On a whiteboard in Room 214 at Harbor City High, chemistry teacher Mara Kline drew a thick rectangle around two words — “GIVEN” and “GOAL” — and told a Saturday workshop of students, parents and new hires that most mistakes start when people stop writing units.

Kline’s session, hosted by the Harbor City School District after a string of botched science-lab measurements last fall, centered on a routine that instructors said works as reliably as a checklist: write the units everywhere, build conversion factors as ratios equal to 1, and arrange them so unwanted units cancel.

“Conversion factors aren’t magic numbers,” Kline said, tapping the board. “They’re statements that two measures describe the same amount.”

District science coordinator Daryl Nguyen said the approach is being rolled into middle- and high-school math and science classes this spring, after teachers reported that students could memorize individual conversions but often mixed them up under pressure.

“Dimensional analysis is a setup method,” Nguyen said. “If the setup is correct, the units tell you the answer before you compute it.”

The day’s rule: ratios equal to 1

Kline wrote the anchor idea in marker: if two quantities are equal, their ratio is 1.

• Because 1 m = 100 cm, then (1 m / 100 cm) = 1 and (100 cm / 1 m) = 1.
• Choosing which “1” to multiply by depends on which unit needs to disappear.

She then modeled what she called “paper discipline”: box the given and goal units, label every conversion factor, cancel units in the setup, and do a quick order-of-magnitude check before accepting the final number.

Worked example 1 (length): cm → m

Kline started with a common lab measurement written in centimeters.

[GIVEN] 250 cm
[GOAL] m

Conversion factor (from 1 m = 100 cm):

• CF1: (\dfrac{1\ ext{m}}{100\ ext{cm}}) (equals 1)

Setup (units shown on every line):

[
250\ ext{cm} imes \frac{1\ ext{m}}{100\ ext{cm}}
]

Cancel units (visually):

[
250\ \cancel{ ext{cm}} imes \frac{1\ ext{m}}{100\ \cancel{ ext{cm}}}=2.5\ ext{m}
]

Order-of-magnitude check: 250 cm is a few meters; 2.5 m is plausible.

Senior Maya Singh said she used to “divide by 100 because teachers said so,” but liked seeing the cancellation.

“If centimeters disappear, I know I didn’t flip it,” she said.

Worked example 2 (mass): g → kg

The next example addressed a mistake teachers said they see in recipe scaling and lab balances.

[GIVEN] 3,750 g
[GOAL] kg

Conversion factor (from 1 kg = 1000 g):

• CF1: (\dfrac{1\ ext{kg}}{1000\ ext{g}}) (equals 1)

Setup:

[
3750\ ext{g} imes \frac{1\ ext{kg}}{1000\ ext{g}}
]

Cancel units:

[
3750\ \cancel{ ext{g}} imes \frac{1\ ext{kg}}{1000\ \cancel{ ext{g}}}=3.75\ ext{kg}
]

Order-of-magnitude check: 3,750 g is a few kilograms; 3.75 kg matches.

Nguyen said the district has asked teachers to “refuse unitless work” in early assignments.

“Students turn in 3.75,” he said. “We ask, ‘3.75 what?’”

Worked example 3 (volume): mL → L

Kline moved to volume after noting that medication labels and lab glassware mix units.

[GIVEN] 850 mL
[GOAL] L

Conversion factor (from 1 L = 1000 mL):

• CF1: (\dfrac{1\ ext{L}}{1000\ ext{mL}}) (equals 1)

Setup:

[
850\ ext{mL} imes \frac{1\ ext{L}}{1000\ ext{mL}}
]

Cancel units:

[
850\ \cancel{ ext{mL}} imes \frac{1\ ext{L}}{1000\ \cancel{ ext{mL}}}=0.85\ ext{L}
]

Order-of-magnitude check: 850 mL is a little less than 1 liter; 0.85 L fits.

Parent volunteer Angela Ruiz, who said she works in a clinic, told students the method mirrors how staff double-check doses.

“We read the units out loud,” Ruiz said. “If the units don’t make sense, we stop.”

Worked example 4 (time): min → s

To show the same setup works outside science class, Kline converted minutes to seconds.

[GIVEN] 12 min
[GOAL] s

Conversion factor (from 1 min = 60 s):

• CF1: (\dfrac{60\ ext{s}}{1\ ext{min}}) (equals 1)

Setup:

[
12\ ext{min} imes \frac{60\ ext{s}}{1\ ext{min}}
]

Cancel units:

[
12\ \cancel{ ext{min}} imes \frac{60\ ext{s}}{1\ \cancel{ ext{min}}}=720\ ext{s}
]

Order-of-magnitude check: 10 minutes is about 600 seconds; 12 minutes should be about 720 seconds.

Multi-step chain conversion (length): km → cm

Kline saved a longer problem for the end, telling students that multi-step conversions are “the same game, just longer.”

[GIVEN] 3.4 km
[GOAL] cm

Conversion factors (labeled):

• CF1 (from 1 km = 1000 m): (\dfrac{1000\ ext{m}}{1\ ext{km}})
• CF2 (from 1 m = 100 cm): (\dfrac{100\ ext{cm}}{1\ ext{m}})

Setup (keep units on every line):

[
3.4\ ext{km} imes \frac{1000\ ext{m}}{1\ ext{km}} imes \frac{100\ ext{cm}}{1\ ext{m}}
]

Cancel units (step-by-step):

[
3.4\ \cancel{ ext{km}} imes \frac{1000\ ext{m}}{1\ \cancel{ ext{km}}} imes \frac{100\ ext{cm}}{1\ ext{m}}
]

[
3.4 imes 1000\ \cancel{ ext{m}} imes \frac{100\ ext{cm}}{1\ \cancel{ ext{m}}}=340{,}000\ ext{cm}
]

Order-of-magnitude check: 1 km is 100,000 cm, so 3.4 km should be a few hundred thousand cm; 340,000 cm matches.

‘If the units don’t cancel, don’t calculate’

Kline said she wants students to treat cancellation as the “green light” to pick up a calculator.

“Don’t start punching buttons if the units are still tangled,” she told the room. “If the units don’t cancel to the goal, the setup is wrong.”

Nguyen said the district’s new grading guidance will give partial credit for correct dimensional setups even when arithmetic slips.

“That’s how we’re trying to change behavior,” he said. “We want students to show their thinking in units.”

By the end of the session, students were asked to box their own GIVEN and GOAL, write conversion factors with labels, and circle a quick reasonableness check.

Singh, the senior, said it felt slower than her old approach but less fragile.

“I’m not guessing anymore,” she said. “The units tell me where to put everything.”