by 1, the rule starts with × 2, but there may be a second step.
• Undoing steps in the same order when working backward. Undo the last step first.
• Thinking + 3, then × 2 is the same as × 2, then + 3. Try both with an input of 1.

A 10-minute function machine routine

• Choose Mystery rule with one step. Ask a child to choose the first input.
• Before each new input, ask for a prediction and type it in.
• When the class has three rows, press I know the rule! and vote.
• Switch to Two steps and repeat.
• End with one Work backward puzzle, asking how the class undid the steps.

Where it fits by grade

• Grades 3 and 4: identify arithmetic patterns (3.OA.D.9) and generate patterns that follow a given rule (4.OA.C.5).
• Grade 5: generate two numerical patterns from two rules and compare them (5.OA.B.3).
• Grades 6 and 7: write expressions with a variable (6.EE.A.2), describe how one quantity depends on another (6.EE.C.9), and solve two-step equations by undoing steps.
• England: function machines are a common picture in Years 5 and 6 and Key Stage 3. Year 6 uses simple formulae and expresses missing number problems algebraically.

Learning more

Order of operations, negative numbers and solving equations are skills on Eurekami's skill map, and Daily Practice brings them back when they start to fade. Lessons explain each step in plain words, and the AI tutor asks guiding questions when a rule is hard to spot.