← Back to course
Expressions and Equations 6-8 / Week 09 / Thursday
4/6
Week 09 · Linear Functions

Thursday

The card box
// Rate of change, starting value, and graphs that tell a story
⏱ about 20 min

Thursday: The Card Box

Raven opens the Puzzle Post log to the card box page. "Day 0, 15 cards in the box. Every day we add 4."

"So day 1 is 19, day 2 is 23," Rocket says. "Rate 4, start 15. c = 4d + 15."

"What do you notice if I only tell you two days?" Raven asks. "Day 3 had 27 cards. Day 6 had 39."

Rocket thinks. "Three days apart, 12 more cards. Four a day. Then back to day 0: 27 take away 12 is 15."

Nova hovers over the log, her light moving between the two rows. "A story, a table or two points," she says.

"All three give the same two numbers," Rocket says.

Nova hums. "Then the box can be predicted from any of them. Try day 10."

Rate and start from three forms

From a story: "15 cards, then 4 more each day." The rate of change is 4 and the initial value is 15.

From a table: find the change in cards for one step in days, and read the day 0 row for the start.

From two points: rate = rise ÷ run. Then step back to day 0 by subtracting the rate for each day.

These are the crew's own counts from the card box, written in Raven's log. They are made-up Puzzle Post numbers.

Day dCards cOrdered pair
015(0, 15)
119(1, 19)
223(2, 23)
327(3, 27)
431(4, 31)
535(5, 35)
A table, day and cards: 3 and 27, 6 and 39, 10 and a blank

From (3, 27) to (6, 39): run 3, rise 12, so the rate of change is 4 cards a day.

Back to day 0: 27 - 3 × 4 = 15. The initial value is 15, and the rule is c = 4d + 15.

In the box story, the rate of change means 4 new cards every day. The initial value means 15 cards before day 1.

READ THE CARD BOX
  • Read the question.
  • Tap your answer.
The log shows day 3 with 27 cards and day 6 with 39. What is the rate of change?
The log shows (3, 27) and (6, 39). How many cards were in the box on day 0?
Which equation fits the card box, with x for days and y for cards?
The card box rule is c = 4d + 15. How many cards on day 10?
OTHER BOXES
  • Read the question.
  • Tap your answer.
Nova's box shows (2, 20) and (5, 35). What is its rate of change?
Nova's box shows (2, 20) and (5, 35). How many cards did it start with?
The crew's box adds 4 cards a day. Nova's box went from (2, 20) to (5, 35). Which has the greater rate of change?
The crew's rule is c = 4d + 15. On which day does the box hold 43 cards?
In c = 4d + 15, what is the rate of change? Type the number.
In c = 4d + 15, what is the initial value? Type the number.
How many cards are in the crew's box on day 10? Type the number.
StatementTrue or false?
The rate of change of the card box means 4 new cards each day.?
The initial value is the number of cards on day 1.?
Two points are enough to find a linear rule.?
From (3, 27) to (6, 39) the rate of change is 12.?
WHY THIS EXERCISEReading what the two numbers mean in the story is as important as finding them.
Try it
Count something at home that grows by the same amount each day, like pages read or steps on a chart.
Write the count on two different days and work out the rate and the start from just those two points.
Draw the card box line, days across and cards up, from day 0 to day 6. Mark the start at (0, 15).

Rates and starts from any form. Tomorrow you describe graphs in words and sketch them from stories.

← Wednesday