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Expressions and Equations 6-8 / Week 07 / Monday
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Week 07 · Two Quantities That Change Together

Monday

The rule machine
// Inputs, outputs and the rule machine
⏱ about 20 min

Monday: The Rule Machine

Rocket sets a cardboard box on the Puzzle Post bench. It is painted orange, with a slot cut in each end.

"The rule machine," he says. "A card goes in one side. A new card comes out the other."

Raven slides in a card marked 3. Rocket scribbles behind the box and pushes out a card marked 12.

"What do you notice?" Raven asks. She tries 5. Out comes 20.

"Each number in gives one number out," Rocket says. "The rule is times 4."

Nova hovers over the slot, her light steady. "Which number did you choose?" she asks. "And which one depended on your choice?"

"I chose the input," says Raven. "The output depended on it." Nova hums. "That is the whole week."

Rocket, Raven and Nova at the Puzzle Post bench with an orange rule-machine box, a card going into its slot.

Two quantities that change together

When Raven changes the card she puts in, the card that comes out changes too. Two quantities change together.

The quantity you choose is the independent variable. The quantity that follows from your choice is the dependent variable.

In the rule machine, the input is independent and the output is dependent. The rule connects them: output = 4 × input.

A rule machine labeled × 4, 3 going in, 12 coming out

A function: one output for each input

A rule is a function when every input gives exactly one output. Put in 3 and you always get 12, never 11.

A machine that gave 12 one time and 13 the next for the same card would not be a function. You could not trust it.

We write the rule with letters: y = 4x. Here x is the input and y is the output.

Input xRuleOutput yOrdered pair (x, y)
11 × 44(1, 4)
22 × 48(2, 8)
33 × 412(3, 12)
44 × 416(4, 16)
55 × 420(5, 20)
FEED THE MACHINE
  • Read the question.
  • Tap your answer.
The rule is × 4. What comes out when 3 goes in?
The rule is × 4. What comes out when 7 goes in?
The rule is + 5. What comes out when 9 goes in?
The rule is × 2 + 1. What comes out when 6 goes in?
WHICH DEPENDS ON WHICH?
  • Read the question.
  • Tap your answer.
Raven chooses the input card, and the machine gives the output. Which is the dependent variable?
Rocket picks how many minutes to walk, and the meters he covers follow from that. Which is the independent variable?
The crew chooses how many envelopes to fill, and the beans used follow from that. Which is the dependent variable?
StatementTrue or false?
Every input to a function gives exactly one output.?
The input is the dependent variable.?
In y = 4x, y is the output.?
A machine that gives 12 and then 13 for the same input is a function.?
WHY THIS EXERCISENaming which quantity depends on which is the first step in every rule the crew writes this week.
The rule is × 4 and the input is 6. What is the output? Type the number.
WHY THIS EXERCISEWorking the rule for one input is how you check any table the machine makes.
Try it
Pick a secret rule, like × 3 or + 7. Have someone give you three inputs and tell them each output.
Can they guess your rule? Then swap roles.
Draw the rule machine with three cards going in and their three cards coming out for the rule × 4.

A strong start at the rule machine. Tomorrow you turn a rule into a table, ordered pairs and a graph.