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Geometry and Measurement 6-8 / Week 12 / Friday
5/6
Week 12 · Round Solids, Irrational Numbers and Build Day

Friday

Decimals that repeat, and the course review
// The can tank, the cone roof, the ball, and numbers that never end
⏱ about 20 min

Friday: Decimals That Repeat

Rocket divides 1 by 3 on paper for the floor tiles. "0.333... It goes on forever too! Is one third irrational?"

"What do you notice about the digits?" Raven asks. "They repeat. The same 3, over and over."

"√2 never repeated," Rocket says. "Its digits were all different. So repeating is the difference?"

Nova hovers over the division, her light pulsing once per 3. "Every fraction ends or repeats," she says. "Every time."

"And a repeating decimal can turn back into a fraction," Raven says. "Call it x. Multiply by 10. Subtract."

Rocket writes 10x = 3.333..., then 10x - x = 3. "9x = 3. x is one third!"

Nova hums. "Rational: ends or repeats. Irrational: never. Then review the whole course for Build Day."

Every number has a decimal

A rational number is a fraction of whole numbers. Its decimal either ends, like 0.75, or repeats, like 0.333...

An irrational number cannot be written as a fraction. Its decimal never ends and never repeats: √2, √10, π.

The crew writes a repeating part in parentheses: 0.(3) means 0.333... and 0.(45) means 0.454545...

Turning a repeating decimal into a fraction

  1. Call the number x. For 0.(3), x = 0.333...
  2. Multiply by 10 for each repeating digit. One digit repeats, so 10x = 3.333...
  3. Subtract the first line from the second: 10x - x = 3, so 9x = 3.
  4. Solve: x = 3/9 = 1/3. Check by dividing 1 by 3.
Repeating decimalFraction
0.333...1/3
0.666...2/3
0.454545...5/11
0.1666...1/6
WHICH FRACTION?
  • Read the question.
  • Tap your answer.
Which fraction equals 0.333... (the 3 repeats forever)?
Which fraction equals 0.666... (the 6 repeats forever)?
Which fraction equals 0.454545... (the 45 repeats forever)?
Which fraction equals 0.1666... (the 6 repeats forever)?
RATIONAL OR IRRATIONAL?
  • Read the question.
  • Tap your answer.
Is √2 rational or irrational?
Is 0.333... rational or irrational?
Is √16 rational or irrational?
Is π rational or irrational?

The whole course in twelve lines

Twelve weeks at the Blueprint Table built one model clubhouse. Put the build back in order.

OUR COURSE, IN ORDER
  • ?We taped a coordinate grid on the floor and drew the walls as a polygon.
  • ?We drew the garage to scale.
  • ?We measured the roof angles and crossed parallel sticks.
  • ?We packed the storage bin with cubes and measured its volume.
  • ?We built straw triangles with rules and sliced a clay block.
  • ?We slid, flipped and turned tiles for the floor pattern.
  • ?We unfolded a box into a net and found the cardboard for the walls.
  • ?We cut paper shapes and found the floor area by composing and decomposing.
  • ?We enlarged the plan with a flashlight shadow and a dilation.
  • ?We found the volume of the tank, the roof and the ball, and met √2.
  • ?We wrapped string around a can and met π for the round window.
  • ?We braced a wall and proved a² + b² = c².
WHY THIS EXERCISEEvery week's math built one part of the clubhouse. Seeing the order shows how the parts depend on each other.
Week reviewTrue or false?
A cone holds one third of the cylinder with the same base and height.?
√2 can be written as a fraction.?
Every repeating decimal equals a fraction.?
0.75 is irrational because it has a decimal.?
√10 is between 3 and 4.?
WHY THIS EXERCISEThe number system now has a place for every length the crew measured, even the brace of a tile.
Try it
Divide 2 by 11 on paper until you see the pattern. Write the repeating part in parentheses.
Then turn it back into a fraction with the x method and check that you get 2/11.
Draw a number line from 0 to 2. Mark 1/3, 0.75, √2 and √3 as closely as you can.
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