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Expressions and Equations 6-8 / Week 12 / Wednesday
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Week 12 · Big Numbers, Small Numbers and Puzzle Night

Wednesday

Puzzle Lab: the bean jar
// Powers, roots and Puzzle Night
⏱ about 20 min

Wednesday: Puzzle Lab, The Bean Jar

Rocket holds up a full jar of beans. "Puzzle Night question: how many beans? Nobody wants to count them all."

"Then count a spoonful," Raven says. She counts one level spoon. "Forty beans."

Rocket spoons the beans back in, counting spoonfuls. "Fifty spoons to fill it."

"What do you notice when you multiply?" Raven asks.

"40 × 50 = 2,000 beans," Rocket says. "That is 2 × 10³."

Nova hovers over the jar, her light on the lid. "And 2,000 jars?"

"2 × 10³ times 2 × 10³. Multiply the fronts, add the exponents: 4 × 10⁶. Four million beans."

"Scientific notation," Raven says. "Big numbers made short. Write it in the log."

What you need

  • A jar of dried beans or buttons with a lid, and a tablespoon.
  • A tray to work over, so spilled beans are easy to sweep up.
  • A stack of paper, a ruler marked in centimeters, and a tape measure.
  • Your Puzzle Notebook and a pencil.
Safety first
Beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
Work over a tray. Sweep up any bean that falls so no one slips on it.
Beans are counters this week, never food.

Powers of ten and scientific notation

Powers of ten are the easy powers: 10³ = 1,000, 10⁶ = 1,000,000, and 10⁻² = 1 ÷ 100 = 0.01.

Scientific notation writes a number as a number from 1 to 10 times a power of ten. 2,000 is 2 × 10³.

The exponent counts how many places the decimal point moves. 2,190,000 becomes 2.19 × 10⁶: six places.

Tiny numbers use negative exponents. 0.01 is 1 × 10⁻², and 0.0032 is 3.2 × 10⁻³.

Run the lab

  1. Count the beans in one level spoonful. Write the count.
  2. Count how many spoonfuls fill the jar. Multiply to estimate the beans in the jar.
  3. Write the estimate in scientific notation.
  4. Measure a stack of 200 sheets of paper in centimeters. Divide by 200 for one sheet.
  5. Write the thickness of one sheet in scientific notation, in centimeters.
  6. Record every count and measurement in the lab table.

The crew's lab log

The crew's counts and measurements from the lab, written for the Puzzle Post. They are the crew's own numbers, not facts about the world.

MeasurementThe crew's countPlain numberScientific notation
Beans in a spoonful40404 × 10¹
Spoonfuls in the jar50505 × 10¹
Beans in the jar40 × 502,000?
Rocket's steps in a year6,000 a day × 365 days2,190,000?
One sheet of paper2 cm ÷ 200 sheets0.01 cm?
WRITE THE LAB NUMBERS
  • Read the question.
  • Tap your answer.
The jar holds about 2,000 beans. Which is that in scientific notation?
Rocket's year of steps came to 2,190,000. Which is that in scientific notation?
One sheet of paper measured 0.01 cm thick. Which is that in scientific notation?
A drip measured 0.0032 of a cup. Which is that in scientific notation?
BACK TO PLAIN NUMBERS
  • Read the question.
  • Tap your answer.
What number is 4 × 10⁶?
What number is 2.19 × 10⁶?
What number is 1 × 10⁻²?
What number is 5 × 10⁻³?
The jar holds about 2,000 beans, which is 2 × 10 to what power? Type the exponent.
WHY THIS EXERCISEThe exponent is the whole point of scientific notation: it tells you the size at a glance.
Try it
Estimate the grains of rice or the beans in a different container the same way: one spoon, then count spoons.
Write the answer in scientific notation. Then write how many times bigger it is than your jar.
Draw your jar and your spoon. Write the spoonful count, the spoon count and the total in scientific notation.

Great lab work. Tomorrow the crew compares huge distances with a few real planet numbers and does arithmetic in scientific notation.

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