Week 01 · Quantities and the Shape of an Expression

Tuesday

Choose the quantity, pick the accuracy

// Units that cancel, numbers that mean something

⏱ about 20 min

Tuesday: Choose the Quantity, Pick the Accuracy

Comet sets the wooden cart at the top of the plank and lets go. Wren clicks the stopwatch. "One point six."

"One point six what?" Comet says, grinning. "Seconds. And the plank is 240 centimeters. So how fast was it?"

"Fast is not a quantity," Wren says. "What do you notice we measured? A distance and a time."

Nova projects both numbers in the air between them. "Would you like a hint? Divide one by the other."

"240 centimeters divided by 1.6 seconds," Comet says. "150 centimeters per second. That is the cart's speed."

"Centimeters per second is our quantity," Wren says. "And the stopwatch only shows tenths, so no extra digits."

"No pretend digits," Comet agrees. "What can we make? A timing table."

Define the quantity that answers the question

The crew wanted to know how fast the cart went. Fast is a feeling. A quantity is a measurement.

They chose distance divided by time, in centimeters per second. The unit is built from the two units they measured.

240 centimeters ÷ 1.6 seconds = 150 centimeters per second.

A different question needs a different quantity. "Is the ramp steep?" might use centimeters of height per meter of plank.

Question about the build	Quantity that answers it	Unit
How fast does the cart go?	distance ÷ time	centimeters per second
How steep is the ramp?	height ÷ length	centimeters per meter
How long is each test run?	time	seconds
How much tape is on the plank?	length	centimeters

Pick a sensible level of accuracy

The tape measure is marked in centimeters, so the plank is 240 centimeters, not 240.000 centimeters.

The stopwatch shows tenths of a second, so run 4 is 1.7 seconds, not 1.700 seconds.

Divide 240 by 1.7 and a calculator shows 141.1765. Those extra digits are not real.

The crew reports 141 centimeters per second. Round to the nearest whole, matching the tools.

Report a measurement to the smallest mark on the tool, and no finer.
When you divide or multiply measurements, round the result to about as many digits as the roughest input.
Write the unit every time. 150 by itself is not a speed.

Two ways to the same speed

Way one: 240 centimeters ÷ 2 seconds = 120 centimeters per second.

Way two: change to meters first. 2.4 meters ÷ 2 seconds = 1.2 meters per second.

Both are right. 1.2 meters per second × 100 = 120 centimeters per second. The units agree.

COMPUTE THE SPEED

Read the question.
Tap your answer.

Run 1: the cart covered 240 centimeters in 1.6 seconds. What was its speed in centimeters per second?

Run 2: the cart covered 240 centimeters in 2 seconds. What was its speed in centimeters per second?

Run 3: the cart covered 240 centimeters in 1.5 seconds. What was its speed in centimeters per second?

WHICH QUANTITY FITS?

Read the question.
Tap your answer.

The crew wants to compare how steep two ramps are. Which quantity fits best?

Wren wants one number that says how fast the cart moved. Which unit fits?

Run 4: 240 centimeters in 1.7 seconds gives 141.1765 on a calculator. What should the crew report, in whole centimeters per second?

FIND A SPEED IN METERS PER SECOND

?Divide the meters by the seconds
?Change centimeters to meters by dividing by 100
?Round to match the tools and write the unit: meters per second
?Measure the distance in centimeters and the time in seconds

WHY THIS EXERCISEEach step keeps the units straight. The unit at the end tells you the quantity you built.

Statement	True or false?
Centimeters per second is a quantity built from two measured units.	?
A tape measure marked in centimeters can read 240.37 centimeters.	?
240 ÷ 2 = 120	?
A speed of 1.2 meters per second equals 120 centimeters per second.	?
More decimal places always make a measurement more accurate.	?

WHY THIS EXERCISEChoosing the quantity and the accuracy is half of modeling. The arithmetic is the other half.

Try it

Walk a measured hallway and time yourself. Write your speed in meters per second with honest digits.
Then write the same speed in centimeters per second. Check that the two agree.

Draw the timing table for three cart runs: distance, time and speed, each column labeled with its unit.

Loft Log: write one question about a build at home and the quantity, with its unit, that would answer it.Write why 141.1765 centimeters per second is not an honest report for run 4.Which way to the speed did you like better, centimeters first or meters first? Write why.

Good thinking. Tomorrow is Loft Lab: you measure a ramp of your own and time a run.

Grown-up checked my answers

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