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Expressions and Equations 6-8 / Week 07 / Friday
5/6
Week 07 · Two Quantities That Change Together

Friday

Rules all around
// Inputs, outputs and the rule machine
⏱ about 20 min

Friday: Rules All Around

Rocket counts the chairs around the kitchen table. "Six chairs. Twenty-four legs. The rule is legs = 4 × chairs."

"What do you notice about which number you count first?" Raven asks.

"Chairs," Rocket says. "I choose the chairs. The legs follow. Chairs are the input."

Nova hovers by the window, her light soft. "Find a rule with a head start," she says.

Raven points at the stack of paper cups. "One cup is 8 centimeters tall. Each extra cup adds 1 centimeter."

"Height = 1 × cups + 7," Rocket says slowly. "The 7 is the head start."

"Every rule this week fits one of those two shapes," Raven says. "Let us review." Nova hums.

Functions in everyday life

Chairs and legs: legs = 4 × chairs. You choose the chairs, so chairs are the independent variable.

Pages and minutes: a reader at 2 pages a minute has read 2 × minutes. Minutes are the input.

A stack of cups: 8 cm for one cup and 1 cm more for each extra cup. Height = cups + 7, a rule with a head start.

Each of these is a function. One input, one output, every time.

The week in four lines

  • The quantity you choose is independent. The quantity that follows is dependent.
  • A function gives exactly one output for every input.
  • A table, ordered pairs, a graph and an equation are four views of one rule.
  • Read a rule from a table: how much the output changes per step, plus where it starts.
EVERYDAY RULES
  • Read the question.
  • Tap your answer.
Legs = 4 × chairs. How many legs do 6 chairs have?
A reader finishes 2 pages a minute. How many pages in 15 minutes?
A stack of cups is c + 7 centimeters tall. How tall is a stack of 10 cups?
Legs = 4 × chairs. How many chairs have 36 legs?
WEEK REVIEW
  • Read the question.
  • Tap your answer.
A reader picks how many minutes to read, and the pages read follow from that. Which is the independent variable?
A cup stack table shows (1, 8), (2, 9), (3, 10). Which equation fits?
A table from the Puzzle Post: (1, 4), (2, 8), (1, 5). Is the output a function of the input?
On the graph of l = 4c, which ordered pair has c = 6?
Week reviewTrue or false?
The dependent variable is the one you choose.?
In legs = 4 × chairs, the chairs are the input.?
A function can give two different outputs for the same input.?
The ordered pair (3, 12) means an input of 3 and an output of 12.?
A rule with a head start, like c = 3m + 6, is still a function.?
WHY THIS EXERCISEThese four ideas carry straight into slope next week.
OUR WEEK, IN ORDER
  • ?We fed the rule machine and named the independent and dependent variables.
  • ?We turned d = 4t into a table, ordered pairs and a graph.
  • ?We built a rule machine and guessed a secret rule from its cards.
  • ?We read two folding rules from the log and graphed them side by side.
WHY THIS EXERCISEEach day added one more view of the same idea: a rule that turns inputs into outputs.
Try it
Find a rule at home: wheels and bikes, legs and chairs, or minutes and pages.
Write it as an equation, make a table with three rows, and say which quantity is the input.
Draw a graph of legs = 4 × chairs for 1 to 5 chairs. Label the axes chairs and legs.

A whole week of rules. Tomorrow is Puzzle Day: the family plays guess my rule with the orange box.

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