Thursday
Thursday: The Shape of an Expression
Wren sets a clear jug under a slow tap and starts the stopwatch. "Fifteen milliliters already in it."
"And it gains 3 milliliters every second," Comet says, reading the marks. "So after t seconds, 3t + 15."
"What do you notice about the two parts?" Wren asks.
"The 15 is water that was there at the start. The 3t is water that arrived while we waited."
Nova draws the expression in the air and makes the 3 glow. "Would you like a hint? What does the 3 do?"
"It multiplies the seconds," Comet says. "Three milliliters for each second. The coefficient is the fill rate."
"So every piece of the expression is a piece of the Loft," Wren says. "Let's read a few more."
Terms, coefficients and factors
The jug holds 3t + 15 milliliters after t seconds. This expression has two terms, 3t and 15.
A term is a part added or subtracted. The 15 is the constant term: water in the jug at the start.
In the term 3t, the 3 and the t are factors multiplied together. The 3 is the coefficient of t.
In the Loft, the coefficient 3 is milliliters per second, the fill rate. The letter t is seconds.
| Piece of 3t + 15 | Name | What it means in the Loft |
|---|---|---|
| 15 | constant term | milliliters in the jug before the tap ran |
| 3t | variable term | milliliters added while the tap ran t seconds |
| 3 | coefficient of t | the fill rate, 3 milliliters each second |
| t | the variable | how many seconds the tap has run |
The crew's jug readings
The crew checked the jug at several times. These are their own readings, made up for the story, and each matches 3t + 15.
| Seconds (t) | Milliliters (3t + 15) |
|---|---|
| 0 | 15 |
| 5 | 30 |
| 10 | 45 |
| 20 | ? |
| 30 | ? |
On the graph, the line starts at 15 on the up axis. That is the constant term.
It climbs 3 for every 1 across. That is the coefficient, the fill rate.
A frame for the data poster
Comet builds a frame for the Fair Day poster. Its edges measure 2w + 2l, where w is the width and l is the length.
This expression has two terms, 2w and 2l. Each has coefficient 2 because a frame has two of each edge.
With w = 30 centimeters and l = 45 centimeters, the edges total 150 centimeters.
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| 3t + 15 has two terms. | ? |
| In 3t + 15, the constant term is 3. | ? |
| 3 × 10 + 15 = 45 | ? |
| In 2w + 2l, the 2 in front of w means the frame has two width edges. | ? |
| A coefficient is the number added on at the end of an expression. | ? |
Write a start amount and a fill rate, then write your own expression like 3t + 15.
Keep the water over a sink or a tray, and wipe up spills right away.
Sharp reading. Tomorrow you treat a whole chunk of an expression as one thing, then review the week.