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Algebra 1 9-12 / Week 01 / Monday
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Week 01 · Quantities and the Shape of an Expression

Monday

Units ride along
// Units that cancel, numbers that mean something
⏱ about 20 min

Monday: Units Ride Along

Comet drags a long plank up the stairs into the Loft and drops it across two sawhorses. "Ramp course, step one."

Wren pulls the tape measure along it and reads the end. "Two hundred forty." He pauses. "Two hundred forty what?"

"Centimeters," Comet says. "The plans say the ramp should be 2.4 meters. Is that the same?"

Nova drifts down from the rafters, her light settling on the tape. "Would you like a hint? How many centimeters make one meter?"

"One hundred," Wren says. "So 240 centimeters divided by 100 is 2.4 meters. It fits."

"The number changed but the plank did not," Comet says. "What can we make with that idea?"

"A rule," Wren says. "Units ride along with every number, and they cancel like numbers do."

Comet and Wren measure a long plank with a tape measure in the Loft while Nova hovers nearby.

A number with a unit is a quantity

In the Loft, no number stands alone. 240 means nothing until you say 240 centimeters.

A quantity is a number with its unit. The unit tells you what was measured and how.

The crew's plank is 240 centimeters long. That is the same length as 2.4 meters.

Both quantities describe one plank. Only the unit, and so the number, changed.

A solved problem to study

Here is how Wren changed centimeters to meters. Read each line and ask why it is allowed.

  1. Start with the quantity: 240 centimeters.
  2. Write a conversion factor equal to 1: (1 meter) ÷ (100 centimeters). Both parts are the same length.
  3. Multiply: 240 centimeters × (1 meter ÷ 100 centimeters).
  4. The centimeters on top and bottom cancel, like a number over itself.
  5. What is left: 240 ÷ 100 = 2.4 meters.

Multiplying by a fraction equal to 1 never changes the quantity. It only changes the unit it is written in.

Going the other way, meters to centimeters, you multiply by (100 centimeters) ÷ (1 meter). The meters cancel instead.

QuantityConversion factorResult
240 centimeters1 meter ÷ 100 centimeters2.4 meters
75 centimeters1 meter ÷ 100 centimeters0.75 meters
1.5 meters100 centimeters ÷ 1 meter150 centimeters
150 seconds1 minute ÷ 60 seconds2.5 minutes
CHANGE THE UNIT
  • Read the question.
  • Tap your answer.
The plank is 240 centimeters long. How many meters is that? (100 centimeters make 1 meter.)
A sawhorse is 75 centimeters tall. How many meters is that? (100 centimeters make 1 meter.)
A test run lasted 150 seconds. How many minutes is that? (60 seconds make 1 minute.)
The plans call for a 1.5 meter strip of tape. How many centimeters is that?
WHICH FACTOR CANCELS THE UNIT?
  • Read the question.
  • Tap your answer.
You have 320 centimeters and want meters. Which conversion factor cancels the centimeters?
You have 4 minutes and want seconds. Which conversion factor cancels the minutes?
Why is 1 meter ÷ 100 centimeters allowed as a multiplier?
StatementTrue or false?
A quantity is a number together with its unit.?
Multiplying by 1 meter ÷ 100 centimeters changes the length of the plank.?
240 centimeters and 2.4 meters describe the same plank.?
240 ÷ 100 = 2.4?
To change meters into centimeters, you divide by 100.?
WHY THIS EXERCISEUnits ride along and cancel like numbers. That rule guides every multi-step problem this year.
Try it
Measure one long thing at home in centimeters. Write the quantity with its unit.
Write the conversion factor as a fraction equal to 1, cancel the unit and write the length in meters.
Draw the plank with its length written twice, once in centimeters and once in meters. Show the conversion factor beside it.

Strong start. Tomorrow the cart runs down the ramp and you choose the quantity that describes it.